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Abstract
In this study, the realization of multiplexed volume Bragg gratings (VBGs) working as angle amplifiers in high power beam scanning system is theoretically and experimentally investigated. The design of the multiplexed VBG for the working wavelength of 1064 nm is described. We propose a cascaded multiplexed VBGs scheme that consists of 12 grating channels. Three 4-channel multiplexed VBGs were fabricated inside photo-thermo-refractive (PTR) glasses by multiple exposures and subsequent heat treatment. The test results show that this angle amplifier can achieve discrete angle deflection ranging from −45° to + 45°. The relative diffraction efficiency of all the grating channels is more than 80% and is almost polarization independent.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nonmechanical beam steering offers substantial benefits to optoelectronic systems such as laser radar, free-space laser communication, laser weapons, and laser countermeasures. In order to achieve nonmechanical beam control, multiple approaches have been explored, including microlens arrays [1], optical phased arrays based on dynamic liquid crystal gratings (LCOPA) [2,3], liquid crystal polarization gratings [4], liquid optical elements modulated via electro-wetting [5], etc. Significant advances have now been made in key performance areas such as size, agility, and power. However, for various reasons, it is difficult to achieve wide-angle capability in a single electro-optic element. A few different techniques for large angle discrete steering have been investigated [6]. One technique is based on birefringent prisms. Wide-angle capability can be achieved by using a binary set of birefringent prisms. But inherent disadvantages include that the size of the prisms cannot be made very large in diameter and the greater thickness of prisms when large angles are required. A second wide-angle step-steering approach is based on liquid crystal polarization gratings. Multiple stages may be cascaded to implement a coarse, binary beam steerer. However, with the increasing number of polarization grating (PG) stages within the beam steering stack, the overall efficiency is significantly affected by steering efficiency, scatter, absorption and Fresnel loss.
Volume Bragg Grating (VBG) recorded in photo-thermo-refractive (PTR) glass has advantages of high diffraction efficiency (DE), excellent wavelength selectivity as well as angle selectivity, high angle magnification and flexible design, and high power tolerance, making it a good candidate as an angle amplifier in high power beam scanning system [7–11]. Raytheon pursued this approach and demonstrated continuous beam steering over a field of regard greater than 45° [12]. However, to our knowledge, no detailed study has been reported on careful designing and experimental demonstration of multiplexed volume Bragg gratings as angle amplifiers. Hence, in this study, the realization of multiplexed volume Bragg gratings working as angle amplifiers in a high power beam scanning system is theoretically and experimentally investigated. We explored the possibility of multiplexing several grating channels in a single PTR glass first. Matrix-based algorithm was used for determining diffraction efficiencies of significant coupled waves in multiplexed VBG. The cross-talk problem in multiplexed VBGs was analyzed, and a scheme for cross-talk optimization is proposed. It was found that the index modulation dynamic range of PTR glasses and divergence of incoming light are the main reasons that restrict the maximum efficiency. To solve this problem, we proposed a cascaded multiplexed VBGs solution. Multiplexed VBGs were fabricated inside PTR glasses by multiple exposures and subsequent heat treatment. The test results show that this angle amplifier can achieve discrete angle deflection ranging from −45° to + 45°. The relative diffraction efficiency of each of the grating channels can reach a value more than 80% and is almost polarization independent.
2. Design theory
The angle amplification scheme based on multiplexed VBGs is shown in Fig. 1. The desired large angle scanning range can be divided into several sub-angle ranges each of which is within the deflection capability of the LCOPA. With the introduction of multiplexed VBG as the angle amplifier, the control and amplification of beam angle follows two steps. First, the output beam of the first stage addressing LCOPA is deflected to each sub-angle range by multiplexed VBG. Then the second LCOPA finely controls and fills the beam in each sub-angle range. Finally, the quasi-continuous emerging beam angle is realized in a large angle range.
The design flexibility of incident angle and exit angle makes VBG well suitable for beam deflection application domain. Figure 2(a) typically shows the diffraction schematic for a single transmission VBG, where the beam angle is deflected from to . According to Kogelnik’s theory [13], VBG’s diffraction characteristics for a fixed wavelength are mainly determined by four parameters: grating thickness (), grating period (),slant angle (), and amplitude of refractive index modulation (). The grating period and slant angle determine the Bragg incident angle and Bragg emergent angle, and vice versa. Thus, the grating period and slant angle can be calculated when the desired incident angle and emergent angle are determined. It can be analyzed using momentum or K-space diagram method. The k-space diagram is depicted in Fig. 2(b). Here, the radius of the circle is , where n is the bulk index of refraction of the material and the wavelength in free space. Two plane waves are present and are referred as the incident wave and diffracted wave, with wave vectors and , respectively. The grating vector is oriented perpendicular to the fringe planes and is of length . The Bragg condition
can be expressed as . Therefore, for each channel grating of multiplexed VBG, the grating period and slant angle can be calculated individually according to the incident and emergent angle requirement.
Fig. 2 (a) Typical diffraction schematic of single channel VBG. (b) momentum or K-space diagram for reconstruction of single channel VBG.
Further, it is necessary to design the thickness and index modulation amplitude of the grating. To achieve maximum diffraction efficiency, the optimal refractive index modulation of each channel grating needs to match the thickness of the grating. Besides this, we mainly consider the following factors: (1) In order to obtain more channels, the refractive index modulation should be as small as possible. (2) The grating thickness should be as small as possible to lower loss, reduce the light walking distance, and improve the uniformity of the grating in the thickness direction. (3) The diffraction efficiency should be as high as possible to improve the energy transfer efficiency. (4) Interference or cross-talk effect between multiple channel grating should be minimum. (5) The influence of laser source’s beam divergence on the ultimate diffraction efficiency should be taken into consideration.
Note that traditional Kogelnik’s theory is not applicable in analyzing multiplexed VBG directly. In this study, the matrix-based algorithm developed by G. B. Ingersoll was used for the diffraction efficiency calculation of the multiplexed VBG [14]. Rigorous coupled wave method was not used here because the approximations employed in the matrix-based algorithm can allow for an accurate calculation for our application and this efficient algorithm, in turn, provides efficient optimization algorithms for systems of multiplexed volume gratings.
3. Optimal design and fabrication of multiplexed VBG
As mentioned earlier, we should first determine the incident angle and emergent angle as the laser beam passes through the amplifier. The target angle range for our beam scanning system is set as ± 48°, and the system works at a wavelength of 1064 nm. Considering that a typical LCOPA can deflect a beam in the range of ± 3°~5° [2], we divided the scanning range into 13 sub-angle ranges. The sub-angle range which is emergent around 0° does not need to be zoomed by the multiplexed VBG, therefore, a 12-channel multiplexed VBG as angle amplifier is required.
Table 1 shows characteristic parameters of each channel VBG. The incident angles and emergent angle range from −4.8° to + 4.8° and −45° to + 45°, respectively. The interval of incident angle and emergent angle between adjacent channels is set as 0.8° and 7.5°. The grating channels for 1# to 6# are designed to be symmetrical with 7# to 12#, respectively. The grating period is identical and the slant angle is supplementary for the mutually symmetrical grating channel (e.g. 1# and 7#). It is illustrated that the grating period decreases with the increase of deflection angle () and the grating fringe is becomes more inclined at the same time. Considering that the index modulation dynamic range for traditional PTR glass is around 2 × 10−3 [15], the grating thickness was designed as 3 mm. The refractive index modulation (RIM) for each channel grating was almost the same and of value around 1.7 × 10−4.
Table 1. Characteristic parameters of each grating channel
The cross-talk effect on the maximum diffraction efficiency under different thickness conditions was investigated by using matrix-based algorithm mentioned in section 2 and the result is depicted in Fig. 3(a). The analysis results for channels 7# to 12# are omitted due to the symmetry of the channels. It should be noted that the RIM of each channel is simultaneously changed with the thickness in order to achieve the maximum diffraction efficiency of each channel grating as a single VBG. It can be seen that there exists a finite cross-talk effect between different channels when the grating thickness is less than 2 mm and the maximum diffraction efficiency significantly decreases with the decrease of thickness when the grating thickness is less than 1 mm. This could be explained as follows: The decrease of thickness would cause an increase in the full width at half maximum (FWHM) of angular selectivity curve, which is shown in Fig. 3(b), and as a result, the partial overlapping of the angular selectivity curve would cause a diffraction energy distribution in multiple grating channels. Thus, keeping the thickness greater than 2 mm is an effective way to avoid cross-talk effect.
Fig. 3 Dependence of maximum diffraction efficiency (a) and FWHM (b) on grating thickness for each channel grating inside a multiplexed VBG. The RIM of each channel is simultaneously changed with the thickness in order to achieve the maximum diffraction efficiency of each channel grating as a single VBG.
I. V. Ciapurin et al presented a detailed model depicting the diffraction of Gaussian beams on a plane uniform VBG [16]. They indicated that increase in beam divergence results in dramatic decrease in the diffraction efficiency especially when the FWHM value is close to or less than the beam divergence. Therefore, the dependence of each channel’s maximum diffraction efficiency on the beam divergence in multiplexed grating was theoretically analyzed and is depicted in Fig. 4(b). Here, the grating thickness is 3 mm as tabulated in Table 1, and in this case, the cross-talk effect between different channels can be neglected. Figure 4(a) shows the dependence of transmitted minus first order (−1T) diffraction efficiency of each channel on the incident angle for different channel grating when the plane wave is incident. The wavelength of the incident beam is 1064 nm and the polarization is TE mode. From channel 1# to 6#, the FWHM value is decreased and thus the influence of the divergence angle on the maximum diffraction efficiency becomes greater. Especially for channel 6#, the maximum diffraction efficiency is only 63.2% when the beam divergence is 0.6 mrad. However, the divergence of most commercial lasers is in mrad order of magnitude, and in this case, the beam divergence becomes the main factor limiting the maximum achievable diffraction efficiency. Moreover, the complexity of the production technology makes it difficult to realize a perfect 12-channel multiplexed VBG in a whole block PTR glass. For a transmitted VBG, the maximum diffraction efficiency is as a sinc-squared function of RIM, it is a challenge to make all channel grating’s RIM of the optimal value at the same time.
Fig. 4 (a) Dependence of −1T diffraction efficiency on incident angle for each channel grating of multiplexed VBG when the plane wave is incident. (b) Dependence of maximum diffraction efficiency on the beam divergence when the divergent beam is incident.
In order to lower the influence of beam divergence to maximum diffraction efficiency and reduce the difficulty of process, a cascaded multiplexed VBG system is proposed in this study. The schematic is shown in Fig. 5. The multiplexed VBG G1 includes channels 1#, 4#, 7#, and 10#, G2 includes channels 2#, 5#, 8#, and 11#, and G3 includes channels 3#, 6#, 9#, and 12#. For each multiplexed VBG, a larger channel space was chosen to avoid the cross-talk effect. Grating channel with a small angular deflection is placed in front, and this allows minimizing the aperture to a size required for the system. In Fig. 6(a)–6(c), the theoretical angle selectivity curves for different channels in G1, G2, and G3 are presented. The thickness of each multiplexed grating is 1 mm. When the plane wave is incident, maximum diffraction efficiency of 99.6% and above can be achieved. If the divergence of the incident beam is 0.6 mrad, the maximum diffraction efficiency for channel 1# to 6# is 99.8%, 99.2%, 98.3%, 96.9%, 95.2%, and 93.3%, respectively.
Fig. 6 Theoretical angle selectivity curves for different channels in multiplexed VBG G1 (a), G2 (b), and G3 (c).
4. Fabrication and performance test
For experimental demonstration of the proposed cascaded multiplexed grating system, gratings G1, G2, and G3 were fabricated and tested for performance. The fabrication process of each multiplexed VBG mainly contains two steps: sequential exposure and subsequent heat treatment. In the exposure process, home-made PTR glass was placed in an asymmetric dual beam interference light field. He-Cd laser (Kimmon Electric Model IK3501R-G) with a wavelength of 325 nm and TE polarization was used as recoding light source. Shear interference method was employed for fine adjustment of the parallelism of the two expanded beam. The intensities of the two collimated beams were kept equal as required for obtaining maximum interference fringe contrast. The specific values of the incident angles of the two recording beam were calculated according to the interference principle and refraction effect; more details can be found in [17]. It is worth noting that the symmetrical gratings can share the same light path setup. Take grating channels 1# and 7# for example, after the exposure of 1#, we only need to rotate the grating by 180° along the normal of the grating surface, and then perform the second exposure as before to record channel 7#. Therefore, two light path setups with different interference angle are needed for exposure of G1, G2, and G3. Process parameters are optimized according to [18]. The exposure dosage of PTR glass samples for each channel was set to 150 mJ/cm2. The PTR glasses were first made to undergo sequential UV exposure for four times, they were all heat-treated at 500 °C for 6.5 h. The PTR glasses were finally repolished to remove defects, from the surface, caused by heat treatment.
The scheme of DE measurement of multiplexed VBGs is shown in Fig. 7. The analyzed sample is positioned on a motorized rotational stage. The wavelength of the laser source is 1064 nm and the divergence is 0.6 mrad. A linear polarizer is used to select TE or TM polarization. Intensities of transmitted zero order beam (I0T) and minus first order diffracted beam of different grating channel (Ii#) are measured using several power meters (Thorlabs, S120VC). One power meter is placed in the direction of the transmitted zero order beam and the others are placed in the directions of beam diffracted from different grating channels. The angle of the rotator and intensity data of all the power meters are simultaneously collected via a computer. Relative DEs are then calculated as follow.
Here i and j are the sequence numbers of the grating channels, N is the total number of the grating channels. For test of N multiplexed VBG, N + 1 power meters are needed.Volume Bragg gratings recorded in PTR glass have dual-channel characteristics, which is depicted in Fig. 8. When the incident beam R1 fulfills the Bragg condition and the diffracted beam is S1, then when incident beam R2 is in the same direction as the diffracted beam S1, the Bragg condition is still satisfied and the diffracted beam S2 has the same direction as that of R1. Therefore, two valleys will appear in the intensity curve of transmitted zero order beam as a function of incident angle. These two valleys correspond to the two incident angles, one is the Bragg incident angle of the VBG and the other is the Bragg exit angle. Owing to this characteristic, we could precisely and conveniently determine the Bragg incident and exit angles of each grating channel in multiplexed VBGs from the angular curve of transmitted zero order beam.
5. Results and discussions
First, the respective performances of the fabricated multiplexed VBG G1, G2, and G3 were tested individually. Total five power meters were used in the test setup. Figure 9(a)–9(c) presents the angular selectivity curves of G1, G2, and G3, respectively. Relative diffraction efficiency up to over 90% could be achieved for all the grating channels both for TE and TM modes. The effect of the beam divergence on the angular selectivity curve manifests itself in the side lobes. From 1# to 6#, the side lobes are flattened while the FWHM of the grating channel is decreased; this is in line with the theoretical analysis. For experimental demonstration of our cascaded solution, these three multiplexed VBGs are integrated into one whole using a refractive index matching liquid; total thirteen power meters are used in the test setup. Figure 10 shows the normalized intensity curve of transmitted zero order beam. The specific values of the incident angle and exit angle of all the channels are extracted from the curve; these are summarized in Table 2.
Fig. 9 Experimental angle selectivity curves for different channels in multiplexed VBG G1 (a), G2 (b), and G3 (c).
Table 2. Experimental incident angle and exit angle of each channel VBG and deviation of them from expectant
Table 2 reveals that there is a slight difference between the experimental and expected Bragg incident and exit angles for all the grating channels. We experimentally proved that the deviation mainly comes from the sample positioning error in the exposure process. In detail, the positioning error of sample will cause deviation of incident angle of the two interference beam synchronously, resulting in the deviation of the incident angle and the exit angle of the fabricated grating channel. Sample positioning system can be applied to solve this problem; and we have confidence that the deviation of the incident angle and the exit angle of the grating channel could be controlled within 0.1° or even less.
Figure 11 shows the experimental angle selectivity curve of different channels for the cascaded multiplexed VBGs. No significant difference has been observed between test results for TE and TM polarization on comparing Fig. 11(a) and Fig. 11(b). Besides, the relative diffraction efficiency of each channel is slightly lower than the results of individual tests for each multiplexed VBG, especially for channels 3# and 9#. The main reason for this can be explained as follows. The incoming light is partially diffracted to other grating channels when passing through different multiplexed gratings in sequence. Take channel 3# as an example, supposing that the beam is incident at 2.34°, the beam passes through G1 first, and about 5.5% of the light energy gets diffracted by channel 2#. Then the beam passes through G2 and about 2.2% of the light energy is diffracted by channel 4#. Therefore, the incoming light has already lost some energy when it reaches G3, so the ultimate maximum diffraction efficiency of channel 3# is reduced. This cross-talk effect mainly exists between adjacent channels. In this study, the interval of incident angle is set as 0.8°. However, if we set the Bragg incident angle of one channel such that it corresponds to the position of higher order minima of another channel’s diffraction efficiency, the cross-talk can be eliminated and higher diffraction efficiency can be expected.
Fig. 11 Experimental angle selectivity curves of different channels for the cascaded multiplexed VBGs, (a) is the test result of TE polarization, and (b) is of TM polarization.
6. Conclusions
Owing to the limitation of index modulation dynamic range for traditional PTR glasses, when we multiplex 12 channels in a single PTR glass, a larger grating thickness is needed. In this case, the maximum diffraction efficiency that each channel can achieve is mainly limited by the divergence of the incident laser beam. In addition, too many channels will also create significant challenges to the fabrication process. To solve this problem, cascaded multiplexed VBGs solution could be an alternative way. We have experimentally demonstrated the effectiveness of this method. The results reveal that relative diffraction efficiency of all the grating channels can be over than 80% and is almost polarization independent. By optimizing the Bragg incident angle of one channel corresponding to the position of higher order minima of adjacent channel’s diffraction efficiency, the cross-talk can be eliminated. We believe that the deviation of the Bragg incident and exit angles from the expected values could be controlled to within 0.1° or an even smaller value if a sample positioning system can be employed in the exposure process.
Funding
National Natural Science Foundation of China (11604352, 10704079, U1630140, U1430121); Shanghai Science and Technology Committee Program (16JC1420600).
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