Optimization of phase-only computer-generated holograms using

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1.0 - 0.5 0 0 Amp ii tude (a) (b) Fig. 2 Transmittance functions of a binary amplitude gray grating (c). 1260 / OPTICAL ENGINEERING / June 1992 / Vol. 31 No. 6 x BOLSTAD, YATAGAI, and SEKI 0.5 Phase Phase 3 Diffraction Efficiency of Ideal Phase-Only Holograms The diffraction efficiencies of gray and binary gratings were 14 originally described by Brown and First we will describe these and then the diffraction efficiencies of their phase-only counterparts. For simplicity, consider the diffraction efficiency difference between a binary grating and its phase version, as shown in Fig. 2. The transmittance function of a binary grating with a period of L\v is given 0 Amp] itude Amp] itude grating (a), its phase version (b), and a bleached 0 4.) >., u C a) a 4- 4w 1.e 0.5 0 x x Phase x x x by 0 211' Phase Difference _1 m1 . Tb (2irnx 2 7T'fl Lv ) ' (1) Fig. 3 Theoretical plot of diffraction efficiency ratio R and R' versus phase difference 0. where m denotes the modulation level of the grating and O
OPTIMIZATION OF PHASE-ONLY COMPUTER-GENERATED HOLOGRAMS Fig. 4 Flow diagram of computer simulation for phase profile estimation. This is also plotted in Fig. 3 for the case m =1 . The maximum diffraction efficiency is not obtained at 0 =ir, but 0=3.6 rad or 220 deg. 4 Phase Profile Simulation It is possible to simulate some of the processes described so far, diffusion, phase-shift, and reconstruction, by using a digital computer. The flow diagram of the simulation is shown in Fig. 4. First we consider the calculation of the ion-diffusion profile (ion-concentration profile). The resulting ion-concentration after ion exchange can be calculated by solving the 2-D diffusion equation numerically 12 using the finite difference An actual profile can be calculated/simulated from the given data for the experiment. This was done using a VAX 8200 computer. In order to simplify programming, we assumed the diffusion constant to be concentration independent. This is a rough approximation, if the ion concentration is very high. Necessary inputs are time of diffusion, aperture size, and diffusion constant. A relation between ion concentration in a diffused area and the change in refractive index has been found empirically, and can be expressed as follows: n=aC+b, where n is the index change in the local area, C is the local concentration of ions of thallium exchanged with sodium, and a and b are constants. Using an experimental relation,9 the final index increase in a local area can be found using a linear relation. The concentration of ions (thallium) exchanged with sodium! potassium ions is 90% near the surface of the aperture opening, and gradually decreasing to 0% in substrate material. To find the actual index change over the full depth of the diffusion area, we calculate as follows. Consider the coordinate in the depth direction as x. A small element along (9) V 4.) a) 0 E '1e C o8 4., .; 6 0 0_4 2 0 —2 —4 —6 —6 —10 0 2 4 6 0 10 Diffusion Depth (urn) Fig. 5 Simulation result of index change. this direction is called zx. The ion concentration of thallium is found in each small element according to Eq. (9). This is actually done in 20 elements. We can now find the total phase change by the following equation: (pi= , (10) where and L1x1 denote the local change of the refractive index in the position of the (i,j)'th element and a small distance element, respectively. The result of inserting the experimental data for 120 mm of diffusion time and a mole percentage of 90% thallium is shown in Fig. 5. For this case, the maximum index change along the depth direction is n =0.01, and this corresponds to a total phase change of 240 deg when using 633-nm laser light. The diffusion length of thallium is measured to be 6 xm. 5 Phase-Only Grating and Their Diffraction Efficiency There has been no earlier report on holograms made by this technique, and making such an element was also considered as a test of the limitations of ion-diffusion-made elements. In our experiment, the smallest aperture was about 3 rim. Because of sideways diffusion, a good result was achieved mainly for apertures larger than 5 jim and more than 20 pm apart. With a smaller spacing between the apertures, they tended to ''fuse' ' together. By knowing the index increase in a local area, we can find the phase profile of a plane wave passing through the aperture in a vertical direction. The optical path length is an integral over the index distribution multiplied by the length. Then, multiplying the difference in optical path length with the wave number k gives the phase profile after a single pass. We have here neglected refraction due to the small depth (5 to 10 iim) of an aperture. The integral mentioned above can be found by summing the local index values multiplied a small distance x. Doing this for all columns gives the final profile. OPTICAL ENGINEERING / June 1992 / Vol. 31 No. 6 / 1261
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