## Abstract

Computer generated holograms (CGHs) are used for testing aspherical surfaces. They are also used for certifying null lens systems that are in turn used to test large aspheric mirrors. We demonstrate that, in order to minimize size, a null-corrector-certifying CGH can be located at a ray caustic. This placement minimizes the required size of the CGH and so makes it possible to certify some null correctors that otherwise, due to current size limitations of high-precision CGHs, could not be certified. Fabrication limits of precision CGHs are in the size range of 200–250 mm and one part in 10^{5} in feature placement. Our proof of principle involves a CGH that is about 20mm in diameter with a fabrication tolerance of one part in 10^{4}.

© 2005 Optical Society of America

## 1. Introduction

Computer generated holograms (CGHs) are often used for testing aspherical surfaces and for certifying null-lens systems [1–3]. The design of a CGH requires calculating the phase of the test beam in question. To minimize computational complexity, rather than using wave-optical propagation, we employed ray-tracing techniques to determine the optical-path difference and in turn the phase of the beam encoded in a CGH. Normally, CGHs are located in regions where rays do not intersect one another so that a one-to-one mapping can be established. In a caustic, rays intersect; the intersection causes ray-directions to become ambiguous [4]. Avoiding the caustic forces placement of the CGH away from the beam-waist and so incurs a CGH-size penalty. In this paper we show that, despite the ray-direction ambiguity, it is possible to place a multiplex [5–10] CGH at a caustic and, by using two diffraction orders, still reconstruct a single desired test beam. Thus, we demonstrate that certifying a null corrector by placing a CGH in a caustic region is not only theoretically feasible, but it actually functions experimentally.

## 2. CGH multiplexing

Figure 1 shows a CGH that is located in a caustic region formed by rays from a tilted spherical mirror. At the caustic the beam folds into itself and rays overlap. Two rays coming from two different points on the test surface coincide at the CGH plane.

The function of a certifying-CGH is to retro-reflect rays at the CGH along the ray-normals of the surface under test. To achieve this behavior we used two diffracted orders from a single multiplex CGH. Multiplexing allows us to embed the equivalent of two independent CGHs into one physical device. Each embedded CGH retro-reflects one section of the folded beam.

We chose a simple form of multiplexing: multiplication of two binary CGH reflectances. We use squarewaves [11] as the carriers. The mathematical description begins with the Fourier-series expansion of the squarewave with argument “*p*” and duty-cycle one half:

The general form of the encoded complex-amplitude reflectance is *R=Ae ^{iϕ}* but the surfaces that are encoded in the CGH are absorption-free so that i R=eϕ. The position on the CGH is given by the vector

**r**=(

*x, y*) so that the two reflectances of the encoded surfaces are

*R*=exp[(

_{J}*iϕ*(

**r**)] with

*j*=1, 2. The two carrier frequencies are the components of the vector constant

**ρ**

_{j}=(

*ξ*). Encoding is described by making the substitutions.

_{j},η_{j}${p}_{j}={\mathbf{\rho}}_{j}\bullet \mathbf{r}+\frac{{\varphi}_{j}\left(r\right)}{2\pi}$

The two embedded CGHs have reflectances

The reflectance of the multiplex CGH is the product

$$=\frac{1}{4}\sum _{m=-\infty}^{+\infty}\sum _{n=-\infty}^{+\infty}\mathrm{sinc}\left(\frac{m}{2}\right)\mathrm{sinc}\left(\frac{n}{2}\right)\xb7\mathrm{exp}\left[2\pi i\left(m{\mathbf{\rho}}_{1}+n{\mathbf{\rho}}_{2}\right)\bullet \mathbf{r}\right]\mathrm{exp}\left[i\left(m{\varphi}_{1}+n{\varphi}_{2}\right)\right]$$

In Eq. (3), if *m*=1 and *n*=0, then the first-order diffraction from the first CGH is obtained. If *m*=0 and *n*=1 then the first-order diffraction from the second CGH is obtained. This means that the multiplex CGH can produce two diffraction orders that create the necessary retro-reflected beam. Thus, when a multiplexed CGH is designed, a combination of the first and zero-order or the zero and first-order are used for testing. All other order-combinations can be separated out by adding phase tilt or defocus in the CGH design. For example, the **ρ**
_{i}•**r** term in the above equations describes phase tilt in the CGH. The product of squarewaves in Eq. (3) is the formula used for CGH fabrication; the *ϕ _{j}* are calculated by ray-tracing. The Fourier expansion is used for analysis. For example, if the CGH is fabricated as an binary-amplitude CGH, as we did for the experiments, then the ideal irradiance diffraction-efficiency of the (

*m, n*) diffraction order is the square of the series coefficient ${\left[\frac{1}{4}\mathrm{sinc}\left(\frac{m}{2}\right)\mathrm{sinc}\left(\frac{n}{2}\right)\right]}^{2}$. For the diffraction orders that we are using this number is

$\frac{1}{16}{\mathrm{sinc}}^{2}\left(0\right){\mathrm{sinc}}^{2}\left(\frac{1}{2}\right)=\frac{1}{4{\pi}^{2}}=0.025\phantom{\rule{.2em}{0ex}}\mathrm{or}\phantom{\rule{.2em}{0ex}}2.5\%$. When fabricated as a phase grating, the CGH’s efficiency quadruples to 10%.

## 3. Proof of concept

Using the multiplexing technique, we have designed a binary-amplitude CGH to test an off-axis beam. We have experimentally verified the theoretical prediction with the setup shown in Fig. 2. A spherical beam from an interferometer was incident on 30 mm diameter spherical mirror tilted 10° and working at F/3.5 to produce a ray caustic. For convenience, the reflected beam was folded by a plane mirror and then made incident on a multiplexed CGH.

The CGH is shown in Fig. 3; it was designed to retro-reflect the incident beam along its incoming path. In practice we combine the zero-order of one embedded CGH with the first-order of the second embedded CGH to create half of the folded test beam. We use the first-order of the first embedded CGH and the zero-order of the second embedded CGH to create the other half of folded test beam. The other diffraction orders are physically blocked.

In Fig. 2, the illumination beam travels to the right through the pinhole, traverses the system to the CGH.

The reflective Fourier CGH’s diffraction pattern forms in the plane of the pinhole. It is there that the unwanted orders are blocked. Figure 4 shows the diffracted orders that are blocked; the desired orders pass through the pinhole.

For this experiment a mask-less lithographic photo-tool [12] was used to fabricate the CGH. The photo-tool has less than four micrometers positioning uncertainty in feature placement. The size of the CGH is approximately 20mm×6.3mm. To obtain more than 0.1 wave accuracy the minimum pitch of the CGH was designed to be 40 microns. To help separate unwanted diffraction orders from the orders of interest a tilt of 0.2° was added in the CGH. The error analysis in Table 1 shows that, with the CGH fabricated and the components available, an overall test accuracy of 0.1 waves was achieved.

## 4. Experimental results

Test interferograms of the experimental system are shown in Fig. 5. Figure 5(a) shows interference fringes and a significant amount of zero-order light entering the interferometer before a pinhole is used to filter this order. Figure 5(b) shows that, when the CGH was not properly aligned, the interference fringes corresponding to the two different beams break apart and appear as independent separated fringes near the vertical diameter of the figure. Figure 5(c) shows the interference fringes as continuous when the CGH is aligned. Figure 5(d) shows essentially null fringes indicating an error on the order of about 0.1 wave rms.

## 5. Conclusions

Our motivation has been to minimize the size of a certifying hologram by placing it at a beam waist where rays overlap and the beam is smallest in size. Theoretical and experimental results show that a multiplex CGH can be located at a ray caustic and create a retro-reflected test beam. We have explained how the multiplexed hologram works and how to mathematically define it. We used ray tracing to find the phase function of the dual-beam that needs to be retro-reflected and then encoded each half of the beam in a binary-amplitude form. Future work will include minimizing the impact of unwanted orders and maximizing fringe contrast.

## References and links

**1. **J. H. Burge, Advanced techniques for measuring primary mirrors for astronomical telescopes, PhD Dissertation, Univ. of Arizona, Optical Sciences Center, Tucson, AZ (1993).

**2. **Taehee Kim, James H. Burge, and Yunwoo Lee, “Measurement of Highly Parabolic Mirror using Computer Generated Hologram,” Proc. SPIE **4778**, 119–126 (2002). [CrossRef]

**3. **Iwona A. Palusinski and Jose M. Sasian, “Sag and phase descriptions for null corrector certifiers,” Opt. Eng. **43(3)**, 697–701 (2004). [CrossRef]

**4. **Stephan Reichelt, Christof Pruβ, and Hans J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE **4778**, 158–168 (2002). [CrossRef]

**5. **E.N. Leith and J. Upatneiks, “Wavefront reconstruction with diffused illumination and three-dimensional objects,” J. Opt. Soc. Am. **54**, 1295–1301 (1964). [CrossRef]

**6. **J.D. Armitage and A.W. Lohmann, “Theta modulation in optics,” Appl. Opt **4**, 399–403 (1965). [CrossRef]

**7. **J.R. Fienup, “Color images from computer-generated holograms,” Proc. SPIE **48**, 101–105 (1974).

**8. **H. Bartelt, “Space multiplexing with computer holograms,” Opt. Commun. **23**, 203–206 (1977). [CrossRef]

**9. **W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, E.D. B.R. Frieden, (Springer-Verlag, Heidelberg, 1980).

**10. **Stephan Reichelt, Christof Pruss, and Hans J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt **42**, No. 22, 4468–4479(2003). [CrossRef]

**11. **Jack D. Gaskill, “Harmonic Analysis”, in Linear Systems, Fourier Transforms, and Optics, (John Wiley and Sons, New York, 1978).

**12. **John M. Tamkin, Brett Bagwell, Bradley T. Kimbrough, Ghassan Jabbour, and Michael R. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE **4984**, 210–218 (2003). [CrossRef]