## Abstract

In this paper, we present a method for achieving precise evaluation of amplitude of refractive index modulation (RIM) inside the volume Bragg grating (VBG) recorded in photo-thermo-refractive (PTR) glasses. The Gaussian divergence characteristics of the incident beam is theoretically considered when calculating the angular selectivity of VBG, and the profiles of experimental angular selectivity curves are utilized to determine the value of RIM with one step. The effectiveness of our proposed method is experimentally verified. This method is applicable even if the full width at half maximum (FWHM) of VBG’s angular selectivity curve has the same order of magnitude as or is less than the beam divergence.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Volume Bragg gratings (VBGs) recorded in photo-thermo-refractive (PTR) glasses has been widely used in various applications, such as angular and wavelength filter [1–3], longitudinal and transverse mode selection in diode [4–6], angular beam deflectors/magnifiers [7,8], spectral beam combining [9], stretching and compression of laser pulses [10,11], etc. High interest in the application of VBGs recorded in PTR glasses arises from the fact that these glasses combine relatively large permanent refractive index modulation, with low optical losses and low coefficient of thermal expansion, which endows VBGs with high efficiency and high power tolerance [12]. Refractive index modulation (RIM) inside a VBG is controlled by three interconnected parameters: the dosage of UV-exposure and the thermal treatment temperature and duration [13–15]. Controlling the final refractive index change in VBG by changing these technological conditions is a key point for obtaining the desired diffraction efficiency and bandwidth. To achieve this goal, the value of RIM inside a VBG should be exactly determined first. Julien Lumeau has developed a physical model for describing the refractive index change after exposure to ionizing ultraviolet (UV) radiation and thermal development [15]. However, this model predicts the refractive index change with an accuracy of about 10%, and the formalism may differ for different PTR glasses. Moreover, this model cannot evaluate the RIM inside the VBG directly. To our knowledge, no efficient and highly accurate method has been revealed to solve this problem to data.

In this paper, we present an effective method for the precise evaluation of the amplitude of RIM inside the VBG recorded in PTR glasses. The angular selectivity curve (around Bragg angle) was utilized to estimate the amplitude of RIM. The divergence of the incident beam that could be approximated by a Gaussian function was taken into account. On this basis, angular selectivity curve under different RIM amplitude is numerically analyzed. By comparing the experimental angular selectivity curve with theoretical angular selectivity curve under different amplitude of index modulation, the quantitative value of RIM could be evaluated with an accuracy of near 10^{−5} order of magnitude. We have obtained similar values of RIM for each VBG at test wavelengths of 632.8 nm and 1064 nm for both TE and TM modes, which validates the effectiveness of our proposed method.

## 2. Theoretical basis

The analysis of the angular selectivity curves of the recorded diffraction gratings provides quantitative information of thickness and index modulation. For VBG recorded in PTR glass, the thickness is of the order of millimeters and, therefore, is easy to measure. In this case, the index modulation can be obtained by using Kogelnik’s theory [16]. The specific expression for calculation of transmission VBG’s diffraction efficiency (DE) is given as

*n*is the value of RIM,

*d*is the grating thickness,

*K*is the grating vector module,

*ϕ*is the slanted angle of the grating vector,

*λ*is the wavelength of the incident beam, and

*θ*and

_{r}*θ*are the angles of the incident and diffracted beams in the medium, respectively. Kogelnik’s theory is widely used to analyze VBGs, because it has analytical solution and is mathematically simple. However, Kogelnik’s theory is an approximate theory. S. Gallego et al. have studied the limits of applicability of Kogelnik’s theory to volume Bragg gratings [17]. They indicated that for spatial frequencies over 1000 lines/mm there is good agreement between the experimental data of the angular responses of the efficiency of the first diffracted order and the theoretical model proposed by Kogelnik. If Kogelnik’s theory cannot be applied, then it is necessary to use a rigorous theory such as the rigorous coupled wave analysis (RCW) [18]. For VBGs designed for this study, Kogelnik’s theory is applicable to use. Its accuracy was examined with RCW theory; the results of two theories showed a good agreement.

_{s}The divergence characteristic of incident beam must be considered when the full width at half maximum (FWHM) of VBG has the same order of magnitude as or is less than the beam divergence. According to [19], for a Gaussian incident beam, the normalized function of the beam intensity in the angular space can be written as

*θ*is the center incident angle of the Gaussian beam, and

_{i}*b*is the beam divergence. It can be determined aswhere

*λ*

_{0}and

*D*are the central wavelength and the beam diameter at the level of e

^{−2}[half width at e

^{−2}of the maximum (HWe

^{−2}M)] at central wavelength, respectively. The diffraction efficiency of Bragg grating for divergent beam could be given by

*η*(

*θ*) is the DE of Bragg grating for ideal plane monochromatic incident wave, which is given by Eqs. (1)-(3). After substitution of the numerical value of a Gaussian function integral, Eq. (6) could be written as

Figures 1(a)-1(d) correspond to a 2-mm-thick non-tilting transmittance VBG with a 1200 lines/mm spatial frequency. The Bragg incident wavelength is 1064 nm for TE polarization. When the RIM value corresponds to the first order maximum diffraction efficiency, the theoretical FWHM of angle selectivity curve outside the PTR glass is 0.58 mrad, which is close to the beam divergence of most lasers. The effect of index modulation to angular selectivity curve under different divergence is calculated based on Eq. (7). Figure 1(a) shows the DE of Bragg grating for ideal plane monochromatic incident wave, and that the angular selectivity curve changes continuously with the increase of RIM value. In other words, different RIM values correspond to different angular selectivity curves; thus, the experimental angular selectivity curves can be used to determine the index modulation amplitude. As shown in Figs. 1(b)-1(d), for identical index modulations, different beam divergence would also cause the change of angular selectivity curves; therefore, the divergence characteristics of the incident beam must be taken into account when the FWHM of VBG is close to or less than the beam divergence. Similar characteristics exist at different incident wavelengths and polarization states. The maximum DE of VBG is decrease with the increase of divergence, but the index modulation value corresponding to the maximum DE of VBG under Bragg condition is invariable.

In Fig. 2, the effect of index modulation on the angle selectivity curve for different incident wavelengths and polarizations is presented. The divergence for 632.8 nm and 1064 nm lasers is 0.7 mrad and 0.4 mrad, which is obtained from measurement. No matter TE or TM mode, the smaller the incident wavelength is, the more dramatic is the change of the angle selectivity curve with the index modulation. For TE mode, the change of angle selectivity curve profile with index modulation is faster than TM mode. The degree of angle selectivity curve’s differentiation for TE and TM modes is influenced by incident wavelength and VBG’s parameters such as period and slant angle, and this can be analyzed based on Eqs. (1)-(3). The value of index modulation corresponding to the maximum DE of VBG under Bragg condition is changed with the variation of incident wavelength and polarization mode.

For the same VBG whose RIM value is a constant, different incident wavelength and polarization state will cause the change of angle selectivity curve. As a result, the effect of incident wavelength and polarization on the angle selectivity curve can be used to verify the effectiveness of our method, because the RIM value evaluated at different test conditions should be theoretically the same.

In order to estimate the value of RIM precisely and efficiently, a Root-Mean-Square (RMS) error function is introduced to evaluate the coefficient of coincidence (CC) of the experimental and theoretical diffraction efficiency angle selectivity curves.

*DE*

_{exp}(

*θ*) and

_{i}*DE*

_{the}(

*θ*) are the experimental and the theoretical DE under the condition of incident at the angle

_{i}*θ*respectively, and

_{i}*N*is the number of collected incident angle points. When CC gets the minimum value, the corresponding index modulation amplitude is regarded as the actual numerical value of RIM inside the VBG.

## 3. Experimental setup

For the purpose of experimental demonstration of the proposed method, Non-tilting transmitting VBGs in PTR glass were recorded with a He-Cd laser (Kimmon Electric Model IK3501R-G) at a wavelength of 325 nm and an output power of 50 mW. The grating thickness and spatial frequency is 2 mm and 1200 lines/mm, respectively. In order to avoid the imbalance of the interfering beam phases caused by temperature gradients in the environment including vibrations in the air and surroundings that consequently deteriorates the interference pattern, Lloyd’s mirror scheme was applied to perform our interference experiments. Shear interference method was used to finely adjust the parallelism of the expanded beam. The exposure dosage of PTR glass samples was set to 100 mJ/cm^{2} (VBG-1#) and 500 mJ/cm^{2} (VBG-2#). The samples were then heat-treated at 500 °C for 120 minutes. During thermal treatment, samples were heated from room temperature to the development temperature at a rate of about 2 K/min and then, at the end of the development, they were cooled down to room temperature, in the furnace following the natural decrease of the furnace temperature. Finally, PTR glasses were repolished to remove defects on the surface caused by heat treatment.

The scheme of DE measurement of VBGs is shown in Fig. 3. The analyzed sample is positioned on a motorized rotational stage which is controlled via a computer. For a selection of polarization (TE or TM) of the incident beam with 632.8/1064 nm wavelength, a linear polarizer is used. Intensity of transmitted zero order beam (I_{0T}) and minus first order diffracted beam (I_{-1T}) are simultaneously measured by two power meters (Thorlabs, S120VC). Relative DEs are then calculated as in [12]. The angular selectivity curve is depicted directly after the measurement.

## 4. Results and discussions

Owing to the special characteristic of VBG, the Bragg condition provides a corresponding angle for different incident wavelength. The value of beam divergence is a key parameter that impacts the accuracy of RIM evaluation. Therefore, the M^{2} measurement system (Spiricon, M2-200s) is used to determine the divergence and intensity profile of the incident beam. For 632.8 nm and 1064 nm laser (homemade), the divergences are 0.7 mrad and 0.4 mrad, and M^{2} factors are 1.16 and 1.12, respectively. It shows that the actual beam profile can be approximated by a Gaussian function. The refraction effect of the incident beam between air and PTR glass should be considered to avoid the inconsistency between experimentally collected angle and the angle used for theoretical calculation, because experimentally collected angle is incident angle that outside of the PTR glass, yet the angle used for theoretical calculation is incident angle that inside of the PTR glass. The external incident angle of the PTR glass was converted to the internal incident angle of the PTR glass before calculating diffraction efficiency. The incident angle corresponding to the angle selectivity curve given in this paper is the external angle of the PTR glass.

Figure 4 shows the CC evaluation curves under different experimental conditions for VBG-1# and VBG-2#, the minimum difference between experimental and theoretical values of diffraction efficiency was determined as most probable RIM. The specific evaluated RIM values for VBG-1# and VBG-2# under four measurement conditions are summarized in Table 1: 3.04 × 10^{−4}, 3.03 × 10^{−4}, 3.20 × 10^{−4}, 3.12 × 10^{−4} and 4.78 × 10^{−4}, 4.65 × 10^{−4}, 4.17 × 10^{−4}, 4.37 × 10^{−4} respectively. The evaluated RIM values are very similar for each VBG. It indicates that the method proposed in this paper is effective experimentally. Slight difference exists between the evaluated RIM values for each VBG. It could be attributed to the following reasons. Firstly, there is a slight difference in VBG’s characteristic at different position, which was introduced during the fabrication of VBG. Testing at different positions could result in differences in evaluated RIM values. Besides, laser’s beam quality may also have influence on the evaluated RIM value. Ensuring the same testing position for each measurement and using lasers with better beam equality could be effective ways to improve the accuracy of evaluation. To improve the RIM homogeneity of VBG, solutions such as improving the intensity uniformity during the exposure process and temperature uniformity during the heat treatment process will be applied in the future work.

The experimental angle selectivity curves and theoretical angle selectivity curves according to the evaluated RIM values are shown in Fig. 5. The red dotted line and blue solid line corresponds to the experimental and theoretical data, respectively. It is shown that the experimental curves agree well with the theoretical curves. The shape of the experimental angle selectivity curve changes with the change of incident wavelength and polarization mode. If two or more peaks are present in the angular selectivity curve, then we can judge that the actual RIM value is larger than the theoretical RIM value that corresponds to the first order maximum diffraction efficiency directly. This method can be also applied to reflective VBGs recorded in PTR, and the process is similar to that of transmission VBGs.

## 5. Conclusions

In conclusion, a method to precisely determine the RIM in VBG recoded in PTR glasses with one step was proposed. The theoretical basis of the method was described in detail. The designed VBG was fabricated inside PTR glasses. The RIM value of VBGs are determined using our method. The results show that the evaluation precision of RIM value by this method can achieves to near approximately 10^{−5} order of magnitude. This method can be further applied to investigate the influence of UV-exposure dosage and the thermal treatment temperature and duration on the value of RIM inside of VBG recorded in PTR. Finally, we believe that this method is also applicable to other holographic recording material, like photopolymer, if we have a full understanding of these material’s properties such as recording and absorption characteristics.

## Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11604352, U1630140 and U1430121).

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