## Improvements for determining the modulation transfer function of charge-coupled devices by the speckle method

Optics Express, Vol. 14, Issue 13, pp. 5928-5936 (2006)

http://dx.doi.org/10.1364/OE.14.005928

Acrobat PDF (166 KB)

### Abstract

We present and evaluate two corrections applicable in determining the modulation transfer function (MTF) of a charge-coupled device (CCD) by the speckle method that minimize its uncertainty: one for the low frequency region and another for the high frequency region. The correction at the low-spatial-frequency region enables attenuation of the high power-spectral-density values that arise from the field and CCD response non-uniformities. In the high-spatial-frequency region the results show that the distance between the CCD and the aperture is critical and significantly influences the MTF; a variation of 1 mm in the distance can cause a root-mean-square error in the MTF higher than 10%. We propose a simple correction that minimizes the experimental error committed in positioning the CCD and that diminishes the error to 0.43%.

© 2006 Optical Society of America

## 1. Introduction

1. J. C. Feltz and M. A. Karim, “Modulation transfer function of charge-coupled devices,” Appl. Opt. **29**, 717–722 (1990). [CrossRef] [PubMed]

2. S. K. Park, R. Schowengerdt, and M. A. Kaczynski, “Modulation-transfer-function analysis for sampled image system,” Appl. Opt. **23**, 2572–2582 (1984). [CrossRef] [PubMed]

3. A. Daniels, G. D. Boreman, A. D. Ducharme, and E. Sapir, “Random transparency targets for modulation transfer function measurement in the visible and infrared regions,” Opt. Eng. **34**, 860–868 (1995). [CrossRef]

4. S. M. Backman, A. J. Makynen, T. T. Kolehmainen, and K. M. Ojala, “Random target method for fast MTF inspection,” Opt. Express **12**, 2610–2615 (2004). [CrossRef] [PubMed]

5. E. Levy, D. Peles, M. Opher-Lipson, and S. G. Lipson, “Modulation transfer function of a lens measured with a random target method,” Appl. Opt. **38**, 679–683 (1999). [CrossRef]

7. G. D. Boreman, Y. Sun, and A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. **29**, 339–342 (1990). [CrossRef]

8. A. M. Pozo and M. Rubiño, “Optical characterization of ophthalmic lenses by means of modulation transfer function determination from a laser speckle pattern,” Appl. Opt. **44**, 7744–7748 (2005). [CrossRef] [PubMed]

7. G. D. Boreman, Y. Sun, and A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. **29**, 339–342 (1990). [CrossRef]

9. M. Sensiper, G. D. Boreman, A. D. Ducharme, and D. R. Snyder, “Modulation transfer function testing of detector arrays using narrow-band laser speckle,” Opt. Eng. **32**, 395–400 (1993). [CrossRef]

10. A. M. Pozo and M. Rubiño, “Comparative analysis of techniques for measuring the modulation transfer functions of charge-coupled devices based on the generation of laser speckle,” Appl. Opt. **44**, 1543–1547 (2005). [CrossRef] [PubMed]

9. M. Sensiper, G. D. Boreman, A. D. Ducharme, and D. R. Snyder, “Modulation transfer function testing of detector arrays using narrow-band laser speckle,” Opt. Eng. **32**, 395–400 (1993). [CrossRef]

*et al*. [7

7. G. D. Boreman, Y. Sun, and A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. **29**, 339–342 (1990). [CrossRef]

_{output}) at low spatial frequencies are due to the spatial non-uniformity in the CCD response and the non-uniformity of the field without considering the speckle. Furthermore, we show in this work that it is possible to attenuate the high values of the PSD

_{output}, due to this lack of uniformity, enabling a fit of the PSD

_{output}to a polynomial function. To date, the influence of a correction of non-uniformity of response of this type has not been investigated in the determination of the MTF by the speckle method.

**29**, 339–342 (1990). [CrossRef]

## 2. Theoretical background

**29**, 339–342 (1990). [CrossRef]

*ξ, η*are the spatial frequencies corresponding to the horizontal and vertical directions, respectively;

*PSD*is the PSD determined from the speckle pattern captured with the CCD, and it is proportional to the squared magnitude of the Fourier transform of the speckle pattern;

_{output}*PSD*is the theoretical PSD, known for a single slit and given by

_{input}*tri*(

*X*)=1-|

*X*| for |

*X*|≤1 and zero elsewhere, 〈

*I*〉

^{2}is the square of the average speckle irradiance,

*δ*(

*ξ,η*) is a delta function,

*l*and

_{1}*l*are, respectively, horizontal and vertical dimensions of the single-slit, λ is wavelength of the laser, and z is the distance between the slit and the CCD. By virtue of the geometry of the single slit, the MTF can be determined independently for the directions

_{2}*x*and

*y*. In the present work, we have only determined the horizontal MTF to show the corrections that we propose. This can be done in a similar way for the vertical direction.

11. J. R. Janesick, *Scientific Charge-Coupled Devices* (SPIE Press, Bellingham, Washington, 2001), Chap. 4. [CrossRef]

_{I}), diffusion MTF (MTF

_{D}), and charge transfer efficiency MTF (MTF

_{CTE}), attending to the processes that originate them:

*MTF*is a fixed modulation loss that is specific to the geometry of the pixel.

_{I}*MTF*is dependent on the depth of the pixels and occurs because charge generated under a pixel diffuse to a neighbouring one.

_{D}*MTF*is a consequence of the inefficiency in charge transfer from pixel to pixel. The

_{CTE}*MTF*is given by

_{I}*x*that is repeated with periodicity or pixel pitch

*x*, ξ is the spatial frequency, and

*ξ*is the Nyquist frequency of the CCD. In the CCD that we used

_{Ny}*Δx=x*. The MTF measured is affected by the components

*MTF*,

_{I}*MTF*and

_{D}*MTF*, being

_{CTE}*MTF*the fundamental limit.

_{I}## 3. Experimental device and data processing

### 3.1 Experimental device

*ξ*, is given by

_{Ny}*ξ*=1/(2

_{Ny}*x*), 50.5 cycles/mm, in our case.

### 3.2 Correction at low spatial frequencies

12. A. Ferrero, J. Campos, and A. Pons, “Correction of photoresponse nonuniformity for matrix detectors based on prior compensation for their nonlinear behavior,” Appl. Opt. **45**, 2422–2427 (2006). [CrossRef] [PubMed]

15. D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. **32**, 1854–1859 (1993). [CrossRef]

17. E. Schröder, “Elimination of granulation in laser beam projections by means of moving diffusers,” Opt. Commun. **3**, 68–72 (1971). [CrossRef]

*i,j*are the coordinates of each pixel in the directions

*x*and

*y*, respectively. By ‘

*Speckle*’, we refer to the final corrected frame, ‘

_{corrected}*Dark*’ is the image captured with the CCD occluded and which characterizes the FPN, ‘

*Speckle*’ is the image taken with the rotating diffuser stopped, and ‘

_{raw}*Flat*’ is the frame captured with the rotating diffuser in movement which jointly characterizes the PRNU and the non-uniformity produced by the slit.

_{output}.

_{output}is determined. Finally, applying Eqs. (1–2), it is possible to determine the MTF of the CCD.

### 3.3 Correction at high spatial frequencies

*l*is the slit width,

_{1}*λ*the wavelength of the laser, and

*ξ*the Nyquist frequency of the CCD. In our case, we get

_{Ny}*z*=32.3 mm. This value of

*z*really corresponds to the distance between the slit and the CCD plane where the pixels are found. However, the surface of the pixels is not accessible because it is protected by a window. Besides, neither the distance from the window to the surface, neither the thickness of the window and its refractive index are known. Therefore, it is difficult to place the CCD at exactly the correct distance from the aperture, this being the cause of a systematic error in the high-frequency range of the PSD

_{output}. Studying this error, we have placed the external face of the protective window at distance slightly shorter than

*z*from the aperture and have measured the PSD

_{output}at this distance and another around it. It is worth remarking that, for the CCD under study, the protective window is very close to the surface of the pixels.

_{output}.

## 4. Results and discussion

_{output}determined for the speckle pattern without correcting the response non-uniformity is shown against the spatial frequency normalized to the Nyquist frequency. It bears noting the high values obtained at low spatial frequencies for the PSD

_{output}that impede to fit a low order polynomial to the experimental data. These high values are found below the normalized frequency 0.0125, which is equivalent to 0.63 cycles/mm.

_{output}on applying the correction of the response non-uniformity [Eq. (5)] to the speckle pattern. Comparing Figs. 2(a) and 2(b), we do not get high PSD

_{output}values at low frequencies when the correction of the response non-uniformity is used.

_{output}of varying the distance between the CCD and the slit, we measured the PSD

_{output}at 3 different distances between the window of the CCD and the aperture. The first distance chosen between the window and the aperture was 31.8 mm, very close to the value of

*z*obtained from Eq. (6) for the CCD under study. Figure 3 provides a representation of the replicated PSD

_{output}which clearly shows the aliasing. For this distance, the Nyquist frequency is still not reached because the overlap has still not occurred, and therefore the cut-off frequency of the PSD

_{output}is lower than the Nyquist frequency of the CCD. It would thus be necessary to further reduce the distance between the CCD and the aperture and because of that two more measurements were done at 30.8 mm and 29.8 mm.

_{output}begins to increase due to the overlap of the two branches of the spectrum. The correction that we propose here consists of subtracting this minimum value (corresponding to the Nyquist frequency) that can be considered as a baseline value, to the fitted curve of the PSD

_{output}. This subtraction should be done when the overlap of the two branches begins.

_{output}, following the procedure of Boreman

*et al*. [7

**29**, 339–342 (1990). [CrossRef]

_{output}. Figure 5 shows the MTFs calculated after applying correction for the distances of 29.8 mm and 30.8 mm, and the MTF calculated for 31.8 mm are compared. We can see how both MTFs are much closer now than before correcting them. The rms error between the corrected MTFs is only 0.43%, while without correction it was 10.6%. In view of these results, we conclude that the correction proposed minimizes the influence of the error committed on positioning the CCD with respect to the aperture. In addition, this correction presents the advantage that eliminates the growing part in the MTF near the Nyquist frequency (compare Figs. 4 and 5) and therefore permits the characterization of the CCD up to the Nyquist frequency.

_{output}depends on the method used to measure the MTF, since the PSDoutput depends on the pattern used as the input signal. In our case, as shown in Fig. 3, the main effect that the aliasing causes in the PSD

_{output}due to the error committed experimentally on positioning the CCD is equivalent to an increase in the baseline level. In effect, on diminishing the distance between the CCD and the aperture, the speckle pattern contains frequencies higher than the Nyquist frequency, which are aliased to lower frequencies. Because of this, the baseline level of the PSD

_{output}increases. The results shown in Fig. 4 confirm experimentally that the increase in the baseline level due to the aliasing is the main cause of the dispersion of the MTF values at high frequencies. By applying the correction that we propose here, the MTF is not affected by small variations in the position of the CCD with respect to the aperture, as shown in Fig. 5. This does not mean that the aliasing of the speckle pattern has been removed on applying this correction, but rather that the influence of the aliasing has been minimized in order to measure the MTF with low uncertainty.

_{I}, as was necessary, given that the MTF

_{I}sets the theoretical limit for MTF performance. The differences between the experimental MTFs and the MTFI were due to the components MTF

_{D}and MTFCTE of Eq. (3). It might be expected for MTF

_{D}to be noticeable in this CCD, since pixels are small and charge may diffuse to neighboring pixels. It might also be expected for the MTF

_{CTE}to be noticeable, since the CCD has a large number of pixels and there are many charge transfers involved.

## 5. Conclusions

_{output}are due to a non-uniformity of response. By correcting the non-uniformity of the response the PSD

_{output}peak at low frequency disappears and a low order polynomial can be fitted to the PSD

_{output}to calculate the MTF. Secondly, we have analysed the region of high spatial frequencies. Our results show that the distance between the CCD and the aperture is critical. The imprecision in experimentally establishing the distance between the CCD and the aperture influences the fit of the MTF. A slight variation (on the order of 1 mm) in the position of the CCD with respect to the aperture causes a root-mean-square error of 10.6% between MTFs. After a simple correction consisting of subtracting the baseline level in the fitted PSD

_{output}, the root-mean-squared error between MTFs determined at different positions is reduced up to 0.43%. This correction minimizes therefore the influence of the error committed experimentally on positioning the CCD with respect to the aperture, and permits furthermore to characterize the CCD up to the Nyquist frequency. These two corrections should be taken into account when determining the MTF of CCDs by the single-slit speckle method with low uncertainty.

## Acknowledgments

## References and links

1. | J. C. Feltz and M. A. Karim, “Modulation transfer function of charge-coupled devices,” Appl. Opt. |

2. | S. K. Park, R. Schowengerdt, and M. A. Kaczynski, “Modulation-transfer-function analysis for sampled image system,” Appl. Opt. |

3. | A. Daniels, G. D. Boreman, A. D. Ducharme, and E. Sapir, “Random transparency targets for modulation transfer function measurement in the visible and infrared regions,” Opt. Eng. |

4. | S. M. Backman, A. J. Makynen, T. T. Kolehmainen, and K. M. Ojala, “Random target method for fast MTF inspection,” Opt. Express |

5. | E. Levy, D. Peles, M. Opher-Lipson, and S. G. Lipson, “Modulation transfer function of a lens measured with a random target method,” Appl. Opt. |

6. | G. D. Boreman and E. L. Dereniak, “Method for measuring modulation transfer function of charge-coupled devices using laser speckle,” Opt. Eng. |

7. | G. D. Boreman, Y. Sun, and A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. |

8. | A. M. Pozo and M. Rubiño, “Optical characterization of ophthalmic lenses by means of modulation transfer function determination from a laser speckle pattern,” Appl. Opt. |

9. | M. Sensiper, G. D. Boreman, A. D. Ducharme, and D. R. Snyder, “Modulation transfer function testing of detector arrays using narrow-band laser speckle,” Opt. Eng. |

10. | A. M. Pozo and M. Rubiño, “Comparative analysis of techniques for measuring the modulation transfer functions of charge-coupled devices based on the generation of laser speckle,” Appl. Opt. |

11. | J. R. Janesick, |

12. | A. Ferrero, J. Campos, and A. Pons, “Correction of photoresponse nonuniformity for matrix detectors based on prior compensation for their nonlinear behavior,” Appl. Opt. |

13. | A. F. Milton, F. R. Barone, and M. R. Kruer, “Influence of nonuniformity on infrared focal plane array performance,” Opt. Eng. |

14. | M. Schulz and L. Caldwell, “Nonuniformity correction and correctability of infrared focal plane arrays,” Infrared Phys. Technol. |

15. | D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. |

16. | T. S. McKechnie and J. C. Dainty, ed. (Springer-Verlag, New York, 1984). |

17. | E. Schröder, “Elimination of granulation in laser beam projections by means of moving diffusers,” Opt. Commun. |

18. | G. D. Boreman, “Fourier spectrum techniques for characterization of spatial noise in imaging arrays,” Opt. Eng. |

**OCIS Codes**

(040.1520) Detectors : CCD, charge-coupled device

(110.4100) Imaging systems : Modulation transfer function

(110.6150) Imaging systems : Speckle imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 11, 2006

Revised Manuscript: May 24, 2006

Manuscript Accepted: June 19, 2006

Published: June 26, 2006

**Citation**

A. M. Pozo, A. Ferrero, M. Rubiño, J. Campos, and A. Pons, "Improvements for determining the modulation transfer function of charge-coupled devices by the speckle method," Opt. Express **14**, 5928-5936 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-5928

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### References

- J. C. Feltz and M. A. Karim, "Modulation transfer function of charge-coupled devices," Appl. Opt. 29,717-722 (1990). [CrossRef] [PubMed]
- S. K. Park, R. Schowengerdt, and M. A. Kaczynski, "Modulation-transfer-function analysis for sampled image system," Appl. Opt. 23,2572-2582 (1984). [CrossRef] [PubMed]
- A. Daniels, G. D. Boreman, A. D. Ducharme, and E. Sapir, "Random transparency targets for modulation transfer function measurement in the visible and infrared regions," Opt. Eng. 34,860-868 (1995). [CrossRef]
- S. M. Backman, A. J. Makynen, T. T. Kolehmainen, and K. M. Ojala, "Random target method for fast MTF inspection," Opt. Express 12,2610-2615 (2004). [CrossRef] [PubMed]
- E. Levy, D. Peles, M. Opher-Lipson, and S. G. Lipson, "Modulation transfer function of a lens measured with a random target method," Appl. Opt. 38,679-683 (1999). [CrossRef]
- G. D. Boreman and E. L. Dereniak, "Method for measuring modulation transfer function of charge-coupled devices using laser speckle," Opt. Eng. 25,148-150 (1986).
- G. D. Boreman, Y. Sun, and A. B. James, "Generation of laser speckle with an integrating sphere," Opt. Eng. 29,339-342 (1990). [CrossRef]
- A. M. Pozo and M. Rubiño, "Optical characterization of ophthalmic lenses by means of modulation transfer function determination from a laser speckle pattern," Appl. Opt. 44,7744-7748 (2005). [CrossRef] [PubMed]
- M. Sensiper, G. D. Boreman, A. D. Ducharme, and D. R. Snyder, "Modulation transfer function testing of detector arrays using narrow-band laser speckle," Opt. Eng. 32,395-400 (1993). [CrossRef]
- A. M. Pozo and M. Rubiño, "Comparative analysis of techniques for measuring the modulation transfer functions of charge-coupled devices based on the generation of laser speckle," Appl. Opt. 44,1543-1547 (2005). [CrossRef] [PubMed]
- J. R. Janesick, Scientific Charge-Coupled Devices (SPIE Press, Bellingham,Washington, 2001), Chap. 4. [CrossRef]
- A. Ferrero, J. Campos, and A. Pons, "Correction of photoresponse nonuniformity for matrix detectors based on prior compensation for their nonlinear behavior," Appl. Opt. 45,2422-2427 (2006). [CrossRef] [PubMed]
- A. F. Milton, F. R. Barone, and M. R. Kruer, "Influence of nonuniformity on infrared focal plane array performance," Opt. Eng. 24,855-862 (1985).
- M. Schulz and L. Caldwell, "Nonuniformity correction and correctability of infrared focal plane arrays," Infrared Phys. Technol. 36,763-777 (1995). [CrossRef]
- D. L. Perry and E. L. Dereniak, "Linear theory of nonuniformity correction in infrared staring sensors," Opt. Eng. 32,1854-1859 (1993). [CrossRef]
- T. S. McKechnie, "Speckle reduction," in Laser speckle and related phenomena, Vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
- E. Schröder, "Elimination of granulation in laser beam projections by means of moving diffusers," Opt. Commun. 3,68-72 (1971). [CrossRef]
- G. D. Boreman, "Fourier spectrum techniques for characterization of spatial noise in imaging arrays," Opt. Eng. 26,985-991 (1987).

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