## Abstract

High resolution wavefront sensors are devices with a great practical interest since they are becoming a key part in an increasing number of applications like extreme Adaptive Optics. We describe the optical differentiation wavefront sensor, consisting of an amplitude mask placed at the intermediate focal plane of a 4-*f* setup. This sensor offers the advantages of high resolution and adjustable dynamic range. Furthermore, it can work with polychromatic light sources. In this paper we show that, even in adverse low-light-level conditions, its SNR compares quite well to that corresponding to the Hartmann-Shack sensor.

©2003 Optical Society of America

## 1. Introduction

Wavefront sensing is a technique that has been successfully applied in many different fields like optical quality testing, Adaptive Optics, etc. In recent years the number of potential fields has been extended since the development of low-cost devices allowed the application of Adaptive Optics in other fields such as lasers, confocal microscopy or human vision [1–3]. Furthermore, there are specific fields where an accurate description of the incoming wavefront is of particular relevance such as Adaptive Optics systems for very large telescopes, and especially in the search for exoplanets, [4] or wavefront sensing for monitoring LASIK surgery.

We propose a new high resolution wavefront sensor which consists of a telescopic system with a mask at the intermediate focal plane. The incoming field is Fourier transformed by the first lens, then is multiplied by the mask and, finally, is Fourier transformed again. The mask amplitude increases linearly along a certain direction. The wavefront phase derivative is obtained from the detected light intensity at the telescopic system image plane. A rotating filter can be used to provide the derivatives in two orthogonal directions. In practice, the sensor performs an optical differentiation process (that resembles the Foucault knife-edge test principles).

The optical differentiation is a relatively old technique occasionally used to retrieve phase information. It is worth mention the paper of Bortz *et. al.* [5] where a phase retrieval technique similar to the one described here was presented. However, the Bortz’s method requires three measurements with different filters and, as a consequence, a complicated expression for the phase derivative is needed.

In this paper, we analyze the actual implementation of the sensor and the parameters that characterize the mask, we develop an expression for the signal-to-noise ratio of the technique as a function of the mask characterizing parameters for read noise and photon noise and we compare with the Hartmann-Shack SNR under different conditions. In order to perform a complete comparison between the features of both sensors we also compare their dynamic range. Finally, we have developed a computer simulation procedure to compare the Optical Differentiation (OD) sensor performance with that of the Hartmann-Shack (H-S) sensor.

The main advantages of the new sensor are its high (and adjustable) spatial resolution, the easily adjusting of the dynamic range and that it is able to work with polychromatic sources [6]. Currently, there are other adjustable resolution sensors, [7,8] but they present limitations that this sensor overcomes. The main drawback is the energy loss due to the mask absorption, although the Optical Differentiation sensor presents a SNR comparable, or higher, than that of the Hartmann-Shack provided a proper election of the sensor parameters.

## 2. The optical differentiation sensor

To describe the theoretical principles of the OD sensor, let us consider the electric field *E*(*x,y*)=*A* e^{jϕ(x,y)}, where *A* is the constant amplitude and *ϕ*(*x,y*) is the wavefront phase. The arrangement consists of a pair of achromatic lenses forming a telescopic system and a mask *M* placed at the intermediate focal plane (Fig. 1). The first lens performs the FT of the input field on the mask, then the product of the transformed field times the mask is Fourier transformed again onto a detection system (CCD). Two separated measurements are required to obtain the wavefront phase slope in two orthogonal directions.

The masks that perform the differentiation along the *x* (or *y*) direction, *M _{x}* (or

*M*), have linearly increasing amplitude along the derivative direction and can be described as:

_{y}where *λ* is the wavelength, *f* the focal distance of the first lens and *r _{x}* and

*r*represent real distances in the mask plane. The mask can also be expressed in terms of the spatial frequencies of coordinates

_{y}*x*and

*y*in the pupil plane,

*u*and

_{x}*u*, where

_{y}*b*=

*λfb*. In addition,

_{r}*a*and

*b*(or

_{r}*b*) are two constant parameters that determine the mask behaviour.

When a mask of this kind is placed at the intermediate plane of a telescopic system, due to the differentiation property of the FT, [9] the intensity at the CCD is related to the field derivative along the corresponding mask direction:

$${I}_{y}(x,y)={\mid {\mathrm{FT}}^{-1}\left[\mathrm{FT}\left(E(x,y)\right)\xb7{M}_{y}\right]\mid}^{2}={\mid -jb\frac{\partial E(x,y)}{\partial y}+aE(x,y)\mid}^{2}$$

Then, by substituting the field expression, the derivatives of the wavefront phase along orthogonal directions can be obtained from the intensities,

$${\alpha}_{y}=\frac{\partial \varphi (x,y)}{\partial y}=\frac{\frac{\sqrt{{I}_{y}}}{A-a}}{b}=\frac{\frac{\sqrt{{I}_{y}}}{A-a}}{{b}_{r}\lambda f}$$

Note that the wavefront slope can be obtained as the product of the wavefront phase slope, *α*, times *λ*/2π, and thus, it is independent on wavelength. It can be seen that the values of *b _{r}* and

*f*control the dynamic range of the derivative estimate. In contrast with the H-S sensor [10], which is based on the measurement of a centroid position, this is a photometric sensor. Thus, the phase derivative is estimated at each pixel of the detector by comparing the intensity with that corresponding to a flat wavefront portion

*I*

_{0}=(

*aA*)

^{2}. The wavefront phase is sampled by the pixels contained in the CCD illuminated area providing very high spatial resolution without limitations of the dynamic range.

This sensor can also be explained using a ray tracing picture. Note that if achromatic lenses are used each small area of the sensor entrance pupil is directly mapped in one area of the detection plane. In addition, parallel rays (wavefront regions with the same slope) will go to the same point at the filter plane, and thus, will suffer the same attenuation. The intensity at each area of the detection plane provides an average of the wavefront phase slope for the area. When using polychromatic sources, the sensor also provides an average over the whole source bandwidth.

The mask defined in Eq. (1) is a filter with variable transmittance given by (2π*b _{r}r*+

*a*)

^{2}. We will take the parameter

*a*=0.5, which means that only amplitude filters are considered. Finally, the size of the filter is determined by the values of its parameters. Assuming that the maximum value of the mask is equal to one (in order to minimize the lost energy), its width can be easily derived as

*W*=1/2π

*b*=

_{r}*λf*/(2π

*b*), where it is assumed that the centre of the filter lies on the optical axis.

## 3. Signal-to-noise ratio for the OD sensor

The two main sources of error in the OD sensor are the CCD read noise and the photon noise.

#### 3.1. CCD read noise

The magnitude measured is the slope *α* as expressed in Eq. (3). The total intensity at each detector area of the sensor, *I _{x}* (or

*I*) can be expressed as the sum of the intensities,

_{y}*I*, of the

_{i,j}*N*

_{p}pixels of that detector area. Its variance, using the standard error propagation formula, can be written as:

where *σ*
_{r} is the read noise error of the CCD. Consequently, the signal-to-noise ratio for the OD sensor when only read-noise is considered can be expressed as:

where *n _{OD}* is the number of photons arriving at the corresponding area in the entrance pupil of the sensor and <…> means ensemble average. The sampling at the CCD plane can be easily changed using a zoom lens. Then, if a sampling of one pixel per sensor area is considered (

*N*

_{p}=1), it is easy to show that:

where *D*
_{lens} is the diameter of the lens used to evaluate the first Fourier transform, and *b* has been expressed in terms of the number of Airy rings covered by the filter, *N*
_{A} (*W*=1.22·*N*
_{A}
*λf*/*D*
_{lens}).

#### 3.2. Photon noise

From the expression of the slope *α* given by Eq. (3), its variance, using the standard error propagation formula, can be easily developed as:

and the SNR, when detection is affected only by Poisson noise, will be:

Introducing the expression of *b* the SNR is expressed as:

To maximize the SNR an actual filter should have a *b* value as large as possible. This is carried out taking *a*=0.5 and making the filter size as small as possible, although a compromise between the energy loss and the filter size is necessary.

To compare the SNR _{OD} with that of the Hartman-Shack when only photon noise is considered, we set, as a particular case, the filter radius equal to the size of the 15th Airy ring, obtaining the following expression:

## 4. Signal-to-noise ratio for the H-S sensor

#### 4.1.CCD read noise

The wavefront phase slope measured by the Hartmann-Shack (in the *x* direction) is:

where *f*
_{l} is the focal length of the microlens, *λ* is the incoming wavelength and *x*
_{c} is the *x*-position of the spot centroid. The corresponding SNR for read noise is: [11,12]

where *n*
_{H-S} is the number of photons per microlens in the H-S sensor, *N*
_{t} is the spot size in pixels, *d* is the diameter of the microlens and *N*
_{w} is the width of the corresponding subaperture area in the CCD in pixels. As an example, in the case of a quad cell the parameters take the values: *N*
_{w}=2 y *N*
_{t}=1.

#### 4.2. Photon noise

In the case of photon noise the corresponding SNR is: [11,12,13]

for circular microlenses, where *d* is the diameter of the microlens.

It is necessary to state that both Eq. (12) and Eq. (13), due to the approximations used to develop them [11,12,13], only determine an upper limit to the SNR of the H-S.

The ratio between the SNR due to photon noise of both sensors is obtained from Eqs. (10) and (13) and:

In this expression the ratio *n*
_{OD}/*n*
_{H-S} is set to ½ because the light of the OD sensor must be split in two channels.

The SNR ratio dependence on *D*
_{lens}/*d* will be analyzed later using a computer simulation. This dependence is similar in the read noise and in the photon noise cases. For this reason, and as most analyses in the literature, we will only consider the photon noise case in the next sections.

## 5. Comparison of dynamic range

The range of wavefront phase slopes that can be measured also depends on the size of the filter, and, consequently, on *b*. Thus, the maximum slope that can be measured is *α*
_{M}=(2π/*λ*) (*W*/2*f*)=1/(2*b*). This relationship between the parameters of the mask and the wavefronts to be measured enables the election of the appropriate mask. Moreover, different masks can be implemented using a LCD. As a result, the dynamic range of the OD sensor can be easily adjusted. If we define the dynamic range as DR_{OD}=2*α*
_{M}, we obtain:

In contrast with H-S, this equation shows that high dynamic range can be attained without loss of the spatial resolution. Now we consider the Hartmann-Shack dynamic range:

Hence, a high resolution H-S sensor will present a DR _{H-S} smaller than DR _{OD} when *d*<(*Wf*
_{l})/*f*. Furthermore, when the lens size decreases, the size of the PSF at the microlens image plane increases, reducing even more the actual H-S sensor dynamic range.

## 6. Advantages of high resolution sensing

We have perfomed a computer simulation to show the advantages of the OD sensor over the H-S sensor, which are especially relevant in high resolution. Four hundred distorted wavefronts following Kolmogorov statistics with *D*/*r*
_{0}=1 were generated using Roddier’s technique [14]. The number of Zernike modes that we used in the simulation of the wavefront was *N*=560, and the three first modes (piston, tip and tilt) are assumed to be corrected. The number of samples in the wavefront was (π/4)×291×291. Then, the phase derivative was estimated both using a H-S sensor and our technique.

The first step in the comparison was to reproduce the ratio SNR _{OD}/SNR _{H-S}. An analysis of this ratio can be carried out as a function of number of sensing areas covering the sensor entrance pupil. The number of sampling areas (π/4)×*N*
_{s}×*N*
_{s} is the number of microlenses used in the Hartmann-Shack and the number of areas in which the light intensity is detected in the Optical Differentiation sensor where *N*
_{s}=*D*
_{lens}/*d*. From Eq. (14) we see that the ratio SNR_{OD}/SNR _{H-S} depends on the value of *N*
_{s}. Fig. 2 shows the behaviour of SNR _{OD}/SNR _{H-S} obtained from computer simulation for *N*
_{s}=27, 30 and 35 as a function of the number of photons in each sensing area of the OD sensor. We see that for *N*
_{s}=27 the SNR _{H-S} is larger than the SNR _{OD}. However, for *N*
_{s}=30 and 35 the Optical Differentiation sensor provides a SNR better than that corresponding to the H-S sensor. It can be seen that Eq. (14) predicts that the ratio SNR _{OD}/SNR _{H-S} will be larger than one for *N*
_{s}=60. However, the simulation shows that a value over *N*
_{s}=30 is enough to attain this value. The reason of this discrepancy is that the SNR _{H-S} given in Eq. (14) has been evaluated using an approximated procedure so that it only provides an upper limit of the SNR.

From this analysis it is straightforward to deduce that a high resolution Hartman-Shack sensor provides a SNR necessarily lower than that of the Optical Differentiation sensor. The physical reason of this behaviour is explained. As resolution increases, the SNR in both sensor decreases because less photons arrive at each detector area. However, for the H-S sensor there is an additional error source: the resolution increasement means a reduction of the lenslet size, which implies a wider and noisier centroid. This additional error source, that the OD sensor overcomes, explains the advantage of the OD sensor in high resolution despite the better light efficiency of the H-S sensor.

The second step is to reconstruct the wavefront phase from a certain number, *k*, of coefficients obtained from the slopes. The error in the whole process is estimated using the residual phase variance of the reconstructed wavefront phase, defined as:

where *a _{i}* are coefficients of the corresponding Zernike polynomials,

*a*

_{i rec}=0 for

*i*>

*k*. Fig. 3 compares the residual variance obtained using the OD and the H-S sensor as a function of the number of reconstructed modes

*k*. The main conclusion is that the accuracy of our technique is very similar to that of the Hartmann-Shack sensor even in adverse conditions (

*N*

_{s}=8).

Finally, the third step is to take advantage of the high spatial resolution attainable by our sensor. Fig. 3 also shows the residual variance when the OD sensor number of sampling areas is increased. Since a high resolution sensor samples the wavefront in a considerable number of points, the reconstruction of the incoming wavefront can be performed more accurately. In such a case, the OD sensor not only provides better accuracy in all cases but also allows the estimate of higher order modes.

## 7. Simulated experiment

In order to analyse the behaviour of the OD sensor in wavefront compensation, we performed a simulated experiment. An atmospherically distorted wavefront has been generated (Fig. 4(a)). The local phase slopes have been estimated using the OD sensor (Fig. 4(b)), then the phase is reconstructed using a standard procedure [15]. The wavefront is reconstructed after 65 Zernike modes have been compensated (Fig. 4(c)). These figures show that the OD sensor can be a useful tool for those applications in which wavefront sensing is required.

## 8. Conclusions

We present the Optical differentiation wavefront sensor that consists of a linearly increasing amplitude mask placed at the focal plane of a telescopic system. The main advantages of this sensor are that the dynamic range and sampling can be easily adjusted. Furthermore it can work with polychromatic sources. This allows us to attain high resolution, and consequently to estimate a large number of wavefront modes, without loss of dynamic range. Moreover, we have shown that for high resolution sensing it presents better SNR and dynamic range than those of the standard Hartmann-Shack sensor, even in adverse photon noise conditions.

## Acknowledgments

This research was supported by Ministerio de Ciencia y Tecnología grant AYA2000-1565-C02.

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