## Frequency analysis of wavefront curvature sensing: optimum propagation distance and multi-z wavefront curvature sensing

Optics Express, Vol. 17, Issue 5, pp. 3707-3715 (2009)

http://dx.doi.org/10.1364/OE.17.003707

Acrobat PDF (275 KB)

### Abstract

In this paper we determine the optimum propagation distance between measurement planes and the plane of the lens in a wavefront curvature sensor with the diffraction optics approach. From the diffraction viewpoint, the measured wavefront aberration can be decomposed into Fourier harmonics at various frequencies. The curvature signal produced by a single harmonic is analyzed with the wave propagation transfer function approach, which is the frequency analysis of wavefront curvature sensing. The intensity of the curvature signal is a sine function of the product of the propagation distance and the squared frequency. To maximize the curvature signal, the optimum propagation distance is proposed as one quarter of the Talbot length at the critical frequency (average power point at which the power spectrum density is the average power spectrum density). Following the determination of the propagation distance, the intensity of the curvature signal varies sinusoidally with the squared frequencies, vanishing at some higher frequency bands just like a comb filter. To cover these insensitive bands, wavefront curvature sensing with dual propagation distances or with multi-propagation distances is proposed.

© 2009 Optical Society of America

## 1. Introduction

1. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. **27**, 1223–1225 (1988). [CrossRef] [PubMed]

*et al*. [1

1. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. **27**, 1223–1225 (1988). [CrossRef] [PubMed]

*et al*. [2

2. M. Soto, E. Acosta, and S. Ríos, “Performance analysis of curvature sensors: optimum positioning of the measurement planes,” Opt. Express **11**, 2577–2588 (2003). [CrossRef] [PubMed]

3. O. Guyon, “High-performance curvature wavefront sensing for extreme AO,” SPIE **6691**, 66910G (2007). [CrossRef]

*et al*. [4

4. O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, and M. Watanabe, “Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature,” PASP. **120**, 655–664 (2008). [CrossRef]

4. O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, and M. Watanabe, “Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature,” PASP. **120**, 655–664 (2008). [CrossRef]

5. D. Johnston, B. Ellerbroek, and S. Pompeat, “Curvature sensing analysis,” SPIE **2201**, 528–538 (1994). [CrossRef]

7. F. Roddier, *Adaptive Optics in Astronomy* (Cambridge University Press, 1999), Chap. 5. [CrossRef]

8. G. Yang, B. Gu, and B. Dong, “Theory of the amplitude-phase retrieval in an any linear transform system and its applications,” SPIE **1767**, 457–478 (1992). [CrossRef]

## 2. Analysis of wavefront curvature sensing with the transfer function approach

### 2.1 Talbot effect in wavefront curvature sensing

1. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. **27**, 1223–1225 (1988). [CrossRef] [PubMed]

*L*with focal length

*f*. The normalized difference of intensities on two defocus planes

_{L}*M*

_{1}and

*M*

_{2}with defocus distance

*l*is proportional to the local wavefront curvature. Based on the imaging principle, two defocus planes

*M*

_{1}and

*M*

_{2}correspond to two measurement planes

*P*

_{1}and

*P*

_{2}symmetrically located before and after the lens with a propagation distance of

**27**, 1223–1225 (1988). [CrossRef] [PubMed]

9. M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. **73**, 1434–1441 (1983). [CrossRef]

10. F. Roddier, “Wavefront sensing and equation the irradiance transport,” Appl. Opt. **29**, 1402–1403 (1990). [CrossRef] [PubMed]

*P*

_{1}and

*P*

_{2}are linear functions of the local wavefront curvature ∇

^{2}

*φ*on the entrance pupil plane (i.e. the plane of the lens),

*A*is unity,

*λ*is the wavelength, and ∇

^{2}is the Laplacian or curvature operator.

*φ*(

*ξ*,

*η*) may be expressed as a summation of sine wave functions at various spatial frequencies where the (

*ξ*,

*η*) plane is the entrance pupil plane of lens

*L*. The general component is

*a*cos (

*k*+

_{ξ}ξ*k*) +

_{η}η*b*sin (

*k*+

_{ξ}ξ*k*), which allows for any orientation on the (

_{η}η*ξ*,

*η*) plane and any amplitude where

*k*or

_{ξ}*k*are the angular spatial frequency in radians per unit of

_{η}*ξ*, or

*η*.

*ξ*,

*η*) axes, we can return to sin(

*k*) with any amplitude, and this expression will suffice for the following discussion:

_{ξ}ξ*f*

_{0}is the fundamental frequency;

*B*and

_{n}*θ*are the coefficient and the initial phase of the

_{n}*n*order harmonic, respectively;

^{th}*n*is an integer; and the piston term (

*n*= 0) of phase is omitted. The Fourier harmonics or the sine wave functions are the eigenmodes of the Laplacian or the curvature operator [11

11. T. Gureyev and K. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A. **13**, 1670–1682 (1996). [CrossRef]

**B**=

*B*sin (2

*πfξ*+ θ) is given by

*P*

_{1}and

*P*

_{2}conform to the wavefront map

### 2.2 Curvature signal calculated with transfer function approach

13. V. Arrizón and J. Ojeda-Castaneda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A **9**, 1801–1806 (1992). [CrossRef]

*L*is the period, 2

*B*is the modulated amplitude,

*J*is the

_{q}*q*order Bessel function of the first kind,

^{th}*q*is an integer, and

*j*is √-1 . With the transfer function approach, we obtain the complex amplitude on plane

*P*

_{1}to be given by

*x*is the axis on plane

*P*

_{1}parallel to the

*ξ*axis.

*z*satisfies

*P*

_{1}is equal to unity, which suggests the sinusoidal phase component remains invisible at multiples of one half of the Talbot length. No intensity information converted from the phase distortion exists at those positions.

*P*

_{1}can be given as

*else*contains higher order and more complex harmonic patterns. The weighting factor of each harmonic component in Eq. (7) is determined by the product of

*J*(

_{q}*B*). In the range of β≪1, the magnitudes of

*J*

_{0}(

*B*) and

*J*

_{0}(

*B*) are much larger than other higher-order magnitudes. A comparison of the coefficients of the first 5 harmonic components is shown in Fig. 2. Accordingly, the

*else*term in Eq. (7) can be ignored in the small perturbation approximation (i.e.

*B*≪ 1).

*n*-1)

*Z*/4 planes can be expressed only for the sinusoidal phase distribution as

_{T}*P*

_{2}with distance -

*z*can be written as

*S*is given as

*z*and the squared frequency

*f*. The sinusoidal modulation predicted by the transfer function approach cannot be obtained by geometrical optics. Equation (11) is the core of this paper, from which we deduce the optimum propagation distance and the frequency filter of curvature sensing.

### 3. Selection of optimum propagation distance for wavefront curvature sensing

*f*, the curvature signals have the largest amplitude or the highest sensitivity at the propagation distances

*z*= (2

*n*- 1)

*Z*/4, which can be expressed as

_{T}*z*=

*nZ*/2, the curvature signal is null visibility for frequency

_{T}*f*. In other words, the curvature sensor cannot detect phase distortions at the spatial frequency of

*f*with propagation distances

*z*=

*nZ*/2.

_{T}10. F. Roddier, “Wavefront sensing and equation the irradiance transport,” Appl. Opt. **29**, 1402–1403 (1990). [CrossRef] [PubMed]

*z*

_{opt}and the frequency

*f*is given by

*f*can be chosen as the average power point at which the power spectrum density is equal to the average power spectrum density in the measured or compensated frequency band. The wavefront curvature sensor designed with the critical frequency will cover a broader frequency spectrum and receive more power. Following the selection of critical frequency

_{k}*f*, the propagation distance

_{k}*z*of curvature sensing is determined by Eq. (13).

_{k}### 4. Filter model of wavefront curvature sensing

*z*, the sensitivity of curvature sensing depends on the sine function of the squared frequency. Substituting

_{k}*z*in Eq. (11) with

*z*, the curvature signal for phase distortions at various spatial frequencies yields

_{k}*z*, curvature sensing is quite sensitive to phase distortions near frequencies √2

_{k}*n*-1

*f*. On the contrary, it is insensitive to phase distortions near frequencies √2

_{k}*n*

*f*. We call this sensitivity spectrum a comb filter. It implies that the curvature sensing technique does not impose the upper bound for the transferable spatial frequency. Physically, its upper bound is just limited by diffraction, i.e.,

_{k}*f*< λ

^{-1}. In addition, considering the blurring effect induced by the wavefront aberration, diffraction effects limit spatial resolution of the measurements to √

*λz*[14

14. M. van Dam and R. Lane, “Extended analysis of curvature sensing,” J. Opt. Soc. Am. A **19**, 1390–1397 (2002). [CrossRef]

^{-1}to 30 m

^{-1}, the critical frequency

*f*is selected as 10 m

_{k}^{-1}, the determined optimum propagation distance

*z*is 7901 m, the modulated amplitude of phase distortion is 0.2 rad, the wavelength is 0.6328

_{k}*μ*m , and the shade of color in the image stands for the magnitude of the curvature signals.

16. S. Woo s and A. Greenaway, “Wavefront sensing by use of a Green’s function solution to the intensity transport equation,” J. Opt. Soc. Am. A. **20**, 508–512 (2003). [CrossRef]

## 5. The design of dual-z or multi-z wavefront curvature sensing

3. O. Guyon, “High-performance curvature wavefront sensing for extreme AO,” SPIE **6691**, 66910G (2007). [CrossRef]

*z*is the conventional optimum propagation distance, determined by

_{k}*f*and Eq. (13). Following the determination of

_{k}*z*, curvature sensing is not sensitive to the frequencies of √2

_{k}*f*, √4

_{k}*f*, √6

_{k}*f*, √8

_{k}*f*, √10

_{k}*f*, √12

_{k}*f*but to the frequencies of

_{k}*f*, √3

_{k}*f*, √5

_{k}*f*, √7

_{k}*f*, √9

_{k}*f*, √11

_{k}*f*, √13

_{k}*f*. To measure the distortions at frequencies insensitive for

_{k}*z*, a shorter propagation distance

_{k}*z*should be included in the same curvature sensing. For example,

_{k2}*z*

_{k2}could be conveniently chosen as the optimum distance for the frequency of √2

*f*by Eq. (13), which is given by

_{k}*z*is sensitive to the frequencies of √2

_{k2}*f*, √6

_{k}*f*, √10

_{k}*f*, √14

_{k}*f*, √18

_{k}*f*.

_{k}*z*and

_{k}*z*

_{k2}, such as √4

*f*, √8

_{k}*f*, √12

_{k}*f*, √16

_{k}*f*. A shorter propagation distance

*z*

_{t3}sensitive to the frequency of √4

*f*can be introduced into the same curvature sensor, and therefore the insensitive frequencies just leave √8

_{k}*f*, √16

_{k}*f*, √24

*f*. Similarly,

_{k}*z*is given by

_{k3}*z*,

_{k}*z*/2 ,

_{k}*z*/4, and

_{k}*z*/8, respectively. From Fig. 4 it can be seen that the extra propagation distances can fill up the insensitive frequency bands of the first propagation distance. Due to the spreading width of the passband, the covering effect of only two propagation distances, such as

_{k}*z*and

_{k}*z*/8, can go beyond the prospects.

_{k}3. O. Guyon, “High-performance curvature wavefront sensing for extreme AO,” SPIE **6691**, 66910G (2007). [CrossRef]

4. O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, and M. Watanabe, “Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature,” PASP. **120**, 655–664 (2008). [CrossRef]

17. J. Graves and D. McKenna, “University of Hawaii adaptive optics system: III. Wavefront curvature sensor,” SPIE **1542**, 262–272 (1991). [CrossRef]

18. P. Blanchard and A. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt. **38**, 6692–6699 (1999). [CrossRef]

19. P. Blanchard, D. Fisher, S. C. Woods, and A. H. Greenaway, “Phase-diversity wavefront sensing with a distorted diffraction grating,” Appl. Opt. **39**, 6649–6655 (2000). [CrossRef]

20. R. W. Lambert, R. Cortés-Martínez, A. J. Waddie, J. D. Shephard, M. R. Taghizadeh, A. H. Greenaway, and D. P. Hand, “Compact optical system for pulse-to-pulse laser beam quality measurement and applications in laser machining,” Appl. Opt. **43**, 5037–5046 (2004). [CrossRef] [PubMed]

21. F. Xi, Z. Jiang, X. Xu, and Y. Geng, “High-diffractive-efficiency defocus grating for wavefront curvature sensing,” J. Opt. Soc. Am. A **24**, 3444–3448 (2007). [CrossRef]

*M*

^{2}factor measurement. Phase defocus gratings can adjust the photons flowing into different defocus spots.

## 6. Conclusion

## References and links

1. | F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. |

2. | M. Soto, E. Acosta, and S. Ríos, “Performance analysis of curvature sensors: optimum positioning of the measurement planes,” Opt. Express |

3. | O. Guyon, “High-performance curvature wavefront sensing for extreme AO,” SPIE |

4. | O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, and M. Watanabe, “Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature,” PASP. |

5. | D. Johnston, B. Ellerbroek, and S. Pompeat, “Curvature sensing analysis,” SPIE |

6. | J. Goodman, |

7. | F. Roddier, |

8. | G. Yang, B. Gu, and B. Dong, “Theory of the amplitude-phase retrieval in an any linear transform system and its applications,” SPIE |

9. | M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. |

10. | F. Roddier, “Wavefront sensing and equation the irradiance transport,” Appl. Opt. |

11. | T. Gureyev and K. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A. |

12. | J. Cowley and A. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. London Sect. B |

13. | V. Arrizón and J. Ojeda-Castaneda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A |

14. | M. van Dam and R. Lane, “Extended analysis of curvature sensing,” J. Opt. Soc. Am. A |

15. | R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

16. | S. Woo s and A. Greenaway, “Wavefront sensing by use of a Green’s function solution to the intensity transport equation,” J. Opt. Soc. Am. A. |

17. | J. Graves and D. McKenna, “University of Hawaii adaptive optics system: III. Wavefront curvature sensor,” SPIE |

18. | P. Blanchard and A. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt. |

19. | P. Blanchard, D. Fisher, S. C. Woods, and A. H. Greenaway, “Phase-diversity wavefront sensing with a distorted diffraction grating,” Appl. Opt. |

20. | R. W. Lambert, R. Cortés-Martínez, A. J. Waddie, J. D. Shephard, M. R. Taghizadeh, A. H. Greenaway, and D. P. Hand, “Compact optical system for pulse-to-pulse laser beam quality measurement and applications in laser machining,” Appl. Opt. |

21. | F. Xi, Z. Jiang, X. Xu, and Y. Geng, “High-diffractive-efficiency defocus grating for wavefront curvature sensing,” J. Opt. Soc. Am. A |

**OCIS Codes**

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

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 9, 2008

Revised Manuscript: December 16, 2008

Manuscript Accepted: December 19, 2008

Published: February 24, 2009

**Citation**

Xi Fengjie, Jiang Zongfu, Xu Xiaojun, Hou Jing, and Liu Zejin, "Frequency analysis of wavefront curvature
sensing: optimum propagation distance and
multi-z wavefront curvature sensing," Opt. Express **17**, 3707-3715 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3707

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### References

- F. Roddier, "Curvature sensing and compensation: a new concept in adaptive optics," Appl. Opt. 27, 1223-1225 (1988). [CrossRef] [PubMed]
- M. Soto, E. Acosta, and S. Ríos, "Performance analysis of curvature sensors: optimum positioning of the measurement planes," Opt. Express 11, 2577-2588 (2003). [CrossRef] [PubMed]
- O. Guyon, "High-performance curvature wavefront sensing for extreme AO," SPIE 6691, 66910G (2007). [CrossRef]
- O. Guyon, C. Blain, H. Takami, Y. Hayano, M. Hattori, M. Watanabe, "Improving the Sensitivity of Astronomical Curvature Wavefront Sensor Using Dual-Stroke Curvature," PASP. 120, 655-664 (2008). [CrossRef]
- D. Johnston, B. Ellerbroek, and S. Pompeat, "Curvature sensing analysis," SPIE 2201, 528-538 (1994). [CrossRef]
- J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 2.
- F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999), Chap. 5. [CrossRef]
- G. Yang, B. Gu, and B. Dong, "Theory of the amplitude-phase retrieval in an any linear transform system and its applications," SPIE 1767, 457-478 (1992). [CrossRef]
- M. Teague, "Deterministic phase retrieval: a Green’s function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983). [CrossRef]
- F. Roddier, "Wavefront sensing and equation the irradiance transport," Appl. Opt. 29, 1402-1403 (1990). [CrossRef] [PubMed]
- T. Gureyev and K. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A. 13, 1670-1682 (1996). [CrossRef]
- J. Cowley and A. Moodie, "Fourier images IV: the phase grating," Proc. Phys. Soc. London Sect. B 76, 378-384 (1960).
- V. Arrizόn and J. Ojeda-Castaňeda, "Irradiance at Fresnel planes of a phase grating," J. Opt. Soc. Am. A 9, 1801-1806 (1992). [CrossRef]
- M. van Dam and R. Lane, "Extended analysis of curvature sensing," J. Opt. Soc. Am. A 19, 1390-1397 (2002). [CrossRef]
- R. Gerchberg and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
- S. Woods and A. Greenaway, "Wavefront sensing by use of a Green’s function solution to the intensity transport equation," J. Opt. Soc. Am. A. 20, 508-512 (2003). [CrossRef]
- J. Graves and D. McKenna, "University of Hawaii adaptive optics system: III. Wavefront curvature sensor," SPIE 1542, 262-272 (1991). [CrossRef]
- P. Blanchard and A. Greenaway, "Simultaneous multiplane imaging with a distorted diffraction grating," Appl. Opt. 38, 6692-6699 (1999). [CrossRef]
- P. Blanchard, D. Fisher, S. C. Woods, and A. H. Greenaway, "Phase-diversity wavefront sensing with a distorted diffraction grating," Appl. Opt. 39, 6649-6655 (2000). [CrossRef]
- R. W. Lambert, R. Cortés-Martínez, A. J. Waddie, J. D. Shephard, M. R. Taghizadeh, A. H. Greenaway, and D. P. Hand, "Compact optical system for pulse-to-pulse laser beam quality measurement and applications in laser machining," Appl. Opt. 43, 5037-5046 (2004). [CrossRef] [PubMed]
- F. Xi, Z. Jiang, X. Xu, and Y. Geng, "High-diffractive-efficiency defocus grating for wavefront curvature sensing," J. Opt. Soc. Am. A 24, 3444-3448 (2007). [CrossRef]

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