**1. Volume Bragg gratings**

#### **1.1. Description and properties**

Volume grating as its name suggest is a grating that occupies the volume of a medium. Typically, for such gratings the term volume Bragg gratings (VBGs) is used in relation to Sir William Bragg who in 1915 used diffraction of light propagating through a crystal to determine the crystal's lattice structure [1]. What he found was that at certain conditions the light is strongly diffracted by the crystal. Such condition is called "resonant condition" or also "Bragg condition." Here is also the place to make the distinction between surface and volume Bragg gratings. If we start with a surface grating and start increasing its thickness at some point the different diffraction orders will reduce to a moment where there will be only one order. This defines the transition to a volume grating behavior [2].

There are two basic types of VBGs which is shown in **Figure 1**. The first one is a transmission Bragg grating (TBG) for which if the incident light satisfies the Bragg condition it is not transmitted but also diffracted. The second type is a reflection Bragg grating (RBG) which behaves like a mirror for incoming light that matches the Bragg condition.

**Figure 1.** Beam geometries for transmission Bragg grating (a) and for reflective Bragg grating (b).

For simplicity, in **Figure 1**, for both types of volume gratings, the angle of incidence is the same *θ<sup>i</sup>* and the tilt of the volume grating inside the medium is *θ*tilt. Resonant diffraction from each of these VBGs occurs upon satisfaction of their Bragg conditions which are defined by Eqs. (1) and (2).

*λ*TBG = 2*Λ* sin(*θ<sup>i</sup>* + *θ*tilt ) (1)

$$
\lambda\_{\text{nuc}} = \mathcal{D}\Lambda \cos(\theta\_i + \theta\_{\text{nuc}}) \tag{2}
$$

Here, *Λ* is the period of the VBG and *λ*VBG is the wavelength of the incident light which for the particular *θ<sup>i</sup>* and *θ*tilt satisfies the Bragg condition. The gratings depicted in **Figure 1** are uniform VBGs that can be recorded in a photosensitive material by simple interference of two collimated laser beams. The recording wavelength, angle of interference, and the refractive index of the material determine the grating's parameters. There are techniques capable of recording more complex volume gratings which have nonuniform period and for which the Bragg condition will be different depending on the space coordinates [2]. Regardless, if the variation of the period is negligible when compared to the probe beam size used for characterization of the VBG; Eqs. (1) and (2) can still provide the resonance wavelength.

**1. Volume Bragg gratings**

52 Holographic Materials and Optical Systems

**1.1. Description and properties**

same *θ<sup>i</sup>*

Eqs. (1) and (2).

the particular *θ<sup>i</sup>*

defines the transition to a volume grating behavior [2].

like a mirror for incoming light that matches the Bragg condition.

Volume grating as its name suggest is a grating that occupies the volume of a medium. Typically, for such gratings the term volume Bragg gratings (VBGs) is used in relation to Sir William Bragg who in 1915 used diffraction of light propagating through a crystal to determine the crystal's lattice structure [1]. What he found was that at certain conditions the light is strongly diffracted by the crystal. Such condition is called "resonant condition" or also "Bragg condition." Here is also the place to make the distinction between surface and volume Bragg gratings. If we start with a surface grating and start increasing its thickness at some point the different diffraction orders will reduce to a moment where there will be only one order. This

There are two basic types of VBGs which is shown in **Figure 1**. The first one is a transmission Bragg grating (TBG) for which if the incident light satisfies the Bragg condition it is not transmitted but also diffracted. The second type is a reflection Bragg grating (RBG) which behaves

For simplicity, in **Figure 1**, for both types of volume gratings, the angle of incidence is the

**Figure 1.** Beam geometries for transmission Bragg grating (a) and for reflective Bragg grating (b).

*λ*TBG = 2*Λ* sin(*θ<sup>i</sup>* + *θ*tilt ) (1)

*λ*RBG = 2*Λ* cos(*θ<sup>i</sup>* + *θ*tilt ) (2)

Here, *Λ* is the period of the VBG and *λ*VBG is the wavelength of the incident light which for

uniform VBGs that can be recorded in a photosensitive material by simple interference of two

and *θ*tilt satisfies the Bragg condition. The gratings depicted in **Figure 1** are

 and the tilt of the volume grating inside the medium is *θ*tilt. Resonant diffraction from each of these VBGs occurs upon satisfaction of their Bragg conditions which are defined by **Figure 2** exhibits one example of beam geometry for recording uniform transmitting and reflecting volume gratings. In this recording approach, two plane waves (purple beams) illuminate the sample from one side at a half-angle of interference *φ*. Depending on the direction from which the grating is used, it can either work as a TBG (green beams) or as an RBG (orange beams). Before recording, the grating parameters such as period and modulation need to be calculated so the recording is carried out accordingly.

**Figure 2.** Recording geometry used for recording reflective and transmitting Bragg gratings. The recording beams are shown in purple and the half-angle of interference *φ* determines the properties of the grating. The VBG is utilized as an RBG if probed as the orange beams or as TBG if probed as the green beams.

The main model describing volume Bragg gratings was introduced by Kogelnik [3] in 1969. His model describes that diffraction from a VBG is based on coupled wave theory (CWT) and provides analytical solutions for RBGs and TBGs including tilted ones. **Figures 3(a**, **b)** and **4(a**, **b)** present examples of the wavelength and angular responses of an RBG and a TBG correspondingly, calculated using Kogelnik's theoretical approach.

**Figures 3** and **4** give good overview of the main properties of TBGs and RBGs. In particular, RBGs are much more suited for implementation as narrow wavelength filters. For example, the full-width half-maximum (FWHM) wavelength selectivity of the reflective grating simulated in **Figure 3(a)** is around 225 pm but it can reach down to 15–20 pm if designed accordingly. TBGs, in contrast, have much wider wavelength acceptance starting at a few hundred picometers and reaching several nanometers. **Figure 4(a)** shows the spectral response of 1.5 mm thick TBG with nodulation of 330 ppm. For these parameters, its FWHM of 2.3 nm is close to an order magnitude larger if compared to the RBG one.

**Figure 3.** Wavelength (a) and angular (b) response for an RBG. The VBG is 5.5 mm thick, it is 20° tilted, and has a 240 ppm refractive index modulation.

**Figure 4.** Wavelength (a) and angular (b) response for a TBG. The VBG is 1.5 mm thick, it is 20° tilted, and has a 333 ppm refractive index modulation.

TBGs, alternatively, can be used as narrow angular filters with acceptance values as low as 0.1 mrad, whereas RBGs have typical angular acceptance of more than 10 mrad all the way to 100 mrad. These properties of the two types of VBGs define their use in different applications. For example, the narrow angular selectivity of the TBGs makes them a great angular filter that can be used to suppress higher order modes generation in laser cavities while keeping them very compact [4]. Alternatively, RBGs with their narrow wavelength selectivity can be used for narrow wavelength beam combining where the diffracted by and the transmitted through an RBG beams can be separated by only a few hundred picometers [5]. Regardless of the close wavelength separation, the RBG does not diffract the transmitted beam even though both beams have a common propagating direction.

**Table 1** summarizes the TBG's and RBG's characteristics and their typical range. Until now, we have discussed wavelength and angular selectivity of VBGs but the third very important parameter is the VBG's efficiency. There are two generally accepted ways to define a VBG's efficiency and the more widely used on is the so-called "relative diffraction efficiency." It is defined as normalization of the diffracted to the transmitted by a VBG power. Its advantage is that it removes any losses introduced by the medium. The second way is called "absolute diffraction efficiency," where the diffracted power is normalized to the incident power.


**Table 1.** TBG and RBG characteristics.

#### **1.2. Recording materials**

lated in **Figure 3(a)** is around 225 pm but it can reach down to 15–20 pm if designed accordingly. TBGs, in contrast, have much wider wavelength acceptance starting at a few hundred picometers and reaching several nanometers. **Figure 4(a)** shows the spectral response of 1.5 mm thick TBG with nodulation of 330 ppm. For these parameters, its FWHM of 2.3 nm is close

TBGs, alternatively, can be used as narrow angular filters with acceptance values as low as 0.1 mrad, whereas RBGs have typical angular acceptance of more than 10 mrad all the way to 100 mrad. These properties of the two types of VBGs define their use in different applications. For example, the narrow angular selectivity of the TBGs makes them a great angular filter that can be used to suppress higher order modes generation in laser cavities while keeping them very compact [4]. Alternatively, RBGs with their narrow wavelength selectivity can be used

**Figure 4.** Wavelength (a) and angular (b) response for a TBG. The VBG is 1.5 mm thick, it is 20° tilted, and has a 333 ppm

**Figure 3.** Wavelength (a) and angular (b) response for an RBG. The VBG is 5.5 mm thick, it is 20° tilted, and has a 240

to an order magnitude larger if compared to the RBG one.

ppm refractive index modulation.

54 Holographic Materials and Optical Systems

refractive index modulation.

Recoding of a volume grating requires the use of a material that is photosensitive. The modulation of the recording light intensity should create a corresponding refractive index change in the recording material which on a macroscopic level will be in fact the volume Bragg grating. There is a wide variety of photosensitive materials that can be used for recording VBGs [6, 7]. The main requirement that they need to fulfill is to have enough spatial resolution that will allow the recording of gratings with particular periods. The other two factors are the photosensitivity of the material and its dynamic range. The photosensitivity determines the exposure duration and given that VBGs are most commonly recorded by interference of light, it is of great benefit to keep the exposure time as short as possible. The material's dynamic range provides the maximum refractive index change that can be achieved. This property affects the VBGs thickness and the maximum number of volume gratings that can be multiplexed together in the same volume. Other properties that depending on the particular application may be important are the optical damage threshold, the maximum physical dimensions of the material, its losses, its environmental sensitivity, and others.

The most common recording materials are dichromated gelatins, photopolymers, photorefractive crystals, photosensitive fibers, and photothermo refractive glasses. We will not discuss in detail the properties of each of these materials because they have been well investigated in the literature [8–16]. The applications and experimental results shown further in the chapter are based on using photothermo-refractive glass (PTR) due to its capabilities of handling highpower laser radiation because of its low losses, its environmental robustness, and extremely high resolution [14–16].

Photothermo-refractive glass is a relatively new photosensitive material well suited for phase hologram recording. It combines high sensitivity achieved by two-step hologram formation process and high-optical quality resulting from its technological development. The PTR glass is a Na2 O-ZnO-Al<sup>2</sup> O3 -SiO2 glass doped with silver (Ag), cerium (Ce), and fluorine (F). It is transparent from 350 to 2500 nm. The chain of processes, which produce refractive index variation, is as follows: the first step is the exposure of the glass to UV radiation, somewhere in the range from 280 to 350 nm. This exposure results in photoreduction of silver ions Ag<sup>+</sup> to atomic state Ag<sup>0</sup> . This stage is similar to formation of a latent image in a conventional photo film because no significant changes in the optical properties of the glass occur. The final formation of the holographic recording is secured by subjecting the glass to thermal development. During this step, at elevated temperatures, a number of silver containing clusters are formed in the exposed regions of the glass due to the increased mobility of Ag<sup>0</sup> atoms. These silver-containing clusters serve as nucleation centers for the growth of NaF crystals. Interaction of these nanocrystals with the surrounding glass matrix causes the decrease of refractive index. Refractive index change Δ*n* of about 1.5 × 10−3 (1500 ppm) can be achieved and is enough to allow the recoding of high-efficiency hologram into glass wafers with thickness exceeding several hundred microns.

The second consequence of the crystalline phase precipitation in PTR glass is related to its physical properties and is extremely valuable. The NaF crystalline particles in the glass matrix are almost impossible to destroy by any type of radiation which makes PTR holograms stable under exposure to IR, visible, UV, X-ray, and gamma-ray irradiation. For example, laser damage threshold for 8 ns laser pulses at 1064 nm is in the range of 40 J/cm2 . Also, the nonlinear refractive index of PTR glass is the same as that for fused silica which allows the use of PTR diffractive elements in all types of pulsed lasers. Another PTR advantage is its very low losses—on the order of 10−5 cm−1. Testing of VBG recorded in the PTR glass performed under irradiation of 9 kW CW with a 6-mm-diameter spot showed heating that did not exceed 15 K [17]. Even though small heating effects lead to thermal variations of the refractive index of the glass (*dn*/*dt* = 5 × 10−8 K−1). In the case of Bragg grating written inside a PTR glass, this feature leads to thermal shift of the Bragg wavelength of around 10 pm/K. It is worth mentioning also that due to the melting temperature of the NaF crystals being almost 1000°C, PTR holograms are stable at elevated temperatures and could tolerate thermal cycling up to 400°C. This temperature is determined by the plasticity point of the glass matrix.

Typically, Bragg gratings in the PTR glass are recorded by an exposure to interference pattern of radiation from a He-Cd laser operating at 325 nm. The spatial frequency of the gratings can vary from 50 up to about 10,000 mm−1, their thickness from 0.5 to 25 mm, and a diffraction efficiency of up to 99.9%.
