**2. Two-wave mixing in photorefractive crystal: Vectorial model**

The photorefractive (PR) effect known as light induced change of material refractive index was discovered in Bell Laboratory in 1966 (Ashkin, et al., 1966). In order to be photorefractive a material should have photoconductive and electro-optic properties. Then, in such a material, photo-induced charges migrate in the presence of light from regions of high optical intensity to regions of low optical intensity and eventually reach a static charge distribution. This charge distribution creates a strong, static electric field (so called spacecharge field), which in turn alters the index of refraction of the material by the first-order electro-optic (Pockels) effect. Since its discovery in LiNbO3 and LiTaO3 the PR effect was observed in a great number of materials, including dielectrics, semiconductors, liquid crystals, organic polymers and even in liquids and gases.

All approaches for multiple hologram recording can be grouped into three classes: spatial, angular and spectral multiplexing techniques. In the first approach, holograms are recorded in different parts of the same crystal. This technique allows one to provide maximal independence of channel performance; the cross-talk is practically precluded. However, the number of multiplexed holograms (and therefore number of channels) is limited by the crystal size, and in most cases does not exceed a few tens (Kulchin, et al., 2000b). Moreover, an equivalent number of individual reference beams is required for recording of each hologram in this case, which also leads to complexity of the measurement system and brings

Angular multiplexing of holograms allows significant increase in the number of channels formed in a single crystal. In this case the same volume of crystal can be used for recording several holograms, signal light beams can partially or even completely overlap in a crystal and a common reference beam can be used for all holograms. However cross-talk between channels becomes more probable in such a scheme. This defines one of the important problems which should be solved. It is necessary to find conditions which preclude a crosstalk between angular multiplexed holograms or the cross-talk level will be below the

The spectral multiplexing approach is based on using different wavelengths for dynamic hologram recording. This approach naturally fits with the WDM-technique of fiber-optical sensors (especially FBG) multiplexing. As a result effective multichannel measurement systems can be created by combined these two principles. However a realization of the spectral multiplexing approach requires keeping a number of peculiarities. In particular, the spectral working range should match the spectral sensitivity of a photosensitive material (e.g. PRC). Moreover, it is necessary to take into account the possible appearance of crosstalk due to overlap of multiplexed channel spectra. The last requirement together with the first one constrains the number of channels which could be realized in a multichannel

Below in the following Sections 4-6 we consider more detailed practical realizations of multichannel adaptive interferometers and measurement systems based on the above listed principles of dynamic hologram multiplexing in a photorefractive crystal. First we give briefly the basics of wave mixing in photorefractive crystal (Section 2) and introduce a relative detection limit as a parameter which characterizes adaptive interferometer

The photorefractive (PR) effect known as light induced change of material refractive index was discovered in Bell Laboratory in 1966 (Ashkin, et al., 1966). In order to be photorefractive a material should have photoconductive and electro-optic properties. Then, in such a material, photo-induced charges migrate in the presence of light from regions of high optical intensity to regions of low optical intensity and eventually reach a static charge distribution. This charge distribution creates a strong, static electric field (so called spacecharge field), which in turn alters the index of refraction of the material by the first-order electro-optic (Pockels) effect. Since its discovery in LiNbO3 and LiTaO3 the PR effect was observed in a great number of materials, including dielectrics, semiconductors, liquid

**2. Two-wave mixing in photorefractive crystal: Vectorial model** 

crystals, organic polymers and even in liquids and gases.

its efficiency down.

inherent noise of the system.

system based on spectral multiplexing.

sensitivity (Section 3).

The first theoretical model of the PR effect was proposed by Kukhtarev etal. in 1979 (Kukhtarev, et al., 1979). Later in 1999, Sturman et.al. developed more general and more rigorous theoretical model of two-wave mixing in PR crystal which takes into account a vectorial nature of light waves, anisotropic properties of photorefractive materials and anisotropy of light diffraction at a dynamic holographic grating (Sturman, et al., 1999).

After decades of studying the PR effect a number of fundamental monographs devoted to wave-mixing in PR materials were written (Petrov, et al., 1983; Petrov, et al., 1991; Solymar, et al., 1996; Sturman & Fridkin, 1992). These books can be recommended to the reader interested in detailed study of PR effect. In this Chapter we give just the main points of the theoretical model of vectorial wave mixing in PR crystal of cubic symmetry (Sturman, et al., 1999) which can help better understanding of further material.

Consider two coherent waves with vectorial amplitudes **A1** and **A2** entering a PR crystal of cubic symmetry under an external electric field **E0** through its (*xy*)-face (Fig.1). In paraxial approximation valid for small angles between wave vectors **k1** and **k2** and axis *z* the wave amplitudes can be considered as 2D-vectors with *x*- and *y*-components.

Fig. 1. Geometry of vectorial wave mixing in a photorefractive crystal

The waves' interference will result in the appearance of space charge field **EK** which alters the crystal refractive index and forms a dynamic holographic grating with wave vector **K** = **k1** – **k2** in plane (*xy*). Diffraction of mixed waves at the dynamic grating provides their coupling which results in change of wave amplitudes at the crystal output. This process is described by the system of coupled-wave equations:

$$\begin{cases} \left(\frac{\partial}{\partial z} + \frac{\alpha}{2} - i\hat{\mathbf{G}}\right) \mathbf{A}\_1 = iE\_K \hat{\mathbf{V}} \mathbf{A}\_{2'}\\ \left(g\frac{\partial}{\partial z} + \frac{\alpha}{2} - i\hat{\mathbf{G}}\right) \mathbf{A}\_2 = iE\_K^\* \hat{\mathbf{V}} \mathbf{A}\_{1'} \end{cases} \tag{1}$$

where *g* is the parameter which takes into account relative direction of wave propagation in the crystal (*g* = +1 in transmission geometry where waves propagate in same direction, and g = –1 in reflection geometry where waves propagate in opposite directions); is the light absorption coefficient; *EK* is the amplitude of space-charge field **EK** . Using the conventional one-trap–one-band model for the charge transfer (Petrov, et al., 1991) and assuming

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 109

determined by the orientation of holographic grating vector, **K,** and the directions of wave propagating (vectors **k1** and **k2**) with respect to crystallographic axes of the crystal

It is worth noting that the particular type of the matrix **H**ˆ is closely related to the type of light diffraction at a dynamic hologram. Thus, in the case of pure isotropic diffraction which preserves the polarization of the diffracted light, the coupling matrix is proportional to the unit matrix, ˆ ˆ **H 1** *const* . In all other cases the diffraction is anisotropic and is accompanied by change of diffracted light polarization state. Particularly if the polarization plane of diffracted wave is rotated by 90 degrees with respect to the incident light the matrix is

**H** . The last case is considered as most efficient for providing a linear regime of

phase demodulation in the adaptive interferometer based on a dynamic hologram recorded in PR crystal in diffusion mode, i.e. without application of external electric field (Kamshilin,

Since in any interferometer a physical parameter, which has to be measured, is finally encoded in the phase modulation of the light wave, its sensitivity to small phase excursions is a parameter serving as a primary criterion for comparison of different systems. It is not only the configuration of the optical scheme but also its particular realisation (the output laser power, the generated wavelength, the type of the crystal, etc.) which determines the sensitivity of an adaptive interferometer. Comparison of different adaptive interferometers is usually done by estimation the extent to which their sensitivity is worse than that of the classical lossless interferometer (Delaye, et al., 1997; de Montmorillon, et al., 1997). It was shown in early papers devoted to analysis of a classical interferometer that its sensitivity can be extremely high if the available light power is not limited and measurements of the phase modulation is carried out within a very narrow frequency band (Bershtein, 1954; Forward, 1978). Evidently, the minimum detectable

several sources of the noise in an optical interferometer: laser noise, thermal and shot noise of the photodetector, and noise of amplifying electronics. When the light power arriving at the photo-detector is high enough, shot noise (which is proportional to the square root of the received light power) of the photo-excited charge carriers prevails over all the other noise levels (Bershtein, 1954; Wagner & Spicer, 1987). The shot noise is the instability of the photodetector current caused by statistical fluctuation of the number of received photons. Its level is primarily defined by the average number of photons and it is

> 2 <sup>2</sup> 2 *shot <sup>D</sup> <sup>e</sup> i P <sup>f</sup> <sup>h</sup>*

where *e* is the electron's charge, *η* is the quantum efficiency of the photo-detector, *hν* is the

energy of the photon, and Δ*f* is the frequency bandwidth of the detection electronics.

is defined by the noise level of the measuring system. There are

, (7)

(Romashko, et al., 2010; Sturman, et al., 1999).

**3. Adaptive ineterferometer sensitivity** 

0 1 <sup>ˆ</sup> 1 0 

et al., 2009).

phase difference *min*

given as:

provisionally that the applied field is parallel to the grating vector the space charge-field can be found, within the linear approximation in the contrast of the interference pattern, as

$$E\_K = im \frac{(E\_D + iE\_0)E\_q}{E\_D + E\_q + iE\_0} \,\text{\AA} \tag{2}$$

where \* 12 0 *m I* 2 / **A A** is the contrast of the interference pattern, *I*0 is the total light intensity, *E*0 is the amplitude of external electric field applied to the crystal, while *ED* and *Eq* are the characteristic fields,

$$E\_D = \frac{Kk\_BT}{e} \text{ (diffusion electric field)}, \ E\_q = \frac{eN\_t}{\varepsilon \varepsilon\_0 K} \text{ (trops saturation field)}.\tag{3}$$

where *e* is the elementary charge, *kB* is the Boltzmann constant, *T* is the absolute temperature, *Nt* is the effective concentration of photorefractive centers (traps), *εε*0 is the dielectric constant. The matrices **G**ˆ and **V**ˆ are given by

$$\mathbf{G}\_{\dot{\mathbf{q}}} = \mathbf{s} \mathbf{E}\_0 \mathbf{H}\_{\dot{\mathbf{q}}}^{(0)} + \mathbf{i} \rho \mathbf{s} \mathbf{S}\_{\dot{\mathbf{q}} \dot{\mathbf{z}} \prime} \qquad \qquad V\_{\dot{\mathbf{q}}} = \mathbf{s} \mathbf{H}\_{\dot{\mathbf{q}} \prime} \tag{4}$$

here <sup>3</sup> *s nr* 0 41 / is a material parameter; *n*0 is the refractive index; *r*41 is the electro-optic coefficient; is the wavelength; *ρ* is the rotatory power; *ijz* is the unit antisymmetric third rank tensor; *E*<sup>0</sup> **E0** ; *EK* **EK** . The Latin subscripts denote Cartesian components, and indices *i* and *j* assume independently the values *x* and *y*.

The tensors ˆ (0) **H** and **H**ˆ responsible for electro-optic contribution to variation of crystal dielectric permittivity caused by external electric field **E0** and internal space-charge field **EK**, are respectively:

$$H\_{ij}^{(0)} = \frac{r\_{ijl}E\_{0l}}{r\_{41}E\_0} \, \prime \qquad H\_{ij} = \frac{r\_{ijl}E\_{Kl}}{r\_{41}E\_K} \, \prime \tag{5}$$

where *rijl* is the electro-optic tensor. Taking into account that the space-charge field appears due to interference of mixed waves as well as a form of coupled-wave equations (1), one can conclude that tensor **H**ˆ has the character of the wave-coupling matrix. The last becomes more clear in the case of waves mixing in non-gyrotropic crystal ( 0 ) without application of external electric field ( <sup>0</sup> *E* 0 ) when the system (1) is reduced to the following

$$\begin{cases} \left(\frac{\partial}{\partial z} + \frac{\alpha}{2}\right) \mathbf{A}\_1 = smE\_K \hat{\mathbf{H}} \mathbf{A}\_{2'}\\ \left(g\frac{\partial}{\partial z} + \frac{\alpha}{2}\right) \mathbf{A}\_2 = -smE\_K \hat{\mathbf{H}} \mathbf{A}\_1. \end{cases} \tag{6}$$

As seen from Eqs.(6) in order for the waves to couple they must interfere ( 0 *m* ) with matrix **H**ˆ being non-zero.

By solving Eqs.(1), one can find amplitudes of mixed waves at the output of the PR crystal. Note that a particular solution depends on wave mixing geometry which, in its turn, is determined by the orientation of holographic grating vector, **K,** and the directions of wave propagating (vectors **k1** and **k2**) with respect to crystallographic axes of the crystal (Romashko, et al., 2010; Sturman, et al., 1999).

It is worth noting that the particular type of the matrix **H**ˆ is closely related to the type of light diffraction at a dynamic hologram. Thus, in the case of pure isotropic diffraction which preserves the polarization of the diffracted light, the coupling matrix is proportional to the unit matrix, ˆ ˆ **H 1** *const* . In all other cases the diffraction is anisotropic and is accompanied by change of diffracted light polarization state. Particularly if the polarization plane of diffracted wave is rotated by 90 degrees with respect to the incident light the matrix is

0 1 <sup>ˆ</sup> 1 0 **H** . The last case is considered as most efficient for providing a linear regime of

phase demodulation in the adaptive interferometer based on a dynamic hologram recorded in PR crystal in diffusion mode, i.e. without application of external electric field (Kamshilin, et al., 2009).
