After a particular filter has been set, the corresponding pattern in the spatial domain is

*pxy F Puv* , exp ,

In order to reduce the remains noise, after the amplitude filter, we use a bilateral filter. It extends the concept of Gaussian smoothing by weighting the filter coefficients with their corresponding relative pixel intensities. Pixels that are very different in intensity from the central pixel are weighted less even though they may be in close proximity to the central pixel. This is effectively a convolution with a non-linear Gaussian filter, with weights based on pixel intensities. This is applied as two Gaussian filters at a localized pixel, one in the spatial domain, named the domain filter, and one in the intensity domain, named the range

All the blocks presented in this chapter are taken in consideration in our simulations. Although the CMOS sensor has the Bayer color sampling and interpolation, we did not take in consideration these blocks because we work with black and white images. The figure 12 presents the propagation of the laser pulse through the singlet, the step index fiber and the CMOS sensor. The figure 13 presents the propagation of a laser pulse through the achromatic doublet, the graded index fiber and the CMOS sensor. The figure 14 presents the propagation of a laser pulse through the apochromat, the self phase modulation fiber and

 a b c d e f Fig. 12. The image at the output of the a) laser resonator, b) singlet, c) step fiber, d) optical

 a b c d e f Fig. 13. The image at the output of the a) laser resonator, b) achromatic doublet, c) graded index fiber, d) optical part of the CMOS, e) electrical part of the CMOS, f) filtered image

part of the CMOS, e) electrical part of the CMOS, f) filtered image

*Guv* , is Fourier transform of the corrupted image *gx*,*y* .

obtained making the inverse Fourier transform:

filter (Toadere, 2010), (Toadere & Mastorakis, 2010).

**7. The result of simulations**

the CMOS sensor

**6.3 The bilateral filter**

*Puv H uv Guv* , , lo g , (86)

. (87)

In this paper we simulate the propagation of a Gaussian laser pulse through different image capture systems in order to find the best configuration that preserve the shape of the pulse during its propagations. We simulate the image characteristics at the output of each block from our different systems configurations. We simulate the functionality of the singlet, the achromatic doublet and the apochromat in order to reduce the chromatic dispersion. We simulate the functionality of the step index fiber, the graded index fiber and the self phase modulation fiber in order to reduce the modal dispersion. We simulate some properties of the CMOS sensor. The sensor suffers of different noises. The purpose of this paper was to put to work together, in the same system, optical and electrical components and to recover the degraded signal. In these types of complex systems, a controlled simulation environment can provide the engineer with useful guidance that improves the understanding of design considerations for individual parts and algorithms.

#### **9. References**

**13**

 *PR China*

*1Changzhou University, Changzhou 2Tsinghua University, Beijing*

**Third-Order Optical Nonlinearities of Novel**

**Phthalocyanines and Related Compounds**

Zhongyu Li1,2, Zihui Chen2, Song Xu1, Xinyu Zhou2 and Fushi Zhang2

The field of nonlinear optics has been developing for a few decades as a promising field with important applications in the domain of optoelectronics and photonics. Materials that exhibit nonlinear optical (NLO) behavior are useful because they allow manipulation of the fundamental properties of laser light beams, and are hence of great technological importance in areas such as photonic switching, optical computing and other optical data processing systems (Perry et al., 1994, 1996; Shirk et al., 2000). NLO activity was first found in inorganic crystals (Zyss, 1994), such as LiNbO3, but the choice of these materials is rather limited. Also, most of them have either low NLO responses or important drawbacks for processing into thin films and being incorporated into micro-optoelectronic devices. By the mid-1980s, organic materials emerged as important targets of choice for nonlinear optical applications because they exhibit large and fast nonlinearities and are, in general, easy to process and integrate into optical devices. Moreover, organic compounds offer the advantage of tailorability: a fine-tuning of the NLO properties can be achieved by rational modification of the chemical structure. Finally, they are ideal to achieve the ultimate goal of

Large optical nonlinearities in organic molecules usually arise from highly delocalized electron systems. During the last two decades, phthalocyanines (Pcs) have been intensively investigated for their third-order NLO properties both in solutions and as thin films because of their extensively delocalized two dimensional 18-electron system (Ho, 1987; Shirk, 1989; Kambara, 1996; Ma, 2003; Zhou, 2004; He, 2007). They also exhibit other additional advantages, namely, exceptional stability, versatility, and processability features. The architectural flexibility of phthalocyanines is well exemplified by the large number of metallic complexes described in the literature, as well as by the huge variety of substituents that can be attached to the phthalocyanine core. Furthermore, some of the four isoindole units can be formally replaced by other heterocyclic moieties, giving rise to different phthalocyanine analogues. All these chemical variations can alter the electronic structure of the macrocyclic core, and therefore, they allow the fine-tuning of the nonlinear response. Aside from their practical interest, Pc-related molecules present very attractive features for fundamental NLO studies. Since the unsubstituted and many substituted compounds are planar (2D, two dimensional), they offer the possibility of investigating of the role of

device miniaturization by going into the molecular level.

**1. Introduction**

### **Third-Order Optical Nonlinearities of Novel Phthalocyanines and Related Compounds**

Zhongyu Li1,2, Zihui Chen2, Song Xu1, Xinyu Zhou2 and Fushi Zhang2 *1Changzhou University, Changzhou 2Tsinghua University, Beijing PR China*

#### **1. Introduction**

252 Laser Systems for Applications

Toadere F. & Mastorakis N. (2009). Imaging a laser pulse propagation trough an image

Toadere F. & Mastorakis N. (2010). Simulation the functionality of a laser pulse image

Toadere F. (2010). Conversion from light in to numerical signal in a digital camera pipeline,

Yzuka K. (2008). *Engineering Optics, Springer*, ISBN 978-0-387-75723-0, New York, USA

(January 2010), pp. 22-31, ISSN 1109-2734

16, 2009

2010

acquisition system, *Recent Advances in Circuits, Systems, Electronics, Control and Signal Processing*, pp. 93-98, ISBN: 978-960-474-139-7, Tenerife, Spain, December 14-

acquisition system, *WSEAS transaction on circuits and systems*, Issue 1, Volume 9,

*Proceedings of SPIE on CD ROM, Volume 7821*, Constanta, Romania, 26-28 August,

The field of nonlinear optics has been developing for a few decades as a promising field with important applications in the domain of optoelectronics and photonics. Materials that exhibit nonlinear optical (NLO) behavior are useful because they allow manipulation of the fundamental properties of laser light beams, and are hence of great technological importance in areas such as photonic switching, optical computing and other optical data processing systems (Perry et al., 1994, 1996; Shirk et al., 2000). NLO activity was first found in inorganic crystals (Zyss, 1994), such as LiNbO3, but the choice of these materials is rather limited. Also, most of them have either low NLO responses or important drawbacks for processing into thin films and being incorporated into micro-optoelectronic devices. By the mid-1980s, organic materials emerged as important targets of choice for nonlinear optical applications because they exhibit large and fast nonlinearities and are, in general, easy to process and integrate into optical devices. Moreover, organic compounds offer the advantage of tailorability: a fine-tuning of the NLO properties can be achieved by rational modification of the chemical structure. Finally, they are ideal to achieve the ultimate goal of device miniaturization by going into the molecular level.

Large optical nonlinearities in organic molecules usually arise from highly delocalized electron systems. During the last two decades, phthalocyanines (Pcs) have been intensively investigated for their third-order NLO properties both in solutions and as thin films because of their extensively delocalized two dimensional 18-electron system (Ho, 1987; Shirk, 1989; Kambara, 1996; Ma, 2003; Zhou, 2004; He, 2007). They also exhibit other additional advantages, namely, exceptional stability, versatility, and processability features. The architectural flexibility of phthalocyanines is well exemplified by the large number of metallic complexes described in the literature, as well as by the huge variety of substituents that can be attached to the phthalocyanine core. Furthermore, some of the four isoindole units can be formally replaced by other heterocyclic moieties, giving rise to different phthalocyanine analogues. All these chemical variations can alter the electronic structure of the macrocyclic core, and therefore, they allow the fine-tuning of the nonlinear response.

Aside from their practical interest, Pc-related molecules present very attractive features for fundamental NLO studies. Since the unsubstituted and many substituted compounds are planar (2D, two dimensional), they offer the possibility of investigating of the role of

(–4;1,2,

where

where *n*(

Third-Order Optical Nonlinearities of Novel Phthalocyanines and Related Compounds 255

susceptibility (3) is directly related to the microscopic averaged hyperpolarizability

(3)(– ; , , ) ( ) ( ) ( ) ( ) (– ; , , ) 4123 4 1 2 3 4123

can be obtained from the product of mass density *D* and Avogadro's constant *A*, divided by

The local field factor at optical frequencies for a pure liquid can be estimated by the

actual electric field acting on the microscopic species in the material. It should be emphasized that Lorentz's local field factor is an approximation since it considers the species to occupy a spherical cavity in the material and the local environment of the species is treated as a continuum. Also note that the indices of refraction used in the local field approximations for optical frequencies are generally assumed to be independent of the

The initial stages in the development of novel third-order NLO materials are the synthesis of molecules with large value and their incorporation into materials with substantial (3) coefficient. Although many additional factors must be considered for practical device applications (e.g., thermal and photochemical stability, processability, etc.) the NLO properties of molecules and molecular materials can be adequately described here by reference to values of and (3). Each of two parameters can be measured by using various experimental techniques. In most such measurements, the effects of resonance must always be considered; and (3) are all wavelength dependent, being strongly resonance enhanced whenever the fundamental or harmonic frequencies are close to an electronic excitation. Although large NLO responses are required, any significant absorption of light is obviously highly undesirable for most application of NLO materials. This aspect has led to the use of expression "transparency/efficiency trade-off" when discussing the merit of

For studies of cubic NLO properties, values of and (3) are usually obtained from thirdharmonic generation (THG) (Hermann et al., 1973) or degenerate four-wave mixing (DFWM) (Linder et al., 1982). The electric-field induced second-harmonic generation (EFISHG) technique may also be used to determine . THG studies simply involve measurement of the light produced at the third harmonic (TH), but are often complicated by the fact that many NLO materials absorb strongly in the TH region, even when using a 1907 nm fundamental which gives a TH at 636 nm. THG occurs almost instantaneously through purely electronic interactions that do not depend on the population of the excited state. The disadvantage of the THG technique is that dynamic nonlinearities are

  *Nf f f f* (4)

3; *N* is the number of molecules per volume unit which

*i*) is the local field factor at irradiation frequency

3 are the input ones, relating with each other

*i*)2 + 2)]/3 (5)

*i*. The local field is the

*i*) at the frequencies of the applied

*i*.

3) through local field correction factors *f*(

1, 2 and

*i*) = [*n*(

*i*) is the linear refractive index for a liquid at frequency

electric fields through the relation,

12

the molar mass *M*, i.e., *N* = *A* × *D*/*M*; *f*(

 *f*(

**3. Measurement of third-order NLO properties**

expression derived by Lorentz

is the output frequency,

by the expression

applied electric field.

such materials.

dimensionality on the NLO response. Moreover, by introducing peripheral substituents or by adding axial substituents, one may obtain three-dimensional (3D) structures with pyramidal shape and examine the effect of a third-dimension on the NLO response. Owing to the extended system, it is well-known that these Pc-related compounds exhibit a high aggregation tendency, and the aggregates usually display outstanding nonlinear optical properties. In order to improve the nonlinear optical properties of materials, the third-order optical nonlinearities of novel phthalocyanines and related compounds continue attracting attention.

This chapter arises from the need to compile the important advances obtained in the field of the NLO properties of phthalocyanines and analogues. Far from giving an exhaustive description of all the work that have been done in this area. We will focus on the third-order optical nonlinearities of some novel phthalocyanines and related compounds which were studied by our group using femtosecond degenerate four-wave mixing (DFWM) technique or picosecond Z-scan method.

#### **2. Theory of nonlinear optics**

Nonlinear optics is a material phenomenon in which intense light induces a nonlinear response in a medium, and in return the medium modifies the optical fields in a nonlinear way. In fact, all media are nonlinear to a certain degree of an applied optical field. The effect of a light wave on a material is usually described through the induced electrical polarization *P*. At lower irradiation intensities, one can assume that this polarization is a linear function of the applied electric field *E*,

$$\mathbf{P} = \mathbf{y}^{(l)} \cdot \mathbf{E} \tag{1}$$

where is the linear susceptibility. However, when the material is subjected to an intense applied electric field (a laser light), one must take into account the deviation of *P* from a linear dependence on *E*. Then the nonlinear polarization can be expressed by

$$\mathbf{P} = \chi^{(1)} \cdot \mathbf{E} + \chi^{(2)} \cdot \mathbf{E} \mathbf{E} + \chi^{(3)} \cdot \mathbf{E} \mathbf{E} \mathbf{E} + \dots \tag{2}$$

where (2) and (3) the quadratic (first-order) and cubic (second-order) susceptibilities, respectively. These two parameters determine the magnitude of the second- and third-order nonlinear optical responses. At the molecular level, a similar equation can be written for the microscopic polarization *P* induced in an atom or a molecule,

$$\mathbf{P} = \mathbf{a} \cdot \mathbf{E} + \mathbf{\beta} \cdot \mathbf{E} \mathbf{E} + \mathbf{\gamma} \cdot \mathbf{E} \mathbf{E} \mathbf{E} + \dots \tag{3}$$

with the coefficients , , and being the linear polarizability, the first (quadratic) hyperpolarizability, and the second (cubic) hyperpolarizability, respectively. The corresponding susceptibilities and hyperpolarizabilities are second rank (*ij*(1) or *ij*), third rank (*ijk*(2) or *ijk*), and fourth rank (*ijkl*(3) or *ijkl*) tensors, respectively. In general, geometrical symmetries may reduce the number of independent nonzero components. The odd rank tensor (2) is zero in centrosymmetric media whereas the even rank tensor (3) does not have any symmetry restrictions and can take place in any materials. (n) is frequency dependent and, as a result, resonant and non-resonant parameters differ significantly depending upon the measurement frequencies. The macroscopic third-order

dimensionality on the NLO response. Moreover, by introducing peripheral substituents or by adding axial substituents, one may obtain three-dimensional (3D) structures with pyramidal shape and examine the effect of a third-dimension on the NLO response. Owing to the extended system, it is well-known that these Pc-related compounds exhibit a high aggregation tendency, and the aggregates usually display outstanding nonlinear optical properties. In order to improve the nonlinear optical properties of materials, the third-order optical nonlinearities of novel phthalocyanines and related compounds

This chapter arises from the need to compile the important advances obtained in the field of the NLO properties of phthalocyanines and analogues. Far from giving an exhaustive description of all the work that have been done in this area. We will focus on the third-order optical nonlinearities of some novel phthalocyanines and related compounds which were studied by our group using femtosecond degenerate four-wave mixing (DFWM) technique

Nonlinear optics is a material phenomenon in which intense light induces a nonlinear response in a medium, and in return the medium modifies the optical fields in a nonlinear way. In fact, all media are nonlinear to a certain degree of an applied optical field. The effect of a light wave on a material is usually described through the induced electrical polarization *P*. At lower irradiation intensities, one can assume that this polarization is a linear function

 P = · E (1) where is the linear susceptibility. However, when the material is subjected to an intense applied electric field (a laser light), one must take into account the deviation of *P* from a

 *P* = (1) · *E* + (2) · *EE* + (3) · *EEE* + … (2) where (2) and (3) the quadratic (first-order) and cubic (second-order) susceptibilities, respectively. These two parameters determine the magnitude of the second- and third-order nonlinear optical responses. At the molecular level, a similar equation can be written for the

 *P* = · *E* + · *EE* + · *EEE* + … (3) with the coefficients , , and being the linear polarizability, the first (quadratic) hyperpolarizability, and the second (cubic) hyperpolarizability, respectively. The corresponding susceptibilities and hyperpolarizabilities are second rank (*ij*(1) or *ij*), third rank (*ijk*(2) or *ijk*), and fourth rank (*ijkl*(3) or *ijkl*) tensors, respectively. In general, geometrical symmetries may reduce the number of independent nonzero components. The odd rank tensor (2) is zero in centrosymmetric media whereas the even rank tensor (3) does not have any symmetry restrictions and can take place in any materials. (n) is frequency dependent and, as a result, resonant and non-resonant parameters differ significantly depending upon the measurement frequencies. The macroscopic third-order

linear dependence on *E*. Then the nonlinear polarization can be expressed by

microscopic polarization *P* induced in an atom or a molecule,

continue attracting attention.

or picosecond Z-scan method.

of the applied electric field *E*,

**2. Theory of nonlinear optics**

susceptibility (3) is directly related to the microscopic averaged hyperpolarizability (–4;1,2,3) through local field correction factors *f*(*i*) at the frequencies of the applied electric fields through the relation,

$$\chi^{(3)}(\!-o\_4;o\_1,o\_2,o\_3) \equiv \text{Nf}(o\_4)f(o\_1)f(o\_2)f(o\_3)\chi(\!-o\_4;o\_1,o\_2,o\_3) \tag{4}$$

where is the output frequency, 1, 2 and 3 are the input ones, relating with each other by the expression 12 3; *N* is the number of molecules per volume unit which can be obtained from the product of mass density *D* and Avogadro's constant *A*, divided by the molar mass *M*, i.e., *N* = *A* × *D*/*M*; *f*(*i*) is the local field factor at irradiation frequency *i*. The local field factor at optical frequencies for a pure liquid can be estimated by the expression derived by Lorentz

$$f(\alpha \eta) = \left[ n(\alpha \eta^2 + 2) \right] / 3 \tag{5}$$

where *n*(*i*) is the linear refractive index for a liquid at frequency *i*. The local field is the actual electric field acting on the microscopic species in the material. It should be emphasized that Lorentz's local field factor is an approximation since it considers the species to occupy a spherical cavity in the material and the local environment of the species is treated as a continuum. Also note that the indices of refraction used in the local field approximations for optical frequencies are generally assumed to be independent of the applied electric field.

#### **3. Measurement of third-order NLO properties**

The initial stages in the development of novel third-order NLO materials are the synthesis of molecules with large value and their incorporation into materials with substantial (3) coefficient. Although many additional factors must be considered for practical device applications (e.g., thermal and photochemical stability, processability, etc.) the NLO properties of molecules and molecular materials can be adequately described here by reference to values of and (3). Each of two parameters can be measured by using various experimental techniques. In most such measurements, the effects of resonance must always be considered; and (3) are all wavelength dependent, being strongly resonance enhanced whenever the fundamental or harmonic frequencies are close to an electronic excitation. Although large NLO responses are required, any significant absorption of light is obviously highly undesirable for most application of NLO materials. This aspect has led to the use of expression "transparency/efficiency trade-off" when discussing the merit of such materials.

For studies of cubic NLO properties, values of and (3) are usually obtained from thirdharmonic generation (THG) (Hermann et al., 1973) or degenerate four-wave mixing (DFWM) (Linder et al., 1982). The electric-field induced second-harmonic generation (EFISHG) technique may also be used to determine . THG studies simply involve measurement of the light produced at the third harmonic (TH), but are often complicated by the fact that many NLO materials absorb strongly in the TH region, even when using a 1907 nm fundamental which gives a TH at 636 nm. THG occurs almost instantaneously through purely electronic interactions that do not depend on the population of the excited state. The disadvantage of the THG technique is that dynamic nonlinearities are

Third-Order Optical Nonlinearities of Novel Phthalocyanines and Related Compounds 257

The third-order NLO properties of phthalocyanines and its analogues have been extensively investigated since 1990, and this area has been reviewed (Torre et al., 2004). The discussion here will be limited to selected highlights of our studies and more recent

The electronic structure of phthalocyanine molecules can be tailored by either metal substitution at the central binding site or by altering the peripheral and axial functionalities, thus affording great versatility in controlling their electro-physical properties. Incorporation of a metal atom in the center of the macrocycle results in two types of charge-transfer transitions: metal-to-ligand and ligand-to-metal. The molecular hyperpolarizability could be enhanced by the metal-ligand bonding through the transfer of the electron density between the metal atom and the conjugated ligand systems. Furthermore, peripheral substituents could also influence the nonlinearity through intermolecular interactions. Therefore, the third-order NLO susceptibility(3) could vary by several orders of magnitude through chemical modification of the macrocycle structure

In general, the absorption spectra of monomeric phthalocyanines are dominated by two intense bands, a Soret band in the near ultraviolet region (at around 350 nm) and a Q band in the visible region (at around 670 nm), with a molar extinction coefficient in the range of 105 M-1 cm-1. When electron-donating alkoxy/aryloxy groups are introduced at the periphery closest to the phthalocyanine ring, a large bathochromic shift occurs in the Q band. These large red-shifted phthalocyanines are very soluble in common organic solvents and can be easily be prepared thickness-controlled thin films for practical applications to photonic devices. However, little attention has been paid to the third-order nonlinear properties of these molecules. Herein, we report the third-order nonlinear optical properties of aryloxy substituted novel diarylethene-phthalocyanine dyads **1**-**4** (the structures are shown in Figure 1) measured by femtosecond DFWM technique at 800 nm under off-

Figure 2 shows the temporal response of the DFWM signal as a function of the probe delay in the diarylethene-phthalocyanine dyads **1**-**4**. The DFWM signal of each sample is wellfitted by a Gaussian function (solid curve), and the half-width of fitting curve is similar to the autocorrelated pulse duration. All the signal profiles almost exhibit symmetry about the maximum signal (the zero time delay), which indicate that the responses time of the thirdorder optical nonlinearities are shorter than the experimental time resolution (50 fs). Such instantaneous response means that the Kerr effect (electronic component) from the distortion of the large -conjugated electron charge distribution of diarylethene-

phthalocyanine dyads **1**-**4** in DMF solution are summarized in Table 1. Among the samples, diarylethene-phthalocyanine dyad **4** possesses the highest e value of 1.12 × 10-30 esu. It was found that the second-order hyperpolarizability values of the diarylethene-phthalocyanine

and nonlinear refractive index for the diarylethene-

phthalocyanine dyad molecules is main reason for generating the DFWM signal.

(3)

**4. Novel phthalocyanines and analogues with third-order NLO properties**

developments.

**4.1 Novel phthalocyanines**

(Nalwa et al., 1993, 1999; He et al., 2007).

**4.1.1 Diarylethene-phthalocyanine dyads**

resonant condition (Li et al., 2007).

The evaluated values of

not probed and no information on the time response of the optical nonlinearity can be given.

DFWM is a convenient method for measuring both electronic and dynamic nonlinearities. It involves three laser beams of the same frequency interacting in a materials to produce a forth degenerate beam, the intensity of which allows determination of (3). DFWM may hence circumvent the effects of resonance, but has the additional aspect that vibrational and orientational mechanisms can contribute substantially to the observed optical nonlinearity. In contrast, only the electronic hyperpolarizability is fast enough to be probed by THG measurements. Hence, THG-derived values are generally considerably smaller than those determined via DFWM, and are more useful for deriving structure activity correlations (SACs) for the purely electronic part of , i.e., the only part of interest for potential practical applications. Polarization and time-dependent DFWM measurements can be used to distinguish the electronic (parametric) part of from other (nonparametric) components, but several experimental subtleties must be considered if this is to be achieved reliably. It is important to remember that (3) values which are most often determined for solution will always be concentration dependent.

Another experimental method that has become popular for characterization of cubic NLO materials is the relatively simple, but highly sensitive Z-scan technique which measures the phase change induced in a single laser beam on propagation through a material (Sheik-Bahae et al., 1989, 1990). The sample is moved along the propagation path (z) of a focused Gaussian beam whilst its transmittance is measured through a finite aperture in the far filed, and the sign and magnitude of the nonlinear refractive index *n2* are deduced from the resulting transmittance curve (Z-scan). If the value of *n2* is positive, then the material has a tendency to shrink a laser pulse and is termed "self-focusing" (SF), whilst a negative *n2* characterized "self-defocusing" (SDF) behaviour. Although the ultrafast, electronic *n2* arises from the real part of (3), published *n2* values often include, and indeed may be dominated by, nonparametric (thermal) contributions. In cases where nonlinear refraction is accompanied by nonlinear absorption, the nonlinear absorption coefficient 2 can be determined by performing a second Z-scan with the aperture removed. Effective (3) and values can also be determined via this method, but must be interpreted with great care in term of their actual origins which may be largely thermal effects.

Ultrafast, time-resolved optical Kerr effect (OKE) measurements (which involve another type of four-wave mixing process) may also be occasionally be used to determine (3) and values. The OKE is the optically-induced birefringence caused by a nonlinear phase shift. There are various mechanisms (electronic deformation, molecular reorientation, molecular libration, molecular redistribution, electrostriction, thermal change) that are responsible for the change of nonlinear refractive index in the OKE and each mechanism has different strengths and response times. In OKE, a strong beam is used as a pump beam and a weak beam is used as a probe beam. The intensity of the probe beam transmitted through a Kerr cell is measured as a function of the delay time between pump and probe.

In each of the THG, DFWM, Z-scan and OKE techniques, is derived from the measured (3) values by using the solute number density of the solution. In a few cases, (3) values have also been determined by using Stark spectroscopy. It should be remembered that comparisons of (3) or values obtained using different techniques with different experimental conditions are generally of little utility.

#### **4. Novel phthalocyanines and analogues with third-order NLO properties**

The third-order NLO properties of phthalocyanines and its analogues have been extensively investigated since 1990, and this area has been reviewed (Torre et al., 2004). The discussion here will be limited to selected highlights of our studies and more recent developments.

#### **4.1 Novel phthalocyanines**

256 Laser Systems for Applications

not probed and no information on the time response of the optical nonlinearity can be

DFWM is a convenient method for measuring both electronic and dynamic nonlinearities. It involves three laser beams of the same frequency interacting in a materials to produce a forth degenerate beam, the intensity of which allows determination of (3). DFWM may hence circumvent the effects of resonance, but has the additional aspect that vibrational and orientational mechanisms can contribute substantially to the observed optical nonlinearity. In contrast, only the electronic hyperpolarizability is fast enough to be probed by THG measurements. Hence, THG-derived values are generally considerably smaller than those determined via DFWM, and are more useful for deriving structure activity correlations (SACs) for the purely electronic part of , i.e., the only part of interest for potential practical applications. Polarization and time-dependent DFWM measurements can be used to distinguish the electronic (parametric) part of from other (nonparametric) components, but several experimental subtleties must be considered if this is to be achieved reliably. It is important to remember that (3) values which are most often determined for solution will

Another experimental method that has become popular for characterization of cubic NLO materials is the relatively simple, but highly sensitive Z-scan technique which measures the phase change induced in a single laser beam on propagation through a material (Sheik-Bahae et al., 1989, 1990). The sample is moved along the propagation path (z) of a focused Gaussian beam whilst its transmittance is measured through a finite aperture in the far filed, and the sign and magnitude of the nonlinear refractive index *n2* are deduced from the resulting transmittance curve (Z-scan). If the value of *n2* is positive, then the material has a tendency to shrink a laser pulse and is termed "self-focusing" (SF), whilst a negative *n2* characterized "self-defocusing" (SDF) behaviour. Although the ultrafast, electronic *n2* arises from the real part of (3), published *n2* values often include, and indeed may be dominated by, nonparametric (thermal) contributions. In cases where nonlinear refraction is accompanied by nonlinear absorption, the nonlinear absorption coefficient 2 can be determined by performing a second Z-scan with the aperture removed. Effective (3) and values can also be determined via this method, but must be interpreted with great care in

Ultrafast, time-resolved optical Kerr effect (OKE) measurements (which involve another type of four-wave mixing process) may also be occasionally be used to determine (3) and values. The OKE is the optically-induced birefringence caused by a nonlinear phase shift. There are various mechanisms (electronic deformation, molecular reorientation, molecular libration, molecular redistribution, electrostriction, thermal change) that are responsible for the change of nonlinear refractive index in the OKE and each mechanism has different strengths and response times. In OKE, a strong beam is used as a pump beam and a weak beam is used as a probe beam. The intensity of the probe beam transmitted through a Kerr

In each of the THG, DFWM, Z-scan and OKE techniques, is derived from the measured (3) values by using the solute number density of the solution. In a few cases, (3) values have also been determined by using Stark spectroscopy. It should be remembered that comparisons of (3) or values obtained using different techniques with different

term of their actual origins which may be largely thermal effects.

cell is measured as a function of the delay time between pump and probe.

experimental conditions are generally of little utility.

given.

always be concentration dependent.

The electronic structure of phthalocyanine molecules can be tailored by either metal substitution at the central binding site or by altering the peripheral and axial functionalities, thus affording great versatility in controlling their electro-physical properties. Incorporation of a metal atom in the center of the macrocycle results in two types of charge-transfer transitions: metal-to-ligand and ligand-to-metal. The molecular hyperpolarizability could be enhanced by the metal-ligand bonding through the transfer of the electron density between the metal atom and the conjugated ligand systems. Furthermore, peripheral substituents could also influence the nonlinearity through intermolecular interactions. Therefore, the third-order NLO susceptibility(3) could vary by several orders of magnitude through chemical modification of the macrocycle structure (Nalwa et al., 1993, 1999; He et al., 2007).

#### **4.1.1 Diarylethene-phthalocyanine dyads**

In general, the absorption spectra of monomeric phthalocyanines are dominated by two intense bands, a Soret band in the near ultraviolet region (at around 350 nm) and a Q band in the visible region (at around 670 nm), with a molar extinction coefficient in the range of 105 M-1 cm-1. When electron-donating alkoxy/aryloxy groups are introduced at the periphery closest to the phthalocyanine ring, a large bathochromic shift occurs in the Q band. These large red-shifted phthalocyanines are very soluble in common organic solvents and can be easily be prepared thickness-controlled thin films for practical applications to photonic devices. However, little attention has been paid to the third-order nonlinear properties of these molecules. Herein, we report the third-order nonlinear optical properties of aryloxy substituted novel diarylethene-phthalocyanine dyads **1**-**4** (the structures are shown in Figure 1) measured by femtosecond DFWM technique at 800 nm under offresonant condition (Li et al., 2007).

Figure 2 shows the temporal response of the DFWM signal as a function of the probe delay in the diarylethene-phthalocyanine dyads **1**-**4**. The DFWM signal of each sample is wellfitted by a Gaussian function (solid curve), and the half-width of fitting curve is similar to the autocorrelated pulse duration. All the signal profiles almost exhibit symmetry about the maximum signal (the zero time delay), which indicate that the responses time of the thirdorder optical nonlinearities are shorter than the experimental time resolution (50 fs). Such instantaneous response means that the Kerr effect (electronic component) from the distortion of the large -conjugated electron charge distribution of diarylethenephthalocyanine dyad molecules is main reason for generating the DFWM signal.

The evaluated values of (3) and nonlinear refractive index for the diarylethenephthalocyanine dyads **1**-**4** in DMF solution are summarized in Table 1. Among the samples, diarylethene-phthalocyanine dyad **4** possesses the highest e value of 1.12 × 10-30 esu. It was found that the second-order hyperpolarizability values of the diarylethene-phthalocyanine

Third-Order Optical Nonlinearities of Novel Phthalocyanines and Related Compounds 259

diarylethene-phthalocyanine dyads (Cu-DE-Pcs) are larger than those of the zinc substituted diarylethene-phthalocyanine dyads (Zn-DE-Pcs). This behavior can explain from the electronic structures of the upper occupied and lower vacant molecular orbitals for the ground states of metal substituted phthalocyanines (MPcs). In the case of Zn-DE-Pcs with completely filled d-shell, we can suppose that the probability for the charge transfer mechanism is very less. The 3d subshell of Zn-DE-Pcs is filled and deep enough to form rather pure molecular orbitals. The Zn-DE-Pcs exhibit a large gap between the HOMO and LUMO. In the case of Cu-DE-Pcs, the unfilled d valence orbital can be split into serials level due to the interaction between the d electrons and -conjugation electrons of Pc ring, this result will lower the transition energy in low-lying d orbitalligand or d-d transition. The existence of excited state with low transition energy will enhance the nonlinear optical susceptibilities of the material. The unfilled d orbit of Cu atoms will couple with the conjugated electrons of Pc ring leading to the extension of conjugated systems. As a result, the Cu-DE-Pcs with larger conjugated systems will show

substituted diarylethene-phthalocyanine dyad **3**/**4** is larger than that of the relative chlorine substituted diarylethene-phthalocyanine dyad **1/2**. This is probably attributed to electron-pushing effect of the methyl group of the diarylethene-phthalocyanine dyad **3**

**1** 4.50 7.60 8.33 **2** 5.05 8.53 9.35 **3** 4.92 8.32 9.12

**4** 6.62 11.2 12.3

In 2008, we reported the photo-responsive J-aggregation behavior of a novel -aryloxysubstituted zinc phthalocyanine (azobenzene substituted zinc phthalocyanine, hereafter, abbreviated as azo-ZnPc dyad) and its third-order optical nonlinearity (Chen et al., 2008). The azo-ZnPc dyad was synthesized through a rather facile route as shown in Scheme 1. The third-order optical nonlinearities of the photo responsive J-aggregates of the azo-ZnPc dyad (before and after irradiation conditions) were measured using a Z-scan technique at 532 nm

In our previous studies, it was found that -aryloxy-substituted Zinc phthalocyanine could form J-type self-aggregate in noncoordinating solvents through the complementary coordination of the peripheral oxygen atom of one phthalocyanine to the central Zn2+ in another phthalocyanine (Zhang et al., 2007a, 2007b). Moreover, the -aryloxy-substituted Zinc phthalocyanine formed J-aggregate showed another characteristic, that is the addition of methanol could break the J-aggregate, as a consequence, typical Q-band

×10-31 esu) *n*2 ×10-13 esu)

and nonlinear refractive index for the diarylethene-

value of the methyl

larger optical nonlinearities than the Zn-DE-Pcs. Moreover, the

×10-14 esu)

and **4**, which leads to large polarization of molecules.

(3)

Sample

phthalocyanine dyads **1**-**4** in DMF solution

**4.1.2 Azobenzene-phthalocyanine dyads**

Table 1. Evaluated values of

with pulse duration of 25 ps.

restored.

Fig. 1. Structures of diarylethene-phthalocyanine dyads **1**-**4**

Fig. 2. Temporal profiles of DFWM signal of the diarylethene-phthalocyanine dyads **1**-**4**

dyads **1**-**4** are 2-3 times larger than that of tetra-aryloxy substituted metal-free phthalocyanine. This enhancement of mainly comes from the d valence orbital contribution of central metal atoms (Zn and Cu) of the diarylethene-phthalocyanine dyads. Furthermore, it is observed that the values of the copper substituted

**1**

**2**

M = Zn R =

M = Cu R =

**3** M = Zn R =

**4** M = Cu R =

N

Cl S S Cl

N

Cl S S Cl

N

S S CH3 H3C

N

S S CH3 H3C

**2**

**4**

**-150 -100 -50 0 50 100 150**

**Delay time (fs)**

**-150 -100 -50 0 50 100 150**

**Delay time (fs)**

mainly comes from the d valence orbital

values of the copper substituted

N N N

M

O

**DFWM intensity (a. u.)**

**DFWM intensity** (**a. u.**)

R

O

R

N N

N

O

R

**1**

**3**

**DFWM intensity (a. u.)**

Fig. 2. Temporal profiles of DFWM signal of the diarylethene-phthalocyanine dyads **1**-**4**

dyads **1**-**4** are 2-3 times larger than that of tetra-aryloxy substituted metal-free

contribution of central metal atoms (Zn and Cu) of the diarylethene-phthalocyanine

**DFWM intensity** (**a. u.**)

N N

O

R

Fig. 1. Structures of diarylethene-phthalocyanine dyads **1**-**4**

**-150 -100 -50 0 50 100 150**

**-150 -100 -50 0 50 100 150**

**Delay time (fs)**

phthalocyanine. This enhancement of

dyads. Furthermore, it is observed that the

**Delay time (fs)**

diarylethene-phthalocyanine dyads (Cu-DE-Pcs) are larger than those of the zinc substituted diarylethene-phthalocyanine dyads (Zn-DE-Pcs). This behavior can explain from the electronic structures of the upper occupied and lower vacant molecular orbitals for the ground states of metal substituted phthalocyanines (MPcs). In the case of Zn-DE-Pcs with completely filled d-shell, we can suppose that the probability for the charge transfer mechanism is very less. The 3d subshell of Zn-DE-Pcs is filled and deep enough to form rather pure molecular orbitals. The Zn-DE-Pcs exhibit a large gap between the HOMO and LUMO. In the case of Cu-DE-Pcs, the unfilled d valence orbital can be split into serials level due to the interaction between the d electrons and -conjugation electrons of Pc ring, this result will lower the transition energy in low-lying d orbitalligand or d-d transition. The existence of excited state with low transition energy will enhance the nonlinear optical susceptibilities of the material. The unfilled d orbit of Cu atoms will couple with the conjugated electrons of Pc ring leading to the extension of conjugated systems. As a result, the Cu-DE-Pcs with larger conjugated systems will show larger optical nonlinearities than the Zn-DE-Pcs. Moreover, the value of the methyl substituted diarylethene-phthalocyanine dyad **3**/**4** is larger than that of the relative chlorine substituted diarylethene-phthalocyanine dyad **1/2**. This is probably attributed to electron-pushing effect of the methyl group of the diarylethene-phthalocyanine dyad **3** and **4**, which leads to large polarization of molecules.

Table 1. Evaluated values of (3) and nonlinear refractive index for the diarylethenephthalocyanine dyads **1**-**4** in DMF solution

#### **4.1.2 Azobenzene-phthalocyanine dyads**

In 2008, we reported the photo-responsive J-aggregation behavior of a novel -aryloxysubstituted zinc phthalocyanine (azobenzene substituted zinc phthalocyanine, hereafter, abbreviated as azo-ZnPc dyad) and its third-order optical nonlinearity (Chen et al., 2008). The azo-ZnPc dyad was synthesized through a rather facile route as shown in Scheme 1. The third-order optical nonlinearities of the photo responsive J-aggregates of the azo-ZnPc dyad (before and after irradiation conditions) were measured using a Z-scan technique at 532 nm with pulse duration of 25 ps.

In our previous studies, it was found that -aryloxy-substituted Zinc phthalocyanine could form J-type self-aggregate in noncoordinating solvents through the complementary coordination of the peripheral oxygen atom of one phthalocyanine to the central Zn2+ in another phthalocyanine (Zhang et al., 2007a, 2007b). Moreover, the -aryloxy-substituted Zinc phthalocyanine formed J-aggregate showed another characteristic, that is the addition of methanol could break the J-aggregate, as a consequence, typical Q-band restored.

Third-Order Optical Nonlinearities of Novel Phthalocyanines and Related Compounds 261

**-40 -20 0 20 40 Z (mm)**

**-40 -20 0 20 40 Z (mm)**

The nonlinear absorption coefficient (, m/W), the nonlinear refraction coefficient (*n*2, m2/W), the third-order nonlinear susceptibility ((3), esu) and the molecular second hyperpolarizability (, esu) are calculated and listed in Table 2. It is found that the studied azo-phthalocyanine dyad shows large second-order molecular hyperpolarizabilities which are of the order of 10-30 esu both before and after irradiation conditions. The value of of

Fig. 5. Normalized transmission for the closed aperture of Z-scan measurement

**Before irradiation After irradiation**

Fig. 4. Normalized transmission without aperture at 532 nm (open aperture) as a function of distance along the lens axis. The filled triangles and open squares are measured data for before irradiation and after irradiation of azo-phthalocyanine, respectively. Each point

corresponds to the average of 5 pulses. The solid line is the theoretical fit.

**Before irradiation After irradiation**

**0.94**

**0.4**

**0.6**

**0.8**

**Normalized transmittance**

**1**

**1.2**

**1.4**

**0.96**

**0.98**

**Normalized transmittance**

**1**

**1.02**

Scheme 1. Synthesis route of a novel -aryloxy-substituted zinc phthalocyanine

The studied azo-ZnPc dyad shows same absorption behaviors as that of the reported aryloxy-substituted Zinc phthalocyanine. Figure 4 shows the spectra change of initial and UV-illuminated solutions of azo-ZnPc in chloroform with the addition of methanol. As methanol was titrated, the absorption at 740 nm of both of the solutions decreased gradually and finally disappeared completely with the increase in the absorption at 698 nm. The unusual red-shifted peak at 740 nm can be attributed to the formation of Jaggregates. For azo-ZnPc, there is a stronger tendency to form J-aggregate when the azobenzene units is in the cis-conformation, and the irradiation of UV light will cause enhancement of J-aggregation.

Fig. 3. Absorption spectral changes of azo-ZnPc (a: before UV light irradiation; b: after UV light irradiation for 3 min, c = 1.19 × 10-5 M) in chloroform .

The open aperture curves (Figure 4) of Z-scan measurement exhibit the normalized valleys, indicating the presence of reverse saturable absorption with a positive coefficient . And the normalized transmission for the closed aperture of Z-scan measurement is shown in Figure 5. The large valley-to-peak configurations of closed aperture curves suggest that the refractive index changes are negative, exhibiting a strong defocusing effect.

DMF/K2CO3

**N**

**N**

**N**

The studied azo-ZnPc dyad shows same absorption behaviors as that of the reported aryloxy-substituted Zinc phthalocyanine. Figure 4 shows the spectra change of initial and UV-illuminated solutions of azo-ZnPc in chloroform with the addition of methanol. As methanol was titrated, the absorption at 740 nm of both of the solutions decreased gradually and finally disappeared completely with the increase in the absorption at 698 nm. The unusual red-shifted peak at 740 nm can be attributed to the formation of Jaggregates. For azo-ZnPc, there is a stronger tendency to form J-aggregate when the azobenzene units is in the cis-conformation, and the irradiation of UV light will cause

Fig. 3. Absorption spectral changes of azo-ZnPc (a: before UV light irradiation; b: after UV

Absorbance

wavelength/nm 600 700 800 0.0 0.2 0.4 0.6 0.8 1.0

wavelength/nm

The open aperture curves (Figure 4) of Z-scan measurement exhibit the normalized valleys, indicating the presence of reverse saturable absorption with a positive coefficient . And the normalized transmission for the closed aperture of Z-scan measurement is shown in Figure 5. The large valley-to-peak configurations of closed aperture curves suggest that the

light irradiation for 3 min, c = 1.19 × 10-5 M) in chloroform .

600 700 800

refractive index changes are negative, exhibiting a strong defocusing effect.

 initial solution after methanol added

a b

CN CN O N

N

 initial solution after methanol added

CN CN NO2

**O**

**N N N**

**N Zn**

**N**

**O**

**N**

**N N N**

**O O**

**N N**

Scheme 1. Synthesis route of a novel -aryloxy-substituted zinc phthalocyanine

HO N

DBU, Zn(OAc)2 reflux

enhancement of J-aggregation.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Absorbance

**N**

**N**

N +

Fig. 4. Normalized transmission without aperture at 532 nm (open aperture) as a function of distance along the lens axis. The filled triangles and open squares are measured data for before irradiation and after irradiation of azo-phthalocyanine, respectively. Each point corresponds to the average of 5 pulses. The solid line is the theoretical fit.

Fig. 5. Normalized transmission for the closed aperture of Z-scan measurement

The nonlinear absorption coefficient (, m/W), the nonlinear refraction coefficient (*n*2, m2/W), the third-order nonlinear susceptibility ((3), esu) and the molecular second hyperpolarizability (, esu) are calculated and listed in Table 2. It is found that the studied azo-phthalocyanine dyad shows large second-order molecular hyperpolarizabilities which are of the order of 10-30 esu both before and after irradiation conditions.  Synthesis of target azobenzene containing water soluble unsymmetrical zinc (II) and copper (II) phthalocyanines and their inclusion complexes with -cyclodextrin (Zn-Pc8,

Z-scan studies show that each azobenzene containing water soluble unsymmetrical phthalocyanine consists of an electro donating phenylazophenoxy group (D) and six electron withdrawing carboxyl (A) forming a D--A alignment along the x axis. As a result of such unique chemical structure, all the samples showed very large molecular cubic hyperpolarizabilities which are of the order 10-30 esu. The Azobenzene moieties of these compounds could reversibly associate with -CD to form inclusion complexes through hostguest interaction in aqueous media upon alternating illumination of UV and visible light, resulting apparent influences to the third-order NLO properties of these phthalocyanines.

 This enhancement may be attributed to the increase of J-aggregation degree of azo-ZnPc dyad after UV light irradiation.

Table 2. Values of The nonlinear absorption coefficient, the nonlinear refraction coefficient, the third-order nonlinear susceptibility and the molecular second hyperpolarizability of Azo-ZnPc dyad.

#### **4.1.3 Azobenzene-containing water soluble unsymmetrical phthalocyanines**

In 2009, we reported the photoswitching of the third-order nonlinear optical properties of unsymmetrical azobenzene-containing metal phthalocyanines (structures are shown in Figure 6) based on reversible host-guest interacions (Chen, 2009).

#### **MPc7**

Fig. 6. Structures of unsymmetrical azobenzene-containing metal phthalocyanines

It has been well established that azobenzene could reversibly assemble with -cyclodextrin through host-guest interaction under suitable external photo-stimuli, and this phenomenon has been exploited as the basis of some molecular shuttles and motors (Breslow & Dong, 1998; Dugave & Demange, 2003). However, their applications in phthalocyanine chemistry have rarely been studied yet. We believe that this reversible host-guest interaction can be used to modulate the NLO properties of phthalocyanines if the phthalocyanines were judicious designed. Therefore, we prepared for the first time two azobenzene containing water soluble unsymmetrical metal phthalocyanines. Their reversible host-guest interaction with -cyclodextrin in aqueous media and the resulting effects on the NLO properties of such molecules were also investigated. Scheme 2 shows the structures and the synthesis of target azobenzene containing water soluble unsymmetrical zinc (II) and copper (II) phthalocyanines (abbreviated as Zn-Pc7 and Cu-Pc7, respectively) and their inclusion complexes with -cyclodextrin (Zn-Pc8, Cu-Pc8)

before irradiation is 1.25 times larger than that of after irradiation condition. This enhancement may be attributed to the increase of J-aggregation degree of azo-ZnPc dyad

*n*2 / × 10-19 m2 W-1 4.26 5.30

 / × 10-9 m W-1 1.48 4.03 (3) / × 10-13 esu 2.26 2.82

 / × 10-30 esu 3.87 4.82 Table 2. Values of The nonlinear absorption coefficient, the nonlinear refraction coefficient, the third-order nonlinear susceptibility and the molecular second hyperpolarizability of

In 2009, we reported the photoswitching of the third-order nonlinear optical properties of unsymmetrical azobenzene-containing metal phthalocyanines (structures are shown in

> N N N

N N

COOH

COOH

N N N

HOOC COOH

 **MPc7**

M

HOOC COOH

**4.1.3 Azobenzene-containing water soluble unsymmetrical phthalocyanines**

O

Fig. 6. Structures of unsymmetrical azobenzene-containing metal phthalocyanines

It has been well established that azobenzene could reversibly assemble with -cyclodextrin through host-guest interaction under suitable external photo-stimuli, and this phenomenon has been exploited as the basis of some molecular shuttles and motors (Breslow & Dong, 1998; Dugave & Demange, 2003). However, their applications in phthalocyanine chemistry have rarely been studied yet. We believe that this reversible host-guest interaction can be used to modulate the NLO properties of phthalocyanines if the phthalocyanines were judicious designed. Therefore, we prepared for the first time two azobenzene containing water soluble unsymmetrical metal phthalocyanines. Their reversible host-guest interaction with -cyclodextrin in aqueous media and the resulting effects on the NLO properties of such molecules were also investigated. Scheme 2 shows the structures and the synthesis of target azobenzene containing water soluble unsymmetrical zinc (II) and copper (II) phthalocyanines (abbreviated as Zn-Pc7 and Cu-Pc7, respectively) and their inclusion

Figure 6) based on reversible host-guest interacions (Chen, 2009).

N N

complexes with -cyclodextrin (Zn-Pc8, Cu-Pc8)

Sample Before irradiation After irradiation

after UV light irradiation.

Azo-ZnPc dyad.

Scheme 2. Synthesis of target azobenzene containing water soluble unsymmetrical zinc (II) and copper (II) phthalocyanines and their inclusion complexes with -cyclodextrin (Zn-Pc8, Cu-Pc8)

Z-scan studies show that each azobenzene containing water soluble unsymmetrical phthalocyanine consists of an electro donating phenylazophenoxy group (D) and six electron withdrawing carboxyl (A) forming a D--A alignment along the x axis. As a result of such unique chemical structure, all the samples showed very large molecular cubic hyperpolarizabilities which are of the order 10-30 esu. The Azobenzene moieties of these compounds could reversibly associate with -CD to form inclusion complexes through hostguest interaction in aqueous media upon alternating illumination of UV and visible light, resulting apparent influences to the third-order NLO properties of these phthalocyanines.

Third-Order Optical Nonlinearities of Novel Phthalocyanines and Related Compounds 265

Recently, we have measured the third-order nonlinear optical properties of two thiophenebearing phthalocyanines (Ni-TPc and Cu-TPc, their structures are shown in Figure 8) using Z-scan technique at 532 nm (Chen et al., 2011). Both Ni-TPc and Cu-TPc were found to show large molecular cubic hyperpolarizabilities whose values are of the order of 10−30 esu. The value of Cu-TPc is 1.5 times larger than that of Ni-TPc, mainly as a result of their notably different nonlinear absorptions. Most notably, the nonlinear absorption and the nonlinear refraction contribute almost equally to their molecular cubic hyperpolarizabilities, while for our previous studied Pcs, the nonlinear refraction always plays an absolutely predominant role. It is assumed that the incorporation of thiophene rings into phthalocyanines could

Fig. 8. Chemical structures of two thiophene-bearing phthalocyanines, Ni-TPc and Cu-TPc

Phthalocyanines are special organic systems in a way that they offer tremendous opportunities in tailoring their photophysical and optical properties over a wide range either by substituting different metal atoms into the central binding site or by altering the peripheral and axial functionalities. It is possible to incorporate a variety of peripheral substituents around the phthalocyanines core as well as replace some of the isoindole units by other heterocyclic moieties, giving rise to different phthalocyanine analogues. For example, in the previous studie (Liu et al., 1999), it was shown that the phthalocyanine may lose a bridging nitrogen atom when complexed with phosphorus which was proved to be a

phthalocyanine analogue−dihydroxy phosphorus (V) tetrabenzotriazacorrole (TBC).

Due to its special three-dimensional -electron structure and features which are different from phthalocyanines, the TBC macrocycle should have a potential application in photonic devices. However, to our knowledge, there is no report on third-order optical

**4.1.5 Novel thiophene-bearing phthalocyanines**

**4.2 Novel phthalocyanine related compounds**

**4.2.1 Dihydroxy phosphorus (V) tetrabenzotriazacorroles**

notably increase the multi-photo absorption cross-section of Pc.

This influence is especially striking for the phthalocyanine whose central metal atom is Cu2+. The molecular cubic hyperpolarizability of its inclusion complex with -CD is 2.10 10-30 esu. When the inclusion complex disassociated under the illumination of 365 nm light, the value was 4.2 10-30 esu, which is an 100% increase. Taking account of the large molecular cubic hyperpolarizabilities of these compounds, our endeavors toward ideal third-order NLO photoswitching systems is very promising, with sufficient room for improvement. This work suggested that reversibly control either the chemical structure or the molecular packing arrangement of excellent third-order NLO materials is an attractive strategy for constructing ideal third-order NLO photoswitching systems. Moreover, the present study emphasized the reversible host-guest interaction between azobenzene and -CD on the packing style of phthalocyanines, which may provide new insights to the host-guest chemistry.

#### **4.1.4 Novel copper phthalocyanine-ferrocene dyad**

González-Cabello (González-Cabello et al., 2003) reported that the interplanar distance between the two ciclopentadiynyl rings combined with the rigid, stereochemically well defined,-conjugated linkers between the ferrocene and the Pc-subunits provide an excellent situation for the cofacial stacking of the Pc macrocycles, thus allowing potential NLOfavorable through space interactions between the individual Pc subunits. Moreover, the combination of an electron acceptor moiety such as phthalocyanine and electron donor unit such as ferrocene may give rise to intramolecular charge transfer that may enhance the nonlinear optical response. Herein, third-order nonlinear optical property of a novel copper phthalocyanine-ferrocene dyad (the structure is shown in Figure 7) measured by femtosecond degenerate four–wave mixing technique under off-resonant condition (Bin et al., 2008).

Fig. 7. Structure of novel copper phthalocyanine-ferrocene dyad

The DFWM measurement shows that the second-order molecular hyperpolarizability of this compound was measured to be 1.74 × 10-30 esu, and its response time was also obtained and no more than 50 fs. This large and ultrafast third-order optical nonlinear response is mainly enhanced by the formation of intramolecular charge-transfer which can enhance the delocalized movements of the large-electrons in the molecules.

#### **4.1.5 Novel thiophene-bearing phthalocyanines**

264 Laser Systems for Applications

This influence is especially striking for the phthalocyanine whose central metal atom is Cu2+. The molecular cubic hyperpolarizability of its inclusion complex with -CD is 2.10 10-30 esu. When the inclusion complex disassociated under the illumination of 365 nm light, the value was 4.2 10-30 esu, which is an 100% increase. Taking account of the large molecular cubic hyperpolarizabilities of these compounds, our endeavors toward ideal third-order NLO photoswitching systems is very promising, with sufficient room for improvement. This work suggested that reversibly control either the chemical structure or the molecular packing arrangement of excellent third-order NLO materials is an attractive strategy for constructing ideal third-order NLO photoswitching systems. Moreover, the present study emphasized the reversible host-guest interaction between azobenzene and -CD on the packing style of phthalocyanines, which may provide new insights to the

González-Cabello (González-Cabello et al., 2003) reported that the interplanar distance between the two ciclopentadiynyl rings combined with the rigid, stereochemically well defined,-conjugated linkers between the ferrocene and the Pc-subunits provide an excellent situation for the cofacial stacking of the Pc macrocycles, thus allowing potential NLOfavorable through space interactions between the individual Pc subunits. Moreover, the combination of an electron acceptor moiety such as phthalocyanine and electron donor unit such as ferrocene may give rise to intramolecular charge transfer that may enhance the nonlinear optical response. Herein, third-order nonlinear optical property of a novel copper phthalocyanine-ferrocene dyad (the structure is shown in Figure 7) measured by femtosecond

degenerate four–wave mixing technique under off-resonant condition (Bin et al., 2008).

 Structure of novel copper phthalocyanine-ferrocene dyad

delocalized movements of the large-electrons in the molecules.

N

N

Cu

N

The DFWM measurement shows that the second-order molecular hyperpolarizability of this compound was measured to be 1.74 × 10-30 esu, and its response time was also obtained and no more than 50 fs. This large and ultrafast third-order optical nonlinear response is mainly enhanced by the formation of intramolecular charge-transfer which can enhance the

O N

H C

host-guest chemistry.

**4.1.4 Novel copper phthalocyanine-ferrocene dyad**

Recently, we have measured the third-order nonlinear optical properties of two thiophenebearing phthalocyanines (Ni-TPc and Cu-TPc, their structures are shown in Figure 8) using Z-scan technique at 532 nm (Chen et al., 2011). Both Ni-TPc and Cu-TPc were found to show large molecular cubic hyperpolarizabilities whose values are of the order of 10−30 esu. The value of Cu-TPc is 1.5 times larger than that of Ni-TPc, mainly as a result of their notably different nonlinear absorptions. Most notably, the nonlinear absorption and the nonlinear refraction contribute almost equally to their molecular cubic hyperpolarizabilities, while for our previous studied Pcs, the nonlinear refraction always plays an absolutely predominant role. It is assumed that the incorporation of thiophene rings into phthalocyanines could notably increase the multi-photo absorption cross-section of Pc.

Fig. 8. Chemical structures of two thiophene-bearing phthalocyanines, Ni-TPc and Cu-TPc

#### **4.2 Novel phthalocyanine related compounds**

Phthalocyanines are special organic systems in a way that they offer tremendous opportunities in tailoring their photophysical and optical properties over a wide range either by substituting different metal atoms into the central binding site or by altering the peripheral and axial functionalities. It is possible to incorporate a variety of peripheral substituents around the phthalocyanines core as well as replace some of the isoindole units by other heterocyclic moieties, giving rise to different phthalocyanine analogues. For example, in the previous studie (Liu et al., 1999), it was shown that the phthalocyanine may lose a bridging nitrogen atom when complexed with phosphorus which was proved to be a phthalocyanine analogue−dihydroxy phosphorus (V) tetrabenzotriazacorrole (TBC).

#### **4.2.1 Dihydroxy phosphorus (V) tetrabenzotriazacorroles**

Due to its special three-dimensional -electron structure and features which are different from phthalocyanines, the TBC macrocycle should have a potential application in photonic devices. However, to our knowledge, there is no report on third-order optical

Third-Order Optical Nonlinearities of Novel Phthalocyanines and Related Compounds 267

In 2008, we (Zhao et al., 2008) have synthesized three novel tri-dimensional phthalocyanine polymers with lanthanum (LaPPc), gadolinium (GdPPc) and ytterbium (YbPPc) as centric atoms from a tetranuclear phthalonitrile (the structures and synthesis route of rare earth polymeric phthalocyanines are shown in Scheme 3). And third-order optical nonlinearities of these compounds in DMF solution were measured by a picosecond Z-sacn technique at 532 nm.

Scheme 3. Structures and synthesis route of rare earth polymeric phthalocyanines

mechanism for the nonlinear optical response of three phthalocyanine polymers.

offer this intermolecular possibility from two dimensions.

phthalocyanine.

Based on the Z-scan measurements, it is found that these phthalocyanine polymers show large third-order nonlinear susceptibilities which are of the order of 10-12 esu. Both the nonlinear absorption β and nonlinear refraction *n2* decreases with the order of LaPPc>GdPPc>YbPPc. For all the compounds, the values of Re(3) are one order of magnitude larger than those of Im(3), which determine the magnitude of third-order nonlinear susceptibilities (3). This indicated that the nonlinear refraction is predominant

Furthermore, Researchers (Manas et al., 1997) have described the effect of intermacrocycle interactions on the second hyperpolarizabilities of phthalocyanine dimer and trimer. It was considered that stack form of phthalocyanine can induce molecular electronic interaction between neighbouring phthalocyanine rings and induce charge transfer between them. As shown in Figure 10, the cage structure of phthalocyanine polymer presents more possibilities of electronic interaction and distortion of electron cloud in three dimensions at intermolecular scale. This would elevate the spatial polarizability of molecular and induces large nonlinear coefficient. However, the trimer or binuclear phthalocyanine could only

Fig. 10. Calculated molecular electronic distribution of binuclear, trimer and polymer of

**4.2.2 Rare earth polymeric phthalocyanines**

nonlinearity of dihydroxy phosphorus (V) tetrabenzotriazacorrole. Herein, we report the third-order nonlinear optical properties of a series phthalocyanine analogues, nonsulfonated {P(OH)2TBC}, sulfonated {P(OH)2TBCSn} and isopropoxyl substituted {P(OH)2TBC(Oi Pr)4} dihydroxy phosphorus (V) tetrabenzotriazacorroles (the structures are shown in Figure 9) measured by femtosecond (50 fs) degenerate four–wave mixing technique (Huang et al., 2008).

Fig. 9. Structural formulae of **(a)** P(OH)2TBC and P(OH)2TBC(Oi Pr)4 ; **(b)** P(OH)2TBCSn.

The evaluated values of(3) and for these phthalocyanine analogues are summarized in Table 3 together with the visible absorption maxima (max) and nonlinear refractive index in DMF solutions. The of samples show high values as large as 10-31 esu, and the highest value of 2.26 × 10-31 esu for the P(OH)2TBC(Oi Pr)4 was observed, which is almost two times larger than that of phthalocyanine analogue P(OH)2TBC. This is probably due to the isopropoxyl group is attached to the benzene rings of the phthalocyanine analogue P(OH)2TBC(Oi Pr)4, and isopropoxyl group is a electron-donating group which leads to large polarization of molecules. Considering molecular structure of invested phthalocyanine analogues, two hydroxyl moieties complexed with central phosphorus of molecule in axial direction, and form a three-dimensional configuration, which can enhance third-order optical nonlinearity of molecule. Moreover, their response times are also obtained and no more than 50 fs, which are commonly accepted to the contribution from the transient motion of the conjugate electron distribution.

Table 3. The evaluated values of (3), , and *n2* for the dihydroxy phosphorus (V) tetrabenzotriazacorroles together with the visible absorption maxima (max) in DMF solution.

#### **4.2.2 Rare earth polymeric phthalocyanines**

266 Laser Systems for Applications

nonlinearity of dihydroxy phosphorus (V) tetrabenzotriazacorrole. Herein, we report the third-order nonlinear optical properties of a series phthalocyanine analogues, nonsulfonated {P(OH)2TBC}, sulfonated {P(OH)2TBCSn} and isopropoxyl substituted

are shown in Figure 9) measured by femtosecond (50 fs) degenerate four–wave mixing

R

Pr)4} dihydroxy phosphorus (V) tetrabenzotriazacorroles (the structures

N

N

N

SO3 n

Pr)4 ; **(b)** P(OH)2TBCSn.

Pr)4 was observed, which is almost two times

(×10-14esu) *n*2(×10-14esu)

N

N

P OH

OH

N

N

(a) (b)

Pr P(OH)2TBCSn

The evaluated values of(3) and for these phthalocyanine analogues are summarized in Table 3 together with the visible absorption maxima (max) and nonlinear refractive index in DMF solutions. The of samples show high values as large as 10-31 esu, and the highest

larger than that of phthalocyanine analogue P(OH)2TBC. This is probably due to the isopropoxyl group is attached to the benzene rings of the phthalocyanine analogue

polarization of molecules. Considering molecular structure of invested phthalocyanine analogues, two hydroxyl moieties complexed with central phosphorus of molecule in axial direction, and form a three-dimensional configuration, which can enhance third-order optical nonlinearity of molecule. Moreover, their response times are also obtained and no more than 50 fs, which are commonly accepted to the contribution from the transient motion

Table 3. The evaluated values of (3), , and *n2* for the dihydroxy phosphorus (V) tetrabenzotriazacorroles together with the visible absorption maxima (max) in DMF

(3) (×10-14esu)

P(OH)2TBC 655 4.40 1.21 8.15 P(OH)2TBCSn 657 4.27 1.11 7.98 P(OH)2TBC(OiPr)4 677 4.91 2.26 9.01

Pr)4, and isopropoxyl group is a electron-donating group which leads to large

{P(OH)2TBC(Oi

P(OH)2TBC(Oi

solution.

technique (Huang et al., 2008).

 Structural formulae of **(a)** P(OH)2TBC and P(OH)2TBC(Oi

N

P OH

OH

N

R R

Pr)4 R=Oi

value of 2.26 × 10-31 esu for the P(OH)2TBC(Oi

of the conjugate electron distribution.

max (nm)

Sample

In 2008, we (Zhao et al., 2008) have synthesized three novel tri-dimensional phthalocyanine polymers with lanthanum (LaPPc), gadolinium (GdPPc) and ytterbium (YbPPc) as centric atoms from a tetranuclear phthalonitrile (the structures and synthesis route of rare earth polymeric phthalocyanines are shown in Scheme 3). And third-order optical nonlinearities of these compounds in DMF solution were measured by a picosecond Z-sacn technique at 532 nm.

Scheme 3. Structures and synthesis route of rare earth polymeric phthalocyanines

Based on the Z-scan measurements, it is found that these phthalocyanine polymers show large third-order nonlinear susceptibilities which are of the order of 10-12 esu. Both the nonlinear absorption β and nonlinear refraction *n2* decreases with the order of LaPPc>GdPPc>YbPPc. For all the compounds, the values of Re(3) are one order of magnitude larger than those of Im(3), which determine the magnitude of third-order nonlinear susceptibilities (3). This indicated that the nonlinear refraction is predominant mechanism for the nonlinear optical response of three phthalocyanine polymers.

Furthermore, Researchers (Manas et al., 1997) have described the effect of intermacrocycle interactions on the second hyperpolarizabilities of phthalocyanine dimer and trimer. It was considered that stack form of phthalocyanine can induce molecular electronic interaction between neighbouring phthalocyanine rings and induce charge transfer between them. As shown in Figure 10, the cage structure of phthalocyanine polymer presents more possibilities of electronic interaction and distortion of electron cloud in three dimensions at intermolecular scale. This would elevate the spatial polarizability of molecular and induces large nonlinear coefficient. However, the trimer or binuclear phthalocyanine could only offer this intermolecular possibility from two dimensions.

Fig. 10. Calculated molecular electronic distribution of binuclear, trimer and polymer of phthalocyanine.

Third-Order Optical Nonlinearities of Novel Phthalocyanines and Related Compounds 269

Shirk, J.S.; Lindle, F.J.; Bartoli, C.A.; Hoffman, C.A.; Kafafi, Z.H. & Snow, A.W. (1989). Off-

Kambara, H.; Muruno, T.; Yamashita, A.; Matsumoto, S.; Hayashi, T.; Konami, H. & Tanaka,

Ma, g.; He, J.; Kang, C. & Tang, S. (2003). Excited State Dynamics Studies of Iron(III)

Zhou, J.; Mi, J.; Zhu, R.; Li, B. & Qian, S. (2004). Ultrafast Excitation Relaxation in Titanylphthalocyanine Thin Film, *Optical Materials*, Vol.27, PP.377-382. He, C.; Wu, Y.; Shi, G.; Duan, W.; Song, W. & Song, Y. (2007). Large Third-Order Optical

Hermann, J.P.; Ricard, D. & Ducuing, J. (1973). Optical Nonlinearities in Conjugated

Linder, R.C.; Steel, D.G. & Dunning, G.J. (1982). Phase Conjugation by Resonantly Enhanced Degenerate Four-Wave Mixing, *Optical Engineering*, Vol.21, PP.190-198. Sheik-Bahae, M.; Said, A.A. & van Stryland, E.W. (1989). High-Sensitivity, Single-Beam n2

Sheik-Bahae, M.; Said, A.A.; Wei, T.H.; Hagan, D.J. & van Stryland, E.W. (1990). Sensitive

Torre, G.; Vazquez, P.; Agullo-Lopez, F. & Torres, T. (2004). Role of Structure Facror in the

Nalwa, H.S.; Kakuta, A. & Mukoh, A. (1993). Third-Order Optical Nonlinearities of Tetrakis-

Nalwa, H.S.; Hanack, M.; Pawlowski, G. & Klaus, E.M. (1999). Third-Order Nonlinear

He, C.; Chen, Y.; Nie, Y. & Wang, D. (2007). Third Order Optical Nonlinearities of Eight-- Octyloxy-Phthalocyanines, *Optics Communications*, Vol.271, PP.253-256. Li, Z.; Chen, Z.; Xu, S.; Niu, L.; Zhang Z.; Zhang, F. & Kasatni, K. (2007). Off-Rsonant Third-

Chen, Z.; Zhong, C.; Zhang, Z.; Li, Z.; Niu, L.; Bin, Y. & Zhang, F. (2008). Photoresponsive J-

Measurement of Optical Nonlinearities Using a Single Beam, *IEEE Journal of*

Nonlinear Optical Properties of Phthalocyanines and Related Compounds, *Chemical*

*n*-Pentoxy Carbonyl Metallo-Naphthalocyanines, *Chemical Physics Letters*, Vol.203,

Optical Properties of Porphyrazine, Phthalocyanine and Naphthalocyanine Germanium Derivatives: Demonstrating the Effect of -Conjugation Length on Third-Order Optical Nonlinearity of Two-Dimensional Molecules, *Chemical Physics*,

Order Optical Nonlinearities of Novel Diarylethene-Phthalocyanine Dyads,

Aggregation Behavior of a Novel Azobenzene-Phthalocyanine Dyad and Its Third-Order Optical Nonlinearity, *Journal of Physical Chemistry B,* Vol.112, PP.7387-7394. Huang, X.; Zhao, F.; Li, Z.; Tang, Y.; Zhang, F. & Tung, C.-H. (2007). Self-Assembled

Nanowire Networks of Aryloxy Zinc Phthalocyanines Based on Zn-O

*Applied Phisics Letters*, Vol.55, PP.1287-1288.

*Letters*, Vol.370, PP.293-299.

Fullerene, *Journal of Applied Physics*, Vol.80, PP.3674-3682.

Phthalocyanine, *Organic Electronics*, Vol.8, PP.198-205.

Measurements, *Optical Letters*, Vol.14, PP.955-957.

*Quantum Electronics,* Vol.26, PP.760-769.

*Chemical Physics Letters*, Vol.447, PP.110-114.

Coordination, *Langmuir*, Vol.23, PP.5167-5172.

*Reviews*, Vol.104, PP.3723-3750.

PP.109-113.

Vol.245, PP.17-26.

Systems: β-Carotene, *Applied Physics Letters*, Vol.23, PP.178-180.

Resonant Third-Order Optical Nonlinearities of Metal-Substituted Phthalocyanines,

N. (1996). Third-Order Nonlinear Optical Properties of Phthalocyanine and

Phthalocyanine Using Femtosecond Pump–Probe Techniques, *Chemical Physics*

Nonlinearites of Ultrathin Films Containing Octacarboxylic Copper

#### **5. Conclusions**

Major advances have been made in the design and synthesis of the novel phthalocyanines and related compounds for third-order optical nonlinearity. They provide useful examples to illustrate the new features of the NLO response of the phthalocyanines and related compounds. This review summarizes our results recently obtained on the correlation between molecular structure and NLO response and offers some strategies for rendering new systems with improved NLO properties.

Owing to the extended system, it is well-known that these Pc-related compounds exhibit a high aggregation tendency, and the aggregates usually display outstanding nonlinear optical properties. Moreover, by introducing peripheral substituents or by adding axial substituents, one may obtain three-dimensional (3D) structures with pyramidal shape and examine the effect of a third-dimension on the NLO response. Some of the four isoindole units can be formally replaced by other heterocyclic moieties, giving rise to different phthalocyanine analogues. All these chemical variations can alter the electronic structure of the macrocyclic core, and therefore, they allow the fine-tuning of the nonlinear response.

In conclusion, one could say that structural variations explored in phthalocyanines, such as metal insertion, introduction of functional groups into the periphery of the macrocycle, extension of conjugation, and variation of the main structure of the macrocycle, allow the tuning of the nonlinear responses. However, a detailed understanding of the factors affecting the nonlinear response is still necessary, and further work should be devoted to this objective.

#### **6. Acknowledgments**

We appreciate the help of Dr. Li Deng and Professor Zhenrong Sun, Key Laboratory of Optical and Magnetic Resonance Spectroscopy, East China Normal University, in carrying out the DFWM experiment. This work was supported by National Natural Science General Foundation of China (Grants 20773077, 20572059 and 20502013) and National Key Fundamental Research Program (2007CB808000).

#### **7. References**

Major advances have been made in the design and synthesis of the novel phthalocyanines and related compounds for third-order optical nonlinearity. They provide useful examples to illustrate the new features of the NLO response of the phthalocyanines and related compounds. This review summarizes our results recently obtained on the correlation between molecular structure and NLO response and offers some strategies for rendering

Owing to the extended system, it is well-known that these Pc-related compounds exhibit a high aggregation tendency, and the aggregates usually display outstanding nonlinear optical properties. Moreover, by introducing peripheral substituents or by adding axial substituents, one may obtain three-dimensional (3D) structures with pyramidal shape and examine the effect of a third-dimension on the NLO response. Some of the four isoindole units can be formally replaced by other heterocyclic moieties, giving rise to different phthalocyanine analogues. All these chemical variations can alter the electronic structure of the macrocyclic core, and therefore, they allow the fine-tuning of the nonlinear response. In conclusion, one could say that structural variations explored in phthalocyanines, such as metal insertion, introduction of functional groups into the periphery of the macrocycle, extension of conjugation, and variation of the main structure of the macrocycle, allow the tuning of the nonlinear responses. However, a detailed understanding of the factors affecting the nonlinear response is still necessary, and further work should be devoted to

We appreciate the help of Dr. Li Deng and Professor Zhenrong Sun, Key Laboratory of Optical and Magnetic Resonance Spectroscopy, East China Normal University, in carrying out the DFWM experiment. This work was supported by National Natural Science General Foundation of China (Grants 20773077, 20572059 and 20502013) and National Key

Perry, J.W.; Mansour, K.; Mander, K.J.; Perry, K.J.; Alvarez, D. & Choong, I. (1994).

Perry, J.W.; Mansour, K.; Lee, I.-Y.S. & Wu, X.-L. (1996). Organic Optical Limiter with a Strong Nonlinear Absorptive Response, *Science*, Vol.273,PP.1533-1536. Shirk, J.S.; Pong, R.F.S.; Flom, S.R.; Heckmann, H. & Hanack, M. (2000). Effect of Axial

Zyss, J. (Ed.). (1994). *Molecular Nonlinear Optics : Materials, phisics and Devices*, Academic

Ho, Z.Z.; Ju, C.Y. & Hetheringto, W.M. (1987). Third Harmonic Generation in

Substituted Phthalocyanines, *Optical Letters*, Vol.19,PP.625-627.

Phthalocyanines, *Journal of Applied physics*, Vol.62, PP.716-718.

*of Physical Chemistry A*, Vol.104,PP.1438-1449.

Enhanced Reverse Saturable Absorption and Optical Limiting in Heavy-Atom-

Substitution on the Optical Limiting Properties of Indium Phthalocyanines, *Journal*

**5. Conclusions**

this objective.

**7. References**

**6. Acknowledgments**

new systems with improved NLO properties.

Fundamental Research Program (2007CB808000).

Press, New York.

**14**

*Russia*

**Broadband Instability of Electromagnetic**

Wave propagation in nonlinear media is usually accompanied by variation of their spacetime spectra (V.I. Bespalov & Talanov, 1966; Lighthill, 1965; Litvak & Talanov, 1967; Benjamin & Feir, 1967; Zakharov & Ostrovsky, 2009; Bejot et al., 2011). The character of this variation is frequently explained in terms of modulation instability which occurs, in particular, for electromagnetic waves in cubic media, where polarization is approximately equal to the cube of electric field intensity. Modulation instability manifests itself in partitioning of the originally uniform packet into separate beams and pulses. It has been studied in ample detail for the perturbations whose frequencies and propagation directions slightly differ from those of intense waves (pump waves), i.e., temporal and spatial spectra of the waves participating in the process are rather narrow (paraxial approximation) (V.I. Bespalov & Talanov, 1966; Litvak & Talanov, 1967; Agraval, 1995; Talanov & Vlasov, 1997). In this case, modulation instability is described by the nonlinear parabolic equation for a

The instability is quite different, if the nonlinearity is higher than the third order of magnitude. It was shown in (Talanov & Vlasov, 1994; Koposova & Vlasov, 2007) that the wave propagating in such a medium may be unstable relative to collinear perturbations at frequencies so high that not only wave packet envelope but the structure of each wave in the packet may change too. The parabolic equation does not hold for description of such phenomena; hence, the methods for solution of wave equations in a wide frequency band developed in (Talanov & Vlasov, 1995; Brabec & Krausz, 1997; V.G. Bespalov et al., 1999; Kolesic & Moloney, 2004; Ferrando et al., 2005; Koposova et al., 2006) are employed. One of such methods, namely, the technique of pseudodifferential operators (Koposova et al., 2006)

The first part of the paper that is an extension of (Talanov & Vlasov, 1994) is concerned with the instability of perturbation waves at combination frequencies noncollinear to pump for the case of nonlinear polarization represented as a polynomial of arbitrary but finite degree. Further, application of the theory to air, for which dielectric permittivity may be represented in the form of a polynomial, is addressed (Loriot et al., 2009). Finally, methods of finding amplitude and phase distribution of electric field in an ultrashort wave packet by analyzing signals from intensity autocorrelator traditionally used for measuring duration of ultrashort

**1. Introduction**

wave train envelope.

is used in the current paper.

laser pulses are considered.

 **Waves in Nonlinear Media**

Sergey Vlasov, Elena Koposova and Alexey Babin *Institute of Applied Physics, Russian Academy of Science*

### **Broadband Instability of Electromagnetic Waves in Nonlinear Media**

Sergey Vlasov, Elena Koposova and Alexey Babin *Institute of Applied Physics, Russian Academy of Science Russia*

#### **1. Introduction**

270 Laser Systems for Applications

Huang, X.; Zhao, F.; Li, Z.; Huang, L.; Tang, Y.; Zhang, F. & Tung, C.-H. (2007). A Novel

Chen, Z.; Dong, S.; Zhong, C.; Zhang, Z.; Niu, L.; Li, Z. & Zhang, F. (2009). Photoswitching

Breslow, R. & Dong, S. (1998). Biomimetic Reactions Catalyzed by Cyclodextrins and Their

Dugave, C. & Demange, L. (2003). Cis−Trans Isomerization of Organic Molecules and

González-Cabello, A.; Claessens C, G.; Martin-Fuch, G.; Ledoux-Rack, I.; Vázquez, P.; Zyss,

Bin, Y.; Xu, S.; Li, Z.; Huang, L.; Zhang, Z. & Zhang, F. (2008). Large Third-Order Optical

Liu, J.; Zhang, F. ; Zhao, Y ; Tang, Y ; Song, X. & Yao, G. (1995). Complexation of Phosphorus

*Journal of Photochemistry and Photobiology A: Chemistry*, Vol.91, PP.99-104. Chen, Z.; Zhou, X.; Li, Z.; Niu, L.; Yi, J. & Zhang, F. (2011). The Third-Order Optical

*Journal of Photochemistry and Photobiology A: Chemistry*, Vol.211, PP.64-68. Huang, L. Li, Z. ; Zhang, F.; Tung, C. & Kasatani, K. (2008). Off-Resonant Optical

Tetrabenzotriazacorroles, *Optics Communication,* Vol.281, PP.1275-1279. Zhao, P.; Xu, S.; Li, Z. & Zhang, F. (2008). Nonlinear Optical Properties of Novel Polymeric

Manas, E.S ; Spano, F.C & Chen, L.X. (1997). Nonlinear Optical Response of Cofacial

Triads for Nonlinear Optics, *Synthsis Metal*, Vol. 137, PP.1487-1488.

Vol.36, PP.108-109.

Vol.25, PP.3257-3259.

*Physics Letters*, Vol.6, PP.2058-2061.

2532.

719.

*Photobiology A: Chemistry*, Vol.206, PP.213-219.

Derivatives, *Chemical Review*, Vol.98, PP.1997-2012.

Self-Aggregates of Phthalocyanine Based on Zn-O Coordination, *Chemistry Letters*,

of Third-Order Optical Properties of Azobenzene-Containing Phthalocyanines Based on Reversible Host-Guest Interactions, *Journal of Photochemistry and*

Biomolecules: Implications and Applications, *Chemical Review*, Vol.103, PP.2475-

J.; Agulló-López, F. & Torres, T. (2003). Phthalocyanine-Ferrocene Dyads and

Nonlinearity of a Novel Copper Phthalocyanine Ferrocene, *Chinese Physics Letters*,

(III) with a Novel Tetrapyrrolic Phthalocyanine-Like Macrocyclic Compound,

nonlinearities of Thiophene-Bearing Phthalocyanines Studied by Z-Scan Technique,

Nonlinearities of Phthalocyanine Analogues: Dihydroxy Phosphorus (V)

Rare Earth Phthalocyanine Studied Using Picosecond Z-scan Technique, *Chinese*

Phthalocyanine Dimers and Trimers, *Journal of Chemical Physics*, Vol.107, PP.707-

Wave propagation in nonlinear media is usually accompanied by variation of their spacetime spectra (V.I. Bespalov & Talanov, 1966; Lighthill, 1965; Litvak & Talanov, 1967; Benjamin & Feir, 1967; Zakharov & Ostrovsky, 2009; Bejot et al., 2011). The character of this variation is frequently explained in terms of modulation instability which occurs, in particular, for electromagnetic waves in cubic media, where polarization is approximately equal to the cube of electric field intensity. Modulation instability manifests itself in partitioning of the originally uniform packet into separate beams and pulses. It has been studied in ample detail for the perturbations whose frequencies and propagation directions slightly differ from those of intense waves (pump waves), i.e., temporal and spatial spectra of the waves participating in the process are rather narrow (paraxial approximation) (V.I. Bespalov & Talanov, 1966; Litvak & Talanov, 1967; Agraval, 1995; Talanov & Vlasov, 1997). In this case, modulation instability is described by the nonlinear parabolic equation for a wave train envelope.

The instability is quite different, if the nonlinearity is higher than the third order of magnitude. It was shown in (Talanov & Vlasov, 1994; Koposova & Vlasov, 2007) that the wave propagating in such a medium may be unstable relative to collinear perturbations at frequencies so high that not only wave packet envelope but the structure of each wave in the packet may change too. The parabolic equation does not hold for description of such phenomena; hence, the methods for solution of wave equations in a wide frequency band developed in (Talanov & Vlasov, 1995; Brabec & Krausz, 1997; V.G. Bespalov et al., 1999; Kolesic & Moloney, 2004; Ferrando et al., 2005; Koposova et al., 2006) are employed. One of such methods, namely, the technique of pseudodifferential operators (Koposova et al., 2006) is used in the current paper.

The first part of the paper that is an extension of (Talanov & Vlasov, 1994) is concerned with the instability of perturbation waves at combination frequencies noncollinear to pump for the case of nonlinear polarization represented as a polynomial of arbitrary but finite degree. Further, application of the theory to air, for which dielectric permittivity may be represented in the form of a polynomial, is addressed (Loriot et al., 2009). Finally, methods of finding amplitude and phase distribution of electric field in an ultrashort wave packet by analyzing signals from intensity autocorrelator traditionally used for measuring duration of ultrashort laser pulses are considered.

Broadband Instability of Electromagnetic Waves in Nonlinear Media 273

generated by each term of the series (2) and related to the nonlinearity of degree 2 1 *N* , the

We will study stability of the solution relative to two-frequency perturbations of the form

,

perturbation of the first degree *u*1 and \* *u*2 we will select the terms of identical degrees *A*0

 2 01 1 exp[ ] *N*

\* \*

0 12

they will have identical frequencies and propagation constants. Sets of frequencies

variations of the wave packet envelope that are slow compared to the carrier.

and the terms whose difference of degrees *A*0 and \* *A*0 is equal to the even integer

0 0 0 02 2 exp[ ( ) ( ) ] exp[ ] *NM NM A A i N M i N M u i ik r*

propagation constants *h* will be different for *M* 1 , *M* 2 , and so on. In other words, solution of the form (5) may have *M* branches, with the characteristics of branch *M* depending on nonlinear polarization terms with indices *M N S* . For *M* 1 , we have perturbations near carrier frequency – the well-known modulation instability describing

Making use of the equality (8) we will express the frequencies 10 20

and frequency *M*

real in the frequency range under consideration, from Eq. (1) we obtain for the functions *u*1

1 \*

*u i ui u z*

{ } ,

*u i ui u z*

( )( ) 1 ( ) ( )2

*M N NL*

2 \*

 2 2 0 0

*Mn N k M k Mh k*

1 2

0 0 0 2 1,

0 0 1

*n M* ,

*N*

*S*

2 1

*M M*

*M M*

0 01 1 2 2 , exp( ) exp[ ] exp[ ] . . 2

in (4) is of the (2 1) *N th* order of magnitude.

2 and their propagation constant *h* will be found from solution

*t hz* . Among the polarization terms with the

*A u i ik r* (6)

, *Mh h* 0 (8)

1,2 and

*c*

is

(9)

1,2 and *h* the terms (6) and (7) will be synchronous, i.e.,

 *M M*

0 . Assuming that ( ) ( ) *n k*

 ,

*A i u i ik r u i ik r c c E* (5)

. (7)

*k* being their transverse wave number.

nonlinearity coefficient 2 1 *N*

Wave frequencies

where 00 0

and \* *A*0 :

2 2 *M N* :

where

If the conditions

are fulfilled, for definite values of

through their difference 1 2

and \* *u*2 the following system of equations

1 1 2 2 exp[ ( ) ] exp[ ( ) ] *u i t hz ik r u i t hz ik r*

1 and

of dispersion equations for perturbations. We will seek solution of Eq. (1) in the form

> *t hz* , 1 1

 *t hz* , 2 2

> 2*M*

2

\*

{ }

#### **2. Plane wave instability in media with polynomial nonlinearity**

Consider a linearly polarized wave packet propagating along the *z* -axis. We will make use of the equations for field *E* in this beam:

$$\frac{\partial E}{\partial z} + i \sqrt{-\frac{1}{c^2} \frac{\partial^2}{\partial t^2} n^2 (-i \frac{\partial}{\partial t}) + \Lambda\_\perp} E = i \frac{1}{2 \sqrt{-\frac{1}{c^2} \frac{\partial^2}{\partial t^2} n^2 (-i \frac{\partial}{\partial t}) + \Lambda\_\perp}} \frac{4\pi}{c^2} \frac{\partial^2 P\_{\text{ML}}}{\partial t^2} ,\tag{1}$$

obtained in (Koposova et al., 2006). In Eq. (1), *n i* ( )*t* is the operator describing linear dispersion of the medium; for the processes exp[ ] *i t* stationary in time, *n*( ) is the index of refraction at circular frequency ; is Laplace operator in *r* coordinates transverse to the propagation direction; *PNL* is nonlinear polarization; and *c* is the velocity of light. Assuming the angular difference between the directions of the interacting waves to be small, we will neglect dispersion of the nonlinear polarization coefficient and the Laplace operator in the right-hand side of (1) and suppose that

$$\frac{1}{\sqrt{-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}n^2\left(-i\frac{\partial}{\partial t}\right) + \Lambda\_\perp}} \approx \frac{c}{a\sqrt{\varepsilon(a)}}$$

where 2 () () *n* is the dielectric permittivity at the same frequency. We will describe the polarization by the polynomial dependence on the magnitude of electric field:

$$P\_{\rm NL} = \sum\_{i=1}^{S} \mathcal{X}^{(2N+1)} E^{2N+1} \, , \tag{2}$$

where (2 1) *N* are (2 1) *N th* order susceptibilities, and 2 1 *S* is the highest degree polarization taken into consideration.

To the first approximation of the asymptotic theory of nonlinear oscillations (Bogolyubov & Mitropol'skii, 1958), a steady-state plane wave

$$E(t,\vec{r}) = \frac{A\_0 \exp\{i o\_0 t - i h\_0 z\} + A\_0 \left[-i o\_0 t + i h\_0 z\right]}{2} \tag{3}$$

having amplitude *A*0 , frequency 0 , and propagation constant 0 0 (1 ) 2 *NL h k* (where 0 0 0 *k n* ( ) *c* is the propagation constant of the wave of the same frequency in a linear medium and *NL* is a nonlinear additive to permittivity) may propagate in the medium with permittivity (2). The nonlinear additive is represented as a sum 2 1, 1 *S NL N NL N* of nonlinear additives

$$\left| \circledast \varepsilon\_{2N+1,\text{NL}} = \frac{4\pi}{\varepsilon(a\_0)} \frac{(2N+1)!}{N!(N+1)!2^{2N}} \mathcal{Z}^{(2N+1)} \left| A\_0 \right|^{2N} = \varepsilon\_{2N+1}' \left| A\_0 \right|^{2N} \tag{4}$$

generated by each term of the series (2) and related to the nonlinearity of degree 2 1 *N* , the nonlinearity coefficient 2 1 *N* in (4) is of the (2 1) *N th* order of magnitude.

We will study stability of the solution relative to two-frequency perturbations of the form 1 1 2 2 exp[ ( ) ] exp[ ( ) ] *u i t hz ik r u i t hz ik r* , *k* being their transverse wave number. Wave frequencies 1 and 2 and their propagation constant *h* will be found from solution of dispersion equations for perturbations.

We will seek solution of Eq. (1) in the form

$$E = \frac{A\_o \exp(i\phi\_o) + \mu\_1 \exp[i\phi\_1 - i\vec{k}\_\perp \vec{r}\_\perp] + \mu\_2 \exp[i\phi\_2 + i\vec{k}\_\perp \vec{r}\_\perp] + c.c.}{2},\tag{5}$$

where 00 0 *t hz* , 1 1 *t hz* , 2 2 *t hz* . Among the polarization terms with the perturbation of the first degree *u*1 and \* *u*2 we will select the terms of identical degrees *A*0 and \* *A*0 :

$$-\left|A\_0\right|^{2N} u\_1 \exp\left[i\varphi\_1 - i\vec{k}\_\perp \vec{r}\_\perp\right] \tag{6}$$

and the terms whose difference of degrees *A*0 and \* *A*0 is equal to the even integer 2 2 *M N* :

$$-A\_0^{N+M} A\_0^{"N-M} \exp[i(N+M)\rho\_0 - i(N-M)\rho\_0] \mu\_2^\* \exp[-i\rho\_2 - i\vec{k}\_\perp \vec{r}\_\perp] \ . \tag{7}$$

If the conditions

272 Laser Systems for Applications

Consider a linearly polarized wave packet propagating along the *z* -axis. We will make use

1 1 4 ( )

*E PNL i n i Ei z ct t c t*

2 2

2 2 2 2 2

2 2 2

*n i ct t*

is the operator describing linear

*r* coordinates transverse to

, (1)

is the index

stationary in time, *n*( )

,

*P E* , (2)

0 , and propagation constant 0 0 (1 ) 2

(2 1) 2 2

(4)

*N N N*

 *NL h k*

of nonlinear additives

(where

*A i t ih z A i t ih z Etr* (3)

1

 *S NL N NL N*

*N A A*

1 2 ()

*c*

are (2 1) *N th* order susceptibilities, and 2 1 *S* is the highest degree

is the propagation constant of the wave of the same frequency in a linear medium

is a nonlinear additive to permittivity) may propagate in the medium with permittivity

2 1, 2 0 210

*N NL N N*

1 ( ) ( )

We will describe the polarization by the polynomial dependence on the magnitude of

(2 1) 2 1

To the first approximation of the asymptotic theory of nonlinear oscillations (Bogolyubov &

 \* 0 00 0 00 exp[ ] exp[ ] (, ) 2

*S N N*

*NL i*

4 (2 1)! ( ) !( 1)!2

*N N*

(2). The nonlinear additive is represented as a sum 2 1,

0

  1

; is Laplace operator in

the propagation direction; *PNL* is nonlinear polarization; and *c* is the velocity of light. Assuming the angular difference between the directions of the interacting waves to be small, we will neglect dispersion of the nonlinear polarization coefficient and the Laplace operator

> 2 2

*n i ct t*

*n* is the dielectric permittivity at the same frequency.

1

2 2

**2. Plane wave instability in media with polynomial nonlinearity**

of the equations for field *E* in this beam:

of refraction at circular frequency

where 2

electric field:

where (2 1) *N*

0 0 0 *k n* ( ) *c*

and *NL*

() ()

polarization taken into consideration.

having amplitude *A*0 , frequency

Mitropol'skii, 1958), a steady-state plane wave

2

obtained in (Koposova et al., 2006). In Eq. (1), *n i* ( )*t*

dispersion of the medium; for the processes exp[ ] *i t*

in the right-hand side of (1) and suppose that

$$2M\alpha\rho\_0 = \alpha\rho\_1 + \alpha\rho\_2 \qquad \qquad M\hbar\mathbf{h}\_0 = \mathbf{h} \tag{8}$$

are fulfilled, for definite values of 1,2 and *h* the terms (6) and (7) will be synchronous, i.e., they will have identical frequencies and propagation constants. Sets of frequencies 1,2 and propagation constants *h* will be different for *M* 1 , *M* 2 , and so on. In other words, solution of the form (5) may have *M* branches, with the characteristics of branch *M* depending on nonlinear polarization terms with indices *M N S* . For *M* 1 , we have perturbations near carrier frequency – the well-known modulation instability describing variations of the wave packet envelope that are slow compared to the carrier.

Making use of the equality (8) we will express the frequencies 10 20 *M M* , through their difference 1 2 2 and frequency *M*0 . Assuming that ( ) ( ) *n k c* is real in the frequency range under consideration, from Eq. (1) we obtain for the functions *u*1 and \* *u*2 the following system of equations

$$\begin{aligned} \frac{\partial \{\boldsymbol{u}\_1\}}{\partial \boldsymbol{z}} &= -i\mathcal{B}\_{\mathcal{M}+} \boldsymbol{u}\_1 - i\boldsymbol{\gamma}\_{\mathcal{M}+} \boldsymbol{u}\_2^\* \\ \frac{\partial \{\boldsymbol{u}\_2^\*\}}{\partial \boldsymbol{z}} &= i\mathcal{B}\_{\mathcal{M}-} \boldsymbol{u}\_2^\* + i\boldsymbol{\gamma}\_{\mathcal{M}-} \boldsymbol{u}\_{1^\*} \end{aligned} \tag{9}$$

where

$$\mathcal{J}\_{M\pm} = \sqrt{k^2 \left(M o o\_0 \pm \Omega \right) - k\_{\perp}^2} - M h\_0 + k\_0 \frac{\left(M o\_0 \pm \Omega \right) n \left(a\_0 \right)}{o\_0 n \left(M o\_0 \pm \Omega \right)} \sum\_{N=1}^{S} \frac{N+1}{\mathfrak{Q}} \delta \varepsilon\_{2N+1, \mathcal{M}\_{\perp}}$$

Broadband Instability of Electromagnetic Waves in Nonlinear Media 275

*d d NII*

that is equal to the product of the derivative of the nonlinear additive to intensity and

The modulation instability occurs (Litvak & Talanov, 1967) at *D* 0 . Its behavior changes as

with temporal scales come to the forefront. Note that the modulation instability increments

 2 1, 1

From the latter condition it follows that the increments produced by different terms in the

*S NL N*

*kM*

*M N N*

The propagation constant *HM* will be a complex one, and instability will occur, given

*N M*

0

2 ( )!( )!

. The increment reaches its maximum near zeros of the expression

2

2 *M k*

 2 2

2 2 *k*

located in one transparency band of the substance, the equality (15) is fulfilled for the hyperbolae (17). There exists in this case a minimal value of the transverse wave number

The expression may become zero at an arbitrary sign of nonlinear additive due to appearance of a transverse component of the perturbation wave number on a certain curve

0 0 ( ) [ 1] () 2 *n M k M n k* |

*NMNM*

*S NL N NL N d I N*

*N NL N NL NL*

2 *Dk k* () () . (12)

0

 2 2 0 *H kk k M MM* ( ( ) ( ) ), (14)

*dI* . (13)

0, 0 *k* , perturbations with spatial scale

0 , (15)

 0 2 2 0

0

*k* (17)

0 and

0

*H k M* 0 . (16)

*M* and frequencies *M*

0, 0 *k* , perturbations

*dI dI*

For *M* 1 (modulation instability) the radicand (11) is rewritten in the form

are more pronounced ("self-focusing" instability); whereas for

,2 1 !( 1)!

For *M* 2 , in the paraxial approximation we obtain

expansion (2) may "obliterate" each other under certain conditions.

. For

>

*N N*

*S S*

 2 1, 2 1, 1 1

intensity. In the case of cubic nonlinearity, this expression is equal to the additive.

a function of the signs of *k* and

turn to zero at ( )0 *k* and

where

 *kM*

on the ,

with the increment being of order

*kM*

*k* -plane. At normal dispersion

In (12) we have

$$\mathcal{I}\_{M\pm} = k\_0 \frac{(M a\_0 \pm \Omega) n(a\_0)}{a\_0 n(M a\_0 \pm \Omega)} \frac{A\_0^{\pm M}}{A\_0^{\prime \pm M}} \sum\_{N=M}^{S} \frac{\partial \varepsilon\_{N\perp 2N+1} N! (N+1)!}{\Im(N-M)! (N+M)!} \tag{1}$$

Solutions to (9) are sought in the form

$$\begin{aligned} \mu\_1 &= A\_1 \exp[-iH\_1 z] \\\\ \mu\_2^\* &= A\_2^\* \exp[-iH\_1 z] \end{aligned} $$

constants *A*1 and \* *A*2 are found from the system of linear equations with constant coefficients the determinant of which is a characteristic equation for the corrections *HM* to the propagation constants *Mh h* 0 . The determinant has the form

$$\{H\_M - \beta\_{M\ast}\} \{H\_M + \beta\_{M-}\} + \gamma\_{M\ast}\gamma\_{M-} = 0 \tag{10}$$

For the corrections *HM* to the propagation constants we have

$$H\_M = \frac{\mathcal{B}\_{M\*} - \mathcal{B}\_{M-}}{2} \pm \sqrt{\frac{\left(\mathcal{B}\_{M\*} + \mathcal{B}\_{M-}\right)^2}{4} - \mathcal{Y}\_{M\*}\mathcal{Y}\_{M-}}\,. \tag{11}$$

By wave of illustration we present in Fig. 1 the diagram of wave vectors 1 ( ) *k* , 2 ( ) *k* , 0 *k*( ) for the considered effects in the case *M* 2 , when the inequality 0 0 0 00 0 2 2 *k nk n* (2 ) (2 ) 2 ( ) ( ) *c c* or 0 0 *n n* (2 ) ( ) is met.

Fig. 1. Wave vector diagram of converting frequency 0 to the frequency close to the second harmonic for modulation instability and condition 1 0 *k k* ( ) 2( ) .

The latter inequality is satisfied if the frequencies 0 and 0 2 are in one transparency band of the substance (Born & Volf, 1964). The wave vector diagram shows that in this case the waves at frequencies 0 and 0 2 may be synchronized during propagation in the direction of vector 0 *k* , which is accompanied by transformation of four vectors (quanta) of frequency 0 into two vectors (quanta) of frequency 0 2.

Let us study (12) in more detail in the quasiparaxial approximation, omitting everywhere except the expression 2 2 0 *kM k* ( ) . Let us designate mismatches as

$$
\Delta k\_M = \frac{\sqrt{k^2(\mathcal{M}\alpha\mathcal{g}\_b + \Omega\mathfrak{z}) - k\_\perp^2} + \sqrt{k^2(\mathcal{M}\alpha\mathcal{g}\_b - \Omega\mathfrak{z}) - k\_\perp^2}}{2k\_0} - M\_\perp.
$$

For *M* 1 (modulation instability) the radicand (11) is rewritten in the form

$$D = \langle \Delta \mathbf{k} \rangle^2 + \langle \Delta \mathbf{k} \rangle \Delta \varepsilon\_\dots \tag{12}$$

In (12) we have

274 Laser Systems for Applications

0 00 ,2 1

*u A iH z* 11 1 exp[ ] ,

 \* \* 22 1 *u A iH z* exp[ ], constants *A*1 and \* *A*2 are found from the system of linear equations with constant coefficients the determinant of which is a characteristic equation for the corrections *HM* to

0 *k*( ) for the considered effects in the case *M* 2 , when the inequality

 or 0 0 *n n* (2 ) ( )

> *k*

0 and 0 2

*k* , which is accompanied by transformation of four vectors (quanta) of frequency

*k* .

0 *kM k* ( ) . Let us designate mismatches as

of the substance (Born & Volf, 1964). The wave vector diagram shows that in this case the

 . Let us study (12) in more detail in the quasiparaxial approximation, omitting everywhere

> 2 2 2 2 0 0 0

() ()

*kM k kM k k M*

2 4

By wave of illustration we present in Fig. 1 the diagram of wave vectors 1 ( )

2 ( )

*M nA N N k*

*N M*

( )( ) !( 1)! ( ) 2( )!( )! *M S NL N*

{ }{ } *H H M M M M MM* 0 . (10)

*MM MM HM M M* . (11)

is met.

.

may be synchronized during propagation in the direction

 *k* , 2 ( ) *k* ,

0 to the frequency close to the second

2 *k*( )

are in one transparency band

*nM A N M N M* .

the propagation constants *Mh h* 0 . The determinant has the form

For the corrections *HM* to the propagation constants we have

0 0 0 00 0 2 2 *k nk n* (2 ) (2 ) 2 ( ) ( ) *c c*

Fig. 1. Wave vector diagram of converting frequency

The latter inequality is satisfied if the frequencies

0 and 0 2

> 2 2

2 *M*

0 into two vectors (quanta) of frequency 0 2

harmonic for modulation instability and condition 1 0 *k k* ( ) 2( )

0 *k*( )

1 *k*( )

0 \* 00 0

*M M*

Solutions to (9) are sought in the form

>

waves at frequencies

except the expression

of vector 0

$$\Delta \mathfrak{E} = \sum\_{N=1}^{S} \mathrm{N} \mathfrak{E} \mathfrak{E}\_{2N+1, \mathrm{NL}} = \sum\_{N=1}^{S} I \frac{d}{dI} \mathfrak{E} \mathfrak{E}\_{2N+1, \mathrm{NL}} = I \frac{d}{dI} \, \mathrm{\mathcal{S}} \mathfrak{E}\_{\mathrm{NL}}$$

that is equal to the product of the derivative of the nonlinear additive to intensity and intensity. In the case of cubic nonlinearity, this expression is equal to the additive.

The modulation instability occurs (Litvak & Talanov, 1967) at *D* 0 . Its behavior changes as a function of the signs of *k* and . For 0, 0 *k* , perturbations with spatial scale are more pronounced ("self-focusing" instability); whereas for 0, 0 *k* , perturbations with temporal scales come to the forefront. Note that the modulation instability increments turn to zero at ( )0 *k* and

$$
\Delta \mathfrak{E} = I \frac{d}{dI} \delta \mathfrak{E}\_{\text{NL}} = \sum\_{N=1}^{S} N \delta \mathfrak{E}\_{2N \times 1, \text{NL}} = \mathbf{0} \tag{13}
$$

From the latter condition it follows that the increments produced by different terms in the expansion (2) may "obliterate" each other under certain conditions.

For *M* 2 , in the paraxial approximation we obtain

$$H\_M = k\_0 \{ \Delta k\_M \pm \sqrt{\left( \Delta k\_M + \Delta \varepsilon \right)^2 - \left( \Delta \overline{\varepsilon} \right)^2} \}\,\tag{14}$$

where

$$\Delta \overline{\varepsilon} = \frac{M}{2} \sum\_{N=M}^{S} \frac{\delta \varepsilon\_{Nl,2N+1} N! (N-1)!}{(N-M)! (N+M)!}$$

The propagation constant *HM* will be a complex one, and instability will occur, given *kM* . The increment reaches its maximum near zeros of the expression

$$
\Delta k\_M + \Delta \varepsilon = 0 \tag{15}
$$

with the increment being of order

$$H\_M = k\_0 \Delta \overline{\varepsilon} \,\tag{16}$$

The expression may become zero at an arbitrary sign of nonlinear additive due to appearance of a transverse component of the perturbation wave number on a certain curve

$$
\Delta k\_M + \Delta \varepsilon \approx M [\frac{n(M\alpha\_0)}{n(o\_0)} - 1] + \frac{\Omega^2}{2k\_0} \frac{\partial^2 k}{\partial \phi^2} \parallel \varepsilon\_{o=Ma\_0} - \frac{k\_\perp^2}{2k\_0^2} + \Delta \varepsilon = 0 \tag{17}
$$

on the , *k* -plane. At normal dispersion 2 2 *k* | 0 0 *M* and frequencies *M*0 and 0

located in one transparency band of the substance, the equality (15) is fulfilled for the hyperbolae (17). There exists in this case a minimal value of the transverse wave number

Broadband Instability of Electromagnetic Waves in Nonlinear Media 277

The instability regions on the plane of transverse wave numbers and frequencies within the 5-octave band for air, where the frequency dependence of refractive index is known (Grigor'ev & Melikhov, 1991), are shown in Fig. 3 for two values of intensity: 2 *I TW cm* 15 / and 2 *I TW cm* 19 / for 14 1 0

radiation wavelength 0

the normalized increment

*M* 1,2,3,4 .

min 0,0

1.0

0.0

2.0

3.0

*M* 1

*M* 2

4.0

5.0

1,2 0 /

of

0 2 0.8 *m k*

> *M H H*

at maximum nonlinear additive to permittivity and 0

0 *M*

is larger than its value at maximum nonlinear additive to permittivity and 0

*M* 4

Fig. 3. Instability increment *HM* in air on the plane of parameters *k k* /, / 0 0

max 2,3 10

*increment* min

*M* 2,3 attains its maximum near the frequencies 0 2

0.01 0.02 0.03 0.04 0.05

*M* 3

becomes worse near the boundaries of the frequency interval

branch. For the branch *M* 1 , the increment becomes small when

for 2 *I TW cm* 15 / (a) and 2 *I TW cm* 19 / (b) at 0

6

 0 *k k*

> 0.8

a) b)

0,0

*M* 1

1.0

0.0

2.0

3.0

4.0

5.0

1,2 0 /

For the values of intensities given above, the increments are different for all branches. At small intensities, 2 *I TW cm* 15 / , the increment is largest for *M* 1 (modulation instability); at large intensities it is largest for *M* 2 . The increment for perturbations with

satisfying the equality (15). It should be born in mind that accuracy of estimates

Fig. 4, that supplements Fig. 3., demonstrates maximal increments as a function of intensity, with the maximum attained for each branch and each value of intensity at definite values of frequency and transverse wave number that are also functions of intensity and number of the

other branches, the increments grow with increasing intensity in the 2 *I TW cm* 10 / region.

 and 0 3

>

 7.85 10 *s*

, which corresponds to the

. In the second case, the intensity

.

*M* 4

 0 *k k*

6

max 8,64 10

, respectively, for the values

vanishes to zero; for the

*m* for different instability branches

0.01 0.02 0.03 0.04 0.05

*increment*

*M* 3

0 , 0 5.

. The shadow density is proportional to the value of

*k* . In the first case, the intensity is smaller than its value

*M* 2

$$\frac{k\_{\perp}}{k\_0} \approx \sqrt{M[\frac{n(Moo\_0)}{n(o\_0)} - 1]} \Big|\_{\text{\tiny{}}} \quad M \ge 2 \tag{18}$$

at which instability occurs at arbitrary weak nonlinearity. Fulfillment of the equality (18) indicates the presence of conic radiation.

#### **3. Intense plane wave instability in air**

modulation (self-focusing) instability occurs at *M* 1 .

Consider the effects in air. The dependence at normal pressure of the nonlinear index of refraction 2 0 21 0 2 1 *S N NL N N n A* on intensity 2 0 8 *c I A* : , *S N n nI* (19)

*NL NL N*

$$^{N=1}$$
 taken from (Loriot et al., 2009). is shown in Fig. 2. Curve 1 is plotted taking into consideration four (all known from (Loriot et al., 2009)) terms, curve 2 taking into consideration two terms, and curve 3 taking into consideration only the first term, when purely cubic nonlinearity occurs. Note that for curve 1 there exist unstable branches at  $M=1,2,3,4$ , and for curve 2, despite its qualitative coincidence with curve 3 (one maximum), the instability branches exist only at  $M=1,2$ ; for curve 3 instability known as

Fig. 2. Nonlinear additive to the index of refraction of air as a function of intensity 1– fourterm approximation, 2– two-term approximation, 3– one-term approximation.

( ) [ 1] ( )

at which instability occurs at arbitrary weak nonlinearity. Fulfillment of the equality (18)

Consider the effects in air. The dependence at normal pressure of the nonlinear index of

on intensity 2

 , 1 *S*

Fig. 2. Nonlinear additive to the index of refraction of air as a function of intensity 1– four-

1

10. 20. 30. 40. 0.0

term approximation, 2– two-term approximation, 3– one-term approximation.

taken from (Loriot et al., 2009). is shown in Fig. 2. Curve 1 is plotted taking into consideration four (all known from (Loriot et al., 2009)) terms, curve 2 taking into consideration two terms, and curve 3 taking into consideration only the first term, when purely cubic nonlinearity occurs. Note that for curve 1 there exist unstable branches at *M* 1,2,3,4 , and for curve 2, despite its qualitative coincidence with curve 3 (one maximum), the instability branches exist only at *M* 1,2 ; for curve 3 instability known as

3

*NL NL N N*

0 8 *c I A*

*N*

*k n* , *M* 2 (18)

:

*n nI* (19)

2

 2 *TW I cm*

 0 0 0

*k nM M*

indicates the presence of conic radiation.

refraction 2 0

2.0

4.0

6 10 *nNL*

6.0

*NL N N n A*

**3. Intense plane wave instability in air**

21 0 2 1

modulation (self-focusing) instability occurs at *M* 1 .

 *S N* The instability regions on the plane of transverse wave numbers and frequencies within the 5-octave band for air, where the frequency dependence of refractive index is known (Grigor'ev & Melikhov, 1991), are shown in Fig. 3 for two values of intensity: 2 *I TW cm* 15 / and 2 *I TW cm* 19 / for 14 1 0 7.85 10 *s* , which corresponds to the radiation wavelength 0 0 2 0.8 *m k* . The shadow density is proportional to the value of the normalized increment 0 *M M H H k* . In the first case, the intensity is smaller than its value at maximum nonlinear additive to permittivity and 0 . In the second case, the intensity is larger than its value at maximum nonlinear additive to permittivity and 0 .

Fig. 3. Instability increment *HM* in air on the plane of parameters *k k* /, / 0 0 for 2 *I TW cm* 15 / (a) and 2 *I TW cm* 19 / (b) at 0 0.8*m* for different instability branches *M* 1,2,3,4 .

For the values of intensities given above, the increments are different for all branches. At small intensities, 2 *I TW cm* 15 / , the increment is largest for *M* 1 (modulation instability); at large intensities it is largest for *M* 2 . The increment for perturbations with *M* 2,3 attains its maximum near the frequencies 0 2 and 0 3 , respectively, for the values of satisfying the equality (15). It should be born in mind that accuracy of estimates becomes worse near the boundaries of the frequency interval 0 , 0 5.

Fig. 4, that supplements Fig. 3., demonstrates maximal increments as a function of intensity, with the maximum attained for each branch and each value of intensity at definite values of frequency and transverse wave number that are also functions of intensity and number of the branch. For the branch *M* 1 , the increment becomes small when vanishes to zero; for the other branches, the increments grow with increasing intensity in the 2 *I TW cm* 10 / region.

Broadband Instability of Electromagnetic Waves in Nonlinear Media 279

Naganuma et al., 1989; Iaconis & Walmsley, 1998), are devoted to interference measurements. Another group of fairly recent publications (Naganuma et al., 1989; Iaconis & Walmsley, 1998; Delong et al., 1996; Koumans & Yariv, 1999; Kane, 1999) take advantage of the mathematically proven fact that a multi-dimensional (noninterference) auto- or cross-correlation function allows retrieving the phase distribution of a short pulse. A two-dimensional signal distribution on the frequency-delay plane is usually measured in experiment, and then an iterative processing algorithm yields the desired parameters. The most advanced to-date technique of this group is referred to as FROG in the literature (Kane, 1999). The third group of methods (Naganuma et al., 1989; Kane, 1999; Peatross et al., 1998; Nicholson et al., 1999), that are based on the results of simultaneous spectral and correlation energy measurements and some iterative algorithm, also make it possible to determine phase and amplitude characteristics of an

The interference methods of retrieving ultrashort wave packet field parameters are simpler for implementation and in terms of processing than the methods of the second group. However, despite the fact that the speed of obtaining qualitative information for the latter techniques often does not meet the needs of experiment, such techniques are clearer, since the form of phase modulation of an optical pulse can qualitatively be deduced immediately from a two-dimensional distribution of the obtained signal. In this respect, methods of the third group are much faster, as the corresponding iterative retrieval procedure is based on operations with one-dimensional data files. However, the problem of accuracy of retrieval of the desired parameters remains open for such methods. In this part we demonstrate that in certain cases, the interference methods can also ensure clear, simple, and fast acquisition of information on the amplitude and phase of an optical pulse. In our opinion, the present part is methodical to a significant extent, although it has practical applications related to measuring dispersion characteristics of optical materials. It can easily be shown that the output signal from the photodetector of an interferometric intensity autocorrelator, whose scheme can be found in a number of

optical pulse.

publications (e.g., Krukov, 2008), has the form:

1 is satisfied, i.e., if the envelope

*Et t* ( ) ( ) exp[ ( )] . .

Equation (20) is known (Akhmanov, Vysloukh & Chirkin, 1988) to be valid as long as the

respectively. Note that, strictly speaking, the signal (20) contains the intensity

 *j*

2 2 2 *U t t t t t t dt* ( ) ( ) ( ) 2 ( ) ( ) cos[ ( ) ( )] (20)

( )*t* , respectively, are the slowly varying amplitude (envelope) and phase

are the central frequency and duration of the wave packet envelope,

2 2 *G t t dt* () ( ) (22)

*t cc* (21)

( )*t* contains at least a few optical cycles.

where

Here,

( )*t* and

inequality 0 0

of the optical-pulse field

0 and 0

autocorrelation function

Fig. 4. Maximum value of increment *HM*,max versus intensity.

The studied instability may be one of the mechanisms of conic beam generation (Nibbering et al., 1996; Dormidontov et al., 2010).

#### **4. Analysis of characteristics of ultrashort wave packet phase modulation by means of autocorrelation intensity function**

It has been demonstrated above that the originally spectrum-limited powerful short wave packet is phase modulated during propagation in a nonlinear medium. In this Section we will show that some characteristics of the acquired phase modulation may be measured by analyzing the intensity autocorrelator signal usually used for measuring duration of ultrashort laser pulses. In addition, it is possible to qualitatively assess by the shape of this signal the presence of phase modulation in the studied light signal. Basically, solution of this problem allows retrieving amplitude and phase distribution of electric field in an ultrashort wave packet.

Methods for retrieving the total field of an optical pulse may conventionally be divided into three groups. The first group includes techniques based on interference measurements. They have a long history and, consequently, have been developed most comprehensively. It is clear that, based on the interference of an ultrashort pulse in the temporal or spectral domain, one can in principle derive information on the phase distribution of the studied field. A great number of publications, from which we cite only the key papers (Kuznetsova, 1968; Verevkin et al., 1971; Sala et al., 1980; Diels et al., 1985;

Fig. 4. Maximum value of increment *HM*,max versus intensity.

**means of autocorrelation intensity function**

et al., 1996; Dormidontov et al., 2010).

0.4

0.8

1.2

5 *HM*,max 10

wave packet.

The studied instability may be one of the mechanisms of conic beam generation (Nibbering

 2 *TW I cm*

*M* 4

*M* 1

*M* 3

*M* 2

0.0 4.0 8.0 12.0 16.0 20.0

**4. Analysis of characteristics of ultrashort wave packet phase modulation by**

It has been demonstrated above that the originally spectrum-limited powerful short wave packet is phase modulated during propagation in a nonlinear medium. In this Section we will show that some characteristics of the acquired phase modulation may be measured by analyzing the intensity autocorrelator signal usually used for measuring duration of ultrashort laser pulses. In addition, it is possible to qualitatively assess by the shape of this signal the presence of phase modulation in the studied light signal. Basically, solution of this problem allows retrieving amplitude and phase distribution of electric field in an ultrashort

Methods for retrieving the total field of an optical pulse may conventionally be divided into three groups. The first group includes techniques based on interference measurements. They have a long history and, consequently, have been developed most comprehensively. It is clear that, based on the interference of an ultrashort pulse in the temporal or spectral domain, one can in principle derive information on the phase distribution of the studied field. A great number of publications, from which we cite only the key papers (Kuznetsova, 1968; Verevkin et al., 1971; Sala et al., 1980; Diels et al., 1985; Naganuma et al., 1989; Iaconis & Walmsley, 1998), are devoted to interference measurements. Another group of fairly recent publications (Naganuma et al., 1989; Iaconis & Walmsley, 1998; Delong et al., 1996; Koumans & Yariv, 1999; Kane, 1999) take advantage of the mathematically proven fact that a multi-dimensional (noninterference) auto- or cross-correlation function allows retrieving the phase distribution of a short pulse. A two-dimensional signal distribution on the frequency-delay plane is usually measured in experiment, and then an iterative processing algorithm yields the desired parameters. The most advanced to-date technique of this group is referred to as FROG in the literature (Kane, 1999). The third group of methods (Naganuma et al., 1989; Kane, 1999; Peatross et al., 1998; Nicholson et al., 1999), that are based on the results of simultaneous spectral and correlation energy measurements and some iterative algorithm, also make it possible to determine phase and amplitude characteristics of an optical pulse.

The interference methods of retrieving ultrashort wave packet field parameters are simpler for implementation and in terms of processing than the methods of the second group. However, despite the fact that the speed of obtaining qualitative information for the latter techniques often does not meet the needs of experiment, such techniques are clearer, since the form of phase modulation of an optical pulse can qualitatively be deduced immediately from a two-dimensional distribution of the obtained signal. In this respect, methods of the third group are much faster, as the corresponding iterative retrieval procedure is based on operations with one-dimensional data files. However, the problem of accuracy of retrieval of the desired parameters remains open for such methods. In this part we demonstrate that in certain cases, the interference methods can also ensure clear, simple, and fast acquisition of information on the amplitude and phase of an optical pulse. In our opinion, the present part is methodical to a significant extent, although it has practical applications related to measuring dispersion characteristics of optical materials. It can easily be shown that the output signal from the photodetector of an interferometric intensity autocorrelator, whose scheme can be found in a number of publications (e.g., Krukov, 2008), has the form:

$$\mathcal{U}(\tau) \approx \int\_{-\eta}^{\eta} \left[ \rho^2(t) + \rho^2(t+\tau) + 2\rho(t) \ast \rho(t+\tau) \ast \cos[\phi(t) - \phi(t+\tau)] \right]^2 dt \tag{20}$$

where ( )*t* and ( )*t* , respectively, are the slowly varying amplitude (envelope) and phase of the optical-pulse field

$$E(t) = \rho(t) \* \exp[j\rho(t)] + c.c.\tag{21}$$

Equation (20) is known (Akhmanov, Vysloukh & Chirkin, 1988) to be valid as long as the inequality 0 0 1 is satisfied, i.e., if the envelope ( )*t* contains at least a few optical cycles. Here, 0 and 0 are the central frequency and duration of the wave packet envelope, respectively. Note that, strictly speaking, the signal (20) contains the intensity autocorrelation function

$$\mathcal{G}(\tau) = \bigcap\_{\tau=0}^{\infty} \rho^2(t) \* \rho^2(t+\tau)dt \tag{22}$$

Broadband Instability of Electromagnetic Waves in Nonlinear Media 281

autocorrelation function (22) even for a transform limited optical signal and contains several

( )*t* and is determined by the function 2 2

side of Eq. (24), and smooth wings with shape determined by the function *G*( )

operation of a detecting system, then the signal *U G* ( )~1 2 ( )

 1 0 ~ 2

 0 3 2 1/2 2 2 (2 ) *L* ;

are determined by the third and fourth terms on the right-hand side of Eq. (24). Finally, the

the larger the value of *L* , i.e., the greater deviations of an optical pulse from a transform limited one, the more prominent the localization of the oscillatory component of such a