**3. Two-wave mixing and optical gain amplification in photorefractive media**

to the wave vectors k, k ± **K**; k ± 2**K**; k ± m**K**, where k is the wave number of light beam in the medium and **K** is the grating wave vector. The diffraction efficiency for the Raman-Nath

*<sup>m</sup>*(*δ*) is the amplitude of the *m*th order diffracted beam, expressed by Bessel's function (*δ* is expression of modulation index corresponding to the multiple scattered orders), *n* is the

In a Bragg regime, after interacting with the grating, only one diffracted wave is produced, and the diffraction occurs only when the incident angle satisfies the Bragg conditions [1–4] (both

The simplified diffraction efficiency (in case of transmission type, sinusoidal phase grating) is

( \_ *πnL*

Generally, Klen-Cook dimensionless parameter has been accepted as distinguishing factor

Although this parameter has been extensively used as a criterion which regime to apply (*Q* ≤ 1 for Raman-Nath or *Q* > ~1 for the Bragg-matched regime), it requires several limitation of the

*<sup>λ</sup>cosθ*) (2)

*<sup>λ</sup>cosθ*) (3)

*<sup>n</sup>* Λ2 (4)

*m* <sup>2</sup>(*δ*) = *J m* 2 ( \_2*πLn*

refractive index of the medium and *L* is the interaction thickness [3, 4].

the energy and momentum are conserved)—see **Figure 1(b)**.

**Figure 1.** Raman-Nath (a) and (b) Bragg matched regime of diffraction [1].

between the Raman-Nath or Bragg regimes of operations, defined by [5]:

*η* = *si n*<sup>2</sup>

*Q* = \_\_\_\_\_ <sup>2</sup>*πλ<sup>L</sup>*

diffraction is given by [1]

482 Holographic Materials and Optical Systems

where *J*

given by [2]

grating strength [6].

*η<sup>m</sup>* = *J*

The two-wave mixing appears in variety of non-linear media, and owing to its wide range of applications, it has been extensively studied in a past [7–10]. In general, the two-wave mixing is described by two beam interactions inside the photosensitive material, forming a light interference pattern (index grating). As a result, the two beams diffracted by the index grating they created in a way that in one direction, the diffracted and transmitted intensities provide constructive interference (with higher resultant intensity), whereas in the other direction, the beams experience destructive interference (with lower resultant intensity). Thus, the most significant importance of the two-wave mixing is the energy exchange between the two interacting beams.

The light illumination in a photosensitive material causes generation of free charge carriers and their redistribution from the regions of high intensity to those of low intensity (see **Figure 2**). This net migration leads to inhomogeneous charge distribution and accumulation of an internal electric field known as a space charge field *Esc*. Actually, this space charge field is of key importance for the photorefractive effect and can play significant role for the LC molecules reorientation as discussed further.

Briefly, the space charge field is expressed by the following set of well-known equations [1, 7, 8]:

$$\frac{\partial N}{\partial t} - \frac{\partial N\_{\text{D}}^{i}}{\partial t} = \frac{1}{\varepsilon} \left| \nabla \cdot J \right| \tag{5}$$

$$\frac{\partial N\_D^i}{\partial t} = \left(N\_D - N\_D^i\right) \text{sI} - \mathcal{V}\_R \, N N\_D^i \tag{6}$$

$$\mathbf{J} = J\_{\text{dry}} + J\_{\text{diffusion}} = \text{ eN}\mu \, E\_{\text{sc}} + \text{k\_\text{p}} \, T\mu \, \nabla \cdot \text{N} \tag{7}$$

$$\nabla \cdot \mathcal{E} \, E\_{\varkappa} = \, \rho(r) = -q \left( \mathcal{N} + \mathcal{N}\_{\Lambda} - \mathcal{N}\_{\mathcal{D}}^{\vee} \right) \tag{8}$$

where *e* is the electron charge, *N* is density of main charge carriers, *ND* is total donor density, *ND i* is ionized donor density, *NA* is density of acceptors, *s* is absorption cross section of excitation, *γR* is ionized trap recombination rate, *μ* is mobility, *J* is current density, *ρ* is charge density, *ε* is dielectric constant, *kB* is Boltzmann constant and *T* is the temperature. In Eqs. (5)–(8), Eq. (5) is the rate equation of the main carriers density; Eq. (6) is the rate equation of ionized donors (the first term is the rate of main carriers generation and the second term is the rate of the trap capture); Eq. (7) is the current density equation (if neglect the photovoltaic effect); and Eq. (8) is the Poisson equation.

The first term in Eq. (7) is expressed by the drift of the charge carriers due to the space charge field *Esc* and the second term is a diffusion, due to the gradient of the charge carrier density, expressed by the diffusion length *LD* <sup>=</sup> ( *E* \_\_\_*D <sup>K</sup> μ τD*) 1/2 ; where *τ<sup>D</sup>* <sup>=</sup> *<sup>e</sup>* <sup>Λ</sup><sup>2</sup> \_\_\_\_\_\_\_ 4 *π*<sup>2</sup> *μ k <sup>B</sup> <sup>T</sup>* is the diffusion time and

**Figure 2.** The photorefractive effect: (a) two-beams interference; (b) photo excitation process (intensity pattern); (c) charge transport; (d) space-charge distribution; (e) space-charge field and (f) index grating formation.

*ED* <sup>=</sup> \_\_\_ *KD <sup>μ</sup>* <sup>=</sup> *<sup>K</sup> <sup>k</sup>* \_\_\_\_*<sup>B</sup> <sup>E</sup> <sup>T</sup>* is the diffusion field. The magnitude of the *Esc* depends on several materials parameters, among which the Debye screening length *LD* <sup>=</sup> ( *<sup>ε</sup> <sup>ε</sup>* \_\_\_0 *Neff k <sup>B</sup> <sup>T</sup>* \_\_\_ *<sup>e</sup>* <sup>2</sup> ) \_\_1 2 and effective trap density <sup>N</sup>*eff* <sup>=</sup> (*ND* <sup>−</sup> *ND i* )*ND i* \_\_\_\_\_\_\_\_\_\_ *ND* .

In liquid crystals, the drift is the dominant mechanism for the charge carrier migration due to the small trap density of organics. In contrast, the diffusion is the dominant mechanism for inorganic materials and the rate at which the recombination happens determine how far the main charge carrier diffuse and how strong is the refractive index modulation. For instance, inorganic crystals offer several orders of magnitude higher concentration of effective trap density in contrast to the LCs and therefore are able to support formation of the small grating spacing and Bragg match regime of diffraction as will be discussed later.

As a result of charge migration and redistribution, the space charge field in combination with the electro-optic effect modulates the refractive index of the media via the Pockel's effect [1, 8, 9]:

$$
\Delta \left( \frac{1}{n^2} \right) = r\_{\parallel k} E\_{\kappa} \tag{9}
$$

where *rijk* is the electro-optic coefficient.

**Figure 2.** The photorefractive effect: (a) two-beams interference; (b) photo excitation process (intensity pattern); (c)

charge transport; (d) space-charge distribution; (e) space-charge field and (f) index grating formation.

484 Holographic Materials and Optical Systems

The recorded refractive index grating can diffract light, with the diffraction pattern reconstructing the light-intensity pattern, originally stored in the media. Therefore, the index grating created in the photorefractive material is a "volume phase hologram", which can be written and erased by light, making photorefractive materials fully reversible. Thus, the photorefractive materials have the ability to detect and store spatial distributions of optical intensity in the form of spatial patterns of modulated refractive index.

The most significant consequence from the photorefractive effect is the phase shift between the light-intensity pattern and internal spatial pattern where the later one shifted in respect to the intensity distribution by π/2 period (see **Figure 2**). This π/2 phase shift induces an optical energy exchange between the two interacting beams (beam amplification) and refers the photorefractive effect as non-local, non-linear effect [2, 7, 8]. Therefore, when two interfering beams have different intensities noted as respectively *I* s "signal" beam (with lower intensity) and *I*p "pump" beam (with higher intensity), due to destructive and constructive interference, the unidirectional transfer of optical energy allows a weak beam to grow exponentially with the distance. As a result, at the exit of the medium, the signal beam is not only amplified but also experienced a non-linear phase shift.

The interaction between the two coherent laser beams inside the photosensitive material (assuming the grating wave vector **K** directed along the x axis) gives the total electric field of the two incident beams

$$\mathbf{E} = A\_1 \exp\left[i(k\_1 \cdot r - \omega\_1 t)\right] + A\_2 \exp\left[i(k\_2 \cdot r - \omega\_2 t)\right] \tag{10}$$

(where *A*1,2 are the beam amplitudes; *κ*<sup>1</sup> and *κ*<sup>2</sup> are the wave vectors **K** = |*κ*<sup>2</sup> − *κ*<sup>1</sup> |; *r* and *t* are spatial and temporal coordinates; *ω*<sup>1</sup> and *ω*2 are the angular frequency (∆*ω* = *ω*<sup>2</sup> − *ω*<sup>1</sup> ) and *δ* is the frequency detuning between the two beams) and generates a light-intensity fringe pattern of bright and dark regions (sinusoidal light-intensity pattern) described as [1, 7, 8]:

$$\mathbf{I} = \left| A\_{\mathbf{i}} \right|^2 + \left| A\_{\mathbf{i}} \right|^2 + \left\{ A\_{\mathbf{i}} A\_{\mathbf{i}}^\* \exp[i(-\mathbf{K}\mathbf{x} + \delta\mathbf{t})] + \mathbf{c} \text{ .c.} \right\} \tag{11}$$

In fact, Eq. (11) represents the spatial variation of the intensity pattern inside the photosensitive media that generates and redistributes the charge carriers and accumulates the space charge field.

As a result, the refractive index grating, summarized by Refs. [8, 9], can be written by:

$$\Delta \mathbf{n} = n\_0 + \left\{ \Delta n\_0 A\_1 A\_2^\* \exp[i(-\mathbf{K}\mathbf{x} + \delta t + \mathbf{\mathcal{O}})] + \mathbf{c} \cdot \mathbf{c} \right\} \tag{12}$$

where Φ is the phase difference between the refractive index grating and interference pattern, *n*0 is the refractive index without the light and the <sup>Δ</sup> *<sup>n</sup>*<sup>0</sup> is amplitude of index modulation.

By using the couple-mode theory [1, 11], the quantitative measure of the beam-coupling is expressed by the gain coefficient Γ [1, 8, 9]

$$
\Gamma = \frac{2\pi\Lambda}{\lambda\cos\theta}\sin\phi\tag{13}
$$

Experimentally, the gain coefficient can be measured by the ratio

$$
\Gamma = \frac{1}{L} \ln \left( \frac{I\_s^{\top} I\_p}{I\_s I\_p^{\top}} \right) \tag{14}
$$

where *I*′ *p*(*s*) is the transmitted intensity of the pump (signal) beam with a coupling, and *I p*(*s*) is the transmitted beam intensity without coupling.

The general parameter characterizing the gain is the gain amplification *G*, given by:

$$G = \frac{1}{L} \log\_e |\Gamma|\tag{15}$$

where *L* is the interaction length of the media.

In last decades, the two-beam coupling effect has been widely investigated in varieties of organic and inorganic compounds [1, 7–13]. In terms of organics, the electro-optic response and the build-up of a refractive index grating in liquid crystals arise from the reorientation of LCs molecules due to an induced space charge field. This effect is known as "orientational photorefractive effect" or "photorefractive-like effect" [12]. For most applications, an external electric field needs to be applied along the grating vector direction, since the drift is the dominant mechanism for the charge migration in LC systems. LCs or PLDCs provide very high amplification gain (up to 2600 cm−1 [14]); however, the large grating spacing and small trap density, typical for organics restricted the two-beam coupling to the Raman-Nath regime of diffraction. Therefore, due to the multiple orders of diffracted beams, which accumulate the optical losses, the energy lost limits many of the practical uses.

(where *A*1,2 are the beam amplitudes; *κ*<sup>1</sup>

Δn = *n*<sup>0</sup> + {Δ *n*<sup>0</sup> *A*<sup>1</sup> *A*<sup>2</sup>

expressed by the gain coefficient Γ [1, 8, 9]

<sup>Г</sup> <sup>=</sup> <sup>2</sup>*π*<sup>Δ</sup> *<sup>n</sup>* \_\_\_\_\_0

*Г* = \_\_<sup>1</sup>

the transmitted beam intensity without coupling.

*G* = \_\_<sup>1</sup>

where *L* is the interaction length of the media.

is the refractive index without the light and the <sup>Δ</sup> *<sup>n</sup>*<sup>0</sup>

Experimentally, the gain coefficient can be measured by the ratio

spatial and temporal coordinates; *ω*<sup>1</sup>

I = |*A*1|

486 Holographic Materials and Optical Systems

charge field.

*n*0

where *I*′

*p*(*s*)

and *κ*<sup>2</sup>

+ {*A*<sup>1</sup> *A*<sup>2</sup>

As a result, the refractive index grating, summarized by Refs. [8, 9], can be written by:

of bright and dark regions (sinusoidal light-intensity pattern) described as [1, 7, 8]:

2 + |*A*2| 2

the frequency detuning between the two beams) and generates a light-intensity fringe pattern

In fact, Eq. (11) represents the spatial variation of the intensity pattern inside the photosensitive media that generates and redistributes the charge carriers and accumulates the space

where Φ is the phase difference between the refractive index grating and interference pattern,

By using the couple-mode theory [1, 11], the quantitative measure of the beam-coupling is

*<sup>L</sup>* ln( *I s* ′ *I* \_*p I s I p* ′

The general parameter characterizing the gain is the gain amplification *G*, given by:

is the transmitted intensity of the pump (signal) beam with a coupling, and *I*

*<sup>L</sup> lo ge*

In last decades, the two-beam coupling effect has been widely investigated in varieties of organic and inorganic compounds [1, 7–13]. In terms of organics, the electro-optic response and the build-up of a refractive index grating in liquid crystals arise from the reorientation of LCs molecules due to an induced space charge field. This effect is known as "orientational photorefractive effect" or "photorefractive-like effect" [12]. For most applications, an external electric field needs to be applied along the grating vector direction, since the drift is the dominant mechanism for the charge migration in LC systems. LCs or PLDCs provide very

are the wave vectors **K** = |*κ*<sup>2</sup> − *κ*<sup>1</sup>

\* *exp*[*i*(−*Kx* + *δt*)] + *c* . *c*.} (11)

\* *exp*[*i*(−*Kx* + *δt* + Φ)] + *c* . *c*.} (12)

is amplitude of index modulation.

*<sup>λ</sup>cos<sup>θ</sup>* sin*φ* (13)

) (14)

⌊Г⌋ (15)

and *ω*2 are the angular frequency (∆*ω* = *ω*<sup>2</sup> − *ω*<sup>1</sup>


) and *δ* is

*p*(*s*) is In terms of inorganic crystals, the direction of the optical energy transfer depends on the sign of the electro-optic coefficient and the sign of the main charge carriers. Up to now, the highest gain coefficient (over 100 cm−1) has been reported in Fe-doped LiNbO3 crystal due to its large refractive index modulation Δ*n* ~ 2 × 10−3 [15]. Relatively high beam amplification has been achieved in SBN and BaTiO3 inorganic crystals [7, 10]. In Bi12(M = Si,Ti)O20 sillenite crystals, the gain coupling is much lower due to the smaller values of the electro-optical coefficient, restricted by the cubic symmetry [16]. However, doped sillenite crystals offer the potential for high-carrier mobility (high photoconductivity) and together with the high-trap density can compensate the small-trap density of LCs when combined into a hybrid structure to support the fine grating spacing and to fulfil the requirements for Bragg-matched regime of diffraction. Furthermore, by selecting the photorefractive substrate sensitivity, the operation interval of the proposed hybrid devices can be easily adjusted. Moreover, doping sillenites with transition metal elements significantly improve their sensitivity and response time at nearinfrared spectral range [17–22].
