## Analysis of the signal-to-noise ratio in the optical differentiation wavefront sensor

Optics Express, Vol. 11, Issue 21, pp. 2783-2790 (2003)

http://dx.doi.org/10.1364/OE.11.002783

Acrobat PDF (151 KB)

### 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

3. E. J. Fernandez, I. Iglesias, and P. Artal, “Closed-loop adaptive optics in the human eye,” Opt. Lett. **26**, 746–748 (2001) [CrossRef]

4. V. F. Canales and M. P. Cagigal, “Gain estimate for exoplanet detection with adaptive optics,” Astron. Astrophys. Suppl. Ser. **145**, 445–449 (2000) [CrossRef]

*et. al.*[5

5. J. C. Bortz and B. J. Thompson, “Phase retrieval by optical phase differentiation,” Proceedings of the SPIE **351**, 71–79 (1983) [CrossRef]

7. E. N. Ribak “Harnessing caustics for wave-front sensing,” Opt. Lett. **26**, 1834–1836 (2001). [CrossRef]

8. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. **43**, 289–293 (1996). [CrossRef]

## 2. The optical differentiation sensor

*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.

*x*(or

*y*) direction,

*M*(or

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

_{y}*λ*is the wavelength,

*f*the focal distance of the first lens and

*r*and

_{x}*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.

*α*, times

*λ*/2π, and thus, it is independent on wavelength. It can be seen that the values of

*b*and

_{r}*f*control the dynamic range of the derivative estimate. In contrast with the H-S sensor [10

10. J. M. Geary, *Introduction to wavefront sensors* (SPIE Press, Washington, 1995). [CrossRef]

*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.

*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

### 3.1. CCD read noise

*α*as expressed in Eq. (3). The total intensity at each detector area of the sensor,

*I*(or

_{x}*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:

*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:

*n*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 (

_{OD}*N*

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

*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

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

*b*the SNR is expressed as:

*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.

_{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

*x*direction) is:

*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

11. R. Irwan and R. G. Lane, “Analysis of optimal centroid estimation applied to Shack-Hartmann sensing,” Appl. Opt. **32**, 6737–6743 (1999). [CrossRef]

12. J. Primot, G. Rousset, and J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am A **7**, 1598–1608 (1990) [CrossRef]

*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

11. R. Irwan and R. G. Lane, “Analysis of optimal centroid estimation applied to Shack-Hartmann sensing,” Appl. Opt. **32**, 6737–6743 (1999). [CrossRef]

12. J. Primot, G. Rousset, and J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am A **7**, 1598–1608 (1990) [CrossRef]

13. B. M. Welsh and C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am A **12**, 1913–1923 (1989) [CrossRef]

*d*is the diameter of the microlens.

11. R. Irwan and R. G. Lane, “Analysis of optimal centroid estimation applied to Shack-Hartmann sensing,” Appl. Opt. **32**, 6737–6743 (1999). [CrossRef]

12. J. Primot, G. Rousset, and J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am A **7**, 1598–1608 (1990) [CrossRef]

13. B. M. Welsh and C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am A **12**, 1913–1923 (1989) [CrossRef]

*n*

_{OD}/

*n*

_{H-S}is set to ½ because the light of the OD sensor must be split in two channels.

*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

*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:

_{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

*D*/

*r*

_{0}=1 were generated using Roddier’s technique [14

14. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. **29**, 1174–1180 (1990) [CrossRef]

*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.

_{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.

*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:

*a*are coefficients of the corresponding Zernike polynomials,

_{i}*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).

## 7. Simulated experiment

15. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am **69**, 972–977 (1979) [CrossRef]

## 8. Conclusions

## Acknowledgments

## References and links

1. | G. Vdovin, “Micromachined membrane deformable mirrors” in |

2. | M. A. Neil, M. J. Booth, and T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am A |

3. | E. J. Fernandez, I. Iglesias, and P. Artal, “Closed-loop adaptive optics in the human eye,” Opt. Lett. |

4. | V. F. Canales and M. P. Cagigal, “Gain estimate for exoplanet detection with adaptive optics,” Astron. Astrophys. Suppl. Ser. |

5. | J. C. Bortz and B. J. Thompson, “Phase retrieval by optical phase differentiation,” Proceedings of the SPIE |

6. | O. von der Lühe, “Wavefront error measurement technique using extended, incoherent light sources,” Opt. Eng. |

7. | E. N. Ribak “Harnessing caustics for wave-front sensing,” Opt. Lett. |

8. | R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. |

9. | K. Iizuka, |

10. | J. M. Geary, |

11. | R. Irwan and R. G. Lane, “Analysis of optimal centroid estimation applied to Shack-Hartmann sensing,” Appl. Opt. |

12. | J. Primot, G. Rousset, and J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am A |

13. | B. M. Welsh and C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am A |

14. | N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. |

15. | R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(070.2580) Fourier optics and signal processing : Paraxial wave optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 19, 2003

Revised Manuscript: October 15, 2003

Published: October 20, 2003

**Citation**

Jose Oti, Vidal Canales, and Manuel Cagigal, "Analysis of the signal-to-noise ratio in the optical differentiation wavefront sensor," Opt. Express **11**, 2783-2790 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-21-2783

Sort: Journal | Reset

### References

- G. Vdovin, �??Micromachined membrane deformable mirrors�?? in Adaptive Optics Engineering Handbook, R. Tyson, ed. (Marcel Dekker Inc, New York, 1998)
- M. A. Neil, M. J. Booth, T. Wilson, �??New modal wave-front sensor: a theoretical analysis,�?? J. Opt. Soc. Am A 17, 1098-1107 (2000) [CrossRef]
- E. J. Fernandez, I. Iglesias and P. Artal, �??Closed-loop adaptive optics in the human eye,�?? Opt. Lett. 26, 746-748 (2001) [CrossRef]
- V. F. Canales and M. P. Cagigal, �??Gain estimate for exoplanet detection with adaptive optics,�?? Astron. Astrophys. Suppl. Ser. 145, 445-449 (2000) [CrossRef]
- J. C. Bortz and B. J. Thompson, �??Phase retrieval by optical phase differentiation,�?? Proceedings of the SPIE 351, 71-79 (1983) [CrossRef]
- O. von der Lühe, �??Wavefront error measurement technique using extended, incoherent light sources,�?? Opt. Eng. 27, 1078-1087 (1988).
- E. N. Ribak �??Harnessing caustics for wave-front sensing,�?? Opt. Lett. 26, 1834-1836 (2001). [CrossRef]
- R. Ragazzoni, �??Pupil plane wavefront sensing with an oscillating prism,�?? J. Mod. Opt. 43, 289-293 (1996). [CrossRef]
- K. Iizuka, Engineering optics (Springer-Verlag, Berlin, 1987)
- J. M. Geary, Introduction to wavefront sensors (SPIE Press, Washington, 1995). [CrossRef]
- R. Irwan and R. G.Lane, �??Analysis of optimal centroid estimation applied to Shack-Hartmann sensing,�?? Appl. Opt. 32, 6737-6743 (1999). [CrossRef]
- J. Primot, G. Rousset, and J. C. Fontanella, �??Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,�?? J. Opt. Soc. Am A 7, 1598-1608 (1990) [CrossRef]
- B. M. Welsh and C. S. Gardner, �??Performance analysis of adaptive optics systems using laser guide stars and slope sensors,�?? J. Opt. Soc. Am A 12, 1913-1923 (1989) [CrossRef]
- N. Roddier, �??Atmospheric wavefront simulation using Zernike polynomials,�?? Opt. Eng. 29, 1174-1180 (1990) [CrossRef]
- R. Cubalchini, �??Modal wave-front estimation from phase derivative measurements,�?? J. Opt. Soc. Am 69, 972-977 (1979) [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.