**4. Organic-inorganic hybrid structures operating at Raman-Nath regime of diffraction**

#### **4.1. Organic-inorganic structure design and principle of operation**

This type of organic-inorganic hybrid structure consists of photoconductor substrate (usually an inorganic crystal plate with 0.4–0.6 mm thickness) and electro-optic layer (few microns thickness) arranged into a cell, supported by a glass substrate from another side (see the schematic diagram at **Figure 3(a)**). Selected inorganic crystals stand as photoconductors, whereas a LC layer is used

**Figure 3.** (a) Experimental set-up for voltage-transmittance measurements. The hybrid structure is placed between two crossed polarizes making an angle of 45° with respect to the transmission axis of the polarizer (b) Example of Voltage-Transmittance dependence for BSO:Ru/LC structure with graphene-based electrodes.

as electro-optic material. The proposed configuration is also known as optically addressed spatial light modulator (OASLM) structure. As the read-out process can provide an optical gain, the OASLMs have been also considered as a liquid crystal light valve (LCLV) devices.

Bi12SiO20 (BSO) and Bi12TiO20 (BTO) crystals are among the perfect components for OASLM devices due to their remarkable photoconductivity and high charge carrier mobility. For a first time, non-doped BSO crystal was assembled with a LC layer into a LCLV, working at transmittance mode by Aubourg et al. [23]. Later on, several devices operating at visible spectral range have been demonstrated [24–26]. Other preferable photoconductor substrates are semiconductors as a-Si:H, crystalline silicon, gallium arsenide, indium phosphate and cadmium tellurite [27, 28]. Usually, transparent indium tin oxide (ITO) conductive layers are preliminary deposited on the outer side of the photoconductive plate and the inner side of glass substrate and coated with polyvinyl alcohol (PVA) for planar alignment of the LC molecules. Recently, owing to the ITO limited transparency at near infrared spectral range and increased cost because of the Indium scarcity in the nature, ITO has been successfully replaced with graphene conductive layers, and several graphene-based devices have been successfully demonstrated [29, 30].

The operation principle of OASLM device relays on the electro-optically controlled birefringence of the LCs molecules where the high sensitivity and photoconductivity comes from the photoconductive plate and high birefringence is provided by the LC layer [2, 31]:


Owing to the LC's anisotropy, the output beam obtains a phase shift

$$
\omega \rho = \frac{2\pi}{\lambda} (\boldsymbol{\pi}\_{\boldsymbol{e}} - \boldsymbol{\pi}\_{\boldsymbol{o}}) \boldsymbol{L} \tag{16}
$$

which is a function of the applied voltage *V*<sup>0</sup> and the pump light intensity *I*:

$$\forall \qquad \lambda \quad \lambda \quad \text{(} \mathbf{v}\_{\epsilon} \text{ --} \mathbf{v}\_0\text{)}\text{.} \tag{12}$$

$$\text{which is a function of the applied voltage } V\_0 \text{ and the pump light intensity } \mathbf{h}.$$

$$I = \frac{1}{2} \sin n^2 \frac{\Phi}{2} = \frac{1}{2} \sin n^2 \left[\frac{\pi (n\_r - n\_o)L}{\lambda}\right] \tag{13}$$

where *L* is the LC thickness, *ne* and *n*<sup>0</sup> are refractive indexes for a beam polarized along the long or short molecular axis ∆*n* = *ne* (*Ip* ) − *n*<sup>0</sup> and Φ is the phase retardation [2]. This allows spatial modulation of the amplitude or phase of the incident beam at the exit of device (see **Figure 4(a)**). During the full range, the reorientation angle of the LC molecules can vary from 0 to π/2, producing a phase shift of several π.

as electro-optic material. The proposed configuration is also known as optically addressed spatial light modulator (OASLM) structure. As the read-out process can provide an optical gain, the

Bi12SiO20 (BSO) and Bi12TiO20 (BTO) crystals are among the perfect components for OASLM devices due to their remarkable photoconductivity and high charge carrier mobility. For a first time, non-doped BSO crystal was assembled with a LC layer into a LCLV, working at transmittance mode by Aubourg et al. [23]. Later on, several devices operating at visible spectral range have been demonstrated [24–26]. Other preferable photoconductor substrates are semiconductors as a-Si:H, crystalline silicon, gallium arsenide, indium phosphate and cadmium tellurite [27, 28]. Usually, transparent indium tin oxide (ITO) conductive layers are preliminary deposited on the outer side of the photoconductive plate and the inner side of glass substrate and coated with polyvinyl alcohol (PVA) for planar alignment of the LC molecules. Recently, owing to the ITO limited transparency at near infrared spectral range and increased cost because of the Indium scarcity in the nature, ITO has been successfully replaced with graphene conductive layers, and several graphene-based devices have been successfully dem-

The operation principle of OASLM device relays on the electro-optically controlled birefringence of the LCs molecules where the high sensitivity and photoconductivity comes from the

voltage acts through the photoconductor substrate because of its high resistance. Since the LCs molecules have different polarizability along their long and short axis, the applied voltage induces a dipole moment in the LC layer, which affects the LC molecules orientation, and they follow the direction of the applied electric field. As the LC nematic phase is characterized by a long-range orientation order, all the LC molecules tend to

the crossed polarizers, "on" and "off" illumination states are obtained (**Figure 3**).

Owing to the LC's anisotropy, the output beam obtains a phase shift

ϕ = \_\_\_ <sup>2</sup>*<sup>π</sup>*

<sup>2</sup> *si <sup>n</sup>*<sup>2</sup> \_\_ Φ <sup>2</sup> <sup>=</sup> \_\_<sup>1</sup> <sup>2</sup> *si n*<sup>2</sup> [

(ii) Illumination with the input pump beam activates the photoconductor substrate, and charge carriers are generated at a rate proportional to the pump intensity. Owing to the crystal's high photoconductivity and high dark resistivity, the charge separation decreases the voltage across the photoconductor, and it reduces its impedance. As a consequence, the accumulated voltage is transferred into the LC layer, resulting in a LC molecular reorientation.

) is applied across the device (**Figure 3(a)**), the applied

lc. Therefore, when the structure is placed between

*<sup>λ</sup>* (*ne* − *n*0)*L* (16)

*<sup>λ</sup>* ] (17)

and the pump light intensity *I*:

*π*(*ne* − *n* \_0)*L*

photoconductive plate and high birefringence is provided by the LC layer [2, 31]:

OASLMs have been also considered as a liquid crystal light valve (LCLV) devices.

onstrated [29, 30].

488 Holographic Materials and Optical Systems

(i) when an external AC voltage (V<sup>0</sup>

align along the nematic LC director n̂

which is a function of the applied voltage *V*<sup>0</sup>

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

**Figure 3** illustrates the experimental set-up and the typical Freedericksz transition characteristics of BSO:Ru/LC device with graphene electrodes, operating at 1064 nm. The modulation behaviour supports the LC molecules alignment in the direction of an applied AC voltage across to the cell. **Figure 4** shows the phase modulation set up and phase difference for the same BSO:Ru/LC device at fixed voltage of 4 V (according to the results at **Figure 3(b)**). The probe-pump intensity dependence is presented as inset at **Figure 4(b)**.

**Figure 4.** (a) Experimental set-up for phase shift measurements and (b) Phase difference as a function of a pump light intensity (fixed at 4V according to the results from figure 3(b)) for BSO:Ru/LC device. Inset graph shows the probe-pump intensity dependences.

Obviously, the light-induced modulation formed in the photoconductive substrate is the driving force, which affects the LC molecules realignment and spatially modulates the refractive index of LC layer.

#### **4.2. Two-wave mixing and Raman-Nath regime of diffraction**

For a first time, energy transfer using two-wave mixing has been demonstrated by Brignon et al. [32] in a device assembled by BSO photoconductive crystal and LC layer. The recorded holographic gratings were not typical local dynamic grating as generally supposed to be created from the photoconductive substrate. The two-beam interaction has been described as a diffraction of the two beams on a fixed thin local grating (the index grating formed in LC), which is pinned to the conductivity grating [32]. Since the photoconductive and the electrooptic regions are separate, the photo-induced grating is not modified by the interacting beams during their propagation in the LC layer. Later on, beam amplification in LCLV devices has been verified and reported in several papers [19, 21, 23–26, 33]. Recently, significant gain amplification values have been obtained in hybrid structures based on semiconductor crystals as GaAs doped with Cr [33, 34] or CdTe [35].

The interference between the pump and signal beam, expressed by their amplitudes *Ep* exp  [ *i*(*κ<sup>p</sup>* ⋅ *r* − *ω<sup>p</sup>* ⋅ *t* ) ] and *Es* exp  [ *<sup>i</sup>*(*κ<sup>s</sup>* <sup>⋅</sup> *<sup>r</sup>* <sup>−</sup> *<sup>ω</sup><sup>s</sup>* <sup>⋅</sup> *<sup>t</sup>* ) ], produced an intensity fringe pattern that induces a space-charge distribution and consequently a molecular re-orientation pattern in the LC layer. The interaction creates a refractive index grating with the same wave vector K, and as a result, the two beams diffracted by the photo-induced grating with spatial grating period of Λ = 2π/K.

In these hybrid structures, the active layer is the LC layer; therefore, the interaction length is sufficiently thin to satisfy the energy and momentum conservation before exiting the medium [1, 2]. As a result, the two-beam coupling occurs in a Raman-Nath regime of diffraction (the LC layer is much thinner in comparison with the grating spacing *L* << Λ). Consequently, the phase grating recorded in the LC layer acts as a thin hologram with a multiple output beams.

A detailed analysis of the two-wave mixing and expression of the zero and m-output diffracted orders is given in Ref. [24] where the evolution of the amplitude of the refractive index grating and relaxation dynamic of the LC molecules orientation has been considered, using the couple-wave theory [1, 11]. We note that the output signal beam according to Refs. [24, 36] can be written in a form:

$$E\_{output} = \sqrt{G\_s I\_s} \exp\{i\phi\} \exp\left[i\left(k\_s \cdot r - \omega\_s t\right)\right] + c \text{ .c.}\tag{18}$$

where *G* is the gain amplification and *φ* is the non-linear phase shift.

**Figure 5** demonstrates the two-wave mixing in BTO:Rh/LC hybrid structure using 1064-nm diode laser. The right side of **Figure 5** supports the Raman-Nath diffraction with several diffracted beams (0, ±1, ±2,…) at the output of the hybrid device, detected on the infrared view card.

**Figure 5.** Two-wave mixing at Raman-Nath regime of diffraction (*Es* and *Ep* are the amplitudes of the pump and signal beams) allowing high-gain amplification values (LC layer acts as thin hologram) at 1064 nm. Right side: Raman-Nath orders observed on a view card for BTO:Rh/LC device with graphene electrodes.

Experimentally, the gain parameter *G* has been measured by monitoring the intensities of two interfering beams as *G* = *I* 0 /*I*s , where *I* s is the input signal intensity (without pump light) and *I* 0 is the amplified signal after the device. For example, at an interaction angle of 2.5°*θ* (12-μm grating period), the gain amplification for BTO:Rh/LC structure is *G* = 4.1 at 1064 nm. The gain coefficient of Г ~ 1180 cm−1 has been calculated by the relation *I*(0) = *I* (s)eΓL, which is among the highest value reported until now for hybrid devices, operating at near infrared spectral range. Very recently, a gain amplification of 17 has been achieved at GaAs-based hybrid structure at 1064 nm with optimized thickness of the LC layer [33].

#### **4.3. Applications**

amplification values have been obtained in hybrid structures based on semiconductor crystals

The interference between the pump and signal beam, expressed by their amplitudes

space-charge distribution and consequently a molecular re-orientation pattern in the LC layer. The interaction creates a refractive index grating with the same wave vector K, and as a result, the two beams diffracted by the photo-induced grating with spatial grating period of Λ = 2π/K. In these hybrid structures, the active layer is the LC layer; therefore, the interaction length is sufficiently thin to satisfy the energy and momentum conservation before exiting the medium [1, 2]. As a result, the two-beam coupling occurs in a Raman-Nath regime of diffraction (the LC layer is much thinner in comparison with the grating spacing *L* << Λ). Consequently, the phase grat-

A detailed analysis of the two-wave mixing and expression of the zero and m-output diffracted orders is given in Ref. [24] where the evolution of the amplitude of the refractive index grating and relaxation dynamic of the LC molecules orientation has been considered, using the couple-wave theory [1, 11]. We note that the output signal beam according to Refs. [24, 36]

**Figure 5** demonstrates the two-wave mixing in BTO:Rh/LC hybrid structure using 1064-nm diode laser. The right side of **Figure 5** supports the Raman-Nath diffraction with several diffracted beams (0, ±1, ±2,…) at the output of the hybrid device, detected on the infrared view card.

and *Ep*

beams) allowing high-gain amplification values (LC layer acts as thin hologram) at 1064 nm. Right side: Raman-Nath

are the amplitudes of the pump and signal

ing recorded in the LC layer acts as a thin hologram with a multiple output beams.

\_\_\_ *G I*

where *G* is the gain amplification and *φ* is the non-linear phase shift.

**Figure 5.** Two-wave mixing at Raman-Nath regime of diffraction (*Es*

orders observed on a view card for BTO:Rh/LC device with graphene electrodes.

exp  [ *<sup>i</sup>*(*κ<sup>s</sup>* <sup>⋅</sup> *<sup>r</sup>* <sup>−</sup> *<sup>ω</sup><sup>s</sup>* <sup>⋅</sup> *<sup>t</sup>* ) ], produced an intensity fringe pattern that induces a

*<sup>s</sup>* exp(*iφ*)*exp*[*i*(*ks* . *r* − *ω<sup>s</sup> t*)] + *c* . *c*. (18)

as GaAs doped with Cr [33, 34] or CdTe [35].

*Ep* exp  [ *i*(*κ<sup>p</sup>* ⋅ *r* − *ω<sup>p</sup>* ⋅ *t* ) ] and *Es*

490 Holographic Materials and Optical Systems

can be written in a form:

*Eoutput* = √

Based on the ability to record dynamic phase holograms, the reviewed electro-optically controlled structures found applications as optically addressed spatial light modulator devices, light-valve structures, to control the fast and slow components of light, in adaptive interferometry and metrology, and so forth [33–37].

When the address beam (incoherent image) is projected into photoconductive substrate, due to its high photoconductivity and high dark resistivity, the crystal reduces its impedance and accumulated voltage is transferred to the LC layer, resulting in LC molecular reorientation. Hence, the intensity distribution of the address beam in the photoconductor has a subsequent connection with the voltage distribution in the LC layer. This is the way how the "optical addressing" is realized. Consequently, when the two interfering beams intersect inside the hybrid structure, dynamic phase holograms can be addressed into the liquid crystal layer.

Based on the phase modulation ability, an evolution of image propagating on BTO:Rh/LC device is demonstrated. **Figure 6** shows an image of the character "A" addressed on BTO:Rh/ LC device at the beginning of the process and its time evolution at 50 and 100 ms. The response time of the hybrid device is limited by the response of the LC molecules (100–150 ms) since the response of BTO:Rh crystal is much faster (20–30 ms) at the near infrared spectral range [21].

**Figure 6.** Optical setup for modulated pump light intensity demonstration in BSO:Ru/LC device with graphene electrodes using 1064-nm pump light.

Besides display applications, the Raman-Nath diffraction with multiple output order beams has been successfully applied to control the fast and slow components of light propagation [36]. Different group delays have been obtained depending on the output order and frequency detuning between the pump and signal beam. Varieties of applications in interferometry, optical signal processing, precision metrology and optical sensing are demonstrated [33–37].
