Direct amplitude detection of Zernike modes by computer-generated holographic wavefront sensor: Modeling and simulation

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Abstract

A fast holographic wavefront sensor is proposed using a computer-generated hologram (CGH). This CGH is a multiplexed hologram of different Zernike mode–amplitude combinations, and is designed in such a manner as to get the corresponding spots on the detector according to the presence and strength of a particular aberration. Interference between the aberrated wavefront (with a single mode–amplitude combination) and the Fourier transform of an image with single bright pixel (defined as dot image) is numerically calculated for one hologram. Different mode–amplitude combination and corresponding different positions of bright pixels (dots) are taken to compute various holograms and then all the holograms are multiplexed to get the final hologram. When the aberrated wavefront with a particular mode–amplitude combination is incident onto the multiplexed hologram, the corresponding dot is generated in the Fourier plane. A lens performs the Fourier transform in optical domain and provides the instant detection of amplitude of the respective Zernike mode. The main advantage of the scheme is to avoid the need of any computations, which makes it really fast. The simulation results are presented with the cross-talk analysis for few Zernike terms.

Introduction

Wavefront sensing is an important technique, which is required for many applications like optical testing, adaptive optics for high-resolution imaging, laser diagnostics, etc. It involves the estimation of the phase-errors of the wavefront coming through various sources of aberration, e.g., optical component, atmosphere, bio-medical tissues, etc. In many applications, the aberration of the wavefront is represented in terms of the orthogonal set of polynomials (modes) [1], and the amplitude of the aberration mode is required to be known. There are various techniques reported in the literature [2], [3], [4], [5] for wavefront sensing. These wavefront sensors, such as Shack–Hartmann, curvature, and shearing interferometer are very good but involve a high degree of computation, because they compute the modal content by measuring some properties of the wavefront, e.g., slope, curvature, etc. This limits the speed of the sensor, which is a major issue in some of the applications, where fast wavefront sensing is mandatory.

A modal wavefront sensor has also been reported [6], which reduces the computation drastically. It involves the introduction of phase bias in the incident beam. It generates two spots on detector for a particular mode. The difference of intensities in the two spots is a measure of the amplitude of the respective mode. This sensor has been used in different applications and the problem of cross-talk is minimized by the use of improved algorithms in the design of phase mask [7], [8], [9], [10], [11], [12], [13]. However, some data processing is still required to avoid the cross-sensitivity between certain modes and to calculate the output signal from the sensor.

Recently, a holographic wavefront sensor has been proposed [14], [15], in which a multiplexed hologram acts as a wavefront sensor. This sensor produces various spots on the detector according to the presence and strength of the particular aberration in the incident beam. A very high speed can be achieved with this sensor, as it completely removes the need of any computation. However, optical recording of multiplexed hologram is tedious process. Besides, the cross-talk is a major issue to get a real sensor. The computer-generated hologram (CGH) is the better option to implement this type of sensor, because it provides the design-freedom to optimize the cross-talk.

In this paper, we propose a new scheme to design the hologram, which acts as the wavefront sensor. Interference between the aberrated wavefront (with a single mode–amplitude combination) and the Fourier transform of an image with single bright pixel (defined as dot image) is numerically calculated for one hologram. Different mode–amplitude combination and corresponding different positions of bright pixels (dots) are taken to compute various holograms and then they are multiplexed to get the final hologram. When the aberrated wavefront with a particular mode–amplitude combination is incident onto the multiplexed hologram, the corresponding dot is generated in the Fourier plane. The issue of cross-talk is also discussed and simulation results are presented for CGH with various Zernike modes–amplitudes combinations.

Section snippets

Model for CGH Zernike sensor

To design a CGH, which acts as a wavefront sensor, we use Zernike modal representation for the phase aberrations. This CGH Zernike sensor detects the amplitudes of Zernike modes optically. A simple model is given for describing the operation of this sensor.

Aberrations of a wavefront can be expressed in terms of Zernike polynomials asW(r,θ)=kakZk.

We follow the simple notation in which a single index k is taken [14], [15], [16], [17]. Zk is a particular polynomial (Zernike mode) and ak is the

Simulation

To start with, we take 256×256 pixels to make a CGH for the defocus (k=4). We have not taken first three Zernike terms (piston, tip and tilt), as they are not compatible with the simple straightforward CGH design. A separate analysis is required to check the feasibility for their inclusion in the holographic wavefront sensor.

As shown in Fig. 3, we initially take eight amplitudes values (from −2λ to +2λ), excluding the zero value, which is not required to design the hologram. To check the

Discussions

The proposed sensor works for single wavelength application only as the hologram is a dispersive element. However, the instant detection of the amplitudes shows promise for high-speed laser based wavefront sensing applications.

In conventional wavefront sensors, e.g., Shack–Hartmann, curvature, or interferometric, computationally intensive algorithms are used, whereas in the proposed sensor, there is no need of such algorithms. It almost removes the need of computation completely, as the main

Conclusion

We have proposed a novel scheme to design the hologram, which acts as the wavefront sensor. Interference between the aberrated wavefront (with a single mode–amplitude combination) and the Fourier transform of a single dot is numerically calculated for one hologram. Different mode–amplitude combination and corresponding different positions of dots are taken to compute various holograms and then all the holograms are multiplexed to get the CGH mask. When the aberrated wavefront with a particular

Acknowledgments

The authors are grateful to the Director, IRDE, Dehradun, for granting the permission to publish this paper. Thanks also to Mrs. Lata Mainali, Scientist, and Ms. Alpana Bhagatji, Research fellow, in the Photonics division, IRDE, for their valuable suggestions to improve the work and the manuscript.

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