## Point spread function characteristics analysis of the wavefront coding system

Optics Express, Vol. 15, Issue 4, pp. 1543-1552 (2007)

http://dx.doi.org/10.1364/OE.15.001543

Acrobat PDF (186 KB)

### Abstract

Most of the previous imaging characteristics analysis of the wavefront coding system has been carried out within the frequency domain. In this paper, the stationary phase method is employed to perform the analysis within the spatial domain. The approximate expression of point spread function (PSF) in the presence of defocus aberration is derived for the system with a cubic phase mask, which shows a good agreement with the Fast Fourier Transform (FFT) approach. Based on this, the PSF characteristics are analyzed in terms of the boundaries, oscillations and sensitivities to defocus, astigmatism and coma.

© 2007 Optical Society of America

## 1. Introduction

1. E. R. Dowski and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. **34**,1859–1866 (1995). [CrossRef] [PubMed]

4. S. S. Sherif, E. R. Dowski, and W. T. Cathey, “A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE **4471**,272–280 (2001). [CrossRef]

8. G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE **5817**,34–44 (2005). [CrossRef]

## 2. Derivation of the PSF

*h*(

*x*,

*W*) of an incoherent imaging system is given by Eq. (1),

_{20}*W*is the traditional defocus aberration constant in unit of wavelength,

_{20}*u*is the normalized pupil coordinate,

*q*(

*u*) is the normalized pupil function, and

*k*is the wavenumber 2π/λ. For the wavefront coding system with a cubic phase mask and a bounded aperture, the normalized pupil function can be represented as

1. E. R. Dowski and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. **34**,1859–1866 (1995). [CrossRef] [PubMed]

*kW*| is assumed. Please see Appendix A for a detailed derivation of this result.

_{20}*W*=0, 2.5λ, 5λ are shown in Figs. 1(a), 1(b) and 1(c), respectively. The boundaries in Eq. (4) and the absolute errors between the FFT and approximate PSFs are also shown in Fig. 1. Though there is a big range in the middle part of approximate PSF with very small absolute errors, large absolute errors occur when the reduced spatial coordinate gets closer to the three boundaries in Eq. (4). This phenomenon can be referred to the stationary phase method and the same problem exists in Dowski and Cathey’s approximate expression of OTF [1

_{20}1. E. R. Dowski and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. **34**,1859–1866 (1995). [CrossRef] [PubMed]

*N*in the sampled pupil range [-

*L*,

*L*],

*N*equally spaced points in the reduced spatial range [-

*N*/(4

*L*),

*N*/(4

*L*)] is calculated within the FFT approach. If

*d*stands for the length of the desired reduced spatial range to plot the PSF, the number of calculated sample points that can be used is

*m*=4

*dL*. In order to acquire an accurate plot of the PSF in the middle part, the sampled pupil range should be enlarged, which leads to an increase of the sample point number

*N*to maintain the FFT accuracy.

## 3. PSF characteristics analysis

### 3.1 Boundaries

*u*=

*t*-

*kW*/(3α) can be applied,

_{20}*Ai*(

*x*) is the Airy function [10

10. D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional tomography using a cubic-phase plate extended depth-of-field system,” Opt. Lett. **24**,253–255 (1999). [CrossRef]

*W*=2.5λ. is shown in Fig. 2. A boundary on the left near the origin can be found for both bounded and unbounded aperture, while the boundary on the right only exists for the bounded aperture. The width of the PSF area of the wavefront coding system with a cubic phase mask and a bounded aperture can be approximated by the subtraction between the right and left boundary position in Eq. (4),

_{20}*f*is the diffraction-limited cutoff frequency of the incoherent imaging system. For the working wavelength λ,

_{0}*f*is equal to

_{0}*L*/(λ

*d*), where

_{i}*L*is the aperture diameter, and

*d*is the distance from the pupil to the diffraction-limited image plane [11]. About 99.79%, 99.77% and 99.68% of the PSF energy are constrained in the areas with the width determined by Eq. (6) for Figs. 1(a), 1(b) and 1(c), respectively. Thus Eq. (6) can be used to evaluate the sharpness of the wavefront-coded intermediate image with adequate accuracies, it is seen that a larger α and a smaller

_{i}*f*result in a broader PSF, i.e., a more blurry intermediate image. For an F/5 incoherent imaging system with a working wavelength of 587.6 nm, the diffraction-limited cutoff frequency is

_{0}*f*≈ 340.4 cyc/mm and the PSF area width is

_{0}*D*≈ 0.26 mm when a cubic phase mask of α=30π is applied. Comparing to the width of the PSF area of a traditional imaging system which can be calculated according to 1/

*f*, it can be found that the PSF area of the wavefront coding system with a cubic phase mask is much larger and the intermediate image without decoding is deeply blurred. Besides, Eq. (6) can help to prepare for the experiment of PSF measurements [12

_{0}12. M. R. Arnison, “Phase control and measurement in digital microscopy” (Sydney Digital Theses, Physics, 2006), http://hdl.handle.net/2123/569.

*kW*|/(3α). The derivation is not included in the paper because a much easier way of analysis will be presented in the paper. When a small defocus occurs, this ratio is approaching unity. However when a large defocus occurs, the decrease of the bandwidth becomes obvious. The same result can be found in a previous paper [13

_{20}13. M. Somayji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using cubic phase mask,” Appl. Opt. **45**,2911–2923 (2006). [CrossRef]

### 3.2 Oscillations

*kW*)

_{20}^{2}/(6πα)<

*x*≤ (3α - 2|

*kW*|)/(2π) is a sinusoidal decaying function with the increase of the frequency. Hence the bandwidth can be determined by the largest sinusoidal modulation frequency. The sinusoidal modulation frequency within this range can be expressed as

_{20}*kW*|/(3α)]. The ratio of this bandwidth to the traditional imaging system’s bandwidth, which is 2 when the normalized spatial coordinate for the pupil are used, is equal to the one derived in the subsection 3.1. An FFT PSF and an approximate PSF of a two-dimensional (2D) system with a cubic phase mask in the presence of defocus aberration are shown in Fig. 3 for α=30π and

_{20}*W*=2.5λ. In these 2D images, regions of larger intensity are given by lighter shades. From the previous analysis, it is easy to understand why the grid points with decreasing intensities and spaces exist in the PSF. From Eq. (7) it can be concluded that a larger α leads to a less space between grid points in the PSF, so Eq. (7) can also be used to estimate the strength of the modulation of the cubic phase mask. Compared to the PSF area width approach described in the subsection 3.1, only the middle part of the PSF is needed instead of the whole PSF.

_{20}3. A. Castro and J. O. Castañeda, “Increased depth of field with phase-only filters: ambiguity function,” Proc. SPIE **5827**,1–11 (2005). [CrossRef]

### 3.3 Sensitivities to aberrations

*kW*)

_{20}^{2}/(6πα) when the wavefront coding system suffers from the defocus aberration, which only leads to a linear phase shift

*exp*[-

*j*(

*kW*)

_{20}^{2}

*u*/3α] in OTF. For a bounded aperture, an equation similar to Eq. (5) can be derived from Eq. (3) with a substitution

*u*=

*t*-

*kW*/(3α),

_{20}*kW*)/3α from its original position. If the aperture shift is negligible compared to the spatial shift, the same phase shift will take place in the OTF, i.e., the MTF will not be significantly affected. Thus the wavefront coding system with a cubic phase mask is insensitive to the defocus aberration. PSFs calculated from Eq. (4) for α=30π and

_{20}*W*=0, 1λ, 2λ, 3λ are shown in Fig. 4 respectively, where the PSFs with only spatial shifts can be found.

_{20}*exp*(

*jkW*) is added into the original wavefront, where

_{22}u^{2}*W*is the traditional astigmatism aberration constant in unit of wavelength. It can be noticed that a defocus of

_{22}*kW*has been introduced, so the wavefront coding system with a cubic phase mask is also insensitive to the astigmatism aberration.

_{22}*exp*[

*jkW*(

_{31}*u*

^{2}+

*v*

^{2})

*u*] is added into the original wavefront of the 2D system, where

*W*is the traditional coma aberration constant in unit of wavelength,

_{31}*u*and

*v*are reduced spatial coordinates. For simplicity,

*v*is set to be zero for the case of 1D system, so the PSF in the presence of coma can be given by

*W*/λ). Thus the sensitivity to coma can be derived from Eq. (4). PSFs calculated from Eq. (4) for α=30π and

_{31}*W*=0, 1λ, 2λ, 3λ are shown in Fig. 5. The standard deviation (STD) of PSFs is also shown in Fig. 5 in black line, which indicates large errors between PSFs.

_{31}*n*leads to phase terms of lower order, which can be used not only to produce a phase mask of lower order [14

14. T. Hellmuth, A. Bich, R. Börret, and A. Kelm, “Variable phaseplates for focus invariant optical systems,” in *Optical Design and Engineering II*,
L. Mazuray and R. Wartmann, eds., Proc. SPIE **5962**,596215 (2005). [CrossRef]

*u*=

*t*-β /(4α),

*kW*. Besides the spatial shift and the aperture shift, coma can be treated as defocus effectively. The system with a quartic phase mask is sensitive to defocus [2

_{31}2. S. Mezouari, G. Muyo, and A. R. Harvey, “Amplitude and phase filters for mitigation of defocus and third-order aberrations,” Proc. SPIE **5249**,238–248 (2004). [CrossRef]

*W*is equal to 6.4λ, only a defocus of -2π is introduced when α is equal to 30π. PSFs of the system with a quartic phase mask in the presence of coma are shown in Fig. 6 for α=30π and

_{31}*W*=0, 2λ, 4λ, 6λ, where small STD errors can be found.

_{31}## 4. Conclusion

## Appendix A:

## Derivation of the PSF for cubic phase mask in the presence of defocus For the integral

*u*) = α

*u*

^{3}+

*ψu*

^{2}- 2

*πux*, and

*ψ*=

*kW*

_{20}. The first and second derivatives of

*μ (u)*are respectively,

*u*exists only when the following conditions are satisfied,

_{0}*u*only exists in the range of

_{0}*sign(x)*is the signum function defined as

*ψ*>0,

*u*exists, then

_{01}*u*exists, then

_{02}*h*(

*x*,

*W*) of the incoherent imaging system is equal to the half of the square of modulus of Eq. (A 1), so Eq. (4

_{20}4. S. S. Sherif, E. R. Dowski, and W. T. Cathey, “A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE **4471**,272–280 (2001). [CrossRef]

## References and links

1. | E. R. Dowski and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. |

2. | S. Mezouari, G. Muyo, and A. R. Harvey, “Amplitude and phase filters for mitigation of defocus and third-order aberrations,” Proc. SPIE |

3. | A. Castro and J. O. Castañeda, “Increased depth of field with phase-only filters: ambiguity function,” Proc. SPIE |

4. | S. S. Sherif, E. R. Dowski, and W. T. Cathey, “A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE |

5. | S. Mezouari and A. R. Harvey, “Primary aberrations alleviated with phase pupil filters,” Proc. SPIE |

6. | N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A |

7. | S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A |

8. | G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE |

9. | M. Born and E. Wolf, |

10. | D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional tomography using a cubic-phase plate extended depth-of-field system,” Opt. Lett. |

11. | J. W. Goodman, |

12. | M. R. Arnison, “Phase control and measurement in digital microscopy” (Sydney Digital Theses, Physics, 2006), http://hdl.handle.net/2123/569. |

13. | M. Somayji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using cubic phase mask,” Appl. Opt. |

14. | T. Hellmuth, A. Bich, R. Börret, and A. Kelm, “Variable phaseplates for focus invariant optical systems,” in |

**OCIS Codes**

(110.2960) Imaging systems : Image analysis

(110.4850) Imaging systems : Optical transfer functions

(220.1000) Optical design and fabrication : Aberration compensation

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 8, 2007

Revised Manuscript: January 25, 2007

Manuscript Accepted: January 26, 2007

Published: February 19, 2007

**Citation**

Wenzi Zhang, Zi Ye, Tingyu Zhao, Yanping Chen, and Feihong Yu, "Point spread function characteristics analysis of the wavefront coding system," Opt. Express **15**, 1543-1552 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1543

Sort: Year | Journal | Reset

### References

- E. R. Dowski and W. T. Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866 (1995). [CrossRef] [PubMed]
- S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004). [CrossRef]
- A. Castro and J. O. Castañeda, "Increased depth of field with phase-only filters: ambiguity function," Proc. SPIE 5827, 1-11 (2005). [CrossRef]
- S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001). [CrossRef]
- S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002). [CrossRef]
- N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A 5, 157-163 (2003). [CrossRef]
- S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006). [CrossRef]
- G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1985).
- D. L. Marks, R. A. Stack, and D. J. Brady, "Three-dimensional tomography using a cubic-phase plate extended depth-of-field system," Opt. Lett. 24, 253-255 (1999). [CrossRef]
- J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, 1996), Chap. 6.
- M. R. Arnison, "Phase control and measurement in digital microscopy" (Sydney Digital Theses, Physics, 2006), http://hdl.handle.net/2123/569>.
- M. Somayji and M. P. Christensen, "Enhancing form factor and light collection of multiplex imaging systems by using cubic phase mask," Appl. Opt. 45, 2911-2923 (2006). [CrossRef]
- T. Hellmuth, A. Bich, R. Börret, and A. Kelm, "Variable phaseplates for focus invariant optical systems," in Optical Design and Engineering II, L. Mazuray, R. Wartmann, eds., Proc. SPIE 5962, 596215 (2005). [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.