**8. Possible applications of the holograms**

The main characteristic features of holograms on CaF<sup>2</sup> crystals with color centers are as follows: (1) the opportunity of preparing thick holograms with high angular and spectral selectivities, (2) an extremely high hologram stability with respect to the effects of optical radiation and temperature, and (3) an opportunity to transform the composition of color centers forming the holographic grating via the postexposition photothermal treatment, i.e., to change the type of hologram at the desired readout wavelength. The volume holographic elements with such features and spatial resolution of about 5000 lines mm−1 and more can be useful for solving many problems. Below, two possible applications of such elements are discussed.

#### **8.1. Plane angle measure**

A conventional plane‐angle measure is a regular polyhedral‐fused silica prism, whose angles are set by normals to its faces. Each normal is implemented physically by the autocollimator axis when the cross hairs in its focal plane are aligned with the image arising as a result of collimated beam reflection from the prism face. The angle between two normals is reproduced by rotating the prism around an axis perpendicular to the autocollimator measuring plane. Thus, the set of angles stored by the prism is determined by the mutual positions of its faces and is reproduced using the light beam reflected from them and sample rotation. The prism reproduces angles with effective values close to *m*(360/*n*)°, where *n* is the number of lateral faces and *m* = 1, 2, …, *n* – 1 (actually, *n* ≤ 72). Such a fused silica prism has drawbacks as follows: a large weight and size (1.2 kg and more, dimensions 170 × 20 mm and more); the discreteness of the formed plane angle circular scale that is limited by the number of faces (up to 36 faces); a low production efficiency (the prism preparation is, in principle, the custom‐made, time‐consuming process); and the hazard of spontaneous sharp changes, when storing, in the optical and geometric characteristics (so‐called devitrification).

A new multivalued plane‐angle measure based on the holographic principle that has a number of significant advantages over the fused silica prism was proposed [23–26].

This element (referred to below as sample) is a parallelepiped made of a photochromic CaF<sup>2</sup> crystal in which a system of superimposed holograms is recorded. Their mutual spatial positions form a set of angles (the multivalued holographic measure) stored by this element. The exposure of this sample to a reference laser beam induces a response in the form of several diffracted beams. Depending on the recording method, they arise successively or simultaneously upon rotating the sample and cover a limited range of angles; these beams are recorded by photoelectric detector. The rotation of the sample makes it possible to form a full angular scale.

Angles between directions set by the holograms are the functional analogs of angles between the fused silica prism normals. Hence, this element can be referred to as a holographic prism, HP. For this prism, an angle between holograms forming it can be fairly small (of the order of an arc minute). This circumstance provides a high discreteness of the realized circular scale and, correspondingly, high accuracy of angular measurements.

Earlier, the colloid cubic superlattice with the lattice parameter of *~*20 nm was observed on the

the opportunity of preparing thick holograms with high angular and spectral selectivities, (2) an extremely high hologram stability with respect to the effects of optical radiation and temperature, and (3) an opportunity to transform the composition of color centers forming the holographic grating via the postexposition photothermal treatment, i.e., to change the type of hologram at the desired readout wavelength. The volume holographic elements with such features and spatial resolution of about 5000 lines mm−1 and more can be useful for solving many problems. Below,

A conventional plane‐angle measure is a regular polyhedral‐fused silica prism, whose angles are set by normals to its faces. Each normal is implemented physically by the autocollimator axis when the cross hairs in its focal plane are aligned with the image arising as a result of collimated beam reflection from the prism face. The angle between two normals is reproduced by rotating the prism around an axis perpendicular to the autocollimator measuring plane. Thus, the set of angles stored by the prism is determined by the mutual positions of its faces and is reproduced using the light beam reflected from them and sample rotation. The prism reproduces angles with effective values close to *m*(360/*n*)°, where *n* is the number of lateral faces and *m* = 1, 2, …, *n* – 1 (actually, *n* ≤ 72). Such a fused silica prism has drawbacks as follows: a large weight and size (1.2 kg and more, dimensions 170 × 20 mm and more); the discreteness of the formed plane angle circular scale that is limited by the number of faces (up to 36 faces); a low production efficiency (the prism preparation is, in principle, the custom‐made, time‐consuming process); and the hazard of spontaneous sharp changes, when storing, in the optical and geometric characteristics

A new multivalued plane‐angle measure based on the holographic principle that has a number

This element (referred to below as sample) is a parallelepiped made of a photochromic CaF<sup>2</sup> crystal in which a system of superimposed holograms is recorded. Their mutual spatial positions form a set of angles (the multivalued holographic measure) stored by this element. The exposure of this sample to a reference laser beam induces a response in the form of several diffracted beams. Depending on the recording method, they arise successively or simultaneously upon rotating the sample and cover a limited range of angles; these beams are recorded by photoelectric detector. The rotation of the sample makes it possible to form a full angular scale. Angles between directions set by the holograms are the functional analogs of angles between the fused silica prism normals. Hence, this element can be referred to as a holographic prism,

of significant advantages over the fused silica prism was proposed [23–26].

crystals with color centers are as follows: (1)

surface of electron‐irradiated crystals (see [22] and references therein).

**8. Possible applications of the holograms**

The main characteristic features of holograms on CaF<sup>2</sup>

two possible applications of such elements are discussed.

**8.1. Plane angle measure**

424 Holographic Materials and Optical Systems

(so‐called devitrification).

A holographic prism, as noted above, can be implemented in two modifications, I and II, that produce the set of angles. For modification I, the temperature‐controlled housing with a sample is mounted on the rotation table, their rotation axes coinciding. The interfering beams are in the plane parallel to the table. After recording the first hologram, the sample is turned by the assigned angle to record the next hologram, and so on. Several superimposed holograms form HP‐I. Then, the sample is mounted on the rotation stage and the diffraction responses of the holograms appear successively when rotating the table (**Figure 15**) [23]. The half width of angular selectivity profile of a 14 × 8.5 × 7.7 mm sample with the hologram thickness of 8.5 mm is 1.8′.

**Figure 15.** General view (a) and top view (b) of HP‐I holographic prism: (*1*) is the reference laser, (*2*) is the incident beam, (*3*) is the holographic prism, (*4*) is the rotation stage, (*5*) is the diffracted beam, and (*6*) is the photodetector.

To record an HP‐II, one should use an interferometer wherein the coherent beams cross each other at an angle of 90°. A sample is installed in their interference region on the table of the rotational device. The first beam is aligned with the device axis, whereas the second one, as indicated above, is perpendicular to the first beam. The system of holograms forming HP‐II is recorded successively, and the crystal is rotated by a specified angle after each recording cycle. Holograms thus recorded can be reconstructed simultaneously by the same reference beam that has a direction the same as the first recording beam. The diffracted (signal) beams are oriented perpendicularly to the reference beam. Angles between the beams are the angles of crystal rotation in the course of prism recording. This method for recording the imposed holograms can be used to implement on condition that they are recorded and reconstructed by radiation with the same wavelength, so that the angle of beam convergence specified at recording is exactly reproduced during the reconstruction. The fan of diffracted beams emitted by HP‐II and the reference beam are shown in **Figure 16** [25]. Holograms were reconstructed with the reference beam of *~*1 mm in diameter that is much smaller than the hologram diameter (*~*8 mm); so, the diffracted beams formed extended enough lines on the screen.

**Figure 16.** Fan of diffracted beams emerging from HP‐II.

The uniquely small mass and dimensional characteristics of this angular measure (10 g and 0.5–1 cm<sup>3</sup> , respectively) make it possible to use such HP as a basis for developing devices for measuring/setting rotation angles that will combine two antinomic requirements such as the mobility and high accuracy of angular measurements (see [25] for details).

#### **8.2. Volume holographic elements for mid-IR spectral range**

The specific features of holograms listed above allow for taking the holographic elements based on additively colored CaF<sup>2</sup> crystals to be quite promising as the transmission and reflection filters in the mid‐IR spectral range. Below, the expected characteristics of holographic filters based on CaF<sup>2</sup> crystals with photothermally transformed holograms are discussed.

The absorption spectrum of CaF<sup>2</sup> crystal after special photothermal treatment is shown in **Figure 17**. An absorption band attributed to the short‐wavelength quasi‐colloidal centers (*λ*max ≅ 2 μm) is present in the spectrum.

Under suggestions that (a) the absorption spectrum of the sample with a hologram is similar to that shown in **Figure 17**, (b) the holographic grating plane width is about 0.2 of the grating period *d* (e.g., 1 μm width at 4.5 μm period [13]), and (c) *~*90% of color centers are located within the holographic planes (and, hence, the same fraction of the sample absorption originates from the planes), it is possible to estimate the expected characteristics of transmission and reflection holograms read out with 3.5 μm radiation.

The absorption spectrum shown in **Figure 17** ensures recording of efficient transmission and reflection holograms in the crystal samples of several millimeters thick. When neglecting the absorption of readout radiation, it is possible to use the Kogelnik theory to calculate the phase hologram parameters [27].

**Figure 17.** Absorption spectrum of the photothermally treated sample.

The uniquely small mass and dimensional characteristics of this angular measure (10 g and

measuring/setting rotation angles that will combine two antinomic requirements such as the

The specific features of holograms listed above allow for taking the holographic elements

tion filters in the mid‐IR spectral range. Below, the expected characteristics of holographic

**Figure 17**. An absorption band attributed to the short‐wavelength quasi‐colloidal centers

Under suggestions that (a) the absorption spectrum of the sample with a hologram is similar to that shown in **Figure 17**, (b) the holographic grating plane width is about 0.2 of the grating period *d* (e.g., 1 μm width at 4.5 μm period [13]), and (c) *~*90% of color centers are located within the holographic planes (and, hence, the same fraction of the sample absorption originates from the planes), it is possible to estimate the expected characteristics of transmission

The absorption spectrum shown in **Figure 17** ensures recording of efficient transmission and reflection holograms in the crystal samples of several millimeters thick. When neglecting the

crystals with photothermally transformed holograms are discussed.

mobility and high accuracy of angular measurements (see [25] for details).

**8.2. Volume holographic elements for mid-IR spectral range**

and reflection holograms read out with 3.5 μm radiation.

, respectively) make it possible to use such HP as a basis for developing devices for

crystals to be quite promising as the transmission and reflec-

crystal after special photothermal treatment is shown in

0.5–1 cm<sup>3</sup>

based on additively colored CaF<sup>2</sup>

The absorption spectrum of CaF<sup>2</sup>

(*λ*max ≅ 2 μm) is present in the spectrum.

**Figure 16.** Fan of diffracted beams emerging from HP‐II.

426 Holographic Materials and Optical Systems

filters based on CaF<sup>2</sup>

According to suggestions (b) and (c), the spectral dependence of the modulation amplitude of the absorption coefficient *δα*(*ν*) of the crystal with a hologram can be calculated by multiplication of the absorption spectrum shown in **Figure 17** by coefficient such as (4.5 μm/1 μm) × 0.9. Kramers‐Kronig relation (1) allows the estimation of the corresponding spectral dependence of the modulation amplitude of the refractive index *δn*(*ν*) shown in **Figure 18**.

The modulation amplitude of the refractive index *δn* at the readout wavelength *λ* = 3.5 μm equals to 6.82 × 10-4. The hologram thickness *T* is determined by the phase incursion *ν*<sup>t</sup> = *π*/2 that provides the 100% diffraction efficiency of the phase hologram. To calculate the optimum thickness of the hologram, an expression can be used as follows:

$$T = \nu\_i \lambda \cos \theta\_0 / \pi \delta n,\tag{9}$$

where *θ*<sup>0</sup> is the Bragg angle for the readout radiation inside the holographic medium (*θ*<sup>0</sup> = 15° for the grating period *d* = 4.5 μm). The thickness of a hologram with the above parameters equals to 2.45 mm.

**Figure 18.** Spectral dependence of the modulation amplitude of the refractive index for a crystal with the hologram.

It should be noted that the nonsinusoidal nature of the hologram profile in this crystal results in the appearance of several diffraction orders from the recorded hologram (the diffraction from several spatial harmonic components). The fraction of the first diffraction order that can be used for holographic filtering at the Bragg angle is about 65% of the total diffraction efficiency of the hologram [13].

The spectral selectivity of the transmission hologram, *δλ*, can be approximately calculated using an equation as follows:

$$
\delta\lambda \approx \frac{\lambda \text{dctg}}{T} \tag{10}
$$

At the above values of grating parameters, *δλ* = 24 nm.

Angular selectivity of the hologram δ*θ* is given by

$$
\delta\Theta = \lambda \,\, \xi\_{\eta} \langle \Big(2\pi nT \sin \theta\_{0} \Big),
\tag{11}
$$

where *n* = 1.43 is the refraction index of CaF<sup>2</sup> crystal at the readout wavelength. The misalignment parameter *ξ*<sup>t</sup> is proportional to the deviation of the readout angle (in the medium) from *θ*<sup>0</sup> . The *ξ*t value of ∼2.7 corresponds to DE equal to zero. Under these conditions, the angular selectivity equals to 0.13° according to Eq. (11).

When using the sample with hologram as a reflection‐type filter, the wavelengths of recording (*λ*rec) and reflected (*λ*read) radiations are different but connected—according to the Bragg condition—via the hologram period *d*:

$$d = \frac{\lambda\_{\rm nc}}{2 \sin \theta\_{\rm nc}} = \frac{\lambda\_{\rm rad}}{2 \sin \theta\_{\rm rad}} \tag{12}$$

For the actual case *θ*read = 90° at *λ*read = 3.5 μm, the period of reflection grating *d* equals to 1.22 μm.

Recording of the reflection hologram readout with 3.5 μm radiation can be executed with the 532 nm radiation in the symmetric transmission scheme with the 2*θ*rec = 25.2° angle between the recording beams (**Figure 19**).

The assumption that relation between grating period and holographic plane thickness is the same for both transmission and reflection grating allows using the same *δn*(*ν*) dependence (**Figure 18**) to calculate the phase incursion for the reflection hologram.

The DE of the reflection hologram increases with an increase in the phase incursion, *ν*<sup>r</sup> :

It should be noted that the nonsinusoidal nature of the hologram profile in this crystal results in the appearance of several diffraction orders from the recorded hologram (the diffraction from several spatial harmonic components). The fraction of the first diffraction order that can be used for holographic filtering at the Bragg angle is about 65% of the total diffraction

**Figure 18.** Spectral dependence of the modulation amplitude of the refractive index for a crystal with the hologram.

The spectral selectivity of the transmission hologram, *δλ*, can be approximately calculated

is proportional to the deviation of the readout angle (in the medium) from *θ*<sup>0</sup>

value of ∼2.7 corresponds to DE equal to zero. Under these conditions, the angular selectivity

*<sup>T</sup>* (10)

/(2*πnT* sin *θ*0), (11)

crystal at the readout wavelength. The misalignment

. The

efficiency of the hologram [13].

428 Holographic Materials and Optical Systems

using an equation as follows:

parameter *ξ*<sup>t</sup>

*ξ*t

*δλ* <sup>≈</sup> *<sup>λ</sup>d*ctg *<sup>θ</sup>* \_\_\_\_\_\_\_0

At the above values of grating parameters, *δλ* = 24 nm.

Angular selectivity of the hologram δ*θ* is given by

*δθ* = *λ ξ<sup>t</sup>*

where *n* = 1.43 is the refraction index of CaF<sup>2</sup>

equals to 0.13° according to Eq. (11).

 *η* = th2 (*νr*), (13)

**Figure 19.** The scheme of recording/readout the reflection hologram. *W*<sup>1</sup> and *W*<sup>2</sup> are the recording beams, *R* and *S* are readout and diffracted beams, respectively, *K* = 2*π*/*d* is the grating vector.

where *ν<sup>r</sup>* <sup>=</sup> \_\_\_\_\_\_ *πΔnT λ* sin *θ*<sup>0</sup> . For *δn* = 6.82 × 10-4, the thickness of a hologram with DE = 100% equals to 6 mm. Absorption in such sample does not exceed 3%.

The spectral and angular selectivity of the reflection hologram can be expressed through the misalignment parameter *ξ*<sup>r</sup> . The *ξ*<sup>r</sup> value of 3.5 corresponds to the deviation of pitch angle *θ*<sup>0</sup> from its magnitude of 90°; at the latter, DE ≅ 0 and the *ξ*<sup>r</sup> parameter and the spectral selectivity are connected through a relation as follows:

$$
\delta\lambda = -\frac{\xi\_r \lambda^2}{\left(2\pi m T \sin \theta\_o\right)}\tag{14}
$$

At the above *ξ*<sup>r</sup> value, the spectral |*δλ*| and angular *δϕ* selectivities are equal to 1.3 nm and 0.02°, respectively.

These estimates demonstrate the possibility of using the holographic elements based on CaF<sup>2</sup> crystals with color centers as the volume narrow‐band transmission and reflection filters for the mid‐IR spectral range.
