its Fourier components in phase, while the other half shows phase shifts of in an alternating fashion. In this region, *J*0 is positive and in phase with the Fourier components of the first half above mentioned. For the second case, *J*0 is negative. Taking a value of *A*g such that the Bessel functions with *q* = 1…6 take only positive values, the Fourier spectrum of the resulting grating turns out to be similar to the first case, with *J*0 being negative as the only different phase shift. Within the 3th lobule, it is possible to pick up negative values for *q* = 1, 2, 3, with the resulting Fourier spectrum showing π-shifts accordingly. In Fig.4 , a case of positive values for *q* = 1,2,3 and negative for *q* = 4,5 is shown. This situation corresponds to a region within the 9th lobule of *J*0.

### **4.3 shifts of the Fourier spectra of sinusoidal phase grids**

A rectangular phase grid *G*, can be generated with two phase gratings of equal spatial frequency. Fig. 5(a) depicts the signs of the diffraction orders of a grid made up from two crossed gratings having spectra as those of the one in Fig. 2. Positive signs are denoted by hollow circles, whereas negative signs are marked with crosses. The dashed lines form regions enclosing diffraction orders of index pairs 0,*n* or *m*,0. Then, order 0,0 is found at the intersection of these regions. Two possible window configurations can be considered for a TBPGI; these configurations are shown in Fig. 5(b), and they are denoted by **W**1 and **W**2. The respective displacements of the diffraction patterns are indicated with displaced dashed lines too. For the case of phase grids with windows in configuration **W2**, the image can be written as:

$$\begin{split} O^\*(\mathbf{x}, \mathbf{y}) &= O(\mathbf{x}, \mathbf{y}) \* \tilde{\mathbf{G}}\_2(\mathbf{x}, \mathbf{y}) \\ &= \sum\_{q=-\alpha}^{\alpha} w \Big( \mathbf{x} + \frac{\mathbf{x}\_0}{2}, \mathbf{y} + \frac{\mathbf{x}\_0}{2} \Big) \* I\_q(2\pi A\_{\mathcal{S}}) I\_r(2\pi A\_{\mathcal{S}}) \delta \Big( \mathbf{x} - qF\_0, \mathbf{y} - rF\_0 \Big) \\ &+ \sum\_{q=-\alpha}^{\alpha} w \Big( \mathbf{x} - \frac{\mathbf{x}\_0}{2}, \mathbf{y} - \frac{\mathbf{x}\_0}{2} \Big) \* I\_q(2\pi A\_{\mathcal{S}}) I\_r(2\pi A\_{\mathcal{S}}) \delta \Big( \mathbf{x} - qF\_0, \mathbf{y} - rF\_0 \Big). \end{split} \tag{9}$$

Experimental π Phase-Shifts Observed

´( , )

in

1 0 0 *rxqx* is

 2 1 , 2

  *O xy*

*q r*

where

and cross patterns is to be found in the text.

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 307

1 1

As it is for gratings, fringe modulation of the interference pattern within a window centered

2

*rqrq rrqq*

*JJJJ*

2 11 2

Fig. 5. (a) -phase distribution of diffraction orders of grids. Dashed lines enclose diffraction orders of indexes 0,*n* or *m*,0. b) TBPGI order superposition: Configuration **W**1: interference

displacements of diffraction patterns are highlighted by dashed lines. An explanation of dot

pattern signs for windows displaced along the horizontal axis. Configuration **W**2: interference patterns signs for displaced windows along a line at 45°. Respective

*qr JJJJ*

11

*x*

*x xwyxO* . This result simplifies to

(2 ) (2 ) (2 ) (2 )

*J AJ A J AJ Ae*

*q gr g q gr g*

1 1 , . 2 2

*xw*

0 0

*wx x q y x r*

2 , 2 ´ 2 , 2 ),( 00 *x*00 *y*

*x y*

(11)

0 0 1 1 , 2 2

*i xx q yx r*

*m* (12)

Fig. 4. Fourier spectra of phase gratings: left column, Bessel region (first seven Bessel functions shown); right column, grating spectra.

Again, with 00 *xF*

$$\begin{split} O'(\mathbf{x}, \mathbf{y}) &= \sum\_{q = -\alpha}^{\infty} \sum\_{r = -\alpha}^{\alpha} I\_q(2\pi A\_{\mathcal{g}}) I\_r(2\pi A\_{\mathcal{g}}) \, w \Big( \mathbf{x} + \frac{\mathbf{x}\_0}{2} [1 - 2q]\_r \mathbf{y} + \frac{\mathbf{x}\_0}{2} [1 - 2r] \Big) + \\ &\sum\_{q = -\alpha}^{\infty} \sum\_{r = -\alpha}^{\infty} I\_{q-1}(2\pi A\_{\mathcal{g}}) I\_{r-1}(2\pi A\_{\mathcal{g}}) \, w \Big( \mathbf{x} - \frac{\mathbf{x}\_0}{2} [1 - 2q]\_r \mathbf{y} - \frac{\mathbf{x}\_0}{2} [1 - 2r] \Big) \end{split} \tag{10}$$

Fig. 4. Fourier spectra of phase gratings: left column, Bessel region (first seven Bessel

´( , ) (2 ) (2 ) 1 2 , 1 2 2 2

*x x O xy J A J A w x q y r*

1 1

*q gr g*

*q gr g*

0 0

0 0

(2 ) (2 ) ´ 1 2 , 1 2 , 2 2

*x x J AJ A wx q y r*

(10)

functions shown); right column, grating spectra.

*q r*

*q r*

Again, with 00 *xF*

where 2 , 2 ´ 2 , 2 ),( 00 *x*00 *y x xw x y x xwyxO* . This result simplifies to

$$\begin{split} O^r(\mathbf{x}, y) &= \\ \sum\_{q = -\alpha}^{\infty} \sum\_{r = -\alpha}^{\infty} \Bigg[ I\_q(2\pi A\_{\mathcal{S}}) I\_r(2\pi A\_{\mathcal{S}}) + I\_{q-1}(2\pi A\_{\mathcal{S}}) I\_{r-1}(2\pi A\_{\mathcal{S}}) e^{i\boldsymbol{\theta} \cdot \left[ \left( \mathbf{x} - \mathbf{x}\_0 \left[ q - \frac{1}{2} \right] \right) \left( \mathbf{y} - \mathbf{x}\_0 \left[ r - \frac{1}{2} \right] \right) \right]} \\ & \times w \Big( \mathbf{x} - \mathbf{x}\_0 \left[ q - \frac{1}{2} \right], y - \mathbf{x}\_0 \left[ r - \frac{1}{2} \right] \Big). \end{split} \tag{11}$$

As it is for gratings, fringe modulation of the interference pattern within a window centered in 2 1 , 2 1 0 0 *rxqx* is

Fig. 5. (a) -phase distribution of diffraction orders of grids. Dashed lines enclose diffraction orders of indexes 0,*n* or *m*,0. b) TBPGI order superposition: Configuration **W**1: interference pattern signs for windows displaced along the horizontal axis. Configuration **W**2: interference patterns signs for displaced windows along a line at 45°. Respective displacements of diffraction patterns are highlighted by dashed lines. An explanation of dot and cross patterns is to be found in the text.

Experimental π Phase-Shifts Observed

resulting in similar conclusions.

  2 1*J* , 2 2*J* , 2 3*J* and

( , /2) 2

where

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 309

been simplified to *x*,*y*. The same fringe modulation *mq* results as in Eq. 8. Therefore, the discussion about fringe modulation given in previous sections is retained when introducing the modulation of polarization. Such polarization modulation can also be made for grids,

**5. Experimental testing of the phase-shifts in phase gratings and phase grids**  Figure 6 shows the superposition of Fourier amplitude spectra under window configuration *W1* for a phase grating with 110 lines/mm. The contrast of the corresponding experimental interference patterns can be interpreted as if its first four Fourier coefficients had phase relations as the ones sketched in Fig. 2. They were obtained with the system of Fig.1 before placing the retardation plates QR, QL and the polarizing filters on the image plane. The patterns show the relative phases of the diffraction orders discussed in previous sections. The phase-shifted steps of the experimental fringe patterns can be calculated by applying the algorithm proposed by Kreis, 1986. The resulting mean values are shown in Table 2. It

Table 2. Phase shifts measured from experimental patterns in Fig.6 according to the method

Due to that, the interference patterns are obtained from the interference between the replicas of each beam, centered around each diffraction order. Fig. 7(a) presents the replicas of window A with right circular polarization, and the replicas of window B with left circular polarization; each order is superposed depending on separation 0 *x* . Fig. 7(b) presents the interference pattern generated by the interference of windows AB, due to the cross circular polarization, a linear polarizer is placed to modulate the phase shift on each replicated

are the intensity measurements with the values of

on the coordinates of the centered point has

 , respectively. Note that

, so a good choice for the retarders is quarter-wave

2 4*J*

( ) 0, ( ) /2, ( ) , ( ) 3 /2

can be seen from the table that they depart by small amounts from or 0.

such that 12 3 4

Fig. 6. Experimental results with a phase-grating.

from Kreis, 1986.

interference pattern.

 and *A*( , /2) 1

retarders, as it is well known. Dependence of

Each corresponding fringe modulation depends on the relative phases between *qJ* , *q*1 *J* , *rJ* and *r*1 *J* . These relationships are discussed with the same example of Fig. 5(a). Fig. 5(b) shows the relative phases (and thus, expected signs of *mqr*) of each diffraction order arising from configuration **W**1 or **W**2. If positive signs or negative ones coincide in a diffraction order, fringe modulation will be positive, and this is plotted with one symbol (cross or hollow dot). Only when a cross with a hollow dot appears, modulation is expected to be negative. "Vertical" bands whose sign is equal to that of *mqr* are thus expected in configuration **W**1. Regions with a sign that is equal to that of *mqr* can be seen in configuration **W**2.

## **4.4 Phase shifting interferometry with modulation of polarization**

Turning the attention to gratings, in order to introduce additional phase shifts in the interference pattern centered at ,)2/1( *yqx* 0 for 4-step phase-shifting (Schwieder et al., 1983), each of the windows is illuminated with different polarizations using retarding plates QR and QL (Fig. 1). This arrangement introduces Jones polarization vectors *JR* and *LJ* into the interference terms of Eq. 6 through appropriate modification of Eq. 5. After placing a linear polarizing filter with the transmission axis at an angle , its irradiance results as being proportional to

$$\left\|\vec{f}\_{T}\right\|^{2} = A(\xi, \alpha') \left\langle \left(I\_{q}^{\ 2} + I\_{q-1}{}^{2}\right) + 2J\_{q}J\_{q-1}\cos\left[\xi(\nu', \alpha') - \phi(\mathbf{x}, y)\right] \right\rangle \tag{13}$$

Where is the linear polarizer angle, ' the retardation of each plate, and ( , ') denotes the phase shifting term induced by modulation of polarization given by

$$\xi(\boldsymbol{\nu},\alpha^{\circ}) = \operatorname{ArcTan}\left[\frac{\sin[2\boldsymbol{\nu}]\cdot\sin[\alpha^{\circ}] + \sin^{2}[\boldsymbol{\nu}]\cdot\sin[2\alpha^{\circ}]}{\cos^{2}[\boldsymbol{\nu}] + \sin^{2}[\boldsymbol{\nu}]\cdot\cos[2\alpha^{\circ}] + \sin[2\boldsymbol{\nu}]\cdot\cos[\alpha^{\circ}]}\right],\tag{14}$$

and *A* )',( is defined as

$$A(\boldsymbol{\psi}, \alpha^{\prime}) = \mathbb{I} + \sin[2\boldsymbol{\psi}\,\boldsymbol{\zeta}\cos[\alpha^{\prime}]\,\mathrm{d}\,\mathrm{s}^{\prime}]\,\mathrm{d}\,\mathrm{s}^{\prime}\,\tag{15}$$

For phase-shifting interferometry with four patterns, four irradiances can be used, each one taken at a different angle. Denoting each pattern as

$$\left\|\vec{j}\_{i}\right\|^{2} = A(\xi, \alpha^{\prime}) \left\{ \left(I\_{q}^{\ 2} + I\_{q-1}{}^{2}\right) + 2I\_{q}I\_{q-1}\cos[\xi(\nu\_{i}, \alpha^{\prime}) - \phi(\mathbf{x}, y)]\right\} \tag{16}$$

with *i* = 1…4, the relative phase can be calculated as (J. Schwieder 1983)

$$\tan \phi = \frac{\left\| \mathbf{J}\_1 \right\|^2 - \left\| \mathbf{J}\_3 \right\|^2}{\left\| \mathbf{J}\_2 \right\|^2 - \left\| \mathbf{J}\_4 \right\|^2} \quad \text{{} \text{ } \tag{17}$$

Each corresponding fringe modulation depends on the relative phases between *qJ* , *q*1 *J* , *rJ* and *r*1 *J* . These relationships are discussed with the same example of Fig. 5(a). Fig. 5(b) shows the relative phases (and thus, expected signs of *mqr*) of each diffraction order arising from configuration **W**1 or **W**2. If positive signs or negative ones coincide in a diffraction order, fringe modulation will be positive, and this is plotted with one symbol (cross or hollow dot). Only when a cross with a hollow dot appears, modulation is expected to be negative. "Vertical" bands whose sign is equal to that of *mqr* are thus expected in configuration **W**1. Regions with a sign that is equal to that of *mqr* can be seen in configuration

Turning the attention to gratings, in order to introduce additional phase shifts in the interference pattern centered at ,)2/1( *yqx* 0 for 4-step phase-shifting (Schwieder et al., 1983), each of the windows is illuminated with different polarizations using retarding plates

the interference terms of Eq. 6 through appropriate modification of Eq. 5. After placing a linear polarizing filter with the transmission axis at an angle , its irradiance results as

> 2 2 2 1 1 ( , ') 2 cos[ ( , ') ( , )] *T q q q q J A J J JJ*

*x y*

sin[2 ] sin[ '] sin [ ] sin[2 '] ( , ') cos [ ] sin [ ] cos[2 '] sin[2 ] cos[ ']

]'cos[]2sin[1)',(

For phase-shifting interferometry with four patterns, four irradiances can be used, each one

 2 2 2 1 1 ( , ') 2 cos[ ( , ') ( , )] *i q q q q i J A J J JJ*

> 2 2

*JJ JJ*

1 tan

2

denotes the phase shifting term induced by modulation of polarization given by

2 2

angle. Denoting each pattern as

with *i* = 1…4, the relative phase can be calculated as (J. Schwieder 1983)

*A*

2

*x y* (16)

2 4

2 3

the retardation of each plate, and

, (17)

and *LJ*

into

(13)

( , ')

, (14)

. (15)

**4.4 Phase shifting interferometry with modulation of polarization**

is the linear polarizer angle, '

*ArcTan*

QR and QL (Fig. 1). This arrangement introduces Jones polarization vectors *JR*

**W**2.

being proportional to

Where

and *A*

taken at a different

)',( is defined as

where 2 1*J* , 2 2*J* , 2 3*J* and 2 4*J* are the intensity measurements with the values of such that 12 3 4 ( ) 0, ( ) /2, ( ) , ( ) 3 /2 , respectively. Note that ( , /2) 2 and *A*( , /2) 1 , so a good choice for the retarders is quarter-wave retarders, as it is well known. Dependence of on the coordinates of the centered point has been simplified to *x*,*y*. The same fringe modulation *mq* results as in Eq. 8. Therefore, the discussion about fringe modulation given in previous sections is retained when introducing the modulation of polarization. Such polarization modulation can also be made for grids, resulting in similar conclusions.

## **5. Experimental testing of the phase-shifts in phase gratings and phase grids**

Figure 6 shows the superposition of Fourier amplitude spectra under window configuration *W1* for a phase grating with 110 lines/mm. The contrast of the corresponding experimental interference patterns can be interpreted as if its first four Fourier coefficients had phase relations as the ones sketched in Fig. 2. They were obtained with the system of Fig.1 before placing the retardation plates QR, QL and the polarizing filters on the image plane. The patterns show the relative phases of the diffraction orders discussed in previous sections. The phase-shifted steps of the experimental fringe patterns can be calculated by applying the algorithm proposed by Kreis, 1986. The resulting mean values are shown in Table 2. It can be seen from the table that they depart by small amounts from or 0.

Fig. 6. Experimental results with a phase-grating.

Table 2. Phase shifts measured from experimental patterns in Fig.6 according to the method from Kreis, 1986.

Due to that, the interference patterns are obtained from the interference between the replicas of each beam, centered around each diffraction order. Fig. 7(a) presents the replicas of window A with right circular polarization, and the replicas of window B with left circular polarization; each order is superposed depending on separation 0 *x* . Fig. 7(b) presents the interference pattern generated by the interference of windows AB, due to the cross circular polarization, a linear polarizer is placed to modulate the phase shift on each replicated interference pattern.

Experimental π Phase-Shifts Observed

needed values of

 can be 1

Fig. 9. Polarizing filters array for 90° phase stepping.

**7. Simultaneous phase shifting interferometry**

compared with the similar square of Fig. 8(b) (dotted lines).

our system is not a shearing interferometer of any type.

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 311

Incorporating modulation of polarization, a TBPGI can be used for dynamic interferometry measurements. This system is able to obtain n-interferograms phase-shifted with a single shot. Phase evolving in time can be calculated and displayed on the basis of phase-shifting techniques with four or more interferograms. The system performs as previous proposals to attain four interferograms with a simultaneos capture (Barrientos et al., 1999; Novak et al., 2005). The system uses a grid as a beam splitter in a way that resembles the well-known double-frequency shearing interferometer as proposed by Wyant 1986, but our proposal differs from it not only due to its modulation of polarization, the use of a single frequency and the use of two windows, but also in the phase steps that our system introduces. Besides,

Figure 1 shows the arrangement of a simultaneous phase-shifting grid interferometer including modulation of polarization with retarders for the beams and linear polarizers on the image plane. The system generates several diffraction orders of similar irradiances, but not of equal fringe modulations, as expected (Eq. 12). In order to use Eq. 17 properly, each interferogram image was scaled to the same values of grey levels (from 0 to 255) and normalized. In the previous section, it was shown that a simplification for the polarizing

. For configuration **W2** (Fig. 8), due to the -shifts, only two linear polarizing filters have to be placed (instead of four filters, as when without the -shifts). The transmission axes of filter pairs P1, P3 and P2, P4 can be the same for each as long as they cover two patterns that are 180° phase apart (Fig. 9). Considering the polarizers angles , according with Sec. 4.3, the

sketched in Fig. 9. The square enclosing the 16 window replicas in the same figure is to be

In order to extract phase distributions that evolve in time using phase-shifting interferometry, simultaneous capture of several interferograms with a prescribed shift has to be done. The well-known four-step phase shifting algorithm is presented. Due to the

46.577 (G Rodriguez 2008a). They are

of 0, and

filter array can be attained when using the phase shifts obtain values of

0 and 2

**6. Phase-grid interference patterns with modulation of polarization**

Fig. 7. Replicated windows generated by the Phase grating. (a) Replicas of A and B. (b) Interference pattern obtained through order diffraction superposition. (window configuration **W**1).

Fig. 8. Replicated windows generated by the Phase grating. (a) Replicas of A and B. (b) Interference pattern obtained through order diffraction superposition. (window configuration **W**2).

For the case of the diffraction orders belonging to a phase-grid constructed with two crossed gratings of equal frequency, the corresponding interference patterns are shown in Fig.8 for window configuration **W**2. Each grating gives patterns as in Fig. 6 when placed alone in the system of Fig. 1, with neither plate retarders nor polarizers. In each case, contrasts are in agreement with the conclusions derived from Fig. 5(b). The relative phase values of the 16 patterns within the square drawn in the patterns of Fig. 8(b) employing the method from Kreis, 1986 can be seen in Table 3.

Table 3. Phase shifts measured by means of the Kreis method for **W**2 configuration presented in Fig. 8.

Fig. 7. Replicated windows generated by the Phase grating. (a) Replicas of A and B. (b)

Fig. 8. Replicated windows generated by the Phase grating. (a) Replicas of A and B. (b)

For the case of the diffraction orders belonging to a phase-grid constructed with two crossed gratings of equal frequency, the corresponding interference patterns are shown in Fig.8 for window configuration **W**2. Each grating gives patterns as in Fig. 6 when placed alone in the system of Fig. 1, with neither plate retarders nor polarizers. In each case, contrasts are in agreement with the conclusions derived from Fig. 5(b). The relative phase values of the 16 patterns within the square drawn in the patterns of Fig. 8(b) employing the method from

Table 3. Phase shifts measured by means of the Kreis method for **W**2 configuration presented

Interference pattern obtained through order diffraction superposition. (window

Interference pattern obtained through order diffraction superposition. (window

configuration **W**1).

configuration **W**2).

in Fig. 8.

Kreis, 1986 can be seen in Table 3.

## **6. Phase-grid interference patterns with modulation of polarization**

Incorporating modulation of polarization, a TBPGI can be used for dynamic interferometry measurements. This system is able to obtain n-interferograms phase-shifted with a single shot. Phase evolving in time can be calculated and displayed on the basis of phase-shifting techniques with four or more interferograms. The system performs as previous proposals to attain four interferograms with a simultaneos capture (Barrientos et al., 1999; Novak et al., 2005). The system uses a grid as a beam splitter in a way that resembles the well-known double-frequency shearing interferometer as proposed by Wyant 1986, but our proposal differs from it not only due to its modulation of polarization, the use of a single frequency and the use of two windows, but also in the phase steps that our system introduces. Besides, our system is not a shearing interferometer of any type.

Figure 1 shows the arrangement of a simultaneous phase-shifting grid interferometer including modulation of polarization with retarders for the beams and linear polarizers on the image plane. The system generates several diffraction orders of similar irradiances, but not of equal fringe modulations, as expected (Eq. 12). In order to use Eq. 17 properly, each interferogram image was scaled to the same values of grey levels (from 0 to 255) and normalized. In the previous section, it was shown that a simplification for the polarizing filter array can be attained when using the phase shifts obtain values of of 0, and . For configuration **W2** (Fig. 8), due to the -shifts, only two linear polarizing filters have to be placed (instead of four filters, as when without the -shifts). The transmission axes of filter pairs P1, P3 and P2, P4 can be the same for each as long as they cover two patterns that are 180° phase apart (Fig. 9). Considering the polarizers angles , according with Sec. 4.3, the needed values of can be 1 0 and 2 46.577 (G Rodriguez 2008a). They are sketched in Fig. 9. The square enclosing the 16 window replicas in the same figure is to be compared with the similar square of Fig. 8(b) (dotted lines).

Fig. 9. Polarizing filters array for 90° phase stepping.

## **7. Simultaneous phase shifting interferometry**

In order to extract phase distributions that evolve in time using phase-shifting interferometry, simultaneous capture of several interferograms with a prescribed shift has to be done. The well-known four-step phase shifting algorithm is presented. Due to the

Experimental π Phase-Shifts Observed

**7.2 Case of nine interferograms**

simplifying the filter array (See Fig. 13).

Fig. 13. Interference patterns detected with a polarizing filter at

symmetrical interferograms.

patterns used are enclosed in the dotted rectangle. (b) Polarizer filter array for N= 8, nine

25 . (a)The interference

second.

in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 313

Fig. 12. Moving distributions. Oil drops moving under gravity on a slide. One capture per

To demonstrate the use of several interferograms, we choose the symmetrical n=*N*+1 phase

employed when using these techniques. The phase shifts of p due to the grid spectra allow the use of a number of polarizing filters that is less than the number of interferograms,

/2 *N* is

steps algorithms, for data processing ( case *N*=8). A constant phase shift value of

capability of TBPGI to capture more than four interferograms in a single shot by placing a phase grid, we have the advantage of analyzing the case of more than four interferograms; the case of *Nn* 1 interferograms has been chosen. This method reduces errors in phase calculations when noisy interferograms are involved (Malacara and Servin, 2005). It the next sections of the chapter, we will show experimental results for *n* = 9 interferograms.

## **7.1 Four interferogram case**

A phase dot was placed in the path of beam B, while beam A was used as a reference; the results obtained are shown in Fig. 10. Fig. 10(a) shows the four patterns with 2/ phase shifts obtained in a single shot, and Fig. 10(b) presents the phase profile of the object, in false color coding. However, more than four interferograms could be used, whether for *N*-steps phaseshifting interferometry (Rodriguez, 2008b) or for averaging images with the same shift.

Fig. 10. Static test objects: Phase dot. (a) Four 90° phase-shifted interferograms. (b) Unwrapped phase.

Figure 11 shows the case of moving distributions, corresponding to water flowing on a microscope slide; it shows the phase profile resulting of phase evolution *tyx* ),,( . Fig. 12, by analyzing flow motion by gravity, shows the temporal evolution of oil flow, clearly showing the phase changes induced by the motion of oil; the evolution of the drops as they go by in front of the camera can be shown. Images are presented in 4-D (Wyant 2003).

Fig. 11. Moving distributions. Representative Frames. Evolution of the phase, one capture per second.

Fig. 12. Moving distributions. Oil drops moving under gravity on a slide. One capture per second.

## **7.2 Case of nine interferograms**

312 Interferometry – Research and Applications in Science and Technology

capability of TBPGI to capture more than four interferograms in a single shot by placing a phase grid, we have the advantage of analyzing the case of more than four interferograms; the case of *Nn* 1 interferograms has been chosen. This method reduces errors in phase calculations when noisy interferograms are involved (Malacara and Servin, 2005). It the next

A phase dot was placed in the path of beam B, while beam A was used as a reference; the

obtained in a single shot, and Fig. 10(b) presents the phase profile of the object, in false color coding. However, more than four interferograms could be used, whether for *N*-steps phaseshifting interferometry (Rodriguez, 2008b) or for averaging images with the same shift.

*tyx* ),,( . Fig. 12,

2/ phase shifts

sections of the chapter, we will show experimental results for *n* = 9 interferograms.

Fig. 10. Static test objects: Phase dot. (a) Four 90° phase-shifted interferograms. (b)

microscope slide; it shows the phase profile resulting of phase evolution

Figure 11 shows the case of moving distributions, corresponding to water flowing on a

by analyzing flow motion by gravity, shows the temporal evolution of oil flow, clearly showing the phase changes induced by the motion of oil; the evolution of the drops as they go by in front of the camera can be shown. Images are presented in 4-D (Wyant 2003).

Fig. 11. Moving distributions. Representative Frames. Evolution of the phase, one capture

results obtained are shown in Fig. 10. Fig. 10(a) shows the four patterns with

**7.1 Four interferogram case**

Unwrapped phase.

per second.

To demonstrate the use of several interferograms, we choose the symmetrical n=*N*+1 phase steps algorithms, for data processing ( case *N*=8). A constant phase shift value of /2 *N* is employed when using these techniques. The phase shifts of p due to the grid spectra allow the use of a number of polarizing filters that is less than the number of interferograms, simplifying the filter array (See Fig. 13).

Fig. 13. Interference patterns detected with a polarizing filter at 25 . (a)The interference patterns used are enclosed in the dotted rectangle. (b) Polarizer filter array for N= 8, nine symmetrical interferograms.

Experimental π Phase-Shifts Observed

29, pp. 685-689.

*Journal*, pp. 345-349.

38, pp. 2069-2074.

*Optical Testing.*

16, 7806-7817.

437-451.

*Opt.*, Vol. 44, pp. 6861-6868.

*Opt Letters*, Vol. 33, pp. 2788- 2790.

*Appl. Opt.*, Vol. 22, pp. 3421-3432.

*Opt. Eng.*, Vol. 15, pp. 477.

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in the Fourier Spectra of Phase Gratings and Applications in Simultaneous PSI 315

Investigaciones en Óptica A.C. (CIO)" through projects 290597 (CONACYT-CIO) and 124145 (CONACYT-BUAP) is also acknowledged. Author *NITA* expresses sincere appreciation to *Luisa*, M*iguel* and *Damian-El* for the support provided, and to CONACYT for grant 102137/43055. Author DISG (Grant:227470/31458) is very grateful to CONACyT

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Ronchi V.,(1964). Forty Years of History of a Grating Interferometer, *Appl. Opt.*, Vol. 3, pp.

Schwieder J., Burow R., Elssner K.-E., Grzanna J., Spolaczyk R. and Merkel K., (1983).

Thomas D. A. and Wyant J. C., (1976). High Efficiency Grating Lateral Shear Interferometer,

of a micropolarizer array-based simultaneous phase-shifting interferometer, *Appl.*

Robledo-Sánchez C., (2008). One-Shot Phase-Shifting Phase-Grating Interferometry with Modulation of Polarization: case of four interferograms," *Opt. Express*, Vol.

(2008). One-shot phase-shifting interferometryfive, seven, and nine interferograms,

Digital Wave-Front Measuring Interferometry: some systematic error sources,

Simultaneous Measurement of all Four Stokes Parameters of Light, *Opt. Acta,* Vol.

Phase-Stepped Interferometry using a Holographic Optical Element, *Opt. Eng*. Vol.

for the graduate scholarship granted and expresses sincere appreciation to *Geliztle*.

Dimensional Periodic Filters, *Opt. Lett*. Vol. 29, pp. 141-143.

Cornejo-Rodríguez A. (1992). Ronchi Test, in *Optical Shop Testing*.

Goodman, Jooseph W.; J. Wiley & Sons (1988). *Introduction to Fourier optics.*

Transform Method, *J. Opt. Soc. Am. A,* Vol. 3, pp. 847-855.

Kneubühl F.,(1969). Diffraction Grating Spectroscopy, *Appl. Opt.*, Vol. 8, pp. 505-519.

Fig. 14. Tilted wave-front. (a) Nine 45° phase-shifted interferograms (b) Unwrapped phase data map.

Considering the retarders at disposal, according to Rodriguez 2008b, it can be shown that for symmetrical nine, 0 1 , 989.92 2 , 975.22 3 , 577.46 4 and 5 903.157 . In this case, each resulting step corresponds to a 45° phase-shift. The corresponding results and calculated phases are shown in Fig. 14, the experimental results shows the interference pattern generated by a tilted wavefront. That is said before, this systems are capable of obtaining *n* = *N*+1 interferograms in a single capture (this case, *n* 12 ).

## **8. Final remark**

Theoretical and experimental evidence ofshifts in the Fourier spectra of phase gratings and phase grids were presented; these shifts are not discussed in the literature as far as we know. When the power spectrum is focused, it tends to hide the effect presented, but it can be of considerable relevance when gratings or grids are used for interferometric applications. For example: in phase shift interferometry, these results are convenient since only one grating displacement is necessary to capture four interferograms, and for the case of polarization phase shifting interferometry, the characteristics of the phase gratings used simplify the placement of polarization filters and allow us to obtain *n*-patterns in one shot.

As a final remark the system presented does not use a physical double window; it generates two beams with variable separation according to the characteristics of the grid used. The combination of circular polarization states and the use of phase-grid enable the generation of *n*-interference patterns with independent phase shifts. This characteristic optimizes the interferometric system used, and allows the analysis of static and dynamic phase distributions.

## **9. Acknowledgments**

Authors thank M. A. Ruiz for his contribution in proofreading the manuscript. Partial support from "Consejo Nacional de Ciencia y Tecnología (CONACYT)" and "Centro de Investigaciones en Óptica A.C. (CIO)" through projects 290597 (CONACYT-CIO) and 124145 (CONACYT-BUAP) is also acknowledged. Author *NITA* expresses sincere appreciation to *Luisa*, M*iguel* and *Damian-El* for the support provided, and to CONACYT for grant 102137/43055. Author DISG (Grant:227470/31458) is very grateful to CONACyT for the graduate scholarship granted and expresses sincere appreciation to *Geliztle*.

## **10. References**

314 Interferometry – Research and Applications in Science and Technology

(a) (b)

data map.

for symmetrical nine, 0

**8. Final remark**

distributions.

**9. Acknowledgments**

In this case, each resulting step

1 , 989.92

Fig. 14. Tilted wave-front. (a) Nine 45° phase-shifted interferograms (b) Unwrapped phase

Considering the retarders at disposal, according to Rodriguez 2008b, it can be shown that

results and calculated phases are shown in Fig. 14, the experimental results shows the interference pattern generated by a tilted wavefront. That is said before, this systems are

Theoretical and experimental evidence ofshifts in the Fourier spectra of phase gratings and phase grids were presented; these shifts are not discussed in the literature as far as we know. When the power spectrum is focused, it tends to hide the effect presented, but it can be of considerable relevance when gratings or grids are used for interferometric applications. For example: in phase shift interferometry, these results are convenient since only one grating displacement is necessary to capture four interferograms, and for the case of polarization phase shifting interferometry, the characteristics of the phase gratings used simplify the placement of polarization filters and allow us to obtain *n*-patterns in one shot. As a final remark the system presented does not use a physical double window; it generates two beams with variable separation according to the characteristics of the grid used. The combination of circular polarization states and the use of phase-grid enable the generation of *n*-interference patterns with independent phase shifts. This characteristic optimizes the interferometric system used, and allows the analysis of static and dynamic phase

Authors thank M. A. Ruiz for his contribution in proofreading the manuscript. Partial support from "Consejo Nacional de Ciencia y Tecnología (CONACYT)" and "Centro de

 3 , 577.46

4 and

corresponds to a 45° phase-shift. The corresponding

5 903.157 .

 2 , 975.22

capable of obtaining *n* = *N*+1 interferograms in a single capture (this case, *n* 12 ).

**16**

*3NTT corporation*

**ONU**

*Japan*

**Length Measurement for Optical Transmission**

*2Department of Information and Systems Engineering, Fukuoka Institute of Technology*

More than half a million kilometres of aerial/underground optical telecommunication cable have been installed in Japan. A guaranteed telecommunication service is replacing the besteffort service as real-time streaming delivery increases. Fiber-to-the-home (FTTH) systems are mainly designed by passive optical network (PON) topology (ITU-T Recommendation G.983.1, 1998; Hornung et al., 1990; Sankawa et al., 2006). They are not allowed for even an instantaneous interruption day and night, because numerous customers suffer damage from

Telecommunication cables are required to be relocated in road construction and work on the water supply. Each optical fiber leading from an optical line terminal (OLT) in a telephone office to a customer's optical network unit (ONU) must be cut and reconnected as shown in Fig. 1. Customers expect real-time transmission for high-quality communications to

Several studies have reported on protection of PON system (Xu & Ho, 2001; Tanaka & Horiuchi, 2008). Electrical transmission apparatus can maintain communication without interruption, even when optical cables are temporarily cut. The system is complicated and any transmission delay during O/E conversion is fatal to real-time communication. Although it is desirable to directly switch the transmission medium itself, it had been thought that some data

**1. Introduction**

**O L T**

ONU: Optical Network Unit

Fig. 1. Optical fiber cable replacement.

it as 1 x 109 bits of data will be lost every second.

continue uninterrupted, especially for video transmission services.

New cable OLT : Optical Line Terminal

Takeshi Tsujimura1, Koichi Yoshida2 and Kuniaki Tanaka3

*1Department of Mechanical Engineering, Saga University*

**Line Using Interferometry**

Wyant J. C., (1973). Double Frequency Grating Lateral Shear Interferometer, *Appl. Opt.*, Vol. 12, pp. 2057-2060.

Wyant J. C., (2003). Dynamic Interferometry, *Optics & Photonics News*, Vol. 14, pp. 36-41.

## **Length Measurement for Optical Transmission Line Using Interferometry**

Takeshi Tsujimura1, Koichi Yoshida2 and Kuniaki Tanaka3

*1Department of Mechanical Engineering, Saga University 2Department of Information and Systems Engineering, Fukuoka Institute of Technology 3NTT corporation Japan*

## **1. Introduction**

316 Interferometry – Research and Applications in Science and Technology

Wyant J. C., (1973). Double Frequency Grating Lateral Shear Interferometer, *Appl. Opt.*, Vol.

Wyant J. C., (2003). Dynamic Interferometry, *Optics & Photonics News*, Vol. 14, pp. 36-41.

12, pp. 2057-2060.

More than half a million kilometres of aerial/underground optical telecommunication cable have been installed in Japan. A guaranteed telecommunication service is replacing the besteffort service as real-time streaming delivery increases. Fiber-to-the-home (FTTH) systems are mainly designed by passive optical network (PON) topology (ITU-T Recommendation G.983.1, 1998; Hornung et al., 1990; Sankawa et al., 2006). They are not allowed for even an instantaneous interruption day and night, because numerous customers suffer damage from it as 1 x 109 bits of data will be lost every second.

Telecommunication cables are required to be relocated in road construction and work on the water supply. Each optical fiber leading from an optical line terminal (OLT) in a telephone office to a customer's optical network unit (ONU) must be cut and reconnected as shown in Fig. 1. Customers expect real-time transmission for high-quality communications to continue uninterrupted, especially for video transmission services.

Fig. 1. Optical fiber cable replacement.

Several studies have reported on protection of PON system (Xu & Ho, 2001; Tanaka & Horiuchi, 2008). Electrical transmission apparatus can maintain communication without interruption, even when optical cables are temporarily cut. The system is complicated and any transmission delay during O/E conversion is fatal to real-time communication. Although it is desirable to directly switch the transmission medium itself, it had been thought that some data

Length Measurement for Optical Transmission Line Using Interferometry 319

switching method is necessary to maintain communications completely during optical cable

107 103 105 10 interruption time (ns)

0 2 4 6 8

We propose an optical line switching technique that uses duplex transmission lines as shown in Fig. 4. Normal transmission system consists of an OLT, ONUs, and an optical fiber labeled 'regular line' in Fig. 4. Signals are bidirectionally transmitted along the line through a wavelength independent optical coupler (WIC) and an optical splitter. Our proposed

Number of ONUs

replacement and thus provide a highly reliable network.

0

0

R

**3. Optical fiber line switching 3.1 Optical line switching technique**

e

storatio

n tim

e (s)

10

20

Fig. 2. Relationship between interruption time and restoration time.

Fig. 3. Relationship between number of ONUs and restoration time.

R

e

storatio

n tim

e (s)

10

20

bits would inevitably be lost during the replacement of optical fibers. An optical fiber cable transfer splicing system has been developed to minimize cable replacing time (Watanabe et al., 1992; Tanaka et al., 2002). It disconnects an optical fiber and exchanges it for another in a flash. It takes 30 ms to switch a transmission line, and more than 2 seconds to restore communications with, for example, a gigabit Ethernet passive optical network (GE-PON) (Azuma et al., 2008). Neither the PON protection technique nor the splicing system avoids communication failures because they cannot reduce the disconnecting time absolutely to zero.

The authors have developed an interruption-free replacement method for in-service telecommunication lines, which can be applied to the current PON system equipped with conventional OLTs and ONUs (Tsujimura et al., 2010; Tanaka et al., 2009; Yoshida & Tsujimura, 2010). Two essential techniques were newly proposed; a measurement method and an adjustment system for the transmission line length. The latter continuously lengthens/shortens the transmission line over very long distances without losing transmitted data based on free space optics (FSO) system (Tsujimura et al., 2009b; Yoshida et al., 2009). The former distinguishes the difference between the duplicated line lengths using interferometry (Tsujimura et al., 2009a). Interferometry is the technique of diagnosing the properties of two or more lasers or waves by studying the pattern of interference created by their superposition. It is an important investigative technique in the fields of astronomy, fiber optics, optical metrology and so on. Studies on optical interferometry are reported to improve tiny optical devices (Saunders & Hardcastle, 1994; Cao & Cartledge, 2002; Torregrosa et al., 2007). We have applied the optical interferometry technique to measure length of several kilometers of transmission lines with a 10 mm resolution in this study.

This paper reveals the problems in replacing optical cables first. It proposes a switching method of optical fiber transmission lines next. We design a shunt system for in-service optical fibers. Then an optical line length measurement method is studied to distinguish the difference of two lines by evaluating interfered optical pulses. An optical line switching system is designed, and a line length adjustment system is prototyped. The proposed system is applied to a 15 km GE-PON optical fiber network while adding a 10 m extension to show the efficiency of this approach when replacing in-service optical cables. Finally, we discuss some proposal for improvement of line length measurement.

## **2. Optical cable replacement**

First of all, fundamental investigations were conducted on the effects of transmission interruption on the telecommunication service.

When an optical fiber line was intensionally disconnected in a short time, transmission broke off and was restored by PON transmission system. Figure 2 and 3 show the experimental results of transmission characteristics when GE-PON optical communication line is switched using high-speed line transfer. Figure 2 expresses the relationship between interruption time and optical network unit (ONU) restoration time for 4 ONUs. Even if an optical fiber line was reconnected only in 100 ns, the transmission system took about 6 seconds to restore the telecommunication service. Up to 8 ONUs can be connected to an OLT. It took more than 10 seconds if eight ONUs were connected to an OLT as shown in Fig. 3, which indicates that the restoration time increases in relation to the number of ONUs.

These results suggest that communication service is inevitably interrupted no matter how quick the conventional method reconnects the transmission line. That is why a new

bits would inevitably be lost during the replacement of optical fibers. An optical fiber cable transfer splicing system has been developed to minimize cable replacing time (Watanabe et al., 1992; Tanaka et al., 2002). It disconnects an optical fiber and exchanges it for another in a flash. It takes 30 ms to switch a transmission line, and more than 2 seconds to restore communications with, for example, a gigabit Ethernet passive optical network (GE-PON) (Azuma et al., 2008). Neither the PON protection technique nor the splicing system avoids communication failures because they cannot reduce the disconnecting time absolutely to zero. The authors have developed an interruption-free replacement method for in-service telecommunication lines, which can be applied to the current PON system equipped with conventional OLTs and ONUs (Tsujimura et al., 2010; Tanaka et al., 2009; Yoshida & Tsujimura, 2010). Two essential techniques were newly proposed; a measurement method and an adjustment system for the transmission line length. The latter continuously lengthens/shortens the transmission line over very long distances without losing transmitted data based on free space optics (FSO) system (Tsujimura et al., 2009b; Yoshida et al., 2009). The former distinguishes the difference between the duplicated line lengths using interferometry (Tsujimura et al., 2009a). Interferometry is the technique of diagnosing the properties of two or more lasers or waves by studying the pattern of interference created by their superposition. It is an important investigative technique in the fields of astronomy, fiber optics, optical metrology and so on. Studies on optical interferometry are reported to improve tiny optical devices (Saunders & Hardcastle, 1994; Cao & Cartledge, 2002; Torregrosa et al., 2007). We have applied the optical interferometry technique to measure length of several kilometers of transmission lines with a 10 mm resolution in this study.

This paper reveals the problems in replacing optical cables first. It proposes a switching method of optical fiber transmission lines next. We design a shunt system for in-service optical fibers. Then an optical line length measurement method is studied to distinguish the difference of two lines by evaluating interfered optical pulses. An optical line switching system is designed, and a line length adjustment system is prototyped. The proposed system is applied to a 15 km GE-PON optical fiber network while adding a 10 m extension to show the efficiency of this approach when replacing in-service optical cables. Finally, we

First of all, fundamental investigations were conducted on the effects of transmission

When an optical fiber line was intensionally disconnected in a short time, transmission broke off and was restored by PON transmission system. Figure 2 and 3 show the experimental results of transmission characteristics when GE-PON optical communication line is switched using high-speed line transfer. Figure 2 expresses the relationship between interruption time and optical network unit (ONU) restoration time for 4 ONUs. Even if an optical fiber line was reconnected only in 100 ns, the transmission system took about 6 seconds to restore the telecommunication service. Up to 8 ONUs can be connected to an OLT. It took more than 10 seconds if eight ONUs were connected to an OLT as shown in Fig. 3, which indicates that the restoration time increases in relation to the number of ONUs.

These results suggest that communication service is inevitably interrupted no matter how quick the conventional method reconnects the transmission line. That is why a new

discuss some proposal for improvement of line length measurement.

**2. Optical cable replacement**

interruption on the telecommunication service.

switching method is necessary to maintain communications completely during optical cable replacement and thus provide a highly reliable network.

Fig. 2. Relationship between interruption time and restoration time.

Fig. 3. Relationship between number of ONUs and restoration time.

## **3. Optical fiber line switching**

## **3.1 Optical line switching technique**

We propose an optical line switching technique that uses duplex transmission lines as shown in Fig. 4. Normal transmission system consists of an OLT, ONUs, and an optical fiber labeled 'regular line' in Fig. 4. Signals are bidirectionally transmitted along the line through a wavelength independent optical coupler (WIC) and an optical splitter. Our proposed

Length Measurement for Optical Transmission Line Using Interferometry 321

**Step 3.** Once the lengths of the two lines coincide, the transmission signals are also

**Step 4.** The regular line is cut. Transmission keeps connected because the signals travels

**Step 6.** The test light measures the lengths of the new line and the detour line. The detour

**Step 7.** The LWPF is then removed and the transmission is duplicated. The detour line is

**Step 8.** Replacement of the optical fiber line is completed without any interruption.

through the detour line. A long-wavelength pass filter (LWPF) is temporarily

line is adjusted to the new line while communications are maintained. The LWPF keeps preventing the optical pulses for telecommunication from traveling through

**WIC**

Fig. 5. Part I

OLT ONU

**Detour line** OLT ONU

OLT ONU

OLT ONU

**length adjuster**

**Regular line**

**Regular line** ONU **Optical splitter**

1650nmLD **Optical line**

2x8

2x8

2x8

2x8

ONU

ONU ONU

1650nmLD

ONU ONU

ONU ONU

1650nmLD

launched into the detour line.

**Step 5.** The LWPF passes the test light alone through the new line.

installed in the new line.

the new line.

finally cut off.

Step 1: Set detour line.

Step 2: Adjust detour line to regular line.

Step 3: Multiplex optical signals.

Step 0: Signals travel through regular line.

technique requires a detour optical fiber to temporarily transmit signals. It also uses a test light whose wavelength is different from ones for signal transmission. The test light is created by an optical frequency-chirped pulse light source, and reaches an oscilloscope through the optical splitter, the regular/detour line, and the WIC. An optical line length measuring method detects the optical path difference between the detour and the regular line with these arrangements. An optical line length adjusting method controls the detour line length in virtue of FSO device.

Because either of two lines are managed to keep connection and to transmit signals at any time, the entire transmission system are secured against interruption. One of optical cables can be freely cut or reconnected, while the other cable maintains connected and transmits signals.

In order to avoid suspending transmission, the communication conditions have to be kept in the physical layer, for example, carrier power or signal phase. Thus, the following two conditions must be met while exchanging an in-service line for a spare line,

Fig. 4. Duplex optical fiber transmission system.

The conditions are satisfied by equalizing the transmission times of both signals transmitted through the duplicated lines. We accomplish this by tuning the length of the detour line.

The optical line replacement procedure, illustrated in Fig. 5 where the transmission lines are simplified, is as follows:

technique requires a detour optical fiber to temporarily transmit signals. It also uses a test light whose wavelength is different from ones for signal transmission. The test light is created by an optical frequency-chirped pulse light source, and reaches an oscilloscope through the optical splitter, the regular/detour line, and the WIC. An optical line length measuring method detects the optical path difference between the detour and the regular line with these arrangements. An optical line length adjusting method controls the detour

Because either of two lines are managed to keep connection and to transmit signals at any time, the entire transmission system are secured against interruption. One of optical cables can be freely cut or reconnected, while the other cable maintains connected and transmits signals. In order to avoid suspending transmission, the communication conditions have to be kept in the physical layer, for example, carrier power or signal phase. Thus, the following two

> OLT : Optical Line Terminal ONU: Optical Network Unit O/E : Optic/Electric converter

Optical frequency-chirped pulse light source

2 x 8

ONU

conditions must be met while exchanging an in-service line for a spare line,

Regular line

Optical line length detector

The conditions are satisfied by equalizing the transmission times of both signals transmitted through the duplicated lines. We accomplish this by tuning the length of the detour line.

The optical line replacement procedure, illustrated in Fig. 5 where the transmission lines are

**Step 0.** Signals are ordinarily transmitted through an optical fiber (regular line) between an

**Step 1.** A detour line is established between the WIC and the 2x8 optical splitter. A

transmission system in the telephone office commonly equips with a WIC next to an OLT. A 1650 nm LD is connected to one of the branches of the 2x8 optical splitter. **Step 2.** The detour line length is measured with a 1650 nm test light using an optical line

length measuring method, and is adjusted to the same length as the regular line

Detour line

WIC ONU

Optical splitter New line

line length in virtue of FSO device.

a. No transmission signals must be lost,

Oscilloscope

Fig. 4. Duplex optical fiber transmission system.

OLT

simplified, is as follows:

b. Transmission time in a protocol must be maintained.

O/E Optical line length adjuster

OLT and ONUs through a 2x8 optical splitter.

using an optical line length adjusting method.

Length Measurement for Optical Transmission Line Using Interferometry 323

The proposed method temporarily provides a duplicate transmission line as shown in Fig. 6 to replace optical fiber lines. A detour line is prepared in advance through which to divert

*Telephone office Outdoors Customers*

New line

Detour line

**Optical line length adjuster**

> Test light source

1650nm DFB-LD

2x8 SP

ONU

ONU

・・ ONU

signals while the existing line (regular line) is replaced with a new one.

FSO

Path #0

*L0 2L0 3L0* FS-0

This system transfers signals between the two lines. Signals are duplicated at the moment of changeover to maintain continuous communications. The signals travel separately through the two lines to a receiver. A difference in the line lengths leads to a difference in the signals' arrival times. A communication fault occurs if, as a result of their proximity, the waveforms of the two arriving signals are too blurred for the signals to be identified as discrete. Thus it

We investigated the tolerance of the multiplexed signal synchronicity in advance. The transmission quality is observed by changing the difference between the duplicated line lengths. The results show that the transmission linkage is maintained if the difference is within 80 mm as with GE-PON. A multiplexed signal cannot be perceived as a single bit when the duplicated line lengths have a larger gap for 1 Gbit/s transmission. Because these characteristics depend on the periodic length of a transmission bit, the requirement is assumed to be more severe when the method is applied to higher-speed communication services.

Experiments determined that the tolerance of the difference in line length is 80 mm with

Accordingly, the proposed system controls the adjustment procedure so that the difference

We next constructed a prototype of the optical line shunt system to apply to a GE-PON

in length between the detour and regular lines is adjusted within 80 mm.

optical fiber line replacement according to the procedure described above.

WDM

SW-1 SW-0

Regular line

**Optical line shunt system**

LWPF

Oscilloscope

FS-1

Path #1

SW-2

Fig. 6. Shunt system of optical fiber transmission line.

is important to adjust the lengths of both lines precisely.

regard to the GE-PON transmission system.

1310/1490nm Amplifier

1650nm Amplifier

WIC

OLT

**Optical line length detector**

Part II

Fig. 5. Optical line replacement procedure.

## **3.2 Optical line shunt system**

The authors have designed an optical line shunt system, and a switching procedure for three wavelengths, namely 1310 and 1490 nm for GE-PON transmission, and 1650 nm for measurement. The optical line shunt system is installed in a telephone office. It is composed of an optical line length detector and an optical line length adjuster. Figure 6 shows an individual optical fiber line in a GE-PON transmission system. Optical pulse signals at two wavelengths are bidirectionally transmitted through a regular line between customers' ONUs and an OLT in a telephone office via a wavelength independent optical coupler (WIC) and a 2x8 optical splitter (2x8 SP), respectively.

A test light at a wavelength different from those of the transmission signals is sent from one of the optical splitter's ports to the duplicated lines. We use a 1650 distributed feedback laser diode (DFB-LD) whose pulse width is 200 ns as a test light source. An oscilloscope is connected to the optical coupler to detect the test light through a long-wavelength pass filter (LWPF). The optical line length adjuster is an FSO application (Yoshida et al., 2011; Willebrand et al., 1999). Some optical switches (SW) and optical fiber selectors (FS) control the flow of the optical signals managed by a PC. The optical pulses are compensated by 1650 and 1310/1490 nm amplifiers (Fukada et al., 2008).

Part II

The authors have designed an optical line shunt system, and a switching procedure for three wavelengths, namely 1310 and 1490 nm for GE-PON transmission, and 1650 nm for measurement. The optical line shunt system is installed in a telephone office. It is composed of an optical line length detector and an optical line length adjuster. Figure 6 shows an individual optical fiber line in a GE-PON transmission system. Optical pulse signals at two wavelengths are bidirectionally transmitted through a regular line between customers' ONUs and an OLT in a telephone office via a wavelength independent optical coupler

A test light at a wavelength different from those of the transmission signals is sent from one of the optical splitter's ports to the duplicated lines. We use a 1650 distributed feedback laser diode (DFB-LD) whose pulse width is 200 ns as a test light source. An oscilloscope is connected to the optical coupler to detect the test light through a long-wavelength pass filter (LWPF). The optical line length adjuster is an FSO application (Yoshida et al., 2011; Willebrand et al., 1999). Some optical switches (SW) and optical fiber selectors (FS) control the flow of the optical signals managed by a PC. The optical pulses are compensated by 1650

OLT ONU

OLT ONU

**New line LWPF**

OLT ONU

OLT ONU

OLT ONU

ONU ONU

ONU ONU

1650nmLD

ONU ONU

1650nmLD

2x8

2x8

**LWPF** 2x8

2x8

2x8

1650nmLD

ONU ONU

1650nmLD

ONU ONU

1650nmLD

Step 6: Adjust detour line to new line.

Step 7: Cut off detour line.

Step 4: Cut regular line.

Step 5: Connect new line.

Step 8: Complete replacement.

**3.2 Optical line shunt system**

Fig. 5. Optical line replacement procedure.

(WIC) and a 2x8 optical splitter (2x8 SP), respectively.

and 1310/1490 nm amplifiers (Fukada et al., 2008).

The proposed method temporarily provides a duplicate transmission line as shown in Fig. 6 to replace optical fiber lines. A detour line is prepared in advance through which to divert signals while the existing line (regular line) is replaced with a new one.

Fig. 6. Shunt system of optical fiber transmission line.

This system transfers signals between the two lines. Signals are duplicated at the moment of changeover to maintain continuous communications. The signals travel separately through the two lines to a receiver. A difference in the line lengths leads to a difference in the signals' arrival times. A communication fault occurs if, as a result of their proximity, the waveforms of the two arriving signals are too blurred for the signals to be identified as discrete. Thus it is important to adjust the lengths of both lines precisely.

We investigated the tolerance of the multiplexed signal synchronicity in advance. The transmission quality is observed by changing the difference between the duplicated line lengths. The results show that the transmission linkage is maintained if the difference is within 80 mm as with GE-PON. A multiplexed signal cannot be perceived as a single bit when the duplicated line lengths have a larger gap for 1 Gbit/s transmission. Because these characteristics depend on the periodic length of a transmission bit, the requirement is assumed to be more severe when the method is applied to higher-speed communication services.

Experiments determined that the tolerance of the difference in line length is 80 mm with regard to the GE-PON transmission system.

Accordingly, the proposed system controls the adjustment procedure so that the difference in length between the detour and regular lines is adjusted within 80 mm.

We next constructed a prototype of the optical line shunt system to apply to a GE-PON optical fiber line replacement according to the procedure described above.

Length Measurement for Optical Transmission Line Using Interferometry 325

where *k, W, n* are the number of journeys between the retroreflectors, the retroreflector

where *Wmax* and *Wmin* are the maximum and minimum retroreflector intervals. The retroreflector stroke of our prototype, *Wmax- Wmin*, was 0.3 m, the refractive index, *n*, of the optical fiber was 1.46, the number of journeys, *k* was 10, and the optical line span, *L0*, tuned by the adjuster was 2 m. Our prototype adjusted the detour line length with a resolution of

The limit of the adjustable range is a practical problem when this system is applied to several kilometers of access network. Therefore, we designed an optical line length accumulator. The optical line length adjuster contains two optical paths, #0 and #1, as shown in Fig. 6. An optical switch (SW) and an optical fiber selector (FS) are installed in each path. Optical switches control the optical pulse flow. Each optical fiber selector is equipped with various lengths of optical fiber, for example L0, 2L0 and 3L0. The path length can be

The optical line length adjuster can extend the detour line as much as required using the operation shown in Fig. 9. First, the FSO system lengthens path #0 by L0 by gradually increasing the retroreflector interval. After the optical fiber selector, FS-1, has selected an optical fiber of length L0, the active line is switched from path #0 to path #1 at time t0. The FSO system then returns to the origin, and the optical fiber selector, FS-0, selects an optical fiber of length L0 instead to keep the length of path #0 at L0. The FSO system increases the retroreflector interval again at time 2t0 to repeat the same operation. In this way the adjuster accumulates spans extended by the FSO system. The scanning time of our prototype, t0, was 10 seconds, because the retroreflector moved along the motorized sliding stage at 30 mm/s. The optical line length adjuster enables us to lengthen/shorten the detour line while

On line Off line

t0 2t0 3t0 4t0 Time

5t0

*L0* **=** *k* **(***Wmax- Wmin***)/** *n*, (2)

interval, and the refractive index of optical fiber, respectively.

The FSO lengthens the optical line length up to *L0*.

discretely changed by choosing any one of them.

continuing to transmit optical signals.

L0

0 0

Fig. 9. Accumulation of optics line length.

2L0

Accumulated optical line length

Path #0 Path #1

0.1 mm.

An optical line length adjuster, shown in Fig. 7, was installed along the detour line. The adjuster was equipped with two retroreflectors, which directly faced each other as illustrated in Fig. 8. Optical pulses were transmitted through an optical fiber, divided into three wavelengths by wavelength division multiplexing (WDM) couplers, and discharged separately into the air from collimators. They traveled 10 times between the retroreflectors, and were introduced into the opposite optical fiber. The number of reflections was determined based on the retroreflector arrangement.

Fig. 7. Photos of optics line length adjuster.

Fig. 8. Free-space optics line length adjuster.

The detour line between the retroreflectors consisted of an FSO system. The detour line length could be easily adjusted by controlling the retroreflector interval with a resolution of 0.14 mm with a motorized sliding stage. Optical pulses travel **n**-times faster in the air than in an optical fiber, where **n** is the refractive index of the optical fiber. Thus the optical line length adjuster lengthens/shortens the corresponding optical fiber length, *L*

$$L \equiv k \ W / n\_r \tag{1}$$

where *k, W, n* are the number of journeys between the retroreflectors, the retroreflector interval, and the refractive index of optical fiber, respectively.

The FSO lengthens the optical line length up to *L0*.

324 Interferometry – Research and Applications in Science and Technology

An optical line length adjuster, shown in Fig. 7, was installed along the detour line. The adjuster was equipped with two retroreflectors, which directly faced each other as illustrated in Fig. 8. Optical pulses were transmitted through an optical fiber, divided into three wavelengths by wavelength division multiplexing (WDM) couplers, and discharged separately into the air from collimators. They traveled 10 times between the retroreflectors, and were introduced into the opposite optical fiber. The number of reflections was

> **WDM coupler**

> **WDM coupler**

The detour line between the retroreflectors consisted of an FSO system. The detour line length could be easily adjusted by controlling the retroreflector interval with a resolution of 0.14 mm with a motorized sliding stage. Optical pulses travel **n**-times faster in the air than in an optical fiber, where **n** is the refractive index of the optical fiber. Thus the optical line

length adjuster lengthens/shortens the corresponding optical fiber length, *L*

Collimators

**Collimator**

**To optical splitter**

*L* **=** *k W* **/** *n***,** (1)

determined based on the retroreflector arrangement.

Retroreflectors

**Retroreflector**

**To optical coupler**

Fig. 7. Photos of optics line length adjuster.

Fig. 8. Free-space optics line length adjuster.

$$\mathbf{L}\_0 \equiv \mathbf{k} \text{ (}\mathbf{W}\_{\text{max}}\text{- }\mathbf{W}\_{\text{min}}\text{)}\prime\prime\prime\,,\tag{2}$$

where *Wmax* and *Wmin* are the maximum and minimum retroreflector intervals. The retroreflector stroke of our prototype, *Wmax- Wmin*, was 0.3 m, the refractive index, *n*, of the optical fiber was 1.46, the number of journeys, *k* was 10, and the optical line span, *L0*, tuned by the adjuster was 2 m. Our prototype adjusted the detour line length with a resolution of 0.1 mm.

The limit of the adjustable range is a practical problem when this system is applied to several kilometers of access network. Therefore, we designed an optical line length accumulator. The optical line length adjuster contains two optical paths, #0 and #1, as shown in Fig. 6. An optical switch (SW) and an optical fiber selector (FS) are installed in each path. Optical switches control the optical pulse flow. Each optical fiber selector is equipped with various lengths of optical fiber, for example L0, 2L0 and 3L0. The path length can be discretely changed by choosing any one of them.

The optical line length adjuster can extend the detour line as much as required using the operation shown in Fig. 9. First, the FSO system lengthens path #0 by L0 by gradually increasing the retroreflector interval. After the optical fiber selector, FS-1, has selected an optical fiber of length L0, the active line is switched from path #0 to path #1 at time t0. The FSO system then returns to the origin, and the optical fiber selector, FS-0, selects an optical fiber of length L0 instead to keep the length of path #0 at L0. The FSO system increases the retroreflector interval again at time 2t0 to repeat the same operation. In this way the adjuster accumulates spans extended by the FSO system. The scanning time of our prototype, t0, was 10 seconds, because the retroreflector moved along the motorized sliding stage at 30 mm/s.

The optical line length adjuster enables us to lengthen/shorten the detour line while continuing to transmit optical signals.

Fig. 9. Accumulation of optics line length.

Length Measurement for Optical Transmission Line Using Interferometry 327

(a) Optical path difference 100m.

(b) Optical path difference 20m.

(c) Optical path difference 1m.

Fig. 10. Delay of duplicated received pulse.

## **4. Optical line length measurement**

The proposed system uses laser pulses at a wavelength of 1650 mm to measure the optical path length. They are introduced from an optical splitter, duplicated, and transmitted toward the OLT through the active and detour lines.

They are distributed by an optical coupler just in front of the OLT, and observed with an oscilloscope. The conventional measurement method evaluates the arrival time interval between the duplicated signals, and converts it to the difference between the lengths of the regular line and the detour line at a resolution of 1 m.

The difference in line length, *L* is described as

$$\mathbf{AL} = \mathbf{c} \cdot \mathbf{At} \text{ / n,} \tag{3}$$

where *c* is the speed of light, *t* is the difference between the signal arrival times for the regular and detour lines, and *n* is the refractive index of optical fiber.

Figure 10 shows the received pulses observed with an oscilloscope. When the detour line was 100 m shorter than the regular line, pulses traveling through the detour reached the oscilloscope about 500 ns earlier than through the regular line as shown in Fig 10 (a). The proposed system lengthened the detour line using the optical path length adjuster. Consequently, the former pulse approached the latter. Figure 10 (b) is an example when the gap was shorten to 20 m, where the regular line pulse arrived after a delay of 100 ns. This method failed if the difference between the line lengths was less than 1 m, because the two pulses combined as shown in Fig. 10 (c).

That is why the authors have developed an advanced technique for measuring a difference of less than 1 m between optical line lengths. Interferometry enables us to obtain more detailed measurements even when the optical pulses combine. A chirped light source generates interference in the waveform of a unified pulse.

A 1650 nm laser pulse, discharged from the test light source, is devided by the 2x8 optical splitter. Two pulses travels through the detour and regular line separately toward the oscilloscope.

Both pulses along the regular line, *E (L1, t)*, and the detour line, *E (L2, t)* are expressed as

$$E\left(L\_1, \mathbf{t}\right) = A\_1 \exp\left\{-i\left(\mathbf{k}\cdot\mathbf{n}\cdot\mathbf{L}\_1\cdot\mathbf{a}\mathbf{p}\cdot\mathbf{t} + \boldsymbol{\Phi}\right)\right\},\tag{4}$$

$$E\left(L\_2, t\right) = A\_2 \exp\left\{-i\left(\mathbf{k}\cdot\mathbf{n}\cdot\mathbf{L}\_2\cdot\mathbf{a}\mathbf{p}\cdot\mathbf{t} + \boldsymbol{\Phi}\_0\right)\right\},\tag{5}$$

where *Aj, k, n, Lj, j, t,* and *0* denote amplitude, wavenumber in a vacuum, refractive index of optical fiber, line length, frequency, time, and initial phase, respectively. Subscript *j* represents the regular line, 1, or the detour line, 2.

The intensity of a waveform with interference, *I*, is calculated by taking the square sum as

$$\mathbf{I} = \begin{vmatrix} \mathbf{E} \ (\mathbf{L}\_{I\prime} \ \mathbf{t}) + \mathbf{E} \ (\mathbf{L}\_{2\prime} \ \mathbf{t}) \ \ \end{vmatrix}^2$$

$$\mathbf{I} = A\_1 \mathbf{i}^2 + A\_2 \mathbf{i}^2 + 2 \ A\_1 \mathbf{A}\_2 \cos \ \begin{pmatrix} \mathbf{k} \ \cdot \mathbf{n} \ \cdot \mathbf{d} \ \mathbf{L} - \mathbf{A} \mathbf{p} \cdot \mathbf{t} \end{pmatrix}, \tag{6}$$

where *L* and represent the differences between line lengths and frequencies, respectively.

The proposed system uses laser pulses at a wavelength of 1650 mm to measure the optical path length. They are introduced from an optical splitter, duplicated, and transmitted

They are distributed by an optical coupler just in front of the OLT, and observed with an oscilloscope. The conventional measurement method evaluates the arrival time interval between the duplicated signals, and converts it to the difference between the lengths of the

Figure 10 shows the received pulses observed with an oscilloscope. When the detour line was 100 m shorter than the regular line, pulses traveling through the detour reached the oscilloscope about 500 ns earlier than through the regular line as shown in Fig 10 (a). The proposed system lengthened the detour line using the optical path length adjuster. Consequently, the former pulse approached the latter. Figure 10 (b) is an example when the gap was shorten to 20 m, where the regular line pulse arrived after a delay of 100 ns. This method failed if the difference between the line lengths was less than 1 m, because the two

That is why the authors have developed an advanced technique for measuring a difference of less than 1 m between optical line lengths. Interferometry enables us to obtain more detailed measurements even when the optical pulses combine. A chirped light source

A 1650 nm laser pulse, discharged from the test light source, is devided by the 2x8 optical splitter. Two pulses travels through the detour and regular line separately toward the oscilloscope. Both pulses along the regular line, *E (L1, t)*, and the detour line, *E (L2, t)* are expressed as

of optical fiber, line length, frequency, time, and initial phase, respectively. Subscript *j*

The intensity of a waveform with interference, *I*, is calculated by taking the square sum as

*I = | E (L1, t) + E (L2, t) | 2*

*2 + 2 A1 A2 cos (k ·n ·*

*1 ·t +*

*2 ·t +*

*L –*

represent the differences between line lengths and frequencies,

*0* denote amplitude, wavenumber in a vacuum, refractive index

*E (L1, t) = A1 exp [ -i (k ·n ·L1 –*

*E (L2, t) = A2 exp [ -i (k ·n ·L2 –*

*t* / *n*, (3)

*0 ) ]* , (4)

*0 ) ]* , (5)

 *·t)* , (6)

*t* is the difference between the signal arrival times for the

**4. Optical line length measurement**

The difference in line length,

where *c* is the speed of light,

pulses combined as shown in Fig. 10 (c).

*j, t,* and

where *Aj, k, n, Lj,*

*L* and

respectively.

where

toward the OLT through the active and detour lines.

regular line and the detour line at a resolution of 1 m.

generates interference in the waveform of a unified pulse.

 *= A1*

*2+ A2*

represents the regular line, 1, or the detour line, 2.

regular and detour lines, and *n* is the refractive index of optical fiber.

*L* is described as

*L* = *c ·*

(c) Optical path difference 1m.

Fig. 10. Delay of duplicated received pulse.

Length Measurement for Optical Transmission Line Using Interferometry 329

0 200 400<PHONE_NUMBER> 1200

Frequency (MHz) (a) Optical path difference 1 m.

0 200 400<PHONE_NUMBER> 1200

(b) Optical path difference 0.5 m.

Frequency (MHz)

Time (ns) (d) Optical path difference 0 m.

(c) Optical path difference 0.1 m.

Time (ns)

1.0 0.8 0.6 0.4 0.2 0

> 1.0 0.8 0.6 0.4 0.2 0

Voltage (V)

Fig. 11. Time-domain unified optical pulses.

0

0

0.1

0.2

Voltage (mV)

0.3

0.1

0.2

Voltage (mV)

0.3

Voltage (V)

The waveform with interference depends on the delay between the pulses' arrival times. Experiments were carried out to obtain time-domain waveforms as shown in Fig. 11. When the optical path difference was 1 m, the waveform contained high-frequency waves as shown in Fig. 11 (a). The less the gap became, the lower-frequency the interfered waveform was composed of. Figures 11 (b) and (c) express that phenomenon. When the lengths of two lines coincided, a quite low-frequency waveform was observed as shown in Fig. 11 (d).

A Fourier-transform spectrum reveals the characteristics. When the optical path difference was 1 m, the waveform with interference was composed of the power spectrum shown in Fig. 12 (a). The peak power represented that the major frequency component was around 700 MHz. Figure 12 (b) and (c) indicate that the peak powers for gaps of 0.5 and 0.1 m were around 400 and 70 MHz, respectively. It became difficult to determine the peak for smaller gaps, because the frequency peak became so low that it was hidden by the near directcurrent part of the frequency component. When the lengths of duplicated lines coincided, the power spectrum was obtained as Fig. 12 (d).

An evaluation of the frequency characteristics in the interfered waveforms showed that the peak power frequencies are proportional to the difference between the line lengths from -1 to 1 m as shown in Fig. 13, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively. This result helps us to determine the optimal position for adjustment. The optimal position where the line lengths coincide can be estimated by extrapolating the data.

The waveform with interference depends on the delay between the pulses' arrival times. Experiments were carried out to obtain time-domain waveforms as shown in Fig. 11. When the optical path difference was 1 m, the waveform contained high-frequency waves as shown in Fig. 11 (a). The less the gap became, the lower-frequency the interfered waveform was composed of. Figures 11 (b) and (c) express that phenomenon. When the lengths of two lines coincided, a quite low-frequency waveform was observed as shown in Fig. 11 (d).

A Fourier-transform spectrum reveals the characteristics. When the optical path difference was 1 m, the waveform with interference was composed of the power spectrum shown in Fig. 12 (a). The peak power represented that the major frequency component was around 700 MHz. Figure 12 (b) and (c) indicate that the peak powers for gaps of 0.5 and 0.1 m were around 400 and 70 MHz, respectively. It became difficult to determine the peak for smaller gaps, because the frequency peak became so low that it was hidden by the near directcurrent part of the frequency component. When the lengths of duplicated lines coincided,

An evaluation of the frequency characteristics in the interfered waveforms showed that the peak power frequencies are proportional to the difference between the line lengths from -1 to 1 m as shown in Fig. 13, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively. This result helps us to determine the optimal position for adjustment. The optimal position where the line lengths coincide can be estimated by extrapolating the data.

(b) Optical path difference 0.5 m.

Time (ns)

(a) Optical path difference 1 m.

Time (ns)

the power spectrum was obtained as Fig. 12 (d).

1.0 0.8 0.6 0.4 0.2

Voltage (V)

0

1.0 0.8 0.6 0.4 0.2

Voltage (V)

0

Fig. 11. Time-domain unified optical pulses.

Length Measurement for Optical Transmission Line Using Interferometry 331

We finally applied the prototype of the optical line shunt system to a 15 km GE-PON optical transmission line replacement. Photographs of our experimental setup are shown in Fig. 14. Figure 14 (a), (b), and (c) display a whole system for experimental optical transmission line, the side view, and the perspective view of the optical line length adjuster, respectively.

A 10 m optical fiber extension was added to the transmission line, while optical signals were switched between the duplicated lines during transmission. We have evaluated the frame loss that occurred during optical line replacement by measuring with a SmartBit network performance analyzer (Spilent Communications, 2011). Optical line length difference was purposely set at 0, 50, 80 and 120 mm in order to verify the effects of measurement accuracy. Same optical line replacement operation, illustrated in Fig. 5, was conducted for each case. No frame loss was observed at any stage of the replacement procedure if the difference between the duplicated line lengths was less than 80 mm. If the difference exceeded 80 mm,

We also evaluated communication quality of the transmission signals through duplicated lines. A 1488.6 nm optical pulse was actually transmitted along both detour and regular lines. Figure 15 shows the eye diagrams of the received signals with regard to optical path difference, L, 1, 22, 44, 66, 88, and 110 mm. If two line lengths were close, communication quality was satisfactory as the eye opened wide in the diagram. When the difference became larger, the transmitted signals turned out erroneous because binary bit patterns were breaking. As a result, we confirmed that the optical signals were completely switched between the regular, detour, and new lines on condition that the line length was adjusted with sufficient accuracy. The experimental results proved that our proposed system successfully relocated

We used a 1650 nm DFB-LD as a test light source. Because it was chirped nonlinearly, interference depended on the pulse width. Experiments were conducted to evaluate the effects on the power spectrum of the interfered waveform with regard to several pulse

The results indicated that the test light pulse with narrower width provided higher sensitivity in estimating optical line length coincidence as shown in Fig. 16, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively. Note that in the practical usage of the narrow width pulse with an oscilloscope, measurable range is restricted by the upper limit of the Fourier-transformed frequency. For example, a test light with 20 ns pulse width has a sensitivity of 6 MHz/mm while 1 GHz oscilloscope is necessary to identify an

As discussed in Section 4, it is difficult for the above proposed method to directly identify the point where the lengths of the detour and regular lines coincide because the frequency

signal multiplexing caused frame loss at step 3 in Fig. 5.

an in-service broadband network without any service interruption.

**5.2 Deterministic line length detection using frequency shift method**

**5. Improvement of line length measurement**

widths 20, 50, 100, 200, and 500 ns.

optical path difference of 150 mm.

**5.1 Sensitivity improvement due to narrow pulse width**

Fig. 12. Frequency-domain unified optical pulses.

Fig. 13. Estimation of line length coincidence.

Our investigation has established a technique for distinguishing the difference between line lengths to an accuracy of better than 10 mm by analyzing interfering waveforms created by chirped laser pulses. We have realized a complete length measurement for optical transmission lines from 100 m to 10 mm.

0 200 400<PHONE_NUMBER> 1200

0 200 400<PHONE_NUMBER> 1200

0

Our investigation has established a technique for distinguishing the difference between line lengths to an accuracy of better than 10 mm by analyzing interfering waveforms created by chirped laser pulses. We have realized a complete length measurement for optical

Optical path difference (m)

P

e

a

k fre

q

u

e

n

c

H

z)

y (M

200

400

600

800

Frequency (MHz) (d) Optical path difference 0 m.

Frequency (MHz) (c) Optical path difference 0.1 m.

0

0

0.1

0.2

Voltage (mV)

Fig. 12. Frequency-domain unified optical pulses.

Fig. 13. Estimation of line length coincidence.

transmission lines from 100 m to 10 mm.

0.3

0.1

0.2

Voltage (mV)

0.3

We finally applied the prototype of the optical line shunt system to a 15 km GE-PON optical transmission line replacement. Photographs of our experimental setup are shown in Fig. 14. Figure 14 (a), (b), and (c) display a whole system for experimental optical transmission line, the side view, and the perspective view of the optical line length adjuster, respectively.

A 10 m optical fiber extension was added to the transmission line, while optical signals were switched between the duplicated lines during transmission. We have evaluated the frame loss that occurred during optical line replacement by measuring with a SmartBit network performance analyzer (Spilent Communications, 2011). Optical line length difference was purposely set at 0, 50, 80 and 120 mm in order to verify the effects of measurement accuracy. Same optical line replacement operation, illustrated in Fig. 5, was conducted for each case. No frame loss was observed at any stage of the replacement procedure if the difference between the duplicated line lengths was less than 80 mm. If the difference exceeded 80 mm, signal multiplexing caused frame loss at step 3 in Fig. 5.

We also evaluated communication quality of the transmission signals through duplicated lines. A 1488.6 nm optical pulse was actually transmitted along both detour and regular lines. Figure 15 shows the eye diagrams of the received signals with regard to optical path difference, L, 1, 22, 44, 66, 88, and 110 mm. If two line lengths were close, communication quality was satisfactory as the eye opened wide in the diagram. When the difference became larger, the transmitted signals turned out erroneous because binary bit patterns were breaking.

As a result, we confirmed that the optical signals were completely switched between the regular, detour, and new lines on condition that the line length was adjusted with sufficient accuracy. The experimental results proved that our proposed system successfully relocated an in-service broadband network without any service interruption.

## **5. Improvement of line length measurement**

## **5.1 Sensitivity improvement due to narrow pulse width**

We used a 1650 nm DFB-LD as a test light source. Because it was chirped nonlinearly, interference depended on the pulse width. Experiments were conducted to evaluate the effects on the power spectrum of the interfered waveform with regard to several pulse widths 20, 50, 100, 200, and 500 ns.

The results indicated that the test light pulse with narrower width provided higher sensitivity in estimating optical line length coincidence as shown in Fig. 16, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively. Note that in the practical usage of the narrow width pulse with an oscilloscope, measurable range is restricted by the upper limit of the Fourier-transformed frequency. For example, a test light with 20 ns pulse width has a sensitivity of 6 MHz/mm while 1 GHz oscilloscope is necessary to identify an optical path difference of 150 mm.

## **5.2 Deterministic line length detection using frequency shift method**

As discussed in Section 4, it is difficult for the above proposed method to directly identify the point where the lengths of the detour and regular lines coincide because the frequency

Length Measurement for Optical Transmission Line Using Interferometry 333

30 mV

Voltage

Voltage

Voltage

Voltage

Voltage

Voltage

30 mV

30 mV

30 mV

30 mV

30 mV

Fig. 15. Eyediagram of duplicated transmission signal.

10 ns

10 ns

10 ns

10 ns

10 ns

10 ns

Time

L=22 mm

L1 mm

L=44 mm

L=66 mm

L=88 mm

L=110 mm

peak is hidden by the near direct-current part of the frequency component in the power spectrum.

(a) Whole system for experimental optical transmission line.

(b) Side view of optical line length adjuster.

(c) Perspective view of optical line length adjuster.

Fig. 14. Photographs of optical line replacement experiments.

peak is hidden by the near direct-current part of the frequency component in the power

Regular

Detour line

line

(a) Whole system for experimental optical transmission line.

(b) Side view of optical line length adjuster.

Collimators

Retroreflectors

(c) Perspective view of optical line length adjuster.

Collimators

Sliding stage Retroreflectors

ONUs

Optical line length adjuster

spectrum.

OLTs

Oscilloscope

Sliding stage

Fig. 14. Photographs of optical line replacement experiments.

Fig. 15. Eyediagram of duplicated transmission signal.

Length Measurement for Optical Transmission Line Using Interferometry 335

white circles in Fig. 20, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum,

Black circles in the figure represent the remainders of subtracting 80 MHz from the peak frequency. They also show the linearity as good as we obtained without the frequency filter. If adopting this method, we can determine the optimal position where the line lengths coincide in high accuracy only by finding the specified value of the frequency shifter instead

Fig. 17. Advanced duplex optical fiber transmission system with frequency shifter.

O/E

Optical line length adjuster

Detour (15 km)

Regular line (15 km)

WIC

Optical splitter

Oscilloscope

80 MHz frequency shifter

light source

respectively.

of extrapolation.

Fig. 16. Estimation of line length coincidence.

We are developing an advanced method to resolve the problem. Our new proposal is to deliberately shift the interference frequency. Figure 17 shows an advanced duplex optical fiber transmission system with frequency shifter. It equips an 80 MHz frequency shifter along the detour line. Other devices are the same as those described in Section 3.2.

We have conducted the same experiments with this advanced transmission system as before. We evaluated interference of the duplicated signals by changing the optical path difference with the optical line length adjuster.

Experimental results are shown in Figs. 18 and 19. Time-domain waveform of unified optical pulses indicates in Fig. 18 that signals along the detour and regular lines have interfered. A Fourier-transformation provided the frequency-domain characteristics of the interference as shown in Fig. 19. It reveals that each interfered waveform has the peak of spectrum power. In case the detour line length agrees with the regular line, the frequency of the peak power is 80 MHz which is as much as we have assigned with the frequency shifter.

An evaluation of the frequency of the peak power in terms of the optical path difference confirmed that the frequencies were proportional to the optical path difference as shown by

150 100 50 0 50 100 150

Optical path difference(mm)

along the detour line. Other devices are the same as those described in Section 3.2.

We are developing an advanced method to resolve the problem. Our new proposal is to deliberately shift the interference frequency. Figure 17 shows an advanced duplex optical fiber transmission system with frequency shifter. It equips an 80 MHz frequency shifter

We have conducted the same experiments with this advanced transmission system as before. We evaluated interference of the duplicated signals by changing the optical path

Experimental results are shown in Figs. 18 and 19. Time-domain waveform of unified optical pulses indicates in Fig. 18 that signals along the detour and regular lines have interfered. A Fourier-transformation provided the frequency-domain characteristics of the interference as shown in Fig. 19. It reveals that each interfered waveform has the peak of spectrum power. In case the detour line length agrees with the regular line, the frequency of the peak power is 80 MHz which is as much as we have assigned with the frequency

An evaluation of the frequency of the peak power in terms of the optical path difference confirmed that the frequencies were proportional to the optical path difference as shown by

0

Fig. 16. Estimation of line length coincidence.

difference with the optical line length adjuster.

500

Frequency(MHz)

shifter.

1000

:500 ns :200 ns :100 ns :50 ns :20 ns

Pulse width

white circles in Fig. 20, where the horizontal and vertical axes express the optical path difference and the frequency for the maximum power of Fourier-transformed spectrum, respectively.

Black circles in the figure represent the remainders of subtracting 80 MHz from the peak frequency. They also show the linearity as good as we obtained without the frequency filter.

If adopting this method, we can determine the optimal position where the line lengths coincide in high accuracy only by finding the specified value of the frequency shifter instead of extrapolation.

Fig. 17. Advanced duplex optical fiber transmission system with frequency shifter.

Length Measurement for Optical Transmission Line Using Interferometry 337

0 100 200 300 400 500 Frequency (MHz)

(a) Optical path difference 0.31 m.

173.7 MHz

0 100 200 300 400 500 Frequency(MHz)

(b) Optical path difference 0.14 m.

0 100 200 300 400 500 Frequency(MHz)

(c) Optical path difference 0 m.

285.06 MHz

=205.06 MHz

=93.7 MHz

0

0

0

10

20

Voltage(mV)

Fig. 19. Frequency-domain unified optical pulses.

30

40

50

10

20

Voltage(mV)

30

40

80 MHz

80 MHz

50

10

20

Voltage (mV)

30

40

80 MHz

50

Fig. 18. Time-domain unified optical pulses.

0<PHONE_NUMBER>00

0<PHONE_NUMBER>00

Time (ns)

(b) Optical path difference 0.14 m.

Time (ns)

(a) Optical path difference 0.31 m.

0

100

0

0

20

40

Voltage(mV)

Fig. 18. Time-domain unified optical pulses.

60

80

100

20

40

60

Voltage (mV)

80

20

40

60

Voltage (mV)

80

100

Fig. 19. Frequency-domain unified optical pulses.

Length Measurement for Optical Transmission Line Using Interferometry 339

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Modulators at One Fourth of the Data Rate, *IEEE Photonics Technology Letters*,

one Protection Switching Technique for Optical Fiber Network, *Proceedings of 14th European Conference on Networks and Optical Communications/ 4th Conference on*

measurement for error-free optical fiber line switching system, *Proceedings of*

**7. References**

April 1990

Optical Networks (PON), 1998

*2009*, paper PDPD6, 2009

*Cable Symposium*, November 2002

Vol.19, No.22, (2007), pp.1837- 1839

*Optical Cabling and Infrastructure*, 2009

Spilent Communications. (2011). *SmartBits,* Available from:

http://www.spirent.com/Solutions-Directory/Smartbits.aspx

*Optic Engineers Conference*, pp. 1-6. San Diego, February 2008

Fig. 20. Deterministic identification of line length coincidence.

## **6. Conclusion**

This paper proposed the high-resolution optical line length measurement technique, which was applied to the switching method for optical transmission lines transferring live optical signals. The method exchanged optical fibers instead of using electric apparatus to control transmission speed.

We have disclosed the requirements for replacement of in-service optical transmission cables after investigating problems of the existing methods.

We next proposed the optical line switching technique and the optical cable replacement procedure that used duplex transmission lines. It was determined that the tolerance of the difference in duplex line length is 80 mm as for the GE-PON transmission system.

The optical line shunt system was designed, and its prototype was actually constructed. An optical line length adjuster, designed based on an FSO system, continuously lengthened the optical line up to 100 m with a resolution of 0.1 mm.

An optical line length measurement technique was investigated in detail by evaluating the duplicated test light pulses. As a result, it successfully identified the difference in length between the duplicated lines from 100 m to 10 mm. An interferometry measurement distinguished the difference between line lengths to an accuracy of better than 10 mm by analyzing interfering waveforms created by chirped laser pulses. This system was applied to a 15 km GE-PON network and succeeded in replacing the communication lines without inducing any frame loss.

We also discussed the improvement of line length measurement. It was suggested that the test light with narrow pulse width could give a higher sensitivity. If using the frequency shifter within the detour transmission line, we could identify the optimal position where the line lengths coincide in high accuracy without extrapolation.

## **7. References**

338 Interferometry – Research and Applications in Science and Technology

: measured value : remainder

0 0.1 0.2 0.3 0.4

This paper proposed the high-resolution optical line length measurement technique, which was applied to the switching method for optical transmission lines transferring live optical signals. The method exchanged optical fibers instead of using electric apparatus to control

We have disclosed the requirements for replacement of in-service optical transmission

We next proposed the optical line switching technique and the optical cable replacement procedure that used duplex transmission lines. It was determined that the tolerance of the

The optical line shunt system was designed, and its prototype was actually constructed. An optical line length adjuster, designed based on an FSO system, continuously lengthened the

An optical line length measurement technique was investigated in detail by evaluating the duplicated test light pulses. As a result, it successfully identified the difference in length between the duplicated lines from 100 m to 10 mm. An interferometry measurement distinguished the difference between line lengths to an accuracy of better than 10 mm by analyzing interfering waveforms created by chirped laser pulses. This system was applied to a 15 km GE-PON network and succeeded in replacing the communication lines without

We also discussed the improvement of line length measurement. It was suggested that the test light with narrow pulse width could give a higher sensitivity. If using the frequency shifter within the detour transmission line, we could identify the optimal position where the

difference in duplex line length is 80 mm as for the GE-PON transmission system.

0

Fig. 20. Deterministic identification of line length coincidence.

cables after investigating problems of the existing methods.

optical line up to 100 m with a resolution of 0.1 mm.

line lengths coincide in high accuracy without extrapolation.

200

Peak

**6. Conclusion**

transmission speed.

inducing any frame loss.

frequency (M

Hz)

400

Optical path difference (m)

0.5

**1. Introduction**

standing wave scattering, respectively.

Atomic interferometry is very useful in fundamental studies of coherence, decoherence and phase shifts and for practical precision measurements, with the example of gravimeters, gyroscopes, and gradiometers (Cronin et al., 2009). Bose-Einstein Condensate (BEC) based atomic interferometry provides high contrasts, long integration times and the possible use of small devices (Baumgärtner et al., 2010; Sapiro et al., 2009a; Torii et al., 2000). In such atomic interferometry, coherent momentum manipulation is very effective for splitting and recombining the condensate (Deng et al., 1999; Kozuma, Deng et al., 1999; Ovchinnikov et al., 1999), for realizing the interference. In some precision measurements, the accumulated phase is positively correlated to the atomic velocity, so that the larger the atomic momentum is, the

**High Order Momentum States by**

*School of Electronics Engineering and Computer Science,*

**Light Wave Scattering**

Xiaoji Zhou and Xuguang Yue

*Peking University, Beijing*

*China*

**17**

Based on our previous experimental and theoretical work about the interaction between laser pulses and a BEC, in this chapter we mainly present methods of getting high order of momentum states by resonant superradiance (Zhou et al., 2009) and by multi-pulse Kapitza-Dirac scattering (Xiong et al., 2011), corresponding to travelling wave scattering and

In the superradiant Rayleigh scattering (Bar-Gill et al., 2007; Guo et al., 2008; Inouye et al., 1999; Kozuma, Suzuki et al., 1999; Moore & Meystre, 1999; Pu et al., 2003; Sadler et al., 2007; Schneble et al., 2003; Slama et al., 2007; Zobay & Nikolopoulos, 2006), the spatial and time evolutions of superradiant scattering are studied for a weak pump beam with different frequency components traveling along the long axis of an elongated Bose-Einstein condensate. Through the analysis of the mode competition between the different resonant channels and the local depletion of the spatial distribution, we can get a large number of high-order forward

On the other hand, the atomic diffraction from a standing wave light grating is a primary method of atomic momentum manipulation. Different to usual separation of the Kapitza-Dirac regime, the Bragg regime and the channeling regime (Keller et al., 1999), the scattering process is described by the projection of atomic states between the momentum presentation and Bloch states form by the scattering standing waves. According to this

modes by resonant frequency components of the pump beam (Zhou et al., 2009).

more precise the measurements can be if the measuring time is unchanged.

## **High Order Momentum States by Light Wave Scattering**

Xiaoji Zhou and Xuguang Yue

*School of Electronics Engineering and Computer Science, Peking University, Beijing China*

## **1. Introduction**

340 Interferometry – Research and Applications in Science and Technology

Tsujimura, T., Yoshida, K., Tanaka, K., Katayama, K., & Azuma, Y. (2009b). Interruption-free

Watanabe, I., Saito, H., Kobayashi, H., & Takashima, S. (1992). Subscriber optical fiber cable

Xu, D. J., Yen, W., & Ho, E. (2001). Proposal of a New Protection Mechanism for ATM PON

Yoshida, K., & Tsujimura, T. (2010). Seamless Transmission Between Single-mode Optical

Yoshida, K., Tanaka, K., & Tsujimura, T. (2011). Robotic Waveguide by Free Space Optics, In: *Advances in Mechatronics,* Martínez-Alfaro, H., InTech, ISBN 978-953-307-373-6 Yoshida, K., Tanaka, K., Katayama, K., Tsujimura, T., & Azuma, Y. (2009). Collimator Focus

*Proceedings of ICROS-SICE International Joint Conference,* pp. 1338-1341, 2009

*Application,* Vol. 8, No. 16, (2009), pp. 125-131

2160-2165, 2001

*Technology*, Vol. 10, No. 6, (June, 1992), pp. 720-727 Willebrand, H. et al. (1999), *Free-Space Optics*, Sams Publishing

*System Integration*, Vol. 3, No. 2, (2010), pp.94-100

Shunt System for Fiber Optics Transmission Line, *SICE Transactions on Industrial*

transfer splicing using mechanically transferable connectors, *IEEE Journal of Lightwave*

Interface, *Proceedings of IEEE International Conference on Communications*, Vol. 7, pp.

Fibers Using Free Space Optics System, *SICE Journal of Control Measurement and*

Adjustment for Free Space Optics System Using Single-mode Optical Fiber,

Atomic interferometry is very useful in fundamental studies of coherence, decoherence and phase shifts and for practical precision measurements, with the example of gravimeters, gyroscopes, and gradiometers (Cronin et al., 2009). Bose-Einstein Condensate (BEC) based atomic interferometry provides high contrasts, long integration times and the possible use of small devices (Baumgärtner et al., 2010; Sapiro et al., 2009a; Torii et al., 2000). In such atomic interferometry, coherent momentum manipulation is very effective for splitting and recombining the condensate (Deng et al., 1999; Kozuma, Deng et al., 1999; Ovchinnikov et al., 1999), for realizing the interference. In some precision measurements, the accumulated phase is positively correlated to the atomic velocity, so that the larger the atomic momentum is, the more precise the measurements can be if the measuring time is unchanged.

Based on our previous experimental and theoretical work about the interaction between laser pulses and a BEC, in this chapter we mainly present methods of getting high order of momentum states by resonant superradiance (Zhou et al., 2009) and by multi-pulse Kapitza-Dirac scattering (Xiong et al., 2011), corresponding to travelling wave scattering and standing wave scattering, respectively.

In the superradiant Rayleigh scattering (Bar-Gill et al., 2007; Guo et al., 2008; Inouye et al., 1999; Kozuma, Suzuki et al., 1999; Moore & Meystre, 1999; Pu et al., 2003; Sadler et al., 2007; Schneble et al., 2003; Slama et al., 2007; Zobay & Nikolopoulos, 2006), the spatial and time evolutions of superradiant scattering are studied for a weak pump beam with different frequency components traveling along the long axis of an elongated Bose-Einstein condensate. Through the analysis of the mode competition between the different resonant channels and the local depletion of the spatial distribution, we can get a large number of high-order forward modes by resonant frequency components of the pump beam (Zhou et al., 2009).

On the other hand, the atomic diffraction from a standing wave light grating is a primary method of atomic momentum manipulation. Different to usual separation of the Kapitza-Dirac regime, the Bragg regime and the channeling regime (Keller et al., 1999), the scattering process is described by the projection of atomic states between the momentum presentation and Bloch states form by the scattering standing waves. According to this

Fig. 1. (Color online) Our experimental scheme. A cigar-shape BEC is illuminated by a far off-resonant laser pulse along its long axis **zˆ**. Collective Rayleigh scattering induces superradiance. Two end-fire modes, which are also along **zˆ** axis, form in superradiance

High Order Momentum States by Light Wave Scattering 343

order scattering mode is suppressed at the resonant condition Δ*ω* = 8*ωr* and the forward second order mode is enhanced, resulting from the combination of mode competition effects and spatial distribution of the modes. The case of the three-frequency pump beams for a large number of the forward third order scattering modes, and the higher modes for more resonant frequencies are studied, which supplies a new method to get a large number of atoms in higher

We consider the pump laser, with amplitude E*l*(*t*), polarization **ey**, wave vector **k***l*, frequencies *ωl* and *ωl* − Δ*ωn*, propagating along the long axis **zˆ** of an elongated BEC, **E***l* = E*l*(*t*)**ey**[(1 +

happens, end-fire modes spread along the same axis. The E+ mode has the same direction as the incident light and mainly interacts with the right part of the condensate, and the E− mode overlaps with the left part of the condensate. The atoms are recoiled to some discrete momentum states with momentum 2*mh*¯ **k**, where *m* is an integer and the wave vector of end-fire mode light *k* is approximated as *kl* for energy conservation. The total electric field **E**(**r**, *t*) = **E**(+) + **E**(−) is given by (Bar-Gill et al., 2007; Li, Zhou et al., 2008; Yang et al., 2008;

)E*l*(*t*)*e*−i(*ωlt*−*klz*)

where *ω* = *ck*, **E**(−) = **E**(+)∗, and E+ is ignored because it has the same wave vector as the pump beam but is very small in comparison to E*l*. Δ*ωn* satisfies the condition Δ*ωn* � *ωl* (Bar-Gill et al., 2007) and the initial phases of the different frequency components

Since the BEC is tightly constrained in its short axis (**xˆ**, **yˆ**) in the present superradiance setup and the Fresnel number of the optical field is around 1, one dimensional approximation is usually used (Bar-Gill et al., 2007; Hilliard et al., 2008; Inouye et al., 1999; Li, Zhou et al.,

/2 + E−(*z*, *t*)*e*−i(*ωt*+*kz*)

]**ey** (1)

)*ei*(*klz*−*ωlt*) + *c*.*c*.]/2, as shown in Fig. 1. When supperradiant Rayleigh scattering

process and the 1st-order recoiled atoms obtain a momentum of 2¯*h***k**.

order forward modes. Finally, some discussion and conclusion are given.

*ei*Δ*ωnt*

**2.1 Model for a multiple-frequency end-pumped beam**

Σ*nei*Δ*ωnt*

Zobay & Nikolopoulos, 2006)

are assumed to be zero.

**E**(+)(**r**, *t*) = [(1 + ∑*n*

method, we are able to design and realize several specific high order momentum states for the atomic interferometry (Xiong et al., 2011).

## **2. High-order momentum modes by resonant superradiant scattering**

A typical superradiance experiment consists in a far off-resonant laser pulse traveling along the short axis of a cigar-shaped BEC sample (Inouye et al., 1999), the scattered lights, called end-fire modes, propagate along the long axis of the condensate, and the recoiled atoms are refereed to as side modes.  Several theoretical descriptions of these cooperative scattering in BEC with single-frequency pump have also been presented (Guo et al., 2008; Moore & Meystre, 1999; Pu et al., 2003; Zobay & Nikolopoulos, 2006).

For the long and weak pump beam, we can observe the forward peaks correspond to Bragg diffraction of atoms (Inouye et al., 1999), where the high order scattering is limited by detuning barriers for the end-fire mode radiation (Zobay & Nikolopoulos, 2007). On the other hand, a X-shaped recoiling pattern is demonstrated in a short and strong pulse as Kapitza-Dirac diffraction of atoms (Schneble et al., 2003), where an atom in the condensate absorbs a photon from the pump laser, then emits a photon into an end-fire mode, and recoils forwardly. Meanwhile another atom absorbs a photon from the end-fire modes, emits into the pump beam and finally recoils backwardly. In this case, there is an energy mismatch of four times the one-photon recoil kinetic energy ¯*hωr* in backward scattering, which then remains very weak unless a short pumping pulse with a broad spectrum is used. Hence, two phase-locked incident lasers with the frequency difference Δ*ω* compensating for the energy mismatch has been used (Bar-Gill et al., 2007; Cola et al., 2009; Yang et al., 2008), which is named resonant superradiance, where a large number of backward recoiling atoms can be produced.

Followed that, it is attractive to extent this idea to achieve a high momentum transfer by overcoming the detuning barriers, by a weak and long pump beams with the resonant frequency. It requires to analysis the competition between the different transition channels and the spatial distribution of different modes. Because the above traditional superradiant-scattering configuration involves many atomic side modes coupled together, to simplify it, we chose another configuration where a pump beam travels along the long axis of the BEC. This scheme is widely studied in photon echo (Piovella et al., 2003), decoherence (Fallani et al., 2005), spatial distribution effects (Li, Zhou et al., 2008) and self-organized formation of dynamic gratings (Hilliard et al., 2008). Since the pulse length is far longer than the initial spontaneous process (Zobay & Nikolopoulos, 2006), we choose the semi-classical theory which can well describe the experimental results (Bar-Gill et al., 2007; Yang et al., 2008; Zobay & Nikolopoulos, 2006).

In this section, we first introduce the semi-classical theory for the superradiance scattering with a several-frequency pump in the weak coupling. Then the spatial and time evolutions of scattered modes are analyzed for two-frequency pump beam, we find the backward first 2 Will-be-set-by-IN-TECH

method, we are able to design and realize several specific high order momentum states for

A typical superradiance experiment consists in a far off-resonant laser pulse traveling along the short axis of a cigar-shaped BEC sample (Inouye et al., 1999), the scattered lights, called end-fire modes, propagate along the long axis of the condensate, and the recoiled atoms are refereed to as side modes.  Several theoretical descriptions of these cooperative scattering in BEC with single-frequency pump have also been presented (Guo et al., 2008; Moore & Meystre, 1999;

For the long and weak pump beam, we can observe the forward peaks correspond to Bragg diffraction of atoms (Inouye et al., 1999), where the high order scattering is limited by detuning barriers for the end-fire mode radiation (Zobay & Nikolopoulos, 2007). On the other hand, a X-shaped recoiling pattern is demonstrated in a short and strong pulse as Kapitza-Dirac diffraction of atoms (Schneble et al., 2003), where an atom in the condensate absorbs a photon from the pump laser, then emits a photon into an end-fire mode, and recoils forwardly. Meanwhile another atom absorbs a photon from the end-fire modes, emits into the pump beam and finally recoils backwardly. In this case, there is an energy mismatch of four times the one-photon recoil kinetic energy ¯*hωr* in backward scattering, which then remains very weak unless a short pumping pulse with a broad spectrum is used. Hence, two phase-locked incident lasers with the frequency difference Δ*ω* compensating for the energy mismatch has been used (Bar-Gill et al., 2007; Cola et al., 2009; Yang et al., 2008), which is named resonant superradiance, where a large number of backward recoiling atoms can be

Followed that, it is attractive to extent this idea to achieve a high momentum transfer by overcoming the detuning barriers, by a weak and long pump beams with the resonant frequency. It requires to analysis the competition between the different transition channels and the spatial distribution of different modes. Because the above traditional superradiant-scattering configuration involves many atomic side modes coupled together, to simplify it, we chose another configuration where a pump beam travels along the long axis of the BEC. This scheme is widely studied in photon echo (Piovella et al., 2003), decoherence (Fallani et al., 2005), spatial distribution effects (Li, Zhou et al., 2008) and self-organized formation of dynamic gratings (Hilliard et al., 2008). Since the pulse length is far longer than the initial spontaneous process (Zobay & Nikolopoulos, 2006), we choose the semi-classical theory which can well describe the experimental results (Bar-Gill et al., 2007;

In this section, we first introduce the semi-classical theory for the superradiance scattering with a several-frequency pump in the weak coupling. Then the spatial and time evolutions of scattered modes are analyzed for two-frequency pump beam, we find the backward first

**2. High-order momentum modes by resonant superradiant scattering**

the atomic interferometry (Xiong et al., 2011).

Pu et al., 2003; Zobay & Nikolopoulos, 2006).

Yang et al., 2008; Zobay & Nikolopoulos, 2006).

produced.

Fig. 1. (Color online) Our experimental scheme. A cigar-shape BEC is illuminated by a far off-resonant laser pulse along its long axis **zˆ**. Collective Rayleigh scattering induces superradiance. Two end-fire modes, which are also along **zˆ** axis, form in superradiance process and the 1st-order recoiled atoms obtain a momentum of 2¯*h***k**.

order scattering mode is suppressed at the resonant condition Δ*ω* = 8*ωr* and the forward second order mode is enhanced, resulting from the combination of mode competition effects and spatial distribution of the modes. The case of the three-frequency pump beams for a large number of the forward third order scattering modes, and the higher modes for more resonant frequencies are studied, which supplies a new method to get a large number of atoms in higher order forward modes. Finally, some discussion and conclusion are given.

### **2.1 Model for a multiple-frequency end-pumped beam**

We consider the pump laser, with amplitude E*l*(*t*), polarization **ey**, wave vector **k***l*, frequencies *ωl* and *ωl* − Δ*ωn*, propagating along the long axis **zˆ** of an elongated BEC, **E***l* = E*l*(*t*)**ey**[(1 + Σ*nei*Δ*ωnt* )*ei*(*klz*−*ωlt*) + *c*.*c*.]/2, as shown in Fig. 1. When supperradiant Rayleigh scattering happens, end-fire modes spread along the same axis. The E+ mode has the same direction as the incident light and mainly interacts with the right part of the condensate, and the E− mode overlaps with the left part of the condensate. The atoms are recoiled to some discrete momentum states with momentum 2*mh*¯ **k**, where *m* is an integer and the wave vector of end-fire mode light *k* is approximated as *kl* for energy conservation. The total electric field **E**(**r**, *t*) = **E**(+) + **E**(−) is given by (Bar-Gill et al., 2007; Li, Zhou et al., 2008; Yang et al., 2008; Zobay & Nikolopoulos, 2006)

$$\mathbf{E}^{(+)} (\mathbf{r}, t) = [(1 + \sum\_{\mathbf{n}} e^{i\Lambda \omega\_{\mathbf{n}} t}) \mathcal{E}\_l(t) e^{-\mathbf{i} \left(\omega\_l t - k\_l z\right)}/2 + \mathcal{E}\_-(z, t) e^{-\mathbf{i} \left(\omega t + kz\right)}] \mathbf{e}\_{\mathbf{y}} \tag{1}$$

where *ω* = *ck*, **E**(−) = **E**(+)∗, and E+ is ignored because it has the same wave vector as the pump beam but is very small in comparison to E*l*. Δ*ωn* satisfies the condition Δ*ωn* � *ωl* (Bar-Gill et al., 2007) and the initial phases of the different frequency components are assumed to be zero.

Since the BEC is tightly constrained in its short axis (**xˆ**, **yˆ**) in the present superradiance setup and the Fresnel number of the optical field is around 1, one dimensional approximation is usually used (Bar-Gill et al., 2007; Hilliard et al., 2008; Inouye et al., 1999; Li, Zhou et al., 2008). We expand the wavefunction of the condensate *ψ*(**r**, *t*) in momentum space, *ψ*(**r**, *t*) = ∑*m φm*(*z*, *t*) ×*e*−*i*(*ωmt*−2*mkz*) , where *φm*(*z*, *t*) = *ψm*(*z*, *t*)/ √*A*, *ωm* = 2¯*hm*2*k*2/*M* with *M* the atomic mass, *m* = 0 corresponds to the residual condensates, *m* �= 0 denotes the side modes, and *A* is the average cross area of the condensate perpendicular to **ˆz**. Using the Maxwell-Schrödinger equations, we obtain dynamics equations for *φm*(*z*, *t*),

$$\mathrm{i}\frac{\partial\phi\_{m}}{\partial t} = -\frac{\hbar}{2M}\frac{\partial^{2}\phi\_{m}}{\partial z^{2}} - \frac{2\mathrm{im}\hbar\mathrm{k}}{M}\frac{\partial\phi\_{m}}{\partial z} + \overline{\mathrm{g}}\left[\mathcal{E}\_{-}^{\*}\phi\_{m-1}e^{-4\mathrm{i}(1-2m)\omega\cdot t} + \mathcal{E}\_{-}\phi\_{m+1}e^{-4\mathrm{i}(1+2m)\omega\cdot t}\right],\tag{2}$$

where *ωr* = *hk*¯ 2 *l* /2*M* is the recoil frequency, the coupling between modes is given by

$$\bar{\mathbf{g}}(t) = \mathbf{g} \left( 1 + \sum\_{n} e^{i\Delta\omega\_{n}t} \right) \tag{3}$$

Fig. 2. (Color online) Light-field components of a two-frequency pump laser. The broad arrows are the pump laser and narrow ones are the end-fire mode (scattering optical field). In a spontaneous process, atoms in the condensate absorb photons from the pump laser with frequencies *ωl* and *ωl* − 8*ωr*, are scattered to side mode *m* = 1 and emit end-fire mode photons with frequency *ωl* − 4*ωr* (dashed arrow) and *ωl* − 12*ωr* (dotted arrow), respectively. Meanwhile, atoms in the condensate can also absorb end-fire mode photons with frequency

High Order Momentum States by Light Wave Scattering 345

*ωl* − 4*ωr*, be scattered back to side mode *m* = −1 and emit photons with frequency *ωl* − 8*ωr*(solid arrow), resonating to one of the pump laser components. The side mode *m* = 1 can absorb pump laser photons with frequency *ωl* and be scattered to mode *m* = 2, emitting photons with frequency *ωl* − 12*ωr* resonating to the existing end-fire mode.

by the pump laser.

mode *m* = 2 is enhanced.

due to the energy barrier. However, if we use the two components pump laser with frequency difference 8*ωr*, i.e. resonant frequency difference, the energy mismatch can be compensated

Although the resonant condition for the backward mode is satisfied, it should be noticed that two scattering channels exist almost simultaneously. One is atoms scattered from side mode *m* = 0 to *m* = −1 and the other is from *m* = 1 to *m* = 2, resulting in mode competition. The transition from mode *m* = 1 to *m* = 2 requires absorbtion of photons from pump laser, while the backward transition takes photons from the end-fire mode. Because the intensity of the pump laser is far greater than that of the end-fire mode, the transition from *m* = 1 to *m* = 2 has a bigger probability than the transition from *m* = 0 to *m* = −1. Thus the population of the backward mode *m* = −1 is suppressed even at the resonant condition, while the forward

However, the existence of competition between these two channels may not lead to the suppression of the backward mode. If these two channels happen in different spacial parts

with the coupling factor *g* = 3*πc*3*R*/(2*ω*2 *l AL*), *R* is the Rayleigh scattering rate of the pump components, and *L* is the BEC length.

The first term on the right-hand-side of Eq.(2) describes the dispersion of *φm*, and the second term gives rise to their translation. The terms in square brackets describe the atom exchange between *φm* and *φm*+1 or *φm*−1 through the pump laser and end-fire mode fields. An atom in mode *m* may absorb a laser photon and emit it into end-fire mode E−, and the accompanying recoil drives the atom into *m* + 1 mode, hence atoms with mode *m* + 1 can emerge in forward scattering. On the other hand, in the backward scattering, atoms with mode *m* absorb one E− mode photon, deposit it into the laser mode and go into mode *m* − 1. The envelope function of end-fire mode E− is given by

E− = −i *ωrg*¯ 2*cε*0 +∞ *z* d*z*� ∑ *m φm*(*z*� , *t*)*φ*∗ *m*+1(*z*� , *t*)*e* i4(2*m*+1)*ωrt* , (4)

indicating that the end-fire mode field E− is due to the transition between *m* and *m* + 1 mode and the magnitude of E− depends on the spatial overlap between the two modes. In addition, there is a frequency difference of 8*ωr* between adjacent modes.

### **2.2 Mode competition for a two-frequency pump beam**

In the case of a single-frequency pump in the weak coupling regime, the evolution of the side modes and the end-fire mode indicates that the scattering is a localized process. For this end-pumping configuration, the scattering first starts on the leading edge of the BEC and then moves towards the tailing edge. To investigate the effect of the two-frequency pump beam, the different frequency components of the end-fire mode which indicate the energy change during the scattering are depicted in Fig. 2. The momentum of side mode *m* = *n* is 2*nh*¯**k**, and its kinetic energy is 4*n*2*h*¯ 2**k**2/2*M* = 4*n*2*h*¯ *ωr*. For the pump component with frequency *ωl*, atoms from the condensate are pumped to the side mode *m* = 1 and emit end-fire mode photons with frequency *ωl* − 4*ωr* spontaneously. However, in the backward scattering process, an atom in the condensate absorbs the end-fire mode (*ωl* − 4*ωr*) and emits a photon with frequency *ωl* back into the pump laser. Since energy is not conserved in backward-scattering, the backward side mode is not populated in weak-pulse regime. Side mode *m* = 2 is also not populated 4 Will-be-set-by-IN-TECH

2008). We expand the wavefunction of the condensate *ψ*(**r**, *t*) in momentum space, *ψ*(**r**, *t*) =

atomic mass, *m* = 0 corresponds to the residual condensates, *m* �= 0 denotes the side modes, and *A* is the average cross area of the condensate perpendicular to **ˆz**. Using the

*l* /2*M* is the recoil frequency, the coupling between modes is given by

*ei*Δ*ωnt*