**2. Raman-Nath and Bragg match regimes of diffraction**

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needs to be comparable or smaller than the wavelength of the light. Another approach is to design all optically controlled devices for light control and manipulation. All these features make the combination between outstanding properties as large anisotropy and strong birefringence typical for organics (offering easy processing, large structural flexibility and low cost) with the excellent photosensitivity and photoconductivity of inorganic materials (providing mechanical stability and energy gap manipulation) attractive to design single, compact structures with enhanced functionality. Moreover, an emergent need of devices sensitive at the near-infrared spectral range (the required illumination for biological and medical samples) is of particular importance for biomedical sensing and with significant impact for the society. In that aspect, the holography, due to its unique nature, is expected to play an essential role. The following chapter is focused on the two-wave mixing in organic-inorganic hybrid structures and dynamic holography recording resulting to formation of light-induced gratings. The two-wave mixing (denoted also as a two-beam coupling) takes place in a variety of nonlinear media as photorefractive materials (the photorefractive effect refers to refractive index modulation in response to light), third-order non-linear media (as Kerr media) or semiconductor amplifiers. The two-wave mixing is expressed by two beam interactions, forming an interference pattern, which is characterized by periodic spatial variation of the light-intensity distribution. Its main significance is the ability of unidirectional optical energy exchange between the two beams (gain amplification) allowing weak beam to grow exponentially with the distance, which opens various opportunities for designing a new structures and elements

Depending of the organic-inorganic hybrid structure design and in more particular how the inorganic material control the dynamic grating formation, they can operate at Raman-Nath or

i) Generally, the organic-inorganic structures are assembled by photoconductive material (usually inorganic crystal) and birefringent material (liquid crystal [LC] or polymer dispersed liquid crystal [PDLC]). Their operation principle relays on electro-optically controlled birefringence of the liquid crystal molecules that allows spatial modulation of the amplitude or the phase of incident beam. In such configuration, the inorganic crystal serves as a photoconductive layer, which controls the LC molecules alignment and allows subsequent light modulation. As a result, the two-wave mixing happens in a LC layer with very high amplification values; however, the fringe period of the recorded holograms is limited to few micrometer scales (Raman-Nath regime of diffraction). Usually,

this type of hybrid structures is known as electro-optically controlled devices.

ii) Recently, a novel type of organic-inorganic structures has been proposed assembled by photorefractive material and birefringent layer (LC or PDLC). Their operation principle relays on surface-activated photorefractive effect and more specific on the photo-generated space charge field, acting as a driving force for LC molecules reorientation and subsequently the refractive index modulation. The prime significance is the fact that the twowave mixing happens in both photorefractive and birefringent layers, where the charge

for practical realizations.

480 Holographic Materials and Optical Systems

Bragg match regime of diffraction:

The interaction between two coherent laser beams inside the photosensitive material generates a light-intensity fringe pattern of bright and dark regions (sinusoidal light intensity pattern) expressed by [1, 2]:

$$I(\mathbf{x}) = I\_0 \left[1 + m \cos(\mathbf{K} \mathbf{x})\right] \tag{1}$$

where *I* 1 and *I*<sup>2</sup> are the intensities of both beams, *I* 0 is the total intensity *I* <sup>0</sup> = (*I*<sup>1</sup> + *I*<sup>2</sup> ) and *m* is the light modulation, *m* = 2 (*I* 1 *I* 2 ) 1/2 \_\_\_\_\_\_ *I* .

The created diffraction grating is defined by the two interfering beams and their spatial coordinates, determined by the grating wave vector *K* = 2π/Λ and the spatial fringe period <sup>Λ</sup> <sup>=</sup> \_\_\_\_\_ *<sup>λ</sup>* 2*sinθ*, where *λ* is the wavelength, and *θ* is the external half angle between of the intersection beams. After propagation through the medium, the same beams diffract from the holographic grating, which they formed.

The light diffraction phenomenon from periodic structures has been extensively discussed in the past and defined to two regimes of diffraction: (i) Raman-Nath regime (when several diffracted waves are produced named thin grating) and (ii) Bragg matched regime (when only one diffracted wave is produced named thick or volume grating) [1, 2].

Briefly, in Raman-Nath regime, after interacting with the grating, the incident beam is split into several beams resulting in different orders of diffraction (see **Figure 1(a)**). As a consequence, there are several diffracted waves produced 0, ±1, ±2, ±3,… ±*m*, which correspond 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 diffraction is given by [1]

diffraction is given by [1]

$$
\eta\_m = J\_n^2(\delta) = J\_n^2 \left(\frac{2\pi L\eta}{\lambda \cos \theta}\right) \tag{2}
$$

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

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

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

The simplified diffraction efficiency (in case of transmission type, sinusoidal phase grating) is given by [2]

 $\mathbf{S}\_{\text{new}}$  is  $\eta = \sin n^2 \left(\frac{\pi nL}{\lambda \cos \theta}\right)$ 

Generally, Klen-Cook dimensionless parameter has been accepted as distinguishing factor between the Raman-Nath or Bragg regimes of operations, defined by [5]:

$$Q = \frac{2\pi M}{n\Lambda^2} \tag{4}$$

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 grating strength [6].
