**5. Hologram profile**

Due to the diffusion‐drift recording mechanism, the hologram profile does not reproduce the sinusoidal distribution of light intensity in the fringe pattern. The flows of electrons and vacancies off their maxima to minima result in the compression of holographic planes. As a result, several diffraction orders are observed. Below, the profile of a hologram recorded by 532 nm laser with a moderate DE of *~*10% in the first order (the readout wavelength being the same) is discussed [13].

F**igure 7** shows images of the 15 × 15 μm<sup>2</sup> area of this sample obtained using the confocal laser scanning microscope (LSM) in (i) the light of the crystal luminescence excited by argon ion laser operating at 514.5 nm and (ii) the transmitted excitation light. The luminescence is due to *M* and *M*<sup>A</sup> + color centers (the *M*<sup>A</sup> + -center is the *M*‐center in which the Na<sup>+</sup> ion present in the crystal as a trace impurity is incorporated). The transversal profiles of the grating obtained from the images shown in **Figure 7** are presented in **Figure 8**. The transmittance profile is close to the sine curve, whereas the luminescence profile deviates substantially from this shape.

**Figure 7.** Images of the 15 μm × 15 μm area of the sample with a holographic grating obtained using confocal LSM: (a) in the light of the crystal luminescence excited by an argon ion laser operating at 514.5 nm and (b) in the transmitted excitation light.

Fluorite Crystals with Color Centers: A Medium for Recording Extremely Stable but Broadly Transformable Holograms http://dx.doi.org/10.5772/66114 415

**5. Hologram profile**

414 Holographic Materials and Optical Systems

same) is discussed [13].

+

to *M* and *M*<sup>A</sup>

excitation light.

F**igure 7** shows images of the 15 × 15 μm<sup>2</sup>

color centers (the *M*<sup>A</sup>

+

Due to the diffusion‐drift recording mechanism, the hologram profile does not reproduce the sinusoidal distribution of light intensity in the fringe pattern. The flows of electrons and vacancies off their maxima to minima result in the compression of holographic planes. As a result, several diffraction orders are observed. Below, the profile of a hologram recorded by 532 nm laser with a moderate DE of *~*10% in the first order (the readout wavelength being the

scanning microscope (LSM) in (i) the light of the crystal luminescence excited by argon ion laser operating at 514.5 nm and (ii) the transmitted excitation light. The luminescence is due

crystal as a trace impurity is incorporated). The transversal profiles of the grating obtained from the images shown in **Figure 7** are presented in **Figure 8**. The transmittance profile is close to the sine curve, whereas the luminescence profile deviates substantially from this shape.

**Figure 7.** Images of the 15 μm × 15 μm area of the sample with a holographic grating obtained using confocal LSM: (a) in the light of the crystal luminescence excited by an argon ion laser operating at 514.5 nm and (b) in the transmitted


area of this sample obtained using the confocal laser

ion present in the

**Figure 8.** Transversal profiles of the grating obtained from the images shown in **Figure 7**: the transmittance profile (squares) and its sine‐approximation (dashed curve); the luminescence profile (circles) and its best fit with the sum of the first three harmonic components, the amplitude ratio being 100:50:19 (solid curve).

Due to spatial filtering of optical signals with the pinhole diaphragm, confocal microscopy provides enhanced selectivity and contrast of fluorescent and reflection images: only light emitted (or scattered) within a tiny focal volume is collected at the photodetector. 3D images are constructed of "optical slices" obtained by layer‐by‐layer scanning of an object at different focal positions. Contrastingly, the transmitted‐light images of the same thin layers include signal coming from the light cone passing through entire thickness of the sample, and the major contribution to the resulting picture is given by a number of defocused images of hologram sections. Summation of such patterns results in a nearly sinusoidal distribution, even if the original grating consisted of sharp thin lines.

Higher diffraction orders of a volume holographic grating observed at the Bragg angles *θ*<sup>m</sup> corresponding to spatial frequencies *K*m *=* 2*πm*/*d,* where *d* is the spatial period of the pattern, imply its nonsinusoidal shape. In order to reconstruct the spatial dependences of the refractive index *n*(*x*) and absorption coefficient *α*(*x*) of the grating (spatial profiles), we apply partial Fourier series with harmonic coefficients *δn*m and *δα*m obtained from measured angular dependences of diffraction efficiencies for respective orders and alternating signs by analogy with Fourier expansion of the truncated cosine function: \_2*πmx*

$$
\delta n(\mathbf{x}) = \sum\_{\mathbf{m}} (-1)^{m+1} \delta \, n\_{\mathbf{m}} \cos \left(\frac{2\pi mx}{d}\right),
\tag{2}
$$

$$
\delta a(\mathbf{x}) = \sum\_{\mathbf{m}} (-1)^{m+1} \delta \, a\_{\mathbf{m}} \cos \left(\frac{2\pi mx}{d}\right) \tag{3}
$$

$$
\delta \alpha (\text{x }) = \sum\_{\overline{m}} (-1)^{m+1} \delta \, a\_{\overline{m}} \cos \left( \frac{2 \pi m \overline{x}}{d} \right) \tag{3}
$$

Here, *x* is the spatial coordinate along the grating vector *K*.

To determine the values of harmonic components, the angular dependences of the hologram response for three (most intense) diffraction orders can be used. The angular dependences of the zeroth and ±1st diffraction orders read out at *λ* = 532 nm are symmetric with respect to the normal incidence as well as those of the higher diffraction orders (**Figure 9**).

**Figure 9.** Angular dependences of the zeroth and ±1 diffraction orders for the grating when read out at 532 nm. Circles (*η*0 ) and squares (*η*±1) are referred to the experimental data; dashed, and solid curves correspond to the theoretical approximation using Eqs. (5) and (4), respectively.

Using a criterion given in Refs. [14, 15], namely, that of the effectively equal values of diffraction efficiencies in the +1 and ‐1 diffraction orders, one can conclude that the hologram has an amplitude‐phase nature with the amplitude and refractive index gratings being in phase. Therefore, to fit the experimental angular dependences, the following expressions can be used for the angular dependences of the diffraction efficiencies *η*m, *η*<sup>0</sup> in the *m*th (*m* ≠ 0) and zero orders. \_⁄2) \_⁄2)

## 10.15.

$$\eta\_{\rm in}(\theta) = 2 \exp\left(-\frac{2 \,\mathrm{a}\_{\mathrm{o}} \, t}{\cos \theta}\right) \frac{\mathrm{\boldsymbol{\kappa}\_{1}^{2}} + \mathrm{\boldsymbol{\kappa}\_{2}^{2}}}{\mathrm{\boldsymbol{\varepsilon}}\_{\mathrm{o}}} \left\{ \cosh\left[\frac{\overline{\omega\_{\rm o}} \, t \cos(\theta/\mathfrak{d})}{\cos \theta}\right] - \cos\left[\frac{\overline{\omega\_{\rm o}} \, t \sin(\theta/\mathfrak{d})}{\cos \theta}\right] \right\},\tag{4}$$

$$\begin{aligned} \eta\_{\alpha}(\theta) &= 2 \exp\left(-\frac{2 \,\mathrm{u}\_{0} \, t}{\cos \theta}\right) \frac{\kappa\_{1}^{2} + \kappa\_{2}^{2}}{z\_{\mathrm{u}\_{0}}} \left\{ \cosh\left[\frac{\sqrt{\varpi\_{0}} \, t \cos\left(\psi/\mathfrak{z}\right)}{\cos \theta}\right] - \cos\left[\frac{\overline{\varpi\_{0}} \, t \sin\left(\psi/\mathfrak{z}\right)}{\cos \theta}\right] \right\}, \\\\ \eta\_{\mathrm{u}}(\theta) &= \frac{\exp\left(-\frac{2 \,\mathrm{u}\, t}{\cos \theta}\right)}{z\_{\mathrm{u}}} \left\{ \frac{\beta^{2} + z\_{\mathrm{u}}}{2} \cosh\left[\frac{\overline{\varpi\_{0}} \, t \cos\left(\psi/\mathfrak{z}\right)}{\cos \theta}\right] - \frac{\beta^{2} - z\_{\mathrm{u}}}{2} \cos\left[\frac{\overline{\varpi\_{0}} \, t \sin\left(\psi/\mathfrak{z}\right)}{\cos \theta}\right] \right. \\\\ &+ \mathfrak{s} \sqrt{\varpi\_{0}} \, \sin\left(\psi/\mathfrak{z}\right) \text{sincin}\left[\frac{\overline{\varpi\_{0}} \, t \cos\left(\psi/\mathfrak{z}\right)}{\cos \theta}\right] - \mathfrak{s} \sqrt{\varpi\_{0}} \cos\left(\psi/\mathfrak{z}\right) \text{sincin}\left[\frac{\overline{\varpi\_{0}} \, t \sin\left(\psi/\mathfrak{z}\right)}{\cos \theta}\right] \end{aligned} \tag{5}$$

Here *α*<sup>0</sup> = 3.29 cm-1 is the mean absorption coefficient of the crystal at readout wavelength, *n*<sup>0</sup> = 1.43 is its mean refractive index, *t* is the thickness of the grating in *n*m, and *α*m are the modulation amplitudes of the *m*th harmonics of the refractive index and absorption coefficients, *κ*<sup>1</sup> *= πn*m*/λ, κ*<sup>2</sup> *= α*m/2,

$$\text{(8)} = \frac{4\pi \, n\_{\text{o}} \sin \theta\_{\text{w}}}{\lambda} (\sin \theta - \sin \theta\_{\text{w}}) \, \tag{6}$$

$$\mathbf{z}\_0 = \left[ (\ $^2 + 4(\kappa\_1^2 - \kappa\_2^2))^2 + (\$  \kappa\_1 \kappa\_2)^2 \right]^{1/2} \tag{7}$$

$$
\omega\_0 = \left\{ \left( \mathbb{1}^\nu + \mathbb{1}\left( \kappa\_1 - \kappa\_2 \right) \right) \cdot \left( \mathbb{1}^\nu \kappa\_1 \kappa\_2 \right) \right\} \tag{8}
$$

$$
\psi\_0 = \arccos \left\{ -\frac{\left\{ \mathbb{1}^\nu + 4(\kappa\_1^2 - \kappa\_2^2) \right\} \right}{z\_0} \right\} \tag{8}
$$

Modulation amplitudes for the *m*th harmonics of the absorption coefficient and refractive index found as the fit parameters are shown in **Table 1**.

Using data given in the second and third columns of **Table 1**, one may determine the ratios of amplitudes for the first three harmonic components of the hologram. These ratios are 100:58:22 and 100:42:14 for the absorption coefficient and refractive index, respectively. A difference between the ratios is probably caused by the presence of several types of color centers in the crystal and disproportionality of the spatial distributions of different type centers along the grating vector. Accordingly, the relative magnitudes of modulation amplitudes of the absorption coefficient and refractive index for different spatial harmonics appear to be similar but unequal.


**Table 1.** Grating parameters.

Fourier series with harmonic coefficients *δn*m and *δα*m obtained from measured angular dependences of diffraction efficiencies for respective orders and alternating signs by analogy with

(−1 )*<sup>m</sup>*+1 *δ n<sup>m</sup>* cos(

(−1 )*<sup>m</sup>*+1 *δ αm* cos(

To determine the values of harmonic components, the angular dependences of the hologram response for three (most intense) diffraction orders can be used. The angular dependences of the zeroth and ±1st diffraction orders read out at *λ* = 532 nm are symmetric with respect to the

Using a criterion given in Refs. [14, 15], namely, that of the effectively equal values of diffraction efficiencies in the +1 and ‐1 diffraction orders, one can conclude that the hologram has

**Figure 9.** Angular dependences of the zeroth and ±1 diffraction orders for the grating when read out at 532 nm. Circles

) and squares (*η*±1) are referred to the experimental data; dashed, and solid curves correspond to the theoretical

normal incidence as well as those of the higher diffraction orders (**Figure 9**).

\_2*πmx*

\_2*πmx*

*<sup>d</sup>* ), (2)

*<sup>d</sup>* ) (3)

Fourier expansion of the truncated cosine function:

Here, *x* is the spatial coordinate along the grating vector *K*.

*<sup>δ</sup>n*(*<sup>x</sup>* ) <sup>=</sup> <sup>∑</sup>*<sup>m</sup>*

416 Holographic Materials and Optical Systems

*δα*(*<sup>x</sup>* ) <sup>=</sup> <sup>∑</sup>*<sup>m</sup>*

(*η*0

approximation using Eqs. (5) and (4), respectively.

The hologram profile determined from the luminescence measurements (i.e., the spatial distribution of luminescent color centers) is adequately described by the sum of three harmonic components (amplitude ratio 100:50:19, **Figure 8**). This ratio does not differ strongly from the harmonics ratio for the absorption profile 100:58:22 that follows from the angular dependences of diffraction efficiency and represents the spatial distribution of all color centers forming the grating. This confirms that both the spatial profiles reconstructed from holographic and microscopic measurements are determined by the same spatial distribution of color centers.

The modulation amplitude of absorption coefficient found from the analysis of angular dependences shows that the concentration of colored centers between the holographic planes is small as compared to the average absorption of the crystal with the hologram. At the saturation value of DE, this concentration does not exceed several percents of the total amount of the centers. The overwhelming majority of color centers present in the crystal with hologram are located in the holographic planes.
