In the case of a large signal (pulse modulation) oscillation-relaxation rings form after the leading and trailing edges of the rectangular pulse. Thus, the pulsed bias current modulates the number of electrons and photons in the active region of LD (Agrawal & Dutta, 1993). The number of photons is proportional to the lased optical power (Agrawal & Dutta, 1993). which can be measured. The pulsing number of electrons is responsible for the frequency chirp phenomena inherent to LDs. The chirp is a considerable change in the frequency (wavelength) of the optical mode of the LD during the large signal modulation. This frequency chirp can also be measured. The origin of this effect relates to the other phenomena present even at the continuous-wave (cw) operation of LD. It manifests in the enhanced linewidth of the optical mode of the LDs. The principle of that is a change in the real and imaginary parts of the refractive index *n*′ and *n*′′ as electron density is pulsing (chirp) or fluctuates (linewidth enhancement). The phenomenon and the linewidth enhancement factor (or chirp parameter) *α Δ* = *n /* ′ ′′ *Δn* showing the intensity of the phenomenon itself were correspondingly explained and introduced by C. H. Henry (Henry, 1982, 1983, 1986). Measurements of the frequency chirp during large signal modulation is a well-known technique for the estimation of the optical signal transmission in the fiber communications. The chirp intensity is proportional to the electron density, and thus also features oscillation-relaxation behaviour during the pulse modulation. This can be seen in

There are several chirp parameter extraction methods and the time-resolved chirp method is one of them (Harder et al., 1983; Arakawa & Yariv, 1985; Kikuchi et al., 1984; Kikuchi & Okoshi, 1985; Linke, 1985; Bergano, 1988). In fact, this method is more frequently used for electro-optical modulators, where the time-resolved chirp is a little simpler than that of LD, because there is only a transient term (Jeong & Park, 1997). Time-resolved frequency chirp measurements reveal more detailed information about LDs optical beam frequency behaviour by comparing with ordinary averaged optical beam spectrum measurements, because the time-resolved representation of frequency deviations shows specific chirp details, such as frequency oscillations and the amplitude of these oscillations (Welford, 1985;

A schematic diagram of the finished laser structure (around the mesa) is shown in Fig. 1.

Fig. 1. Schematic diagram of the finished laser structure in the vicinity of the mesa. p-InP p:1018 cm-3, p2-InP p:1018 cm-3, p1-InP p:5·1017 cm-3, n1-InP n:1018 cm-3, n -InP n:1018 cm-3 nB-InP n:1018 cm-3, pB-InP p:6·1017 cm-3. Active region of the InGaAsP MQWs is formed

example, electron lifetime, photon lifetime in the LD cavity, differential gain, nonlinear gain K-factor are only a few parameters which are usually extracted from the small signal modulation measurements (Lu et. al., 1995).

In the case of relative intensity noise (RIN) analysis the situation is similar (Tatham et al., 1992). The similarity of both characterisations becomes obvious if one just notices that the difference is only in the modulation source acting on the same system (LD). First one is the external modulation source probing the response of the system and another one is an internal source of the origin of spontaneous fluctuations of the number of electrons and photons. One can recall the Fluctuation-Dissipation theorem which states that the response of the system at equilibrium to the external small signal perturbation is the same as the response to the spontaneously occurring fluctuations inside the system. However, one applies DC current and thus the system is not at equilibrium. But it means no more that an additional information in the fluctuation spectra could be present in principle. Quantitatively the precision of extracted parameters from RIN measurements should be lower compared with small signal modulation measurements due to the random nature of the detected signal. Qualitatively it may have advantages, since parasitic capacitances are avoided.

Another interesting point is that the equations for small signal modulation and RIN have minor differences in the cases of the LDs of different materials and even structures. One can compare the equations of conventional edge-emitting laser diode (studied here), verticalcavity surface-emitting laser (VCSEL) (Larsson et. al., 2004) or even quantum-dot (QD) (Bhattacharya et. al., 2000) laser.

Here the case of large signal current modulation is studied. Given dynamic parameters of LD extracted from the small signal measurements (relative intensity noise (RIN) of the optical beam intensity) are compared with the parameters extracted from the large signal measurements (optical power and time-resolved frequency chirp). Contrary to the small signal, the rate equations do not have closed-form expressions for the solutions of large signal (pulse) and thus rate equations should be integrated numerically. If the small signal parameter extraction from the experimental results consists of the standard function fitting procedure, then due to the lack of the closed-form function (in the case of a large signal) one needs other fitting means.

In this work rate equations were numerically solved taking the fixed set of input parameters such as the lifetime of photons, the coefficient of optical amplification, the nonlinear amplification factor, the spontaneous radiation and the optical confinement factor etc. The numerical solution obtained is then compared with the pulse from experiment. The procedure is repeated with a different set of parameters until the best coincidence is obtained. The final results of simulation were compared with experimental ones and the matching accuracy was calculated

### **2. Large signal: Optical power modulation and chirp**

In the time domain, the small signal solutions of the rate equations are in the form of decaying oscillations (oscillation-relaxation solutions) in the densities (or number) of electrons and photons. After a small disturbance, coupled subsystems of electrons and photons oscillate (in number), exchange and dissipate the energy while approaching stationary conditions until the next disturbance. Two terms appear: the oscillation-relaxation frequency (showing the rate of exchange) *f*0 and the damping frequency (showing the rate of dissipation) *Γ*0. In the frequency domain this process has the form of resonance spectrum.

example, electron lifetime, photon lifetime in the LD cavity, differential gain, nonlinear gain K-factor are only a few parameters which are usually extracted from the small signal

In the case of relative intensity noise (RIN) analysis the situation is similar (Tatham et al., 1992). The similarity of both characterisations becomes obvious if one just notices that the difference is only in the modulation source acting on the same system (LD). First one is the external modulation source probing the response of the system and another one is an internal source of the origin of spontaneous fluctuations of the number of electrons and photons. One can recall the Fluctuation-Dissipation theorem which states that the response of the system at equilibrium to the external small signal perturbation is the same as the response to the spontaneously occurring fluctuations inside the system. However, one applies DC current and thus the system is not at equilibrium. But it means no more that an additional information in the fluctuation spectra could be present in principle. Quantitatively the precision of extracted parameters from RIN measurements should be lower compared with small signal modulation measurements due to the random nature of the detected signal. Qualitatively it may have

Another interesting point is that the equations for small signal modulation and RIN have minor differences in the cases of the LDs of different materials and even structures. One can compare the equations of conventional edge-emitting laser diode (studied here), verticalcavity surface-emitting laser (VCSEL) (Larsson et. al., 2004) or even quantum-dot (QD)

Here the case of large signal current modulation is studied. Given dynamic parameters of LD extracted from the small signal measurements (relative intensity noise (RIN) of the optical beam intensity) are compared with the parameters extracted from the large signal measurements (optical power and time-resolved frequency chirp). Contrary to the small signal, the rate equations do not have closed-form expressions for the solutions of large signal (pulse) and thus rate equations should be integrated numerically. If the small signal parameter extraction from the experimental results consists of the standard function fitting procedure, then due to the lack of the closed-form function (in the case of a large signal) one

In this work rate equations were numerically solved taking the fixed set of input parameters such as the lifetime of photons, the coefficient of optical amplification, the nonlinear amplification factor, the spontaneous radiation and the optical confinement factor etc. The numerical solution obtained is then compared with the pulse from experiment. The procedure is repeated with a different set of parameters until the best coincidence is obtained. The final results of simulation were compared with experimental ones and the

In the time domain, the small signal solutions of the rate equations are in the form of decaying oscillations (oscillation-relaxation solutions) in the densities (or number) of electrons and photons. After a small disturbance, coupled subsystems of electrons and photons oscillate (in number), exchange and dissipate the energy while approaching stationary conditions until the next disturbance. Two terms appear: the oscillation-relaxation frequency (showing the rate of exchange) *f*0 and the damping frequency (showing the rate of dissipation) *Γ*0. In the frequency domain this process has the form of resonance spectrum.

modulation measurements (Lu et. al., 1995).

advantages, since parasitic capacitances are avoided.

(Bhattacharya et. al., 2000) laser.

needs other fitting means.

matching accuracy was calculated

**2. Large signal: Optical power modulation and chirp**

In the case of a large signal (pulse modulation) oscillation-relaxation rings form after the leading and trailing edges of the rectangular pulse. Thus, the pulsed bias current modulates the number of electrons and photons in the active region of LD (Agrawal & Dutta, 1993). The number of photons is proportional to the lased optical power (Agrawal & Dutta, 1993). which can be measured. The pulsing number of electrons is responsible for the frequency chirp phenomena inherent to LDs. The chirp is a considerable change in the frequency (wavelength) of the optical mode of the LD during the large signal modulation. This frequency chirp can also be measured. The origin of this effect relates to the other phenomena present even at the continuous-wave (cw) operation of LD. It manifests in the enhanced linewidth of the optical mode of the LDs. The principle of that is a change in the real and imaginary parts of the refractive index *n*′ and *n*′′ as electron density is pulsing (chirp) or fluctuates (linewidth enhancement). The phenomenon and the linewidth enhancement factor (or chirp parameter) *α Δ* = *n /* ′ ′′ *Δn* showing the intensity of the phenomenon itself were correspondingly explained and introduced by C. H. Henry (Henry, 1982, 1983, 1986). Measurements of the frequency chirp during large signal modulation is a well-known technique for the estimation of the optical signal transmission in the fiber communications. The chirp intensity is proportional to the electron density, and thus also features oscillation-relaxation behaviour during the pulse modulation. This can be seen in the time-resolved frequency chirp measurements.

There are several chirp parameter extraction methods and the time-resolved chirp method is one of them (Harder et al., 1983; Arakawa & Yariv, 1985; Kikuchi et al., 1984; Kikuchi & Okoshi, 1985; Linke, 1985; Bergano, 1988). In fact, this method is more frequently used for electro-optical modulators, where the time-resolved chirp is a little simpler than that of LD, because there is only a transient term (Jeong & Park, 1997). Time-resolved frequency chirp measurements reveal more detailed information about LDs optical beam frequency behaviour by comparing with ordinary averaged optical beam spectrum measurements, because the time-resolved representation of frequency deviations shows specific chirp details, such as frequency oscillations and the amplitude of these oscillations (Welford, 1985; Olsen & Lin, 1989; Tammela et al., 1997).

### **2.1 Devices under investigation**

A schematic diagram of the finished laser structure (around the mesa) is shown in Fig. 1.

Fig. 1. Schematic diagram of the finished laser structure in the vicinity of the mesa. p-InP p:1018 cm-3, p2-InP p:1018 cm-3, p1-InP p:5·1017 cm-3, n1-InP n:1018 cm-3, n -InP n:1018 cm-3 nB-InP n:1018 cm-3, pB-InP p:6·1017 cm-3. Active region of the InGaAsP MQWs is formed between p1 and n1 layers.

Investigation of High-Speed Transient Processes

The frequency transfer function can be written as:

*f*cw to the point "A" and after – to the point "B".

Transfer Coefficient

Δ*f(t).*

point "B" Fig. 3.

and Parameter Extraction of InGaAsP Laser Diodes 165

An automated polarization controller helps to fix optimal polarization direction. That increases measurement accuracy. By changing the optical path delay time the interferometer can automatically fit the frequency transfer function to a particular carrier frequency of optical signal. The frequency transfer function of the Mach-Zehnder fiber interferometer is shown in Fig. 3. As can be seen, it is a sinusoidal function and its period is called a free spectral range (FSR). The optical signal frequency (chirp) variation should be lower than FSR/4=35 GHz. The frequency transfer function can be arbitrarily shifted with respect to the optical signal carrier frequency. And thus, during the first phase the frequency transfer function is shifted to the carrier frequency at the point "A", during the second phase – at the

> 2 ( ) cos 2 *A f K Kf total FSR*

 β

It follows from the equation (1) that the transfer coefficient depends on carrier frequency.

The last term in equation (2) is a frequency chirp. At first, in order to measure the chirp of the LD or any other device, the optical beam should be passed to the interferometer not modulated, so that the carrier frequency and optical power should be constant. Then the carrier frequency is *f*cw. As was mentioned earlier, the transfer function is shifted to match

β

where *β* is a optical loss in the system and *A* is amplitude of transfer function.

Free Spectral Range

A B

Time

In both cases the transfer function will be shifted and the transfer coefficient will depend on

During the pulse modulation the carrier frequency varies in time as:

Optical Frequency

Fig. 3. Mach-Zehnder interferometer frequency transfer function

π

Optical Intensity

⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ . (1)

*f(t) f* = + *c*w *Δf(t)* . (2)

Time

The laser structure is epitaxially grown on (100) oriented n-doped InP substrates. The residual reflectivity of the front facet coating is within 3 to 8% as measured at the lasing wavelength. The rear facet is coated with a high reflectivity coating of 60 to 80%.

The active region of the investigated laser diodes is buried between two side layers which helps to make the active region bounded in transversal direction. Six strained quantum wells are formed in the active region. In addition, there are periodically truncated quantum wells along the longitudinal axis of the active region, which helps to create a periodical refractive index (due to different refractive indexes of the material in the truncating region) and a periodical optical gain (due to non-uniform carrier density); that forms gain-coupled (plus index-coupled) distributed feedback.

### **2.2 Time-resolved frequency chirp measurement setup and methodology**

Time-resolved frequency chirp measurements setup under pulse operation is shown in Fig. 2. LD bias current is modulated by using a rectangular pulse generator together with a direct current source connected through the wideband bias tee. When the bias current was modulated, the optical beam power and optical wavelength became modulated as well. The optical beam is coupled to the tapered fiber (Shani et al., 1989; Alder et al., 2000), which is connected to a Mach-Zehnder fiber interferometer (Saunders et al., 1994; Jeong & Park, 1997; Laverdiere et al., 2003). An optical amplifier can be used if the interferometer output signal is too weak. Then an optical signal is passed to an optical input of the sampling oscilloscope connected to the PC. The most important part of the setup is the automated Mach-Zehnder fiber interferometer (Fig. 2). The main advantages of the interferometer are a quick timedomain chirp measurement, which takes only 30 seconds or less, high-resolution (< 20 MHz) and high measurement frequency (50 GHz or more). This allows to measure transmission signals greater than 10 Gbps with the measurement repeatability of ~5%.

Fig. 2. Chirp measurement setup. (1) Mach-Zehnder fiber interferometer (Advantest Q7606B Optical Chirpform Test Set), (2) semiconductor laser, (3) direct current source (ILX Lightwave LDC–3900 modular laser diode controller), (4) pulse pattern generator (Agilent HP-70843B error performance analyzer), (5) signal analyzer – digital sampling oscilloscope (Tektronix TDS8000 Digital Sampling Oscilloscope), (6) erbium doped fiber amplifier, (7) personal computer with integrated General Purpose Interface Bus (GPIB) interface

The laser structure is epitaxially grown on (100) oriented n-doped InP substrates. The residual reflectivity of the front facet coating is within 3 to 8% as measured at the lasing

The active region of the investigated laser diodes is buried between two side layers which helps to make the active region bounded in transversal direction. Six strained quantum wells are formed in the active region. In addition, there are periodically truncated quantum wells along the longitudinal axis of the active region, which helps to create a periodical refractive index (due to different refractive indexes of the material in the truncating region) and a periodical optical gain (due to non-uniform carrier density); that forms gain-coupled

Time-resolved frequency chirp measurements setup under pulse operation is shown in Fig. 2. LD bias current is modulated by using a rectangular pulse generator together with a direct current source connected through the wideband bias tee. When the bias current was modulated, the optical beam power and optical wavelength became modulated as well. The optical beam is coupled to the tapered fiber (Shani et al., 1989; Alder et al., 2000), which is connected to a Mach-Zehnder fiber interferometer (Saunders et al., 1994; Jeong & Park, 1997; Laverdiere et al., 2003). An optical amplifier can be used if the interferometer output signal is too weak. Then an optical signal is passed to an optical input of the sampling oscilloscope connected to the PC. The most important part of the setup is the automated Mach-Zehnder fiber interferometer (Fig. 2). The main advantages of the interferometer are a quick timedomain chirp measurement, which takes only 30 seconds or less, high-resolution (< 20 MHz) and high measurement frequency (50 GHz or more). This allows to measure transmission

Fig. 2. Chirp measurement setup. (1) Mach-Zehnder fiber interferometer (Advantest Q7606B

Sampling oscilloscope 5

> Q7606B 1

Optical input Optical output

GPIB

7

GPIB

Optical amplifier 6

Lightwave LDC–3900 modular laser diode controller), (4) pulse pattern generator (Agilent HP-70843B error performance analyzer), (5) signal analyzer – digital sampling oscilloscope (Tektronix TDS8000 Digital Sampling Oscilloscope), (6) erbium doped fiber amplifier, (7) personal computer with integrated General Purpose Interface Bus (GPIB) interface

Optical Chirpform Test Set), (2) semiconductor laser, (3) direct current source (ILX

wavelength. The rear facet is coated with a high reflectivity coating of 60 to 80%.

**2.2 Time-resolved frequency chirp measurement setup and methodology**

signals greater than 10 Gbps with the measurement repeatability of ~5%.

Trigger

4

LD 2

Signal

3

Pulse pattern generator

(plus index-coupled) distributed feedback.

An automated polarization controller helps to fix optimal polarization direction. That increases measurement accuracy. By changing the optical path delay time the interferometer can automatically fit the frequency transfer function to a particular carrier frequency of optical signal. The frequency transfer function of the Mach-Zehnder fiber interferometer is shown in Fig. 3. As can be seen, it is a sinusoidal function and its period is called a free spectral range (FSR). The optical signal frequency (chirp) variation should be lower than FSR/4=35 GHz. The frequency transfer function can be arbitrarily shifted with respect to the optical signal carrier frequency. And thus, during the first phase the frequency transfer function is shifted to the carrier frequency at the point "A", during the second phase – at the point "B" Fig. 3.

The frequency transfer function can be written as:

$$K\_{\text{total}} = \beta K(f) = \beta \frac{A}{2} \cos\left(\frac{2\pi f}{FSR}\right). \tag{1}$$

where *β* is a optical loss in the system and *A* is amplitude of transfer function.

It follows from the equation (1) that the transfer coefficient depends on carrier frequency. During the pulse modulation the carrier frequency varies in time as:

$$f(\mathbf{t}) = f\_{\text{cW}} + \Delta f(\mathbf{t}) \,. \tag{2}$$

The last term in equation (2) is a frequency chirp. At first, in order to measure the chirp of the LD or any other device, the optical beam should be passed to the interferometer not modulated, so that the carrier frequency and optical power should be constant. Then the carrier frequency is *f*cw. As was mentioned earlier, the transfer function is shifted to match *f*cw to the point "A" and after – to the point "B".

Fig. 3. Mach-Zehnder interferometer frequency transfer function

In both cases the transfer function will be shifted and the transfer coefficient will depend on Δ*f(t).*

Investigation of High-Speed Transient Processes

and Parameter Extraction of InGaAsP Laser Diodes 167

because the pulse length or the beginnings of the pulse edges fluctuates. The fiber interferometer and the scope also contribute to the noise level. As was explained, the output signal of the Mach-Zehnder interferometer is displayed on the digital scope screen as well as in the scope's internal memory. It is possible to make inter sample averaging (ensemble averaging); the scope can average up to 1024 samples. That possibility is very important for the periodic signal, because it helps to increase the S/N (signal to noise ratio) (no noise in Fig. 6).

Fig. 4. LD ceramic carrier together with LD, microstrip line and a microwave probe on it

0.0 0.2 0.4 0.6 0.8

Fig. 6. Chirp pulse plotted together with interferometer output pulses corresponding to both

t (ns)

IM-FM

IM+FM

0

chirp

4

Δf (GHz)

8

12

Fig. 5. Eye diagram of a not averaged (noisy) pulse train

0.0

0.2

0.4

0.6

P (mW)

frequency transfer function points

0.8

1.0

$$K\_A = \beta \left[ \frac{A}{2} + \frac{A}{2} \cos \left( \frac{2\pi \Delta f}{FSR} \right) \right] \tag{3}$$

$$K\_B = \beta \left[ \frac{A}{2} - \frac{A}{2} \cos \left( \frac{2 \pi \Delta f}{FSR} \right) \right]. \tag{4}$$

Thus the output signal power in both cases is:

$$P\_{outA}(t) = P\_{in}(t)\beta \left[\frac{A}{2} + \frac{A}{2}\cos\left(\frac{2\pi\Delta f(t)}{FSR}\right)\right],\tag{5}$$

$$P\_{outB}(t) = P\_{in}(t)\beta \left[\frac{A}{2} - \frac{A}{2}\cos\left(\frac{2\pi\Delta f(t)}{FSR}\right)\right].\tag{6}$$

In literature these two equations are usually called IM+FM and IM-FM, because in both of them there are two terms which correspond to power modulation and frequency modulation (chirp). So, in both measurement phases (points "A" and "B") the output power depends not only on the chirp but on the input signal power as well. However, after both measurements are done, the power modulation term can be cancelled:

$$\frac{P\_{outA}(t) - P\_{outB}(t)}{P\_{outA}(t) + P\_{outB}(t)} = \cos\left(\frac{2\pi\Delta f(t)}{FSR}\right) = P\_{calc}(t) \,\, . \tag{7}$$

There *P (t) calc* only contains information about chirp (carrier frequency deviations) but there is no information about power deviations. Further, the chirp can be easily obtained:

$$
\Delta f(t) = \frac{FSR}{2\pi} \arccos\left(P\_{\rm calc}(t)\right). \tag{8}
$$

Or, in case when Δ*f(t)*<<FSR/4, cosine can be treated as a line and the equation (8) becomes:

$$
\Delta f(t) = \frac{FSR}{2\pi} P\_{\text{calc}}(t) \,\text{.}\tag{9}
$$

In fact, the LD length is only 300 μm, so in order to make various measurements handily the LD needs to be put on some device carrier (Fig. 4). The device carrier is useful to easily connect a coaxial microwave probe because the LD is located on the microwave microstrip line. There is a serial resistor implanted in the microstrip line in order to match the microstrip line and LD impedances, so that the reflected parasitic signals are cancelled. The microwave cable is connected to a microstrip line by using the microwave probe (RF probe). The cut-off frequency of the microwave probe is ~ 50 GHz. For high frequency measurements any other DC interfaces composed of any LRC circuits should be disconnected.

The detected signal at the optical input of the oscilloscope is quite noisy Fig. 5. The optical pulse is attenuated and cumulates enough noise while it propagates throughout the system. Noise types and sources are different. There is a white amplitude noise, which continuously distributes through the length of the pulse; there is also a phase noise or jitter, which appears

*2 2 FSR* ⎡ ⎛ ⎞ Δ ⎤ = + ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦

*2 2 FSR* ⎡ ⎛ ⎞ Δ ⎤ = − ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦

> *2 2 FSR* ⎡ ⎛ ⎞ Δ ⎤

*2 2 FSR* ⎡ ⎛ ⎞ Δ ⎤ = − ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦

*calc*

− ⎛ ⎞ Δ = = ⎜ ⎟ + ⎝ ⎠ . (7)

*FSR Δf(t) arccos P (t) 2π* = . (8)

*FSR Δf(t) P (t) 2π* = . (9)

= + ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦

, (3)

. (4)

, (5)

. (6)

*A A 2π f K β cos*

*A A 2π f K β cos*

*A A 2π f (t) P (t) P (t)β cos*

*A A 2π f(t) P (t) P (t)β cos*

In literature these two equations are usually called IM+FM and IM-FM, because in both of them there are two terms which correspond to power modulation and frequency modulation (chirp). So, in both measurement phases (points "A" and "B") the output power depends not only on the chirp but on the input signal power as well. However, after both

*P (t) P (t) 2π f(t) cos P (t) P (t) P (t) FSR*

There *P (t) calc* only contains information about chirp (carrier frequency deviations) but there

Or, in case when Δ*f(t)*<<FSR/4, cosine can be treated as a line and the equation (8) becomes:

In fact, the LD length is only 300 μm, so in order to make various measurements handily the LD needs to be put on some device carrier (Fig. 4). The device carrier is useful to easily connect a coaxial microwave probe because the LD is located on the microwave microstrip line. There is a serial resistor implanted in the microstrip line in order to match the microstrip line and LD impedances, so that the reflected parasitic signals are cancelled. The microwave cable is connected to a microstrip line by using the microwave probe (RF probe). The cut-off frequency of the microwave probe is ~ 50 GHz. For high frequency measurements any other DC

The detected signal at the optical input of the oscilloscope is quite noisy Fig. 5. The optical pulse is attenuated and cumulates enough noise while it propagates throughout the system. Noise types and sources are different. There is a white amplitude noise, which continuously distributes through the length of the pulse; there is also a phase noise or jitter, which appears

( ) *calc*

*calc*

is no information about power deviations. Further, the chirp can be easily obtained:

*A*

*B*

*outA in*

*outB in*

measurements are done, the power modulation term can be cancelled:

*outA outB*

*outA outB*

interfaces composed of any LRC circuits should be disconnected.

Thus the output signal power in both cases is:

because the pulse length or the beginnings of the pulse edges fluctuates. The fiber interferometer and the scope also contribute to the noise level. As was explained, the output signal of the Mach-Zehnder interferometer is displayed on the digital scope screen as well as in the scope's internal memory. It is possible to make inter sample averaging (ensemble averaging); the scope can average up to 1024 samples. That possibility is very important for the periodic signal, because it helps to increase the S/N (signal to noise ratio) (no noise in Fig. 6).

Fig. 4. LD ceramic carrier together with LD, microstrip line and a microwave probe on it

Fig. 5. Eye diagram of a not averaged (noisy) pulse train

Fig. 6. Chirp pulse plotted together with interferometer output pulses corresponding to both frequency transfer function points

Investigation of High-Speed Transient Processes

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

t (ns)

a)

t (ns)

(Šermukšnis et al., 2005)

chirp (GHz)

(Šermukšnis et al., 2005)

a)

0

4

chirp (GHz)

8

12

and Parameter Extraction of InGaAsP Laser Diodes 169

The chirp pulse can be further used for LD analysis. Chirp and power pulses are shown below in Figs. 8a and 9a. The chirp pulse begins earlier than the power pulse, thus there is some time or phase shift between them. As current pulse is applied, the optical beam carrier frequency (as well as the wavelength) starts to change but the power still stays constant. The chirp parameter is extracted from both measured pulses, as explained in detail below.

power (a.u.)

Fig. 8. Chirp parameter extraction in case of a 2.5 Gbit/s modulation rate. Frequency chirp and optical beam power waveform a), linearised time-resolved frequency chirp waveform b)

0.21

Fig. 9. Chirp parameter extraction in case of a 10 Gbit/s modulation rate. Frequency chirp and optical beam power waveform a), linearised time-resolved frequency chirp waveform b)

In optoelectronic devices such as LDs or electro-optical modulators the chirp parameter should preferably be small. Besides, the chirp parameter extraction with good accuracy is not always possible. Dividing both sides of the equation (10) by the photon number *P* and after plotting *Δf /P* versus *2 P/P* , linear dependence of these two terms is expected. It was observed that in most cases such linear dependence actually occurs, furthermore, the linearity is of good quality (Figs. 8b and 9b). From the slope of that linear plot, the chirp parameter can be extracted. Considering that the photon number is proportional to the

experiment conditions not the photon number *P* but the optical beam power *PW* was used.

optical beam power *PW*, the term / *P P* in the equation (10) is equal to *P /P W W*

0.22

0.23

0.24

Power (a.u.)

Chirp/P (a.u.)

0.4 0.6 0.8 1.0 1.2 1.4 1.6

b)

0.25

0.26

b)

chirp/P (a.u.)

012345

(a.u.)

(a. u.)

. So, in real

P'/(P)2

P' / P2

Another effective method is a software high frequency filtering with *f*c=50 GHz which should be applied to the already averaged signal in the scope's memory. Such a filter is admissible because the cut-off frequency of the microwave system is also ~50 GHz, and so, all higher frequencies could be considered as parasitic.

After all these noise-related measurement problems were resolved, it became possible to make high-quality time-resolved chirp measurements. After both power pulses IM+FM and IM-FM were measured, by using the equation (9), the chirp pulse was obtained. All three pulses are shown in Fig. 6.

### **2.3 Time-resolved frequency chirp measurement results**

A typical chirp pulse is shown in Fig. 7a, which is referred to as total chirp. The relatively plain region in the chirp waveform is referred to as adiabatic chirp and oscillations (blue or red shifts) corresponding to both power pulse edges are referred to as transient chirp. There the total chirp is the actually observed chirp, which can be expressed as the sum of transient and adiabatic chirp. While the pulse form of the adiabatic chirp is the same as the form of the optical power pulse, the transient chirp is proportional to the optical power derivative (derivative of ln(P) (Kikuchi, 1990; Koyama & Suematsu, 1985):

$$
\Delta f(t) = \frac{a}{2\pi} \left[ \frac{\dot{P}(t)}{P(t)} + \mathcal{G}\_P P(t) \right],
\tag{10}
$$

where *P* is the photon number, *GP* is the nonlinear gain, α is the chirp parameter. The chirp parameter affects both the transient and the adiabatic chirp, while the nonlinear gain affects only the adiabatic chirp (equation (10)). The presence of the adiabatic term distinguishes the LD from the electro-optical modulators (EOM) due to nonlinear gain (Jeong & Park, 1997). While the electro-optical modulator chirp has only a transient part, the LD chirp has a transient and an adiabatic term (Fig. 7).

Fig. 7. LD time-resolved frequency chirp waveform of LD a) (Šermukšnis et al., 2005) and electro-optical modulator b)

Another effective method is a software high frequency filtering with *f*c=50 GHz which should be applied to the already averaged signal in the scope's memory. Such a filter is admissible because the cut-off frequency of the microwave system is also ~50 GHz, and so,

After all these noise-related measurement problems were resolved, it became possible to make high-quality time-resolved chirp measurements. After both power pulses IM+FM and IM-FM were measured, by using the equation (9), the chirp pulse was obtained. All three

A typical chirp pulse is shown in Fig. 7a, which is referred to as total chirp. The relatively plain region in the chirp waveform is referred to as adiabatic chirp and oscillations (blue or red shifts) corresponding to both power pulse edges are referred to as transient chirp. There the total chirp is the actually observed chirp, which can be expressed as the sum of transient and adiabatic chirp. While the pulse form of the adiabatic chirp is the same as the form of the optical power pulse, the transient chirp is proportional to the optical power derivative

*α P(t) Δf(t) G P(t) 2π P(t)*

parameter affects both the transient and the adiabatic chirp, while the nonlinear gain affects only the adiabatic chirp (equation (10)). The presence of the adiabatic term distinguishes the LD from the electro-optical modulators (EOM) due to nonlinear gain (Jeong & Park, 1997). While the electro-optical modulator chirp has only a transient part, the LD chirp has a

*P*

0

chirp (GHz)

Fig. 7. LD time-resolved frequency chirp waveform of LD a) (Šermukšnis et al., 2005) and

1

2

b)

3

α

, (10)

is the chirp parameter. The chirp

0.0 0.1 0.2 0.3 0.4 0.5 0.6

t (ns)

⎡ ⎤ = + ⎢ ⎥ ⎢⎣ ⎥⎦

all higher frequencies could be considered as parasitic.

**2.3 Time-resolved frequency chirp measurement results**

(derivative of ln(P) (Kikuchi, 1990; Koyama & Suematsu, 1985):

*Total chirp Transient chirp Adiabatic chirp*

where *P* is the photon number, *GP* is the nonlinear gain,

transient and an adiabatic term (Fig. 7).

0.0 0.2 0.4 0.6 0.8

electro-optical modulator b)

t (ns)

chirp (GHz)

a)

pulses are shown in Fig. 6.

The chirp pulse can be further used for LD analysis. Chirp and power pulses are shown below in Figs. 8a and 9a. The chirp pulse begins earlier than the power pulse, thus there is some time or phase shift between them. As current pulse is applied, the optical beam carrier frequency (as well as the wavelength) starts to change but the power still stays constant. The chirp parameter is extracted from both measured pulses, as explained in detail below.

Fig. 8. Chirp parameter extraction in case of a 2.5 Gbit/s modulation rate. Frequency chirp and optical beam power waveform a), linearised time-resolved frequency chirp waveform b) (Šermukšnis et al., 2005)

Fig. 9. Chirp parameter extraction in case of a 10 Gbit/s modulation rate. Frequency chirp and optical beam power waveform a), linearised time-resolved frequency chirp waveform b) (Šermukšnis et al., 2005)

In optoelectronic devices such as LDs or electro-optical modulators the chirp parameter should preferably be small. Besides, the chirp parameter extraction with good accuracy is not always possible. Dividing both sides of the equation (10) by the photon number *P* and after plotting *Δf /P* versus *2 P/P* , linear dependence of these two terms is expected. It was observed that in most cases such linear dependence actually occurs, furthermore, the linearity is of good quality (Figs. 8b and 9b). From the slope of that linear plot, the chirp parameter can be extracted. Considering that the photon number is proportional to the optical beam power *PW*, the term / *P P* in the equation (10) is equal to *P /P W W* . So, in real experiment conditions not the photon number *P* but the optical beam power *PW* was used.

Investigation of High-Speed Transient Processes

equations. The optical amplification coefficient,

that the optical amplification depends on the photon density:

transparency (corresponding to the onset of population inversion) and

emission in the rate equations was supplemented with the factor (1–

τ

γ

τ

γe=1/τ

**3. Theoretical models**

physical parameters.

In the third model, the term *N*/

recombination. The carrier lifetime

and can be used in the rate equations (14).

*dt ed*

e(*N*), because

γ

changed to *N*

1998):

(Carroll et al., 1998):

dispersion, particularly when the data transmission rate is high.

and Parameter Extraction of InGaAsP Laser Diodes 171

regular direct current modulation conditions the frequency chirp variation does not exceed 16 GHz in average. For a semiconductor laser with the wavelength of 1550 nm, that corresponds only to about ~0.1 nm variation in wavelength. Such wavelength variation compared to the total wavelength 1550 nm is negligible but even such small variation can noticeably affect the optical pulse propagation in the optical fiber due to chromatic

In this work four physical models were used. At first, a simplified physical model based on rate equations for carrier and photon densities was used for computer simulation (Šermukšnis et al., 2004a, 2004b). However, though some of the physical factors were not included in this model, the accuracy of modelling results for some DFB lasers was sufficiently good. In the later models, the rate equations were supplemented with additional

For the second physical model the nonlinear amplification factor *ε* was included in the rate

in the earlier model of simplified rate equations (Šermukšnis et al., 2004a, 2004b). Including

where *N* is the density of electrons, *P* is the density of photons, *N*0 is the carrier density at

amplification factor. Then the term of the number of photons generated by spontaneous

γ

( ) 2

where the first term *A* of the equation describes the non-radiative recombination, *B* is the coefficient of radiant interband recombination and *C* is the coefficient of Auger

In the fourth model we used the following rate equation for electron density (Carroll et al.,

*dN J N A BN CN α NN 1 εP P*

( ) ( )( ) *2*

s, where

product of the optical differential amplification, *g* [cm2], and the group velocity, *vg* (

α

*g N,P g 1* () ( ) = −= − − *0 0 εP g*( ) *NN 1*( ) *εP* , (11)

[cm3/s] was assumed to be equal to the

ε

ε*P*).

e(*N*) is the rate of electron recombination

s in the rate equation (Šermukšnis et al., 2004a, 2004b) is

*e N A BN CN* =+ + , (12)

s can be described by these parameters in such a way:

( ) ( )*2 1 s τ N A BN CN* − =+ + , (13)

*0*

=− + + − − − , (14)

α= *vgg*)

is the nonlinear

For simplicity and accuracy, only part of the single pulse, which is shown in Fig. 8a (area in the dash square), was taken for chirp parameter extraction. The other part of the pulse has a considerable noise and distortion at the "0" level of the chirp pulse. Better accuracy is also reached when a 10 Gbit/s bias current pulse sequence is generated and the corresponding chirp waveform is used (Fig. 9). There, in Fig. 9b, good linearity, as well as good chirp parameter extraction precision, can obviously be seen. Though bias current pulses were almost of rectangular form, as can be seen in Fig. 9a, both power and chirp pulses are of semi-sinusoidal form. Of course, that occurs when the modulation rate becomes comparable to the LD direct current modulation speed limits. By comparing these two cases it was observed that the chirp parameter extraction error does not exceed 10 %. Several tens of lasers were measured in such manner and the chirp parameter for all of them was extracted. It was found that this parameter of these lasers is dispersed between 2 and 2.8.

Besides the chirp parameter, time-resolved chirp pulse amplitude is also important. As it was shown above, there are large damped oscillations at the leading and trailing edges of both power and chirp pulses Figs. 8a and Fig. 10b. At a high modulation rate, these oscillations can increase several times, so that the ring amplitude becomes considerably larger than the distance between "1" and "0" levels. In that case, the rings comprise the large part of the total chirp, as well as have the considerable effect on the pulse propagation in the fiber.

Fig. 10. Chirp pulse ring amplitude dependence on damping rate a) (Šermukšnis et al., 2005) and chirp pulse comparison for two lasers with different damping rate b)

It is always desirable to have chirp rings smaller, particularly in these cases, when rings make the largest contribution to the pulse power. In order to find the chirp ring amplitude analytically, one should account for many factors. Besides, the analytical analysis of a large signal is especially complicated. The oscillation frequency and the oscillation amplitude damping rate nearly correspond to the oscillation-relaxation and damping frequency which are used in small signal analysis. Indeed, it was observed that the amplitude (amplitude of the first ring, which largely contributes to the transient chirp) and the damping rate of chirp rings strongly depends on a small signal damping frequency (extracted from RIN measurements) at constant power for different LDs (Fig. 10). It is smaller in lasers with a large damping rate (Fig. 10b).

To summarise, the adiabatic chirp of different LDs was in the region of 4.2 GHz to 5.6 GHz and the dynamic chirp in the region of 3.2 GHz to 10 GHz. Finally, it was observed that in regular direct current modulation conditions the frequency chirp variation does not exceed 16 GHz in average. For a semiconductor laser with the wavelength of 1550 nm, that corresponds only to about ~0.1 nm variation in wavelength. Such wavelength variation compared to the total wavelength 1550 nm is negligible but even such small variation can noticeably affect the optical pulse propagation in the optical fiber due to chromatic dispersion, particularly when the data transmission rate is high.

### **3. Theoretical models**

170 Semiconductor Laser Diode Technology and Applications

For simplicity and accuracy, only part of the single pulse, which is shown in Fig. 8a (area in the dash square), was taken for chirp parameter extraction. The other part of the pulse has a considerable noise and distortion at the "0" level of the chirp pulse. Better accuracy is also reached when a 10 Gbit/s bias current pulse sequence is generated and the corresponding chirp waveform is used (Fig. 9). There, in Fig. 9b, good linearity, as well as good chirp parameter extraction precision, can obviously be seen. Though bias current pulses were almost of rectangular form, as can be seen in Fig. 9a, both power and chirp pulses are of semi-sinusoidal form. Of course, that occurs when the modulation rate becomes comparable to the LD direct current modulation speed limits. By comparing these two cases it was observed that the chirp parameter extraction error does not exceed 10 %. Several tens of lasers were measured in such manner and the chirp parameter for all of them was extracted.

Besides the chirp parameter, time-resolved chirp pulse amplitude is also important. As it was shown above, there are large damped oscillations at the leading and trailing edges of both power and chirp pulses Figs. 8a and Fig. 10b. At a high modulation rate, these oscillations can increase several times, so that the ring amplitude becomes considerably larger than the distance between "1" and "0" levels. In that case, the rings comprise the large part of the total

> -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Fig. 10. Chirp pulse ring amplitude dependence on damping rate a) (Šermukšnis et al., 2005)

It is always desirable to have chirp rings smaller, particularly in these cases, when rings make the largest contribution to the pulse power. In order to find the chirp ring amplitude analytically, one should account for many factors. Besides, the analytical analysis of a large signal is especially complicated. The oscillation frequency and the oscillation amplitude damping rate nearly correspond to the oscillation-relaxation and damping frequency which are used in small signal analysis. Indeed, it was observed that the amplitude (amplitude of the first ring, which largely contributes to the transient chirp) and the damping rate of chirp rings strongly depends on a small signal damping frequency (extracted from RIN measurements) at constant power for different LDs (Fig. 10). It is smaller in lasers with a

To summarise, the adiabatic chirp of different LDs was in the region of 4.2 GHz to 5.6 GHz and the dynamic chirp in the region of 3.2 GHz to 10 GHz. Finally, it was observed that in

and chirp pulse comparison for two lasers with different damping rate b)

chirp (GHz)

0.0 0.2 0.4 0.6 0.8

time (ns)

Γ0 =14 GHz

Γ0 b) =8 GHz

It was found that this parameter of these lasers is dispersed between 2 and 2.8.

chirp, as well as have the considerable effect on the pulse propagation in the fiber.

30 40 50 60 70 80 90

Γ0 (109 rad/s)

large damping rate (Fig. 10b).

Δf

osc(GHz)

In this work four physical models were used. At first, a simplified physical model based on rate equations for carrier and photon densities was used for computer simulation (Šermukšnis et al., 2004a, 2004b). However, though some of the physical factors were not included in this model, the accuracy of modelling results for some DFB lasers was sufficiently good. In the later models, the rate equations were supplemented with additional physical parameters.

For the second physical model the nonlinear amplification factor *ε* was included in the rate equations. The optical amplification coefficient, α [cm3/s] was assumed to be equal to the product of the optical differential amplification, *g* [cm2], and the group velocity, *vg* (α = *vgg*) in the earlier model of simplified rate equations (Šermukšnis et al., 2004a, 2004b). Including that the optical amplification depends on the photon density:

$$\log\left(N\_{\prime}P\right) = \lg\left(1 - \varepsilon P\right) = \lg\left(N - N\_{0}\right)\left(1 - \varepsilon P\right) \tag{11}$$

where *N* is the density of electrons, *P* is the density of photons, *N*0 is the carrier density at transparency (corresponding to the onset of population inversion) and ε is the nonlinear amplification factor. Then the term of the number of photons generated by spontaneous emission in the rate equations was supplemented with the factor (1– ε*P*).

In the third model, the term *N*/τs in the rate equation (Šermukšnis et al., 2004a, 2004b) is changed to *N*γe(*N*), because γe=1/τs, where γe(*N*) is the rate of electron recombination (Carroll et al., 1998):

$$\text{CN}\_{\varepsilon}\text{(N)} = \text{A} + \text{BN} + \text{CN}^{2}\text{ (}\text{}^{\prime}\text{)}\tag{12}$$

where the first term *A* of the equation describes the non-radiative recombination, *B* is the coefficient of radiant interband recombination and *C* is the coefficient of Auger recombination. The carrier lifetime τs can be described by these parameters in such a way:

$$\text{tr}\_s\left(\text{N}\right) = \left(A + BN + CN^2\right)^{-1}\text{ \AA} \tag{13}$$

and can be used in the rate equations (14).

In the fourth model we used the following rate equation for electron density (Carroll et al., 1998):

$$\frac{dN}{dt} = \frac{J}{ed} - N\left(A + BN + CN^2\right) - a\left(N - N\_0\right)\left(1 - \varepsilon P\right)P\,\,\,\tag{14}$$

Investigation of High-Speed Transient Processes

τsτ

p [s2] *I*th

[mA]

Table 1. Experimental results used for simulation (Vasiliauskas et al., 2007)

Fig. 11. Optical power dependence on injection current (Vasiliauskas et al., 2007)

*I*

0 5 10 15 20 25 30 35 40 45 50 55

*I* [mA]

op *I*

1 *I*

th *I*

0

choosing the time step and the number of calculation loops.

*P*1

*P* [mW]

*P*0

Computer simulation was based on the fourth-order Runge-Kutta method for the system of differential equations. The values of photon and electron densities for different time moments were found from initial values of densities, determining other parameters and

No. DFB laser

**5. Simulation results**

and Parameter Extraction of InGaAsP Laser Diodes 173

1. O6j5 1.91⋅10-20 12.31 33.09 2736 10711 4066 2. H3k7 1.63⋅10-20 11.56 33.07 2569 10756 3958 3. G3k4 1.64⋅10-20 11.89 32.94 2642 10638 3982 4. J5k7 1.65⋅10-20 11.15 34.26 2478 11300 3927 5. U2i3 2.26⋅10-20 8.19 30.10 1820 10184 3282 6. R4j5 1.72⋅10-20 11.29 32.33 2509 10542 3827 7. E3k4 1.68⋅10-20 11.36 34.38 2524 11313 3967 8. G1l1 1.60⋅10-20 11.74 32.80 2609 10649 3929 9. P5i3 2.12⋅10-20 8.72 30.65 1938 10309 3311 10. T2s2 2.06⋅10-20 8.85 31.46 1967 10767 3449 11. S2i3 2.37⋅10-20 8.09 28.94 1798 9758 3104 12. R2r3 1.72⋅10-20 11.44 35.80 2542 11842 4069 13. T2i3 2.25⋅10-20 8.02 31.63 1782 10796 3262 14. Q1t1 1.62⋅10-20 11.12 36.82 2471 12282 4082 15. O1k1 1.63⋅10-20 10.50 35.32 2333 11809 3871 16. O2s2 1.99⋅10-20 8.74 32.34 1942 10951 3389

*I*op [mA]

*J*th [A/cm2]

*J*1 [A/cm2]

*J*0 [A/cm2]

where *J* is the density of injection current, *e* is the electron charge, *d* is the width of the semiconductor laser active region, *t* is the time. The first term of the rate equation (14) describes the number of injected carriers that passed the unit volume per unit time interval, the second one describes the spontaneous emission and the third term defines the number of photons generated by stimulated emission per unit time interval. An analogical equation was used for photon density:

$$\frac{dP}{dt} = \Gamma \mathbf{a} \left(\mathbf{N} - \mathbf{N}\_0\right) \left(\mathbf{1} - \varepsilon P\right) P - \frac{P}{\mathbf{r}\_P} + \Gamma \boldsymbol{\beta} \mathbf{B} \mathbf{N}^2 \tag{15}$$

where *τ*p is the lifetime of photons, *β* is the spontaneous emission factor, Γ is the optical confinement factor. The first term of the rate equation (15) is the number of photons generated by stimulated emission per unit time interval, the second term is the number of photons emitted from a resonator (output of the laser). The third term of the equation (15) shows the spontaneous emission contribution to the generated mode.

### **4. Used experimental results**

Experimental results of DFB semiconductor lasers were used for computer simulation. Not all laser parameters needed for simulation were known and some parameters were estimated from experimental results. The product of photon and electron lifetimes, the threshold current and the current at the 5 mW average optical power were known. The extinction ratio (ratio of the optical power level "1" to the level "0") was equal to 8.5 dB:

$$10\lg\left(P\_1/P\_0\right) = 8.5\text{ .}\tag{16}$$

Injection current values at levels "0" and "1" were found from the optical power dependence on injected current by using the condition (6) for the optical power of 5 mW. The active region width was estimated from the TEM image. We also had a possibility to calculate the injection current density from current measurements.

The experimentally obtained parameters of semiconductor DFB lasers, which were used for calculations (see Figs. 11-15 ):

These experimental results for all investigated lasers are shown in Table 1.

where *J* is the density of injection current, *e* is the electron charge, *d* is the width of the semiconductor laser active region, *t* is the time. The first term of the rate equation (14) describes the number of injected carriers that passed the unit volume per unit time interval, the second one describes the spontaneous emission and the third term defines the number of photons generated by stimulated emission per unit time interval. An analogical equation

( )( ) *2*

confinement factor. The first term of the rate equation (15) is the number of photons generated by stimulated emission per unit time interval, the second term is the number of photons emitted from a resonator (output of the laser). The third term of the equation (15)

Experimental results of DFB semiconductor lasers were used for computer simulation. Not all laser parameters needed for simulation were known and some parameters were estimated from experimental results. The product of photon and electron lifetimes, the threshold current and the current at the 5 mW average optical power were known. The extinction ratio (ratio of the optical power level "1" to the level "0") was equal to 8.5 dB:

Injection current values at levels "0" and "1" were found from the optical power dependence on injected current by using the condition (6) for the optical power of 5 mW. The active region width was estimated from the TEM image. We also had a possibility to

The experimentally obtained parameters of semiconductor DFB lasers, which were used for

• mean current (current, when the average optical power is equal to 5 mW), *I*op (different

• injection currents: *I*1 (level "1") and *I*0 (level "0"), found from the optical power

• dimensions of the active region of the DFB laser: *L* = 300 μm (channel length); *w* = 1.5 μm (width); *d* = 0.1 μm (thickness); *S* = *L*⋅*w* = 4.5⋅10-6 cm2 (area) (the same for all lasers);

• experimental pulse characteristics: optical power and chirp (different for each laser).

= − − −+ , (15)

1 0 10 lg(*P P* ) 8.5 = . (16)

Γ

is the optical

*0 dP P Γα NN 1 εP P ΓβBN*

where *τ*p is the lifetime of photons, *β* is the spontaneous emission factor,

shows the spontaneous emission contribution to the generated mode.

calculate the injection current density from current measurements.

• optical powers: *P*1 = 8.59 mW; *P*0 = 1.41 mW (the same for all lasers);

• injection current density (calculated by using active region dimensions);

These experimental results for all investigated lasers are shown in Table 1.

• product of lifetimes, *τ*s*τ*p (different for each laser); • threshold current, *I*th (different for each laser);

dependence on injected current (see Fig. 11);

*dt τp*

was used for photon density:

**4. Used experimental results**

calculations (see Figs. 11-15 ):

for each laser);

Table 1. Experimental results used for simulation (Vasiliauskas et al., 2007)

### **5. Simulation results**

Computer simulation was based on the fourth-order Runge-Kutta method for the system of differential equations. The values of photon and electron densities for different time moments were found from initial values of densities, determining other parameters and choosing the time step and the number of calculation loops.

Fig. 11. Optical power dependence on injection current (Vasiliauskas et al., 2007)

Investigation of High-Speed Transient Processes

λ

region width and *n*ph is the density of photons.

Fig. 13. Simulated electron density (*N*) and experimental chirp (

*P*(simul.) of optical power (Vasiliauskas et al., 2007) b)

(Vasiliauskas et al., 2007) a), and experimental *P*(exp.) and simulated pulse characteristics

the speed of light,

not exceed the limit of 10%.

and Parameter Extraction of InGaAsP Laser Diodes 175

*2 1/2*

− =

− =

is the wavelength of radiated light,

*1 ph 1/2 1/2 1/2 1/2 g 12 1 2 12 hc 1 1 1RR P log Ldwn λμ 2L R R R R 1R R*

*2 1/2*

*2 ph 1/2 1/2 1/2 1/2 g 12 1 2 12 hc 1 1 1RR P log Ldwn λμ 2L R R R R 1R R*

where *R*1 and *R*2 are the reflection coefficients of end mirrors, *h* is the Planck's constant, *c* is

the length of the laser active region, *d* is the thickness of the active region, *w* is the active

The best results were obtained based on the fourth physical model (equations (14) and (15)). The most precise match of experimental and simulation results for all investigated lasers with different characteristics was obtained. The mismatch of analyzed laser parameters does

( )

+ −

( )

+ −

μ

( )( )

( )( )

*2 1*

Δ

*f*) pulse characteristics

*1 2*

, (17)

, (18)

g is the group refraction index, *L* is

In the computer simulation the following unknown parameters were selected: the optical confinement factor Γ, the reflection coefficients from mirrors at the ends, *R*1 and *R*2 (Fukuda, 1999), the lifetime of photons, τp, the coefficient of optical amplification, α, the spontaneous emission factor, β, the carrier density at transparency, *N*0, the nonlinear amplification factor, ε*,* and the coefficients of recombination *A*, *B* and *C*.

The investigation of laser parameter interplay was done. Based on the simulation results obtained, the following technique for the estimation of semiconductor laser parameters was suggested:

The calculations for model verification were done, one of them being the investigation of watt-ampere characteristics (see Fig. 12). The threshold current was found as well. The results obtained were also compared to those of other authors (Agrawal & Dutta, 1993).

Fig. 12. Watt-ampere characteristics of a laser diode near the threshold of injection current for different spontaneous emission factors β(Vasiliauskas et al., 2007)

The output optical power of the semiconductor laser was calculated by using these equations (Agrawal & Dutta, 1993):

In the computer simulation the following unknown parameters were selected: the optical

The investigation of laser parameter interplay was done. Based on the simulation results obtained, the following technique for the estimation of semiconductor laser parameters was

• the frequency of the relaxation oscillation is defined by changing the coefficient of

• the calculated optical power values are approached to the experimental ones by

• by changing the nonlinear amplification factor, *ε1*, and the spontaneous emission factor, *β*, the relaxation oscillation amplitudes can be corrected; in addition, the shape of oscillations can also be corrected by changing the injection current front duration; • the recombination coefficient, *A,* and the density of electrons at transparency, *N*0, was

• the recombination coefficients *B* and *C* were chosen last. These coefficients have insignificant influence on laser dynamics, they only change the shape of the oscillations

The calculations for model verification were done, one of them being the investigation of watt-ampere characteristics (see Fig. 12). The threshold current was found as well. The results obtained were also compared to those of other authors (Agrawal & Dutta, 1993).

Fig. 12. Watt-ampere characteristics of a laser diode near the threshold of injection current

0 1000 2000 3000 4000 5000 6000 7000

Injection current density, A/cm2

The output optical power of the semiconductor laser was calculated by using these

(Vasiliauskas et al., 2007)

β = 10-5 β = 10-4 β = 10-3 β= 10

β

changed to optimize the static characteristics of the semiconductor laser;

, the reflection coefficients from mirrors at the ends, *R*1 and *R*2 (Fukuda,

, the carrier density at transparency, *N*0, the nonlinear amplification factor,

α

, the spontaneous

p, the coefficient of optical amplification,

confinement factor

emission factor,

suggested:

a little.

ε

Γ

τ

*,* and the coefficients of recombination *A*, *B* and *C*.

changing the optical confinement factor *Γ*;

for different spontaneous emission factors

10-5

10-4

10-3

10-2

Optical power, mW

10-1

100

101

equations (Agrawal & Dutta, 1993):

1999), the lifetime of photons,

β

optical amplification, *α*;

$$P\_1 = \frac{hc^2}{\lambda \mu\_\mathcal{g}} \frac{1}{2L} \log \frac{1}{R\_1 R\_2} \text{L} dwn\_{\text{ph}} \frac{\left(1 - R\_1\right) R\_2^{1/2}}{\left(R\_1^{1/2} + R\_2^{1/2}\right) \left(1 - R\_1^{1/2} R\_2^{1/2}\right)}\text{.}\tag{17}$$

$$P\_2 = \frac{hc^2}{\lambda \mu\_{\mathcal{S}}} \frac{1}{2L} \log \frac{1}{R\_1 R\_2} \text{L} dwn\_{\text{ph}} \frac{\left(1 - R\_2\right) R\_1^{1/2}}{\left(R\_1^{1/2} + R\_2^{1/2}\right) \left(1 - R\_1^{1/2} R\_2^{1/2}\right)},\tag{18}$$

where *R*1 and *R*2 are the reflection coefficients of end mirrors, *h* is the Planck's constant, *c* is the speed of light, λ is the wavelength of radiated light, μg is the group refraction index, *L* is the length of the laser active region, *d* is the thickness of the active region, *w* is the active region width and *n*ph is the density of photons.

The best results were obtained based on the fourth physical model (equations (14) and (15)). The most precise match of experimental and simulation results for all investigated lasers with different characteristics was obtained. The mismatch of analyzed laser parameters does not exceed the limit of 10%.

Fig. 13. Simulated electron density (*N*) and experimental chirp (Δ*f*) pulse characteristics (Vasiliauskas et al., 2007) a), and experimental *P*(exp.) and simulated pulse characteristics *P*(simul.) of optical power (Vasiliauskas et al., 2007) b)

Investigation of High-Speed Transient Processes

α

p = 1.5⋅10-12 s;

calculated (see Figs. 14 and 15).

3.91

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

*M* **s [%]**

O6j5

H3k7

3.80

H3k7

G3k4

J5k7

0.00 2.00 4.00 6.00 8.00 10.00 12.00

*M* **p [%]**

O6j5

7.99

4.90

5.93

5.43

6.01

5.03

8.41

8.42

7.78

9.53

8.98

7.81

7.47

9.66

8.37

9.19

5.606.69 6.79 6.57

G3k4

J5k7

τ

0.661; τ

j1 = 54⋅10-12 s;

τ

and Parameter Extraction of InGaAsP Laser Diodes 177

The most precise match of laser characteristics was obtained for the laser with the characteristics shown in Fig. 13a (chirp and density of electrons; the chirp is proportional to

β

The values of parameters for other lasers, the comparison of simulation and experimental results are represented in Tables 1, 2 and 3. In order to evaluate the accuracy of simulation results the following parameters, such as the frequency (*υ*) of relaxation oscillations, the time of oscillation decrement (*T*), "1" and "2" levels proportion (*l*) and the proportion of amplitude of first peak and level "1" (*a*) were compared (see Table 4). The averaged value of the mismatch (*M*) between the simulation and experimental results for these parameters was

Fig. 14. Averaged value of the mismatch of simulated laser parameters for different lasers:

F2k7

G1l1

P5i3

T2s2

S2i3

R2r3

T2i3

Q1t1

O1k1

O2s2

Fig. 15. Averaged value of the mismatch of simulated laser parameters for different lasers:

F2k7

G1l1

P5i3

T2s2

S2i3

R2r3

T2i3

Q1t1

O1k1

O2s2

*M*s is for electron density characteristic (Vasiliauskas et al., 2007)

U2i3

R4j5

E3k4

*M*p is for optical power characteristic (Vasiliauskas et al., 2007)

U2i3

R4j5

E3k4

j2 = 50⋅10-12 s; *A* = 2.5⋅108 cm/s; *B* = 9.1⋅10-12 cm3/s; *C* = 3.80⋅10-29 cm6/s.

5.88 9.09

> 7.06

9.03

7.00

8.41

8.54

7.43

5.02

Γ

= 1⋅10-4; *N*0 = 24.3⋅1017 cm-3; *ε* = 9.5⋅10-18;

= 0.55; *R*1 = 0.041; *R*2 =

the carrier density (Henry, 1982)) and Fig. 13b (optical power pulse characteristic).

Selected values for the matching of unknown parameters were:

= 16.8⋅10-7 cm3/s;

8.73 8.79 8.80

1. O6j5 0.55 0.04 0.664 1.5 16.8⋅10-7 1⋅10-4 24.3⋅1017 2. H3k7 0.49 0.08 0.80 2.0 24.2⋅10-7 1⋅10-3 27.0⋅1017 3. G3k4 0.49 0.08 0.80 2.0 23.6⋅10-7 1⋅10-3 27.0⋅1017 4. J5k7 0.33 0.08 0.80 3.0 29.6⋅10-7 1⋅10-3 26.0⋅1017 5. U2i3 0.37 0.08 0.80 2.0 21.6⋅10-7 1⋅10-4 14.0⋅1017 6. R4j5 0.67 0.08 0.80 1.8 21.0⋅10-7 1⋅10-2 25.0⋅1017 7. E3k4 0.32 0.08 0.80 3.4 29.0⋅10-7 1⋅10-3 27.0⋅1017 8. G1l1 0.42 0.08 0.80 3.0 26.0⋅10-7 1⋅10-2 28.0⋅1017 9. P5i3 0.61 0.08 0.80 2.0 20.0⋅10-7 1⋅10-2 23.0⋅1017 10. T2s2 0.57 0.08 0.80 1.5 17.0⋅10-7 1⋅10-3 19.0⋅1017 11. S2i3 0.38 0.08 0.80 1.9 22.0⋅10-7 1⋅10-3 17.0⋅1017 12. R2r3 0.66 0.08 0.80 2.1 19.0⋅10-7 1⋅10-3 24.0⋅1017 13. T2i3 0.41 0.08 0.80 2.0 29.0⋅10-7 1⋅10-2 15.0⋅1017 14. Q1t1 0.31 0.08 0.80 2.3 29.0⋅10-7 1⋅10-2 25.0⋅1017 15. O1k1 0.48 0.08 0.80 2.5 26.0⋅10-7 1⋅10-2 29.0⋅1017 16. O2s2 0.35 0.08 0.80 1.9 28.0⋅10-7 1⋅10-3 28.0⋅1017

Table 2. Fourth model: computer simulation results (1) (Vasiliauskas et al., 2007

Table 3. Fourth model: computer simulation results (2) (Vasiliauskas et al., 2007)

τ

1. O6j5 9.5⋅10-18 54 50 2.5⋅108 9.1⋅10-12 3.80⋅10-29 2. H3k7 11⋅10-18 56 53 2.8⋅108 8.4⋅10-12 3.25⋅10-29 3. G3k4 11⋅10-18 56 53 2.8⋅108 8.4⋅10-12 3.25⋅10-29 4. J5k7 15⋅10-18 50 50 3.6⋅108 4.7⋅10-12 2.44⋅10-29 5. U2i3 17⋅10-18 66 66 4.1⋅108 5.5⋅10-12 4.00⋅10-29 6. R4j5 16⋅10-18 55 50 4.8⋅108 9.6⋅10-12 3.00⋅10-29 7. E3k4 19⋅10-18 75 60 5.0⋅108 6.0⋅10-12 2.00⋅10-29 8. G1l1 14⋅10-18 60 40 2.0⋅108 3.0⋅10-12 4.00⋅10-29 9. P5i3 15⋅10-18 60 50 3.0⋅108 2.0⋅10-12 4.00⋅10-29 10. T2s2 14⋅10-18 50 50 2.0⋅108 3.0⋅10-12 4.00⋅10-29 11. S2i3 18⋅10-18 60 55 4.0⋅108 7.0⋅10-12 3.00⋅10-29 12. R2r3 11⋅10-18 50 50 4.0⋅108 5.0⋅10-12 4.00⋅10-29 13. T2i3 17⋅10-18 65 45 4.9⋅108 1.0⋅10-12 4.50⋅10-29 14. Q1t1 19⋅10-18 50 40 2.0⋅108 3.0⋅10-12 4.00⋅10-29 15. O1k1 17⋅10-18 50 45 4.0⋅108 2.0⋅10-12 2.00⋅10-29 16. O2s2 19⋅10-18 50 45 2.0⋅108 4.0⋅10-12 2.00⋅10-29

τj1 [ps] τp [ps] α[cm3/s]

j2 [ps] *A* [cm/s] *B* [cm3/s] *C* [cm6/s]

β

*N0* [cm-3]

No. DFB laser *Γ R*1 *R*2

No. DFB laser *ε*

The most precise match of laser characteristics was obtained for the laser with the characteristics shown in Fig. 13a (chirp and density of electrons; the chirp is proportional to the carrier density (Henry, 1982)) and Fig. 13b (optical power pulse characteristic).

Selected values for the matching of unknown parameters were: Γ = 0.55; *R*1 = 0.041; *R*2 = 0.661; τp = 1.5⋅10-12 s; α = 16.8⋅10-7 cm3/s; β = 1⋅10-4; *N*0 = 24.3⋅1017 cm-3; *ε* = 9.5⋅10-18; τj1 = 54⋅10-12 s; τj2 = 50⋅10-12 s; *A* = 2.5⋅108 cm/s; *B* = 9.1⋅10-12 cm3/s; *C* = 3.80⋅10-29 cm6/s.

The values of parameters for other lasers, the comparison of simulation and experimental results are represented in Tables 1, 2 and 3. In order to evaluate the accuracy of simulation results the following parameters, such as the frequency (*υ*) of relaxation oscillations, the time of oscillation decrement (*T*), "1" and "2" levels proportion (*l*) and the proportion of amplitude of first peak and level "1" (*a*) were compared (see Table 4). The averaged value of the mismatch (*M*) between the simulation and experimental results for these parameters was calculated (see Figs. 14 and 15).

Fig. 14. Averaged value of the mismatch of simulated laser parameters for different lasers: *M*s is for electron density characteristic (Vasiliauskas et al., 2007)

Fig. 15. Averaged value of the mismatch of simulated laser parameters for different lasers: *M*p is for optical power characteristic (Vasiliauskas et al., 2007)

Investigation of High-Speed Transient Processes

**7. References**

0013-5194

New York, USA

328-330, ISSN 0003-6951

pp. 1391-1397, ISSN 0018-9197

1997) pp. 605-606, ISSN 0013-5194

45-49, ISSN 0018-9197

and Parameter Extraction of InGaAsP Laser Diodes 179

Agrawal, G.P. & Dutta, N.K. (1993). *Semiconductor Lasers, 2nd Edition*, Van Nostarnd

Agrawal, G.P. (2002). *Fiber-Optic Communication Systems, 3nd Edition,* John Wiley & Sons,

Alder, T.; Stohr, A.; Heinzelmann, R. & Jager, D. (2000). High-Efficiency Fiber-to-Chip

Arakawa, Y. & Yariv, A. (1985). Fermi energy dependence of linewidth enhancement factor

Bergano, N. S. (1988). Wavelenght discriminator method for measuring dynamic chirp in

Bhattacharya, P; Klotzkin, D.; Qasaimeh, O.; Zhou, W.; Krishna, S. & Zhu, D. (2000). High-

Carroll, J.E.; Whiteaway, J.E.A. & Plumb, R.G.S. (1998). *Distributed feedback semiconductors lasers (IEE Circuits, Devices and Systems)*, IEE, ISBN<PHONE_NUMBER>, London, UK Fukuda, M. (1999). *Optical semiconductor devices*, John Wiley & Sons, ISBN 978-0-471-14959-0,

Harder, C.; Vahala, K. & Yariv, A. (1983). Measurement of the linewidth enhancement factor

Henry, C.H. (1982). Theory of the line width of semiconductor lasers. *IEEE Journal of Quantum Electroni*cs, Vol.18, No.2, (February 1982), pp. 259-264, ISSN 0018-9197 Henry, C. H. (1983). Theory of the Phase Noise and Power Spectrum of a Single Mode

Henry, C. H. (1986). Phase Noise in Semiconductor Lasers. *IEEE Journal of Lightwave*

Jeong, J. & Park, Y. K. (1997). Accurate Determination of Transient Chirp Parameter in High

Kikuchi, K. & Okoshi, T. (1985). Estimation of linewidth enhancement factor of AlGaAs

*Quantum Electron*ics, Vol.21, No.6 (June 1985), pp. 669 - 673, ISSN 0018-9197 Kikuchi, K.; Okoshi, T. & Kawai, T. (1984). Estimation of linewidth enhancement factor α of

*Electronics Lett*ers, Vol.20, No.11 (May 1984), pp. 450-451, ISSN 0013-5194 Kikuchi, K. (1990). Static Frequency Chirping in λ/4-Phase-Shifted Distributed-Feedback

Koyama, F. & Suematsu, Y. (1985). Analysis of Dynamic Spectral Width of Dynamic-Single-

*Technology,* Vol.4, No.3, (March 1986), pp. 298-311, ISSN 0733-8724

*Letters,* Vol.12, No.8, (August 2000), pp. 1016-1018, ISSN 1041-1135

*Electronics,* Vol. 6, No. 3, (May 2000), pp. 426-438, ISSN 1077-260X.

Coupling Using Low-Loss Tapered Single-Mode Fiber. *IEEE Photonics Technology*

of GaAlAs buried heterostructure lasers. *Applied Physics Lett*ers, Vol.47, No.9,

DFB lasers. *Electronics Lett*ers, Vol.24, No.20 (September 1988), pp. 1296-1297, ISSN

Speed Modulation and Switching Characteristics of In(Ga)As-Al(Ga)As Self-Organized Quantum-Dot Lasers. *IEEE Journal of Selected Topics in Quantum*

α of semiconductor lasers. *Applied Physics Lett*ers, Vol.42, No.4 (February 1983), pp.

Injection Laser. *IEEE Journal of Quantum Electronics,* Vol.19, No.9, (September 1983),

Speed Digital Lightwave Transmitters. *Electronics Letters,* Vol.33, No.7, (March

lasers by correlation measurement between FM and AM noises. *IEEE Journal of*

CSP-type AlGaAs lasers from measured correlation between AM and FM noises.

Semiconductor Lasers: Influence of Carrier-Density Nonuniformity Due to Spatial Hole Burning. *IEEE Journal of Quantum Electronics,* Vol.26, No.1, (January 1990), pp.

Mode (DSM) Lasers and Related Transmission Bandwidth of Single-Mode. *IEEE Journal of Quantum Electronics,* Vol.21, No.4, (April 1985), pp. 292-297, ISSN 0018-9197

Reinhold Co., ISBN 0-442-01102-4, New York, USA

Inc., ISBN 0-471-21571-6, New York, USA.

(November 1985), pp. 905-907, ISSN 0003-6951

Table 4. Mismatch (%) of simulation data to experimental results (Vasiliauskas et al., 2007)

### **6. Conclusion**

In this work the time-resolved frequency chirp characteristics under pulse operation of DFB LD were investigated. The optical beam carrier frequency (wavelength) deviations were measured by using a fiber Mach-Zehnder interferometer. The pulse current modulation at the 5 GHz and 1.25 GHz rate (10 Gbps and 2.5 Gbps) was achieved and the optical power and time-resolved frequency chirp pulses were investigated. LDs with a higher damping rate are preferable because of lower oscillation amplitudes at pulse edges, both for power pulses and for chirp pulses. It was found that at common operation conditions, i.e. average optical power *P*w=5 mW and extinction ratio of 8.5 dB, the adiabatic chirp does not exceed 4.2-5.6 GHz and the transient chirp (amplitude of the first peak)- 3.2-10 GHz. To summarize, total chirp does not exceed 16 GHz. The chirp parameter (also known as the linewidth enhancement factor) was extracted from time-resolved frequency chirp measurements, which was found to be in the range of 2-2.8 for various devices.

The transient characteristics of semiconductor lasers were investigated. The unknown laser parameters which were not measured in the experiment were obtained by using computer simulation. Based on these simulation results a parameter estimation technique was suggested. The rate equations used for computer simulation were improved by including new terms which were not used in earlier simulations (Šermukšnis et al., 2004a, 2004b). Transient processes were simulated for a large number of similar DFB lasers. The best results were obtained by using the fourth physical model. The mismatch between the simulation and experimental results for all analyzed laser parameters in this case did not exceed the limit of 10%.

### **7. References**

178 Semiconductor Laser Diode Technology and Applications

1. O6j5 2.33 2.02 8.60 7.43 1.33 4.85 3.39 0.90 3.91 3.80 2. H3k7 1.35 7.44 9.72 9.90 9.61 7.33 1.74 7.29 5.60 7.99 3. G3k4 1.49 5.38 9.84 8.22 9.36 4.08 6.04 1.92 6.69 4.90 4. J5k7 2.14 9.98 9.97 8.25 8.37 4.50 6.67 0.98 6.79 5.93 5. U2i3 0.90 4.81 9.75 9.84 9.62 3.94 6.02 3.12 6.57 5.43 6. R4j5 9.32 6.56 9.84 9.21 6.40 6.20 9.35 2.08 8.73 6.01 7. E3k4 5.51 4.96 9.87 9.76 9.85 2.19 9.93 3.19 8.79 5.03 8. G1l1 3.43 8.01 9.78 9.97 1.59 8.96 8.72 6.74 5.88 8.42 9. P5i3 6.59 9.89 9.97 9.68 9.89 4.57 9.93 6.98 9.09 7.78 10. T2s2 9.82 9.33 9.43 9.71 2.89 9.83 6.11 9.26 7.06 9.53 11. S2i3 8.07 7.97 9.51 9.75 9.09 8.92 9.43 9.28 9.03 8.98 12. R2r3 9.96 9.91 6.64 8.74 1.79 3.59 9.62 8.99 7.00 7.81 13. T2i3 9.65 9.82 8.60 9.09 9.63 6.67 5.77 4.30 8.41 7.47 14. Q1t1 9.55 9.93 9.74 9.87 4.95 9.86 9.93 8.99 8.54 9.66 15. O1k1 8.46 9.83 9.12 9.78 9.80 4.36 2.34 9.52 7.43 8.37 16. O2s2 3.88 9.81 9.71 8.31 4.40 9.54 2.10 9.09 5.02 9.19 Table 4. Mismatch (%) of simulation data to experimental results (Vasiliauskas et al., 2007)

In this work the time-resolved frequency chirp characteristics under pulse operation of DFB LD were investigated. The optical beam carrier frequency (wavelength) deviations were measured by using a fiber Mach-Zehnder interferometer. The pulse current modulation at the 5 GHz and 1.25 GHz rate (10 Gbps and 2.5 Gbps) was achieved and the optical power and time-resolved frequency chirp pulses were investigated. LDs with a higher damping rate are preferable because of lower oscillation amplitudes at pulse edges, both for power pulses and for chirp pulses. It was found that at common operation conditions, i.e. average optical power *P*w=5 mW and extinction ratio of 8.5 dB, the adiabatic chirp does not exceed 4.2-5.6 GHz and the transient chirp (amplitude of the first peak)- 3.2-10 GHz. To summarize, total chirp does not exceed 16 GHz. The chirp parameter (also known as the linewidth enhancement factor) was extracted from time-resolved frequency chirp measurements,

The transient characteristics of semiconductor lasers were investigated. The unknown laser parameters which were not measured in the experiment were obtained by using computer simulation. Based on these simulation results a parameter estimation technique was suggested. The rate equations used for computer simulation were improved by including new terms which were not used in earlier simulations (Šermukšnis et al., 2004a, 2004b). Transient processes were simulated for a large number of similar DFB lasers. The best results were obtained by using the fourth physical model. The mismatch between the simulation and experimental results for all analyzed laser parameters in this case did not

which was found to be in the range of 2-2.8 for various devices.

Mismatch of simulation parameters [%] *υ*s *υ*p *T*s *T*p *l*s *l*p *a*s *a*p *M*s *M*p

No. DFB laser

**6. Conclusion**

exceed the limit of 10%.

**10**

*Brazil*

�M� � �

Niklaus Ursus Wetter *Centro de Lasers e Aplicações,*

**Spectral Narrowing and Brightness**

*Instituto de Pesquisas Energéticas e Nucleares de São Paulo,*

**Increase in High Power Laser Diode Arrays**

Diode laser arrays, also called diode bars, are very important light sources that are generally used for pumping of other solid-state lasers and in medical and industrial applications that require high power but low intensity. Although they are very interesting from a commercial point of view, they did not find their way into many important applications because of their

This book chapter will explain why diodes have such a low spectral and spatial beam

In recent years, technology has achieved advances that permit to compare the spatial and spectral brightness of diode lasers to that of crystal solid-state lasers. Recent results of the literature are spatial brightness of 10� MW/cm2str with a beam quality product of M�

20 and narrowing of a diode-arrays' emission bandwidth to the GHz or even MHz range. Diode laser arrays are nowadays the basic building blocks of high power semiconductor lasers. They can continuously emit up to approximately 100 W at specific visible to near infrared wavelengths and can be stacked to give even much higher output powers of the order of several thousand watts. They find their main applications in materials processing, medical applications, printing and optical pumping of solid state lasers. Amongst the many industrial processing applications are welding, hardening, brazing and sintering. Medical and dental applications for diode bars include surgery, hair- and tattoo removal, photodynamic therapy and tooth whitening. Diode bars are also widely used in the military for targeting applications and therefore the distribution of some high power diode arrays is

Diode arrays present small size, efficiency in excess of 50 %, directly convert electricity into light and are the cheapest lasers in terms of watts per dollar on the market. For these reasons it would be of great benefit for commercial and scientific applications if they could substitute the common and expensive gas and crystal lasers that use resonators which are cumbersome to align and generally require highly skilled assembly and maintenance personnel. The reason that diode arrays so far did not substitute gas and crystal lasers in

almost all applications lies in their low spectral and spatial beam quality.

quality and give an overview on the different techniques to improve it.

**1. Introduction**

controlled.

very low beam quality and spectral brightness.

## **Spectral Narrowing and Brightness Increase in High Power Laser Diode Arrays**

Niklaus Ursus Wetter

*Centro de Lasers e Aplicações, Instituto de Pesquisas Energéticas e Nucleares de São Paulo, Brazil*

### **1. Introduction**

180 Semiconductor Laser Diode Technology and Applications

Larsson, A.; Carlsson, C.; Gustavsson, J.; Haglund, A; Modh, P. & Bengtsson, J. (2004). Direct

Laverdiere, C.; Fekecs, A. & Tetu, M. A. (2003). New Method for Measuring Time-Resolved

Linke, R. A. (1985). Modulation Induced Transient Chirping in Single Frequency Lasers.

Lu, H.; Makino T. B & Li, G. P. (1995). Dynamic Properties of Partly Gain-Coupled 1.55-μm

Olsen, C. M. & Lin, C. (1989). Time-Resolved Chirp Evaluations of Gbit/s NRZ and Gain-

Shani, Y.; Henry, C. H.; Kistler, R. C.; Orlowski, K. J. & Ackerman, D. A. (1989). Efficient

Šermukšnis, E.; Vyšniauskas, J.; Vasiliauskas, T. & Palenskis, V. (2004a). Computer

Šermuksnis, E.; Vyšniauskas, J. & Palenskis, V. (2005). Dynamic characteristics and

Tammela, S.; Ludvigsen, H. & Kaivola, M. (1997). Time-Resolved Frequency Chirp

Tatham, M. C.; Lealman, I. F.; Seltzer, C. P.; Westbrook, L. D. & Cooper, D. M. (1992).

Vasiliauskas, T.; Butkus, V.; Šermukšnis, E.; Palenskis, V. & Vyšniauskas, J. (2007).

Welford, D. A. (1985). Rate Equation Analysis for the Frequency Chirp to Modulated Power

No.4, (April 1997), pp. 475-477, ISSN 1041-1135

No.11, (November 1985), pp. 1749-1751, ISSN 0018-9197

No.3, (March 2003), pp. 446-448, ISSN 1041-1135

ISSN 1367-2630

0018-9197

0867-6747

1648-8504

9780819459534

408-414, ISSN 0018-9197

1443-1995, ISSN 0018-9197

1338, ISSN 0013-5194

pp. 2389-2391, ISSN 0003-6951

High-Frequency Modulation of VCSELs and Applications in Fibre Optic RF and Microwave Links. *New Journal of Physics,* Vol. 6, No. 1, (November 2004), pp. 1-17,

Frequency Chirp of High Bit Rate Source. *IEEE Photonics Technology Letters,* Vol.15,

*IEEE Journal of Quantum Electronics,* Vol.21, No. 6, (June 1985), pp. 593-597, ISSN

DFB Lasers. *IEEE Journal of Quantum Electronics,* Vol. 31, No. 8, (August 1995), pp.

Switched DFB Laser Pulses Using Narrowband Fabry-Perot Spectrometer. *Electronics Letters,* Vol.25, No.16, (August 1989), pp. 1018-1019, ISSN 0013-5194 Saunders, R. A.; King, J. P. & Hardcastle, I. (1994). Wideband Chirp Measurement Technique

for High Bit Rate Sources. *Electronics Letters,* Vol.30, No16, (August 1994), pp. 1336-

Coupling of a Semiconductor Laser to an Optical Fiber by Means of a Tapered Waveguide on Silicon. *Applied Physics Letters,* Vol. 55, No.23 (December 1989),

Simulation of High Frequency Modulation of Laser Diode Radiation. *Lithuanian Journal of Physics*, Vol. 44, No.6, (December 2004), pp. 415-420, ISSN 1648-8504 Šermukšnis, E.; Vyšniauskas, J.; Palenskis, V.; Matukas, J. & Pralgauskaitė, S. (2004b).

Dynamic characteristics of gain-coupled InGaAsP laser diodes and their reliability. *Kwartalnik elektroniki i telekomunikacji*, Vol.50, No.4, (April 2004), pp. 591-603, ISSN

reliability of gain-coupled InGaAsP laser diodes. *Proceedings of SPIE, Optical Materials and Applications*, Vol.5946, No.8, (August 2005), pp. 300-308, ISBN

Measurement Using a Silicon-Wafer Etalon. *IEEE Photonics Technology Letters,* Vol.9,

Resonance frequency, damping, and differential gain in 1.5 μm multiple quantumwell lasers. *IEEE Journal of Quantum Electronics,* Vol. 28, No.2, (February 1992), pp.

Computer Simulation of Transient Processes in DBF Semiconductor Lasers. *Lithuanian Journal of Physics*, Vol. 47, No.4, (December 2007), pp. 397-402, ISSN

Ratio of a Semiconductor Diode Laser. *IEEE Journal of Quantum Electronics,* Vol.21,

Diode laser arrays, also called diode bars, are very important light sources that are generally used for pumping of other solid-state lasers and in medical and industrial applications that require high power but low intensity. Although they are very interesting from a commercial point of view, they did not find their way into many important applications because of their very low beam quality and spectral brightness.

This book chapter will explain why diodes have such a low spectral and spatial beam quality and give an overview on the different techniques to improve it.

In recent years, technology has achieved advances that permit to compare the spatial and spectral brightness of diode lasers to that of crystal solid-state lasers. Recent results of the literature are spatial brightness of 10� MW/cm2str with a beam quality product of M� �M� � � 20 and narrowing of a diode-arrays' emission bandwidth to the GHz or even MHz range.

Diode laser arrays are nowadays the basic building blocks of high power semiconductor lasers. They can continuously emit up to approximately 100 W at specific visible to near infrared wavelengths and can be stacked to give even much higher output powers of the order of several thousand watts. They find their main applications in materials processing, medical applications, printing and optical pumping of solid state lasers. Amongst the many industrial processing applications are welding, hardening, brazing and sintering. Medical and dental applications for diode bars include surgery, hair- and tattoo removal, photodynamic therapy and tooth whitening. Diode bars are also widely used in the military for targeting applications and therefore the distribution of some high power diode arrays is controlled.

Diode arrays present small size, efficiency in excess of 50 %, directly convert electricity into light and are the cheapest lasers in terms of watts per dollar on the market. For these reasons it would be of great benefit for commercial and scientific applications if they could substitute the common and expensive gas and crystal lasers that use resonators which are cumbersome to align and generally require highly skilled assembly and maintenance personnel. The reason that diode arrays so far did not substitute gas and crystal lasers in almost all applications lies in their low spectral and spatial beam quality.

Spectral Narrowing and Brightness Increase in High Power Laser Diode Arrays 183

prone to the emission of light. Additionally, the quantum well is flanked by two layers of different p-type materials on one side and two layers of n-type materials on the other side. As one moves farther away from the junction, each layer is made of a material with less high refractive index in order to confine the light generated by the recombination process into a very thin layer around the junction called active area. The refractive index graduation can be achieved for example by co-doping the GaAs semiconductor with small amounts of aluminum (wavelength range 630 nm to 1 μm). Typical dimensions for these layers are 0.01 μm for the quantum well layer and 1 μm for the layers of high index p-type and n-type materials closest to the quantum well. Due to the small dimension of the active area perpendicularly to the junction, all semiconductor diodes are essentially single transverse

As with other lasers, laser action is achieved by giving a preferred direction to the emission of light generated in the active region and by building a resonator. In a laser diode this is generally done by fabricating the metal contact on top of the diode in the form of a long and thin ribbon. Recombination occurs within the semiconductor underneath this contact and preferential emission is achieved in the strip's direction, much as with a solid-state laser rod. Next, the semiconductor crystal is cleaved at its two ends to form a Fabry-Perot resonator. Depending on desired gain and spectral characteristics of the laser diode the end-facets maybe anti-reflection coated or uncoated, in which case they have a reflectance of approximately 30% in the visible region. The width of the active region parallel to the junction is generally of several microns and the length is approximately one millimeter.

Due to the small height of the active region, the stimulated emission perpendicular to the layers is strongly diffracted at the exit facet into an angle of approximately 25 degrees (FWHM) with Gaussian beam quality (fast axis direction). Parallel to the layers, the beam quality depends on the width of the electric contact strip and on the number of strips whereas the divergence angle remains generally around 10 degrees (slow axis direction). The fast axis divergence angle is always larger than in the slow direction because of its smaller dimension and also because of the larger index variation along the fast axis direction which causes larger numerical aperture (NA). Very small strips (width of 2 – 7 μm) lead to single transverse mode with overall diffraction limited beam quality and elliptical beam proportions but have also small power, not suitable for high power beam applications. Furthermore these devices can be made single frequency by limiting the longitudinal modes using suitable wavelength selective devices such as gratings that can be incorporated on chip or externally to the diode. The maximum output power for these devices is around a couple of hundreds of milliwatts for commercial devices although close to one watt has been achieved (Wenzel et al., 2008). Further power enhancement is obtained using an amplifier at the end of the diode in the form of a MOPA (master oscillator and power amplifier), which is a series of more or less independent devices. But also tapered amplifiers are common that employ the whole device including amplifier as resonator. The latter device achieved recently single transverse mode (TEM00) output powers in excess of 10 W at slightly reduced

beam quality, but such devices are mostly only experimental (Muller et al., 2011).

what is more important, spatial or spectral brightness or a combination of both.

Single mode diode lasers have the highest spectral and spatial beam brightness. If higher powers are required the fabrication technology goes different ways and one has to decide

mode in that direction.

A single diode bar is a linear array of approximately 19–70 individual diode emitters. The overall emitting area of a diode bar has generally a width of 1 cm and is 1 μm high. Due to the small height, comparable to the emission wavelength, the emitted beam diverges strongly in this direction (fast divergence axis) whereas in the other direction, given by the array of emitters, the divergence is smaller (slow divergence axis). As a result, the combined output beam is almost diffraction limited in the fast direction but has divergence that is more than thousand times that of a diffraction limited beam in the other direction.

This very poor beam quality hampers the usefulness of diode bars in many applications. Specifically, when trying to focus the beam tightly, very different waist sizes, waist positions and depth of foci are obtained for the fast and slow axis, making it almost impossible to tightly focus the diode's beam and making them resemble more flash light lamps than collimated laser beams.

Another reason that hinders the usefulness of diode beams is their spatial characteristics. From an application point of view it is in most cases interesting that the emission frequency is stable and as narrow as possible. Applications include pumping of solid-state lasers and laser cooling of gases for atomic clocks. Diodes not only change their emission wavelength strongly as a function of temperature but they also show linewidth broadening as a function of pump current. Typical linewidth are of the order of several nanometers and changes with temperature are around 0.25 nm/oC.

### **2. Diode bar fabrication and design**

The main difference between a semiconductor laser and a conventional solid state laser is that the diode laser converts directly electric power into light without the need of an optical pump. It therefore immediately achieves much higher wall-plug efficiency than any other solid-state laser (around 50% wall-plug efficiency, although more than 70 % have been achieved at room temperature (Knigge et al., 2005)). That combined with its ability to be easily fabricated in mass production makes the semiconductor laser the cheapest laser per Watt of output power and the preferred laser candidate whenever its beam and spectral characteristics are apt for the desired application.

The constructive basics of a simple laser diode are well known. A semiconductor diode laser, or simply laser diode, is formed by doping thin layers on a direct band-gap semiconductor crystal wafer to produce n-type and p-type regions on top of each other, resulting in a p-n junction. The favored crystal growth techniques for semiconductors are liquid phase epitaxy and chemical vapor deposition using metal-organic reagents (MOCVD) but also molecular beam epitaxy (MBE) is frequently used. Forward electric bias across the junction forces holes and electrons to enter the depletion region (a region void of carriers that forms close to the junction because of the difference in electric potential between the ntype and p-type semiconductor). There, the carriers recombine generating preferentially spontaneous emission.

However, in order to achieve better efficiency than with this simple model, the commonly preferred constructive method since the 1990s is the separate confinement heterostructure (SCH) quantum well laser diode. This laser has a quantum well layer at the junction center made of high index material that serves to concentrate electrons in energy states that are

A single diode bar is a linear array of approximately 19–70 individual diode emitters. The overall emitting area of a diode bar has generally a width of 1 cm and is 1 μm high. Due to the small height, comparable to the emission wavelength, the emitted beam diverges strongly in this direction (fast divergence axis) whereas in the other direction, given by the array of emitters, the divergence is smaller (slow divergence axis). As a result, the combined output beam is almost diffraction limited in the fast direction but has divergence that is

This very poor beam quality hampers the usefulness of diode bars in many applications. Specifically, when trying to focus the beam tightly, very different waist sizes, waist positions and depth of foci are obtained for the fast and slow axis, making it almost impossible to tightly focus the diode's beam and making them resemble more flash light lamps than

Another reason that hinders the usefulness of diode beams is their spatial characteristics. From an application point of view it is in most cases interesting that the emission frequency is stable and as narrow as possible. Applications include pumping of solid-state lasers and laser cooling of gases for atomic clocks. Diodes not only change their emission wavelength strongly as a function of temperature but they also show linewidth broadening as a function of pump current. Typical linewidth are of the order of several nanometers and changes with

The main difference between a semiconductor laser and a conventional solid state laser is that the diode laser converts directly electric power into light without the need of an optical pump. It therefore immediately achieves much higher wall-plug efficiency than any other solid-state laser (around 50% wall-plug efficiency, although more than 70 % have been achieved at room temperature (Knigge et al., 2005)). That combined with its ability to be easily fabricated in mass production makes the semiconductor laser the cheapest laser per Watt of output power and the preferred laser candidate whenever its beam and spectral

The constructive basics of a simple laser diode are well known. A semiconductor diode laser, or simply laser diode, is formed by doping thin layers on a direct band-gap semiconductor crystal wafer to produce n-type and p-type regions on top of each other, resulting in a p-n junction. The favored crystal growth techniques for semiconductors are liquid phase epitaxy and chemical vapor deposition using metal-organic reagents (MOCVD) but also molecular beam epitaxy (MBE) is frequently used. Forward electric bias across the junction forces holes and electrons to enter the depletion region (a region void of carriers that forms close to the junction because of the difference in electric potential between the ntype and p-type semiconductor). There, the carriers recombine generating preferentially

However, in order to achieve better efficiency than with this simple model, the commonly preferred constructive method since the 1990s is the separate confinement heterostructure (SCH) quantum well laser diode. This laser has a quantum well layer at the junction center made of high index material that serves to concentrate electrons in energy states that are

more than thousand times that of a diffraction limited beam in the other direction.

collimated laser beams.

spontaneous emission.

temperature are around 0.25 nm/oC.

**2. Diode bar fabrication and design**

characteristics are apt for the desired application.

prone to the emission of light. Additionally, the quantum well is flanked by two layers of different p-type materials on one side and two layers of n-type materials on the other side. As one moves farther away from the junction, each layer is made of a material with less high refractive index in order to confine the light generated by the recombination process into a very thin layer around the junction called active area. The refractive index graduation can be achieved for example by co-doping the GaAs semiconductor with small amounts of aluminum (wavelength range 630 nm to 1 μm). Typical dimensions for these layers are 0.01 μm for the quantum well layer and 1 μm for the layers of high index p-type and n-type materials closest to the quantum well. Due to the small dimension of the active area perpendicularly to the junction, all semiconductor diodes are essentially single transverse mode in that direction.

As with other lasers, laser action is achieved by giving a preferred direction to the emission of light generated in the active region and by building a resonator. In a laser diode this is generally done by fabricating the metal contact on top of the diode in the form of a long and thin ribbon. Recombination occurs within the semiconductor underneath this contact and preferential emission is achieved in the strip's direction, much as with a solid-state laser rod. Next, the semiconductor crystal is cleaved at its two ends to form a Fabry-Perot resonator. Depending on desired gain and spectral characteristics of the laser diode the end-facets maybe anti-reflection coated or uncoated, in which case they have a reflectance of approximately 30% in the visible region. The width of the active region parallel to the junction is generally of several microns and the length is approximately one millimeter.

Due to the small height of the active region, the stimulated emission perpendicular to the layers is strongly diffracted at the exit facet into an angle of approximately 25 degrees (FWHM) with Gaussian beam quality (fast axis direction). Parallel to the layers, the beam quality depends on the width of the electric contact strip and on the number of strips whereas the divergence angle remains generally around 10 degrees (slow axis direction). The fast axis divergence angle is always larger than in the slow direction because of its smaller dimension and also because of the larger index variation along the fast axis direction which causes larger numerical aperture (NA). Very small strips (width of 2 – 7 μm) lead to single transverse mode with overall diffraction limited beam quality and elliptical beam proportions but have also small power, not suitable for high power beam applications. Furthermore these devices can be made single frequency by limiting the longitudinal modes using suitable wavelength selective devices such as gratings that can be incorporated on chip or externally to the diode. The maximum output power for these devices is around a couple of hundreds of milliwatts for commercial devices although close to one watt has been achieved (Wenzel et al., 2008). Further power enhancement is obtained using an amplifier at the end of the diode in the form of a MOPA (master oscillator and power amplifier), which is a series of more or less independent devices. But also tapered amplifiers are common that employ the whole device including amplifier as resonator. The latter device achieved recently single transverse mode (TEM00) output powers in excess of 10 W at slightly reduced beam quality, but such devices are mostly only experimental (Muller et al., 2011).

Single mode diode lasers have the highest spectral and spatial beam brightness. If higher powers are required the fabrication technology goes different ways and one has to decide what is more important, spatial or spectral brightness or a combination of both.

Spectral Narrowing and Brightness Increase in High Power Laser Diode Arrays 185

Output power scales with fill factor in part because the maximum power density inside a diode bar is limited by the onset of catastrophic failure on the diodes exit facet. This failure is due to residual absorption at the beam exit facet caused by surface states at the cleavage plane that do not exist within the bulk. This problem hinders the maximum achievable power particularly in GaAs-based semiconductors and a great deal of effort has been spent in recent years to overcome this residual absorption by "passivation" of the exit facets.

The diode bar is normally soldered onto a thin submount which also serves as electrical contact. This submount is then attached to a large heatsink for effective removal of the heat. During this process the very thin (0.1 mm) but large (1 cm) diode array is easily bend by the pressure applied to bond it to the submount and heatsink but also by the differential expansion during the solder process. The overall effect of this process causes a curvature of the array called "smile" which amounts to about 1 -10 μm difference in height at the center of the bar with respect to the bar ends. Some manufactures also bond the bar directly to the copper heat-sink, which is referred to as "direct bonding" and

The problem with array curvature is that it degrades the beam brightness. Once the diode is manufactured, the bar curvature is fixed. Therefore it is of importance to understand the constructive details of diode bar manufacturing to apply effective countermeasures. The deviation from linearity of the smile makes it impossible for a single collimation lens in front of the emitters to be perfectly positioned for all emitters along the bar. This first lens, which serves to collimate the fast-axis (FAC lens), is attached directly in front of the diode bar and is generally a plano-aspherical cylindrical lens for low aberration that resembles approximately a 1 cm long piece of fiber. The focal length of this type of lenses varies from 0.3 to 1.5 mm and the numerical aperture is high, usually between 0.5 and 0.8 to accommodate the strong divergence in the fast axis direction. After the FAC lens, the beam resembles a sheet of light with 1 cm width and of approximately one millimeter height. In the far field the divergence angles are 4-6 degrees along the fast axis and continue 12

After attaching the collimation optics to the diode bar the beam errors are set and the rays emitted by the diode occupy a larger volume than the theoretical minimum. But the errors can be fixed if it is possible to design and introduce a suitable optical element or optical

The beam errors increase the difficulty to design the subsequent optics and reduce the efficiency of the whole optical system in terms of spatial and spectral brightness. It is therefore of great importance to assess the specific nature of these imperfections in order to design effective countermeasures that can in many cases completely eliminate the consequences of these errors. The normal assessment technique for the beam quality of any laser is by means of measuring the quality parameter M2. This is a simple measurement that can give data on beam divergence, focal spot size and confocal parameter but it is not enough information to characterize which is the appropriate correction technique in the case

of beam errors and more elaborate beam assessment methods have to be applied.

causes generally less smile.

degrees along the slow axis.

**2.1 Diode bar beam quality assessment**

system.

Brightness *B* is defined as:

$$B = \frac{P(watt)}{A(cm^2) \cdot \Omega(strand)} = \frac{P}{w\_x w\_y \theta\_x \theta\_y \pi^2} = \frac{P}{\lambda^2 M\_x^2 M\_y^2} \tag{1}$$

Where *P* is the total power, *A* is the area and Ω and θ are the solid angle and half angle divergence, respectively, at 1/e2 of the peak power. λ is the wavelength and *M2* is the beam quality factor, which is 1 for Gaussian beams and gets bigger as quality decreases. Spectral brightness is defined as the brightness divided by the diodes bandwidth.

As the contact strip is made wider the output power increases and with it the locally generated heat (about 50 % of the consumed energy) whereas beam quality degrades. 20 W of output power have been achieved at 980 nm emission wavelength in 100 μm wide strip (Crump et al., 2009). These devices that go by the name of "broad area" diodes (BAL) are commercially available and have output powers of up to more than ten watts with beam quality factors in the directions parallel and perpendicular to the junction of typically 20 and 1, respectively. The width of the broad area can be increased and 500 μm is common but without considerable increase in output power and at a considerable loss of beam quality (M2 parallel to the junction of about 100).

At even larger strip width the transverse direction can develop transverse lasing, amplified spontaneous emission (ASE) or even filamentation. These parasitic phenomena do not only cause losses but also dynamical instabilities and inhomogeneities in the beam profile. It is therefore more convenient to break the active region into a series of parallel strips. The drawback is a strong decrease in beam brightness that is proportional to the amount of "dead space" that is introduced in between the adjacent strips.

Generally there are 19 or 24 emitters in the case of continuous (cw) diode lasers and up to 60 for quasi-continuous (qcw) diode lasers. This high power diode laser (HPDL), which has total dimensions of approximately 1 millimeter length, 10 mm width and about 0.1 mm height is called a "diode array" or "diode bar". Individual emitter width is generally 100 μm to 200 μm and emitter spacing is 500 μm or 400 μm for the 19 and 24 emitter diode bars, respectively, with fill factors of 20% to 30%. The spacing between emitters, which is mainly dictated by cooling requirements, can be made much smaller for qcw diode bars reaching fill factors of up to 90%. Maximum output power for commercial cw diode bars is above 100 W and for qcw diode bars more than 300 W at less than 500 μs pulse duration and less than 10% duty cycle can be achieved. Above values are for conduction cooled diodes. For water cooling the heat removal is more efficient and gets even better with micro-channel water cooling which permits up to 80% fill factor in continuously operated diode bars. As can already be seen and will be shown also further ahead, efficient heat removal is a key factor in obtaining higher brightness for the same amount of output power. Several groups have reported in excess of 700 W cw per bar with production prototypes mainly by increasing the cooling efficiency using different means such as extending the emitter length from 1 mm to more than 3 mm. In all cases the beam quality perpendicular to the junction is close to diffraction limited whereas parallel to the junction the beam quality parameter M2 is well in excess of 1000. At the junction exit the beam is a linear array of individual beamlets that overlap after a couple of millimeters. In the far field this elliptical beams' divergence angles are 30 degrees along the fast axis and 10-12 degrees along the slow axis.

*P watt P P B*

( )

brightness is defined as the brightness divided by the diodes bandwidth.

"dead space" that is introduced in between the adjacent strips.

are 30 degrees along the fast axis and 10-12 degrees along the slow axis.

*A cm sterad w w*

2 2 222

θθπ

λ

 λ*M M* = == ⋅ Ω (1)

are the solid angle and half angle

is the wavelength and *M2* is the beam

( )( ) *x y x y x y*

Ω and θ

quality factor, which is 1 for Gaussian beams and gets bigger as quality decreases. Spectral

As the contact strip is made wider the output power increases and with it the locally generated heat (about 50 % of the consumed energy) whereas beam quality degrades. 20 W of output power have been achieved at 980 nm emission wavelength in 100 μm wide strip (Crump et al., 2009). These devices that go by the name of "broad area" diodes (BAL) are commercially available and have output powers of up to more than ten watts with beam quality factors in the directions parallel and perpendicular to the junction of typically 20 and 1, respectively. The width of the broad area can be increased and 500 μm is common but without considerable increase in output power and at a considerable loss of beam quality

At even larger strip width the transverse direction can develop transverse lasing, amplified spontaneous emission (ASE) or even filamentation. These parasitic phenomena do not only cause losses but also dynamical instabilities and inhomogeneities in the beam profile. It is therefore more convenient to break the active region into a series of parallel strips. The drawback is a strong decrease in beam brightness that is proportional to the amount of

Generally there are 19 or 24 emitters in the case of continuous (cw) diode lasers and up to 60 for quasi-continuous (qcw) diode lasers. This high power diode laser (HPDL), which has total dimensions of approximately 1 millimeter length, 10 mm width and about 0.1 mm height is called a "diode array" or "diode bar". Individual emitter width is generally 100 μm to 200 μm and emitter spacing is 500 μm or 400 μm for the 19 and 24 emitter diode bars, respectively, with fill factors of 20% to 30%. The spacing between emitters, which is mainly dictated by cooling requirements, can be made much smaller for qcw diode bars reaching fill factors of up to 90%. Maximum output power for commercial cw diode bars is above 100 W and for qcw diode bars more than 300 W at less than 500 μs pulse duration and less than 10% duty cycle can be achieved. Above values are for conduction cooled diodes. For water cooling the heat removal is more efficient and gets even better with micro-channel water cooling which permits up to 80% fill factor in continuously operated diode bars. As can already be seen and will be shown also further ahead, efficient heat removal is a key factor in obtaining higher brightness for the same amount of output power. Several groups have reported in excess of 700 W cw per bar with production prototypes mainly by increasing the cooling efficiency using different means such as extending the emitter length from 1 mm to more than 3 mm. In all cases the beam quality perpendicular to the junction is close to diffraction limited whereas parallel to the junction the beam quality parameter M2 is well in excess of 1000. At the junction exit the beam is a linear array of individual beamlets that overlap after a couple of millimeters. In the far field this elliptical beams' divergence angles

Brightness *B* is defined as:

Where *P* is the total power, *A* is the area and

(M2 parallel to the junction of about 100).

divergence, respectively, at 1/e2 of the peak power.

Output power scales with fill factor in part because the maximum power density inside a diode bar is limited by the onset of catastrophic failure on the diodes exit facet. This failure is due to residual absorption at the beam exit facet caused by surface states at the cleavage plane that do not exist within the bulk. This problem hinders the maximum achievable power particularly in GaAs-based semiconductors and a great deal of effort has been spent in recent years to overcome this residual absorption by "passivation" of the exit facets.

The diode bar is normally soldered onto a thin submount which also serves as electrical contact. This submount is then attached to a large heatsink for effective removal of the heat. During this process the very thin (0.1 mm) but large (1 cm) diode array is easily bend by the pressure applied to bond it to the submount and heatsink but also by the differential expansion during the solder process. The overall effect of this process causes a curvature of the array called "smile" which amounts to about 1 -10 μm difference in height at the center of the bar with respect to the bar ends. Some manufactures also bond the bar directly to the copper heat-sink, which is referred to as "direct bonding" and causes generally less smile.

The problem with array curvature is that it degrades the beam brightness. Once the diode is manufactured, the bar curvature is fixed. Therefore it is of importance to understand the constructive details of diode bar manufacturing to apply effective countermeasures. The deviation from linearity of the smile makes it impossible for a single collimation lens in front of the emitters to be perfectly positioned for all emitters along the bar. This first lens, which serves to collimate the fast-axis (FAC lens), is attached directly in front of the diode bar and is generally a plano-aspherical cylindrical lens for low aberration that resembles approximately a 1 cm long piece of fiber. The focal length of this type of lenses varies from 0.3 to 1.5 mm and the numerical aperture is high, usually between 0.5 and 0.8 to accommodate the strong divergence in the fast axis direction. After the FAC lens, the beam resembles a sheet of light with 1 cm width and of approximately one millimeter height. In the far field the divergence angles are 4-6 degrees along the fast axis and continue 12 degrees along the slow axis.

After attaching the collimation optics to the diode bar the beam errors are set and the rays emitted by the diode occupy a larger volume than the theoretical minimum. But the errors can be fixed if it is possible to design and introduce a suitable optical element or optical system.

### **2.1 Diode bar beam quality assessment**

The beam errors increase the difficulty to design the subsequent optics and reduce the efficiency of the whole optical system in terms of spatial and spectral brightness. It is therefore of great importance to assess the specific nature of these imperfections in order to design effective countermeasures that can in many cases completely eliminate the consequences of these errors. The normal assessment technique for the beam quality of any laser is by means of measuring the quality parameter M2. This is a simple measurement that can give data on beam divergence, focal spot size and confocal parameter but it is not enough information to characterize which is the appropriate correction technique in the case of beam errors and more elaborate beam assessment methods have to be applied.

Spectral Narrowing and Brightness Increase in High Power Laser Diode Arrays 187

local absorption at the samples surface and inside the sample which leads to higher temperatures and therefore to more efficient material removal and faster processing speeds. As a result, it increases the range of applications the diode bars can be used for. For instance in medical application, low brightness is enough for hair removal whereas high brightness is necessary for surgery. Both applications may use a laser system composed of the same diode bar component. The difference being that the later application requires additional beam

Most practical beam-shaping techniques used for beam circularization are based on methods that geometrically transform the diode radiation into a rounder spot size. It is convenient for this purpose that the diodes beam consists of individual beamlets which can be rearranged by suitable optical devices. One of the first and still used methods of beamshaping consists of individual fibers, only slightly larger than the individual emitter width, which are precisely positioned in front of each of the diode bar's emitters at a very short distance. The fibers can then be bundled into the desired geometrical form which is for most applications of circular shape. Many other beam-shaping systems consist of optical components that chop the large beam coming from the diode bar into a number of subbeams. These sub-beams are then reflected, diffracted or refracted by clever positioned mirrors, prisms, gratings and micro-lenses in such a way that the sub-beams are stacked on

In figure 3 one can see such a beam-shaper composed of two plane parallel mirrors that are inclined sideways (around a vertical y-axis) and backwards (around the horizontal x-axis). The incoming light sheet from the diode bar passes just above the first mirror where one fifth of the beam is cut-off by the second mirror passing on directly whilst four fifths get reflected slightly downward and to the left (negative x-axis direction) back onto the first mirror. This scheme is repeated until all five parts emerge from the two mirror beam shaper, stacked on top of each other and having ideally a square shape and equal quality factors in the x- and y-directions.

Fig. 3. Two mirror beam shaper used to cut a homogenous light sheet into five sub-beams that are stacked on top of each other. The goal is to achieve a new beam with equal

dimensions and quality factors in the horizontal and vertical directions.

shaping with post-bar additional optical components inside the beam path.

top of each other in order to generate a round spot size.

Fig. 1. Simple imaging system to visualize the beam errors introduced by the diode and the FAC lens: the slow x-axis direction is collimated by two cylindrical lenses.

Although there are nowadays many assessment techniques available, one can obtain most required information by using a simple imaging system that projects the individual beamlets onto a CCD camera (Wetter, 2001). If the diode bar has a FAC lens incorporated and therefore has its fast axis already collimated, this system may comprise simply two cylindrical lenses to image the slow axis direction.

Imprecision in the alignment and positioning of the FAC lens at the micron level further reduces the brightness. Failure to align the lens parallel to the junction causes "skew" that projects the individual beamlets from the emitters at different heights. The problem with skew is that it causes astigmatism and coma as shown in figure 2. Another common problem is to keep the precise distance of the fiber to the array along the whole width of 1 cm. This relative focus error manifests itself in different spot sizes of the beamlets at the image or focus plane, further increasing astigmatism and decreasing the brightness.  These fiber-FAC lenses could easily be bent and introduce further curvature into the highly astigmatic exit beam. Nowadays they have mostly been replaced by bigger glass lenses that have additional support structure to decrease the possibility of bending.

Fig. 2. Two beam errors that are encountered with diode bars; from left to right: smile and skew. The images have been taken with the system in figure 1.

### **3. Beam-shaping techniques**

Beam-shaping techniques applied to diode arrays have two purposes. The first goal is to circularize the highly astigmatic beam of uneven beam quality factors in such a way that it can be used in most practical applications that require generally focusing or collimation and round spot sizes. The second goal is to increase brightness. Brightness increase augments

Fig. 1. Simple imaging system to visualize the beam errors introduced by the diode and the

fx = 100 mm

FAC lens diode

Although there are nowadays many assessment techniques available, one can obtain most required information by using a simple imaging system that projects the individual beamlets onto a CCD camera (Wetter, 2001). If the diode bar has a FAC lens incorporated and therefore has its fast axis already collimated, this system may comprise simply two

Imprecision in the alignment and positioning of the FAC lens at the micron level further reduces the brightness. Failure to align the lens parallel to the junction causes "skew" that projects the individual beamlets from the emitters at different heights. The problem with skew is that it causes astigmatism and coma as shown in figure 2. Another common problem is to keep the precise distance of the fiber to the array along the whole width of 1 cm. This relative focus error manifests itself in different spot sizes of the beamlets at the image or focus plane, further increasing astigmatism and decreasing the brightness.  These fiber-FAC lenses could easily be bent and introduce further curvature into the highly astigmatic exit beam. Nowadays they have mostly been replaced by bigger glass lenses that have additional

Fig. 2. Two beam errors that are encountered with diode bars; from left to right: smile and

Beam-shaping techniques applied to diode arrays have two purposes. The first goal is to circularize the highly astigmatic beam of uneven beam quality factors in such a way that it can be used in most practical applications that require generally focusing or collimation and round spot sizes. The second goal is to increase brightness. Brightness increase augments

FAC lens: the slow x-axis direction is collimated by two cylindrical lenses.

cylindrical lenses to image the slow axis direction.

fx = 25 mm

CCD

support structure to decrease the possibility of bending.

skew. The images have been taken with the system in figure 1.

**3. Beam-shaping techniques**

local absorption at the samples surface and inside the sample which leads to higher temperatures and therefore to more efficient material removal and faster processing speeds. As a result, it increases the range of applications the diode bars can be used for. For instance in medical application, low brightness is enough for hair removal whereas high brightness is necessary for surgery. Both applications may use a laser system composed of the same diode bar component. The difference being that the later application requires additional beam shaping with post-bar additional optical components inside the beam path.

Most practical beam-shaping techniques used for beam circularization are based on methods that geometrically transform the diode radiation into a rounder spot size. It is convenient for this purpose that the diodes beam consists of individual beamlets which can be rearranged by suitable optical devices. One of the first and still used methods of beamshaping consists of individual fibers, only slightly larger than the individual emitter width, which are precisely positioned in front of each of the diode bar's emitters at a very short distance. The fibers can then be bundled into the desired geometrical form which is for most applications of circular shape. Many other beam-shaping systems consist of optical components that chop the large beam coming from the diode bar into a number of subbeams. These sub-beams are then reflected, diffracted or refracted by clever positioned mirrors, prisms, gratings and micro-lenses in such a way that the sub-beams are stacked on top of each other in order to generate a round spot size.

In figure 3 one can see such a beam-shaper composed of two plane parallel mirrors that are inclined sideways (around a vertical y-axis) and backwards (around the horizontal x-axis). The incoming light sheet from the diode bar passes just above the first mirror where one fifth of the beam is cut-off by the second mirror passing on directly whilst four fifths get reflected slightly downward and to the left (negative x-axis direction) back onto the first mirror. This scheme is repeated until all five parts emerge from the two mirror beam shaper, stacked on top of each other and having ideally a square shape and equal quality factors in the x- and y-directions.

Fig. 3. Two mirror beam shaper used to cut a homogenous light sheet into five sub-beams that are stacked on top of each other. The goal is to achieve a new beam with equal dimensions and quality factors in the horizontal and vertical directions.

Spectral Narrowing and Brightness Increase in High Power Laser Diode Arrays 189

and cylindrical shape in the horizontal and vertical direction in order to collimate the fast and slow axis at the same time achieving a divergence of less than 2 degrees in both directions. The focal length is very small, of the order of 100 μm, reducing greatly the size of the macroscopic optic of the beam shaper that follows. Smile reduces significantly the efficiency of such a system and therefore several lenslet arrays are mounted on top of each other with different row shapes that are described by polynomials which match different smile shapes. A lens array row shape can then be assigned to the specific smile of the diode bar (Hamilton et al., 2004). This technique allows the lenslets to be centered in front of each emitter and to carry out their function of beam collimation and aligning the beam

All of above mentioned beam-shaping techniques are intended to generate square beams of equal beam quality factors in both directions. However a square shape is not ideal for coupling into a fiber. To achieve better beam circularization one must change the individual emitter width. If the emitter in the middle of the bar is the largest and if the emitters become progressively smaller as one moves out to the last emitters on opposite ends, stacking as shown in figure 3 will provide for a round spot composed of stacked emitter beamlets.

So far only beam circularization has been achieved. As stated by equation (1), brightness is inversely proportional to the area occupied by the source. In other words, high brightness is achieved with 100% fill factor and the "dead space" between the emitters of the bar has to be removed in order to increase the brightness. As shown earlier, this space cannot be eliminated at the source for cooling reasons and therefore it is necessary to eliminate it afterwards. This can be achieved for instance with the beam-shaper of figure 3 by adjusting the mirrors' positions and angles. In this figure, the light sheet emitted by the bar is

Fig. 5. a) Schematic of a diode bar with 5 emitters. The width of the brackets corresponds to the emitter spacing or emitter pitch. b) Schematic description of the "dead space" being

electrical

emitter beamlets at diode facet

stacked emitter beamlets after beam-shaping a) b)

Each sub-beam is composed of one part useful laser radiation and one part dead space. By moving mirror 2 in figure 3 along the negative x-direction the width of the sub-beam 1 will be decreased until only the emitter beamlet component from the sub-beam remains. If then

into the right direction.

**3.2 Brightness increase**

removed by the beam-shaper.

sectioned into five sub-beams as also shown in figure 5.

emitter contacts

1 2 3 4 5

### **3.1 Correction of diode beam errors**