2.2. Characteristics of HOE

Generally, there are some important properties of a HOE that should be known. They are diffraction efficiency, wavelength selectivity, and angular selectivity. Of the many methods [20] to describe grating behavior, the couple wave theory as presented by Kogelnik [21, 22] will be the primary method used in this study, due to its simplicity and applicability.

Figure 4. Transmission spectra of the unrecorded RGB-sensitive Bayfol HX 102 film.

efficiency, low cost, and excellent signal-to-noise ratio [10, 11]. Furthermore, it does not require any chemical or wet processing after recording the holograms. Because of such advantages, the photopolymer has been used widely in several research fields, which include optical elements

Bayer MaterialScience is developing its photopolymer to be easy to handle, with high diffraction efficiency, polychromatic, durable, and customizable. This material will be simple to expose, with no wet or heat processing. The ease of use and simple processing requirement allow these materials to be amenable to mass production of holographic optical elements

Bayfol HX film 102 consists of a four-layer stack of a backside cover film of the substrate, the substrate itself, the light-sensitive photopolymer, and a protective cover film as shown in Figure 3. A polycarbonate (PC) substrate with a thickness of 175 2 μm and polyethylene (PE) are used as backside cover foil and protective cover foil, which are both 40 μm in thickness. The protective cover film can be removed. The photopolymer layer itself has a thickness

Bayfol HX 102 photopolymer can be used to manufacture reflection and transmission volume-

In Figure 4, the basic product characteristics are depicted. The transmission spectrum of the unrecorded photopolymer film was recorded after removal of the protective cover film. In this material, the dye-related absorption peaks are located at 473, 532, and 633 nm, with associated

Generally, there are some important properties of a HOE that should be known. They are diffraction efficiency, wavelength selectivity, and angular selectivity. Of the many methods [20] to describe grating behavior, the couple wave theory as presented by Kogelnik [21, 22]

will be the primary method used in this study, due to its simplicity and applicability.

phase holograms with appropriate laser light within the spectral range of 440–660 nm.

transmittance of 56%, 45%, and 31%, respectively.

2.2. Characteristics of HOE

Figure 3. Bayfol HX 102 film structure.

[12, 13], holographic storage [14], holographic display [15], etc.

[16–19].

102 Holographic Materials and Optical Systems

(d) of 16.8 μm.

In 1969, Herwig Kogelnik published the coupled wave theory, analyzing the diffraction of light by volume gratings. It assumes that monochromatic light is incident on the volume grating at or near the Bragg angle and polarized perpendicular to the plane of incidence. This theory can predict the maximum possible efficiencies of the various volume gratings and the angular and wavelength dependence at high diffraction efficiencies.

Figure 5 shows the model of a transmission volume hologram grating with slanted fringes. The x-axis is parallel to the recording material on the plane of incidence, the y-axis is perpendicular to the paper, and the z-axis is perpendicular to the surface of the recording material. The grating vector K is oriented perpendicular to the fringe planes and is of length K = 2π/Λ, where Λ is the period of the grating (spatial frequency f = 1/Λ). The angle of incidence measured in the material is θR. The fringe planes are oriented perpendicular to the plane of incidence and slanted with respect to the material boundaries at an angle φ.

Figure 5. Model of a transmission volume grating with slanted fringes.

Figure 6 shows the fringe formations according to the recording process. The fringes are perpendicular to the grating plane for transmission gratings or parallel for reflection gratings.

Figure 6. (a) Volume transmission gratings, (b) volume reflection gratings, and their associated vector diagrams for Bragg condition.

The volume record of the holographic interference pattern usually takes the form of a spatial modulation of the absorption coefficient or of the refractive index n(r) of the material, or both. For the sake of simplicity, here is the analysis restricted to the holographic record of sinusoidal fringe patterns. The grating is assumed dielectric, nonmagnetic, and isotropic. Hence, once the recording process has taken place, the resulting modulation may be described at the first order by the following relations:

$$
\Delta n = n\_0 + \Delta n \cos \left( K \cdot \mathbf{x} \right) \tag{1}
$$

$$
\alpha = \alpha\_0 + \Delta \alpha \cos \left(\mathbf{K} \cdot \mathbf{x}\right) \tag{2}
$$

where x represents the radius vector x = (x, y, z), whereas n<sup>0</sup> is the average reflective index, α<sup>0</sup> is the average absorption coefficient, and Δn and Δα are the amplitudes of the spatial modulations of the index and absorption coefficient, respectively.

Generally, wave propagation in the grating is described by the scalar wave equation:

$$
\nabla E + k^2 E = 0 \tag{3}
$$

where E (x, z) is the complex amplitude of the y-component of the electric field, which is assumed to be independent of y and to have a wavelength λ. The wave number is equal to the average propagation constant β:

Holographic Optical Elements and Application http://dx.doi.org/10.5772/67297 105

$$
\beta = n\_0 k\_0 = \frac{2\pi n\_0}{\lambda} \tag{4}
$$

and the coupling constant κ can be simplified to

Figure 6 shows the fringe formations according to the recording process. The fringes are perpendicular to the grating plane for transmission gratings or parallel for reflection gratings.

The volume record of the holographic interference pattern usually takes the form of a spatial modulation of the absorption coefficient or of the refractive index n(r) of the material, or both. For the sake of simplicity, here is the analysis restricted to the holographic record of sinusoidal fringe patterns. The grating is assumed dielectric, nonmagnetic, and isotropic. Hence, once the recording process has taken place, the resulting modulation may be described at the first order

Figure 6. (a) Volume transmission gratings, (b) volume reflection gratings, and their associated vector diagrams for Bragg

where x represents the radius vector x = (x, y, z), whereas n<sup>0</sup> is the average reflective index, α<sup>0</sup> is the average absorption coefficient, and Δn and Δα are the amplitudes of the spatial modula-

Generally, wave propagation in the grating is described by the scalar wave equation:

∇E þ k 2

where E (x, z) is the complex amplitude of the y-component of the electric field, which is assumed to be independent of y and to have a wavelength λ. The wave number is equal to

tions of the index and absorption coefficient, respectively.

n ¼ n<sup>0</sup> þ Δn cos ð Þ Κ � x (1)

α ¼ α<sup>0</sup> þ Δα cos ð Þ Κ � x (2)

E ¼ 0 (3)

by the following relations:

104 Holographic Materials and Optical Systems

condition.

the average propagation constant β:

$$
\kappa = \frac{\pi \Delta n}{\lambda} - i \frac{\Delta \alpha}{2} \tag{5}
$$

The coupling constant is the central parameter in the couple wave theory as it describes the coupling between the "reference" wave (R) and the "signal" wave (S). If κ = 0, there is no coupling; therefore, there is no diffraction.

The propagation of two coupled waves through the grating can be described by their complex amplitudes: the incoming wave R(z) and the outgoing wave S(z), which vary along the z axis. The total field within the grating is written as follows:

$$E(x, z) = R(z)e^{-jk\_{\circ} \cdot r} + S(z)e^{-jk\_{\circ} \cdot r} \tag{6}$$

where r is the position vector and the symbols ki and ko are the wave vectors of the incoming and outgoing waves, respectively, which are related to each other by

$$\mathbf{k}\_{\mathbf{o}} = \mathbf{k}\_{\mathbf{i}} \cdot \mathbf{K}.\tag{7}$$

The vector relation from Eq. (7) is shown in Figure 7 together with the circle of radius β. Figure 7(b) shows the general case that the length of ko differs from β and the Bragg condition is not met. Figure 7(c) shows the special case that the length of both ki and ko is equal to the average propagation constant β at the Bragg angle θ0. And the Bragg condition is obeyed:

$$\cos\left(\phi - \theta\_R\right) = \frac{K}{2\beta} = \frac{\lambda}{2n\_0\Lambda}.\tag{8}$$

For fixed wavelength, the Bragg condition may be broken by angular deviations Δθ from the Bragg angle θ0. Analogously, for fixed angle of incidence, detuning takes place for changes Δλ with respect to the Bragg wavelength λ0. Differentiating the Bragg condition, we obtain

$$\frac{\Delta\theta}{\Delta\lambda} = \frac{K}{4\pi n\_0 \sin\left(\phi - \Theta\_R\right)} = \frac{f}{2n\_0 \sin\left(\phi - \Theta\_R\right)}\tag{9}$$

that relates the angular selectivity to the wavelength selectivity of a volume hologram grating; small changes in the angle of incidence or the wavelength have similar effects. Highperformance devices, typically, should have a large selectivity and large diffraction efficiency.

Kogelnik introduced the parameter of mismatch constant Γ for evaluating the effects of deviations from the Bragg condition:

$$
\Gamma = K \cos \left( \phi - \theta\_R \right) - \frac{K^2 \lambda}{4 \pi n\_0} \tag{10}
$$

Figure 7. Vector diagram: (a) the relation between the propagation vector and the grating vector, (b) near at Bragg condition, and (c) exact Bragg incidence.

When the Bragg mismatch is due to the angular detuning Δθ and wavelength detuning Δλ, the mismatch constant expressed as

$$
\Gamma = \Delta\theta \cdot \mathbf{K} \sin\left(\phi - \theta\_{\mathbb{R}}\right) - \frac{\Delta\lambda \cdot \mathbf{K}^2}{4\pi n\_0} \tag{11}
$$

Substituting Eqs. (1), (2), (4), and (6) into Eq. (3), R(z) and S(z) must individually satisfy the following equations in order for the wave equation to be satisfied:

$$
\cos\theta\_R \frac{d\mathbf{R}}{dz} + \alpha \mathbf{R} = -j\kappa \mathbf{S} \tag{12}
$$

$$
\cos\theta\_S \frac{d\mathbf{S}}{dz} + (\alpha + j\Gamma)\mathbf{S} = -j\kappa\mathbf{R} \tag{13}
$$

where the obliquity factor cos θ<sup>S</sup> = cos θ<sup>R</sup> � K cos φ/β = � cos(θ<sup>R</sup> � 2φ). Solving Eqs. (12) and (13), the diffraction efficiency η is defined as

$$\eta = \frac{|\cos \theta\_s|}{\cos \theta\_R} SS^\* \tag{14}$$

#### 2.2.1. Transmission HOE

In transmission volume grating, the fringes are perpendicular to the surfaces of the recording material, and the incoming "reference" wave (R) and the outgoing "signal" wave (S) are on the opposite side of the recording material.

In lossless volume gratings, α<sup>0</sup> = Δα = 0, and the coupling constant is κ = πΔn/λ. Diffraction is caused by spatial variation of the refractive index; the diffraction efficiency of the slanted lossless transmission volume grating is as follows:

$$\eta\_T = \frac{\sin^2 \sqrt{v^2 + \xi^2}}{1 + \xi^2/v^2} \tag{15}$$

where ν and ξ are given by

When the Bragg mismatch is due to the angular detuning Δθ and wavelength detuning Δλ,

Figure 7. Vector diagram: (a) the relation between the propagation vector and the grating vector, (b) near at Bragg condition,

Substituting Eqs. (1), (2), (4), and (6) into Eq. (3), R(z) and S(z) must individually satisfy the

where the obliquity factor cos θ<sup>S</sup> = cos θ<sup>R</sup> � K cos φ/β = � cos(θ<sup>R</sup> � 2φ). Solving Eqs. (12) and

<sup>η</sup> <sup>¼</sup> cos <sup>θ</sup><sup>s</sup> j j cos θ<sup>R</sup>

dR

� Δλ � <sup>K</sup><sup>2</sup>

4πn<sup>0</sup>

dz <sup>þ</sup> <sup>α</sup><sup>R</sup> ¼ �jκ<sup>S</sup> (12)

SS<sup>∗</sup> (14)

dz <sup>þ</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>j</sup><sup>Γ</sup> <sup>S</sup> ¼ �jκ<sup>R</sup> (13)

(11)

Γ ¼ Δθ � K sin φ � θ<sup>R</sup>

cos θ<sup>R</sup>

dS

cos θ<sup>S</sup>

following equations in order for the wave equation to be satisfied:

the mismatch constant expressed as

and (c) exact Bragg incidence.

106 Holographic Materials and Optical Systems

(13), the diffraction efficiency η is defined as

$$\nu = \frac{\pi d \Delta n}{\lambda \sqrt{\cos \theta\_{\mathbb{R}} \cdot \cos \theta\_{\mathbb{S}}}} \tag{16}$$

$$\xi = \frac{\Gamma d}{2 \cos \theta\_S} \tag{17}$$

Figure 8 shows the diffraction efficiency of the lossless transmission volume gratings as a function of the parameter ξ for three values of the parameter ν. The diffraction efficiency of the volume grating is 100% for ν = π/2, 50% for ν = π/4 and ν = 3π/4. It can be observed that for a fixed value of ξ the diffraction efficiency drops to zero if there is slight deviation from the Bragg condition.

Figure 8. Transmission grating: diffraction efficiency η of lossless volume grating as a function.

When the wavelength and the angle are gradually out of the Bragg condition, the parameters ξ is obtained as follows:

$$\xi = \frac{\pi f d}{\cos \left( \phi - \theta\_0 \right) - \left( f \lambda\_0 / n\_0 \right) \cos \phi} \left( \Delta \theta \sin \theta\_0 - \frac{f \Delta \lambda}{2 n\_0} \right) \tag{18}$$

From the above Eqs. (16), (17), and (18), it is clear that the diffraction efficiency of the volume grating is influenced by angular deviation Δθ and wavelength deviation Δλ through the parameter ξ.

If there is no slant (φ = π/2) and if the Bragg condition is obeyed, then cos θ<sup>R</sup> = cos θ<sup>S</sup> = cos θ0, and Eq. (15) becomes

$$\eta\_{\pi/2} = \sin^2 \nu = \sin^2 \left(\frac{\pi d \Delta n}{\lambda \cos \theta\_0}\right) \tag{19}$$

As thickness d or the variation of the refractive index Δn increases, the diffraction efficiency increases until the modulation parameter ν = π/2. At this point η = 100 %, and all the energy goes into the diffracted light. When ν increases beyond this point, the energy is back-coupled into the incident wave, and η decreases.

The angular selectivity of un-slanted transmission volume grating could be determined by substituting Eqs. (16), (17), and (18) into Eq. (15) at Δλ = 0:

$$\eta\_T(\Delta\theta) = \frac{\sin^2\left(\pi d \sqrt{\left(\frac{\Delta n}{\lambda \sin \theta\_0}\right)^2 + \left(f\Delta\theta\right)^2}\right)}{1 + \left(\frac{\lambda f \sin \theta\_0 \Delta\theta}{\Delta n}\right)^2} \tag{20}$$

It is important to note that Eq. (20) requires the following criterion for equalizing of diffraction efficiency to zero:

$$
\sqrt{\nu^2 + \xi^2} = j\pi \tag{21}
$$

where j = 1, 2, ⋯n, ⋯. Angular selectivity in the volume grating at the half width at first zero (HWFZ) level, ΔθHWFZ, as the angle between the central maximum and the first minimum at the diffraction efficiency curve. For the volume Bragg grating with 100% diffraction efficiency, the following expression for the HWFZ angular selectivity could be given at j = 1:

$$
\Delta\theta\_T^{\text{HWPZ}} = \frac{\sqrt{3}}{2} \frac{1}{d\_0 f} \approx 0.87 \frac{1}{d\_0 f} \tag{22}
$$

It should be noticed that the HWFZ angular selectivity ΔθHWFZ <sup>T</sup> is slightly lower than widely used grating parameter of HWFZ angular selectivity.

Figure 9 shows the angular selectivity of a transmitting volume Bragg grating. The parameters are varied from more than 100 mrad to less than 0.1 mrad. According to spatial frequency of grating, the value of refractive index modulation Δn can provide 100% diffraction efficiency. And, it should be optimized with equation d<sup>0</sup> = λ cos(φ � θ0)/2Δn.

When the wavelength and the angle are gradually out of the Bragg condition, the parameters

From the above Eqs. (16), (17), and (18), it is clear that the diffraction efficiency of the volume grating is influenced by angular deviation Δθ and wavelength deviation Δλ through the

If there is no slant (φ = π/2) and if the Bragg condition is obeyed, then cos θ<sup>R</sup> = cos θ<sup>S</sup> = cos θ0,

As thickness d or the variation of the refractive index Δn increases, the diffraction efficiency increases until the modulation parameter ν = π/2. At this point η = 100 %, and all the energy goes into the diffracted light. When ν increases beyond this point, the energy is back-coupled

The angular selectivity of un-slanted transmission volume grating could be determined by

It is important to note that Eq. (20) requires the following criterion for equalizing of diffraction

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ν</sup><sup>2</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup>

where j = 1, 2, ⋯n, ⋯. Angular selectivity in the volume grating at the half width at first zero (HWFZ) level, ΔθHWFZ, as the angle between the central maximum and the first minimum at the diffraction efficiency curve. For the volume Bragg grating with 100% diffraction efficiency,

> ffiffiffi 3 p 2 1 <sup>d</sup>0<sup>f</sup> <sup>≈</sup> <sup>0</sup>:<sup>87</sup> <sup>1</sup>

sin <sup>2</sup> πd

q

the following expression for the HWFZ angular selectivity could be given at j = 1:

ΔθHWFZ <sup>T</sup> ¼

It should be noticed that the HWFZ angular selectivity ΔθHWFZ

used grating parameter of HWFZ angular selectivity.

<sup>ν</sup> <sup>¼</sup> sin <sup>2</sup> <sup>π</sup>dΔ<sup>n</sup>

λ cos θ<sup>0</sup> � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

! r

<sup>þ</sup> ð Þ <sup>f</sup> Δθ <sup>2</sup>

� �<sup>2</sup> (20)

¼ jπ (21)

<sup>d</sup>0<sup>f</sup> (22)

<sup>T</sup> is slightly lower than widely

Δn λ sin θ<sup>0</sup> � �<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>f</sup> sin <sup>θ</sup>0Δθ Δn

Δθ sin <sup>θ</sup><sup>0</sup> � <sup>f</sup> Δλ

� �

2n<sup>0</sup>

(18)

(19)

� � � ð Þ <sup>f</sup> <sup>λ</sup>0=n<sup>0</sup> cos <sup>φ</sup>

ηπ=<sup>2</sup> <sup>¼</sup> sin <sup>2</sup>

<sup>ξ</sup> <sup>¼</sup> <sup>π</sup>f d cos φ � θ<sup>0</sup>

ξ is obtained as follows:

108 Holographic Materials and Optical Systems

parameter ξ.

and Eq. (15) becomes

efficiency to zero:

into the incident wave, and η decreases.

substituting Eqs. (16), (17), and (18) into Eq. (15) at Δλ = 0:

ηTð Þ¼ Δθ

Figure 9. Angular selectivity (HWFZ) of transmitting volume gratings at λ = 532 nm and n<sup>0</sup> = 1.5 on spatial frequency for optimal refractive index modulation with grating thickness in 0.5, 2, 5, and 10 mm.

Just as the description for angular selectivity, the wavelength selectivity ΔλHWFZ can be determined as a distance between the central maximum and the first minimum in wavelength distribution of diffraction efficiency. It could be expressed by substitution of Eqs. (16), (17), and (18) into Eq. (15) at Δθ = 0. In the case of un-slanted transmission volume grating, this expression is simplified by the use of Eq. (18) when φ = π/2:

$$\eta\_T(\Delta\lambda) = \frac{\sin^2\left(\frac{nd}{\sin\theta\_0} \left(\sqrt{\left(\frac{\Delta\mu}{\lambda\_0}\right)^2 + \left(\frac{\ell^2\Delta\lambda}{2n\_0}\right)^2}\right)\right)}{1 + \left(\frac{\ell^2\lambda\_0\Delta\lambda}{2n\_0\Delta\pi}\right)^2} \tag{23}$$

Wavelength selectivity has the same structure as angular selectivity due to their linear interrelationship described by Eq. (9). For un-slanted transmitting volume gratings with 100% diffraction efficiency, ΔλHWFZ <sup>T</sup> could be derived by substitution of Eq. (22) into Eq. (9):

$$
\Delta\lambda\_T^{\text{HWPZ}} = \sqrt{3}\frac{n\_0}{d\_0 f^2} \tag{24}
$$

Figure 10 shows dependence of wavelength selectivity on spatial frequency for different grating thicknesses. HWFZ wavelength selectivity of transmitting volume grating could be easily varied from values below 0.1 nm to more than 100 nm by proper choosing of grating parameters.

Figure 10. Wavelength selectivity (HWFZ) of transmitting volume gratings at λ = 532 nm and n<sup>0</sup> = 1.5 on spatial frequency for optimal refractive index modulation with grating thickness in 0.5, 2, 5, and 10 mm.

#### 2.2.2. Reflection HOE

In reflection volume grating, the fringes are more or less parallel to the surfaces of the recording material, and the incoming "reference" wave (R) and the outgoing "signal" wave (S) are on the same side of the recording material. Figure 11 shows the model of a reflection volume hologram grating with slanted fringes. It is expressed in the coupled wave analysis by negative values of the obliquity factor cos θS(cos θ<sup>S</sup> < 0).

The diffraction efficiency of slanted lossless reflection volume grating can be written as

$$\eta\_R = \left[ 1 + \frac{1 - \xi^2/v^2}{\left(\sinh^2\sqrt{v^2 - \xi^2}\right)}\right]^{-1} \tag{25}$$

where ν and ξ are given by

$$\nu = \frac{i\pi d\Delta n}{\lambda\sqrt{\cos\theta\_R \cos\theta\_S}}\tag{26}$$

$$\mathcal{L} = -\frac{\Gamma d}{2\cos\theta\_S} \tag{27}$$

Figure 12 shows the diffraction efficiency of the lossless volume gratings as a function of the ξ, for the values of ν = π/4, π/2 and 3π/4. The figure shows the sensitivity of a grating with ν = π/4 and a peak efficiency of 43%, a grating with ν = π/2 and η = 84 %, and the corresponding values for 3π/4 and η = 96 %. For ν = π/2, the diffraction efficiency drops to zero in all cases when ξ ≈ 3.5.

Figure 11. Model of a reflection volume grating with slanted fringes.

2.2.2. Reflection HOE

110 Holographic Materials and Optical Systems

where ν and ξ are given by

ξ ≈ 3.5.

values of the obliquity factor cos θS(cos θ<sup>S</sup> < 0).

In reflection volume grating, the fringes are more or less parallel to the surfaces of the recording material, and the incoming "reference" wave (R) and the outgoing "signal" wave (S) are on the same side of the recording material. Figure 11 shows the model of a reflection volume hologram grating with slanted fringes. It is expressed in the coupled wave analysis by negative

Figure 10. Wavelength selectivity (HWFZ) of transmitting volume gratings at λ = 532 nm and n<sup>0</sup> = 1.5 on spatial

<sup>1</sup> � <sup>ξ</sup><sup>2</sup> =ν<sup>2</sup> sinh2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>v</sup><sup>2</sup> � <sup>ξ</sup><sup>2</sup> � � <sup>p</sup>

λ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos θ<sup>R</sup> cos θ<sup>S</sup>

2 cos θ<sup>S</sup>

3 5 �1

p (26)

(25)

(27)

The diffraction efficiency of slanted lossless reflection volume grating can be written as

<sup>ν</sup> <sup>¼</sup> <sup>i</sup>πdΔ<sup>n</sup>

<sup>ξ</sup> ¼ � <sup>Γ</sup><sup>d</sup>

Figure 12 shows the diffraction efficiency of the lossless volume gratings as a function of the ξ, for the values of ν = π/4, π/2 and 3π/4. The figure shows the sensitivity of a grating with ν = π/4 and a peak efficiency of 43%, a grating with ν = π/2 and η = 84 %, and the corresponding values for 3π/4 and η = 96 %. For ν = π/2, the diffraction efficiency drops to zero in all cases when

η<sup>R</sup> ¼ 1 þ

2 4

frequency for optimal refractive index modulation with grating thickness in 0.5, 2, 5, and 10 mm.

Figure 12. Reflection grating: diffraction efficiency η of lossless volume grating as a function of the parameter ξ for various values of the parameter ν.

When the wavelength and the angle are gradually out of the Bragg condition, the parameters ξ is obtained as follows:

$$\xi = \frac{\pi f d}{\cos \left( \phi - \theta\_0 \right) - \left( f \lambda\_0 / n\_0 \right) \cos \phi} \left( \Delta \theta \sin \theta\_0 + \frac{f \Delta \lambda}{2 n\_0} \right) \tag{28}$$

For an unslanted grating (φ = 0), the Bragg condition is obeyed; then cos θ<sup>R</sup> = � cos θ<sup>S</sup> = cos θ0, the Eq. (25) becomes to

$$\eta\_0 = \tanh^2 \nu = \tanh^2 \left(\frac{\pi d \Delta n}{\lambda \cos \theta\_0}\right) \tag{29}$$

By increasing of grating thickness d or refractive index modulation Δn, the diffraction efficiency asymptotically approaches the 100% value with the hyperbolic tangent function.

If the diffraction efficiency η<sup>0</sup> could be predetermined at a certain level, the value could be used for designing a reflection volume grating. The interrelationships between refractive index modulation, thickness, and incident Bragg angle θ<sup>0</sup> could be expressed by Eq. (29):

$$
\Delta n = \frac{\lambda \cos \theta\_0 \tanh^{-1} \sqrt{\eta\_0}}{\pi d} \tag{30}
$$

Figure 13 illustrates the interrelation between refractive index modulation, thickness, and predetermined diffraction efficiency η0. The three values of predetermined diffraction efficiency are 90% which correspond to 10 dB transmitted beam attenuation, 99% (20 dB) and 99.9% (30 dB) at λ = 532 nm, respectively. As shown in Figure 13, refractive index modulation Δn is

Figure 13. Dependence of refractive index modulation which secured predetermined diffraction efficiency on the grating thickness. Diffraction efficiency: η<sup>0</sup> = 90 %, 99%, and 99.9%. Normal incidence, λ = 532 nmand n<sup>0</sup> = 1.5.

less than 1000 ppm when the grating thickness is more than 1 mm with η<sup>0</sup> = 99 %. Therefore, reflecting volume gratings should be thick enough with relatively low values of refractive index modulation to secure predetermined diffraction efficiency.

When the wavelength and the angle are gradually out of the Bragg condition, the parameters ξ

For an unslanted grating (φ = 0), the Bragg condition is obeyed; then cos θ<sup>R</sup> = � cos θ<sup>S</sup> = cos θ0,

By increasing of grating thickness d or refractive index modulation Δn, the diffraction efficiency asymptotically approaches the 100% value with the hyperbolic tangent function.

If the diffraction efficiency η<sup>0</sup> could be predetermined at a certain level, the value could be used for designing a reflection volume grating. The interrelationships between refractive index

<sup>Δ</sup><sup>n</sup> <sup>¼</sup> <sup>λ</sup> cos <sup>θ</sup>0tanh�<sup>1</sup> ffiffiffiffiffi

Figure 13 illustrates the interrelation between refractive index modulation, thickness, and predetermined diffraction efficiency η0. The three values of predetermined diffraction efficiency are 90% which correspond to 10 dB transmitted beam attenuation, 99% (20 dB) and 99.9% (30 dB) at λ = 532 nm, respectively. As shown in Figure 13, refractive index modulation Δn is

Figure 13. Dependence of refractive index modulation which secured predetermined diffraction efficiency on the grating

thickness. Diffraction efficiency: η<sup>0</sup> = 90 %, 99%, and 99.9%. Normal incidence, λ = 532 nmand n<sup>0</sup> = 1.5.

modulation, thickness, and incident Bragg angle θ<sup>0</sup> could be expressed by Eq. (29):

<sup>ν</sup> <sup>¼</sup> tanh<sup>2</sup> <sup>π</sup>dΔ<sup>n</sup>

λ cos θ<sup>0</sup> � �

> η0 p

Δθ sin <sup>θ</sup><sup>0</sup> <sup>þ</sup> <sup>f</sup> Δλ

� �

2n<sup>0</sup>

<sup>π</sup><sup>d</sup> (30)

(28)

(29)

� � � ð Þ <sup>f</sup> <sup>λ</sup>0=n<sup>0</sup> cos <sup>φ</sup>

<sup>η</sup><sup>0</sup> <sup>¼</sup> tanh<sup>2</sup>

<sup>ξ</sup> <sup>¼</sup> <sup>π</sup>f d cos φ � θ<sup>0</sup>

is obtained as follows:

112 Holographic Materials and Optical Systems

the Eq. (25) becomes to

The angular selectivity of unslanted reflection volume grating could be determined by substituting Eqs. (26) and (28) to Eq. (25) at Δλ =0:

$$\eta\_R(\Delta\theta) = \left[1 + \frac{1 - \left(\frac{\lambda^f \sin \theta\_0 \Delta\theta}{\Delta n}\right)^2}{\sinh^2 \sqrt{\left(\frac{2\pi n\_0 d \Delta n}{\lambda\_0^2 f}\right)^2 - \left(\frac{2\pi n d \sin \theta\_0 \Delta\theta}{\lambda}\right)^2}}\right]^{-1} \tag{31}$$

To determine angular selectivity ΔθHWFZ at HWFZ level, the diffraction efficiency reaches zero value at multiple points when ν is not equal to ξ:

$$
\sqrt{\nu^2 - \xi^2} = j\pi \tag{32}
$$

where j = 1, 2, ⋯ n, ⋯. The HWFZ angular selectivity could be considerably simplified for unslanted gratings with diffraction efficiency of Eq. (28) at j = 1:

$$
\Delta\Theta\_{\rm R}^{\rm HWEZ} = \frac{\lambda\sqrt{\left(\tan h^{-1}\sqrt{\eta\_0}\right)^2 + \pi^2}}{2\pi n\_0 d \sin \theta\_0} \tag{33}
$$

Figure 14 shows the dependence of angular selectivity on volume grating thickness at different incident Bragg angles θ<sup>0</sup> for a 99% efficiency grating. As one can see, the thicker the grating, the wider the angular selectivity is. For instance, 7 mrad HWFZ selectivity is secured at θ<sup>0</sup> = 2° for 1 mm thick grating or at θ<sup>0</sup> = 10° for 2.01 mm grating thickness.

Figure 14. Angular selectivity (HWFZ) of reflecting volume grating with 99% diffraction efficiency at λ = 532 nm and n<sup>0</sup> = 1.5 on spatial frequency for optimal refractive index modulation with grating thickness in 0.5 mm, 2 mm, 5 mm, and 10 mm.

By the same way as it was described above for angular selectivity, spectral selectivity could be expressed by substitution of Eqs. (26) and (28) to Eq. (25) at Δθ = 0:

$$\eta\_R(\Delta\lambda) = \left[1 + \frac{1 - \left(\frac{\lambda\_0 f^2 \Delta\lambda}{2m\_0 \Delta n}\right)^2}{\sinh^2\sqrt{\left(\frac{2m\_0 d \Delta n}{\lambda\_0^2 f}\right)^2 - \left(\frac{\pi d f \Delta\lambda}{\lambda\_0}\right)^2}}\right]^{-1} \tag{34}$$

The HWFZ wavelength selectivity also could be considerably simplified for un-slanted gratings with diffraction efficiency of Eq. (29):

$$
\Delta\lambda\_R^{HWFZ} = \frac{\lambda\_0 \sqrt{\left(\text{atanh}\sqrt{\eta\_0}\right)^2 + \pi^2}}{\pi df} \tag{35}
$$

Figure 15 shows dependence of wavelength selectivity on spatial frequency for different grating thicknesses. HWFZ wavelength selectivity of reflection volume grating could be easily varied from values below 0.1 nm to more than a dozen nm by proper choosing of grating parameters.

Figure 15. Wavelength selectivity (HWFZ) of reflecting volume grating with 99% diffraction efficiency at λ = 532 nm and n<sup>0</sup> = 1.5 on spatial frequency for optimal refractive index modulation with grating thickness in 0.5, 2, 5, and 10 mm.
