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The degenerated performance of extend depth of field (EDoF) in wavefront coding system which using cubic phase mask is simulated. A periodical rotationally symmetric surface error structure is presented and combined with comparison the similarity of point spread function (PSF). The peak to valley (PV) error of the cubic surface is needed smaller than 15% compare with the sag of the cubic surface for low period error existed.
©2009 Optical Society of America
1. Introduction
Wavefront coding technology can be used for many kinds of optical system just like digital camera, microscope, and range finder [1–4], etc. Some different wavefront coding methods are also presented, One of the most noticed method is Wavefront coding® which was presented by Cathey and Dowski in 1995[5]. Wavefront coding® is low tolerance sensitivity for off-axis aberrations and assembling in optical system were reported [6–10]. But no any research is talked about the manufacturing errors of cubic phase mask and its effect in EDoF system.
The rest of the paper is organized as follows. In Section 2 a periodical rotationally symmetric surface manufacturing error model of cubic phase mask is presented form measurement result and the required manufacturing accuracy of cubic phase mask can be determined by calculating the ratio of EDoF which is based on correlation coefficients of PSF of EDoF system calculating form the surface error model. The results are shown in Section 3. Finally, the conclusions are made in Section 4.
2. Method
2.1 The surface error model
Basing on Fourier series theory, any surface structure can be decomposed into different period (or frequency) with different amplitude [11]. In the study, the surface error can be defined as some kind deform of cubic phase mask which is manufactured by using ultra-precision lathe (ULG-100, Toshiba) and then measured by Ultra Accuracy 3D Profilometer (UA3P, Panasonic). The surface error of the cubic phase mask in the study is shown in Fig. 1.
A periodical rotationally symmetric surface error model of phase mask is presented based on Fourier series which can be expressed as in Eq. (1),
where pvnx, pvmy are peak to valley and φnx and φmy are phase of error structure along x and y direction for different periods n and m on phase mask.
2.2 Cubic phase mask
The most famous phase mask using in Wavefront coding® technology is cubic phase mask [5], which has a non-symmetry surface and can be expressed in Eq. (2),
The higher value of coding parameter (α) in Eq. (2), the more extend depth of field can be achieved, and the minimum of α is 20 at least [5].
2.3 Point spread function calculate
PSF was calculated by using Fraunhofer diffraction formula [12] programmed in Matlab®, where we used unit circle to be aperture stop which illuminating by ideal plane wave with single wavelength 550 nm, and after combine with cubic phase mask , defocus aberration and the error structure, the PSF can be calculated by Eq. (3) and (4),
where circ(x, y) is unit circle aperture, α is the coding parameter described in Eq. (2), z 2 is the defocus aberration coefficient using in simulation, λ is the wavelength and error(x, y) is error structure described in Eq. (1).
2.4 Point spread Function similarity
Correlation coefficient is used to calculate similarity of PSF for different defocus aberration in wavefront coding system. In the study, the correlation coefficient was calculated by Matlab® build-in function, which can be expressed by Eq. (5),
where A and B are the PSF using to compare the similarity in the study, Ā and B̄ are mean value of A and B.
For example, when we calculate the correlation coefficient of PSF in wavefront coding system which surface manufacture error existed. First, we calculate the PSF_A which without defocus aberration, then PSF_B with defocus aberration. After that, correlation coefficient can be got by Eq. (5). And the relationship between correlation coefficient and defocus aberration can be obtained.
And Rayleigh’s criterion (z 2 =0.25 lambda) is used to calculate ratio of EDoF. The ratio can be calculated by the value of correlation coefficient coming from system without wavefront coding when z 2 equals to 0.25 lambda. By using same value of correlation coefficient with error existing in wavefront coding system, the new allowable z′2 can be found and then the ratio and be expressed by Eq. (6),
By using the ratio, the performance of wavefront coding system can be calculated. If the ratio equal to zero that means the performance of EDoF system is as same as the system without wavefront coding.
3. Results and discussion
Figure 1 shows surface error along x and y direction of cubic type phase mask, which the PV and roots mean square (RMS) error of whole surface, x and y direction are listed in Table 1. Some low and high period structures can be observed with different PV error. We notice that error structure along y direction is about 25% compare with error along x direction. In order to simplify the simulation in study, we first only consider an error existed along one direction firstly for different periods and PV for different coding parameter α and then comparing the correlation coefficient results between errors existed along x and y directions at the same time for real phase mask manufacturing describe above.
In the study, used “period” to represent the “numbers of period”. If a surface error which composed only by period 1 and 10, and the amplitude of these two periods is 1 and 0.1, the surface error can be shown in Fig. 2. In Fig. 2, the amplitude of low period just likes the PV error and the amplitude of high period just like the RMS error. The relative error of low period is ten times higher than high period in Fig. 2.
Fig. 2. The demonstration of a surface error which compose by two periods with different amplitude, the low period is 1.0 with amplitude is 1.0, and high period is 10 with amplitude 0.1.
Low period error structure will let the ratio of EDoF decrease seriously because it’s higher relative PV error compare with high period structures. So we only consider 1, 1.5 and 2 periods, with PV error of 1 um for all different coding α and defocus aberration form 0 to 3 lambdas in simulation.
Table 1. PV and RMS error of the phase mask (α= 40) measured by UA3P
Figures 3, 4, 5 and 6 show the correlation coefficient of PSF at different defocus aberration for different coding alpha 20, 40, 60 and 90, with and without error structures. Correlation coefficient decrease rapidly when error structures exist in phase mask. Figure 3 shows that the depths of focus performance close with no wavefront coding system when 1, 1.5 and 2 period with 1 um PV error exists. Surface PV of designed cubic phase mask used in Fig. 2 is 6.36 um (α=20) and then the relative PV error is about 15%.
Figure 7 shows that correlation coefficient result for defocus aberration with same relative PV error in Fig. 3 but change coding parameter α form 20 to 90, and same degeneracy of depth of focus performance can be observed.
Figure 8 shows the correlation result for defocus aberration with same relative PV error which listed in Table 1.
Fig. 3. Correlation coefficient curve of PSF at different defocus aberration for coding parameter (α=20), period error and 1 um PV structure exist and compare with no coding and ideal (no error existed) wavefront coding system.
Fig. 4. Correlation coefficient curve of PSF at different defocus aberration for coding parameter (α=40), period error and 1 um PV structure exist and compare with no coding and ideal (no error existed) wavefront coding system.
Fig. 5. Correlation coefficient curve of PSF at different defocus aberration for coding parameter (α=60), period error and 1u m PV structure exist and compare with no coding and ideal (no error existed) wavefront coding system.
Fig. 6. Correlation coefficient curve of PSF at different defocus aberration for coding parameter (α=90), period error and 1 um PV structure exist and compare with no coding and ideal (no error existed) wavefront coding system.
Fig. 7. Correlation coefficient curve of PSF at different defocus aberration for coding parameter (α=90), with 1.0 period and 15% PV error exist and compare with no coding and ideal (no error existed) wavefront coding system.
Table 2 shows the ratio of EDoF at different coding alpha with certain error structure. We notice that if a signal x direction period error structure exists with 1um PV error, the ratio of the EDoF will decrease proportional to the period of error. And 1 um PV error will let the depth of focus smaller than system without coding phase mask used.
Table 2. Ratio of EDoF system with different period and 1um PV error structure exist in different phase mask
The ratio of the EDoF approach to zero when coding parameter (α equals to 40) and error structure with its period 1.5 along x and y direction, the PV is 1.1998um along x direction and 0.2534um on y direction (same value shown in Table 1). The correlation coefficient result is shown in Fig. 8.
Fig. 8. Correlation coefficient curve of PSF at different defocus aberration for coding parameter (α=40), with period 1.5, 1.1998um PV error and 0.2534um PV error along x and y direction and compare with no coding and ideal (no error existed) wavefront coding system.
4. Conclusion
In this paper, a periodical rotationally symmetric surface error model is presented based on measuring result and required manufacturing accuracy of the cubic phase mask can determined form correlation coefficient of PSF. From the simulation result the relative PV error of the phase mask need much smaller than 15% for low period error structure when coding parameter alpha between 20 to 90 when only one direction error structure existing. The depth of focus range will also smaller than system without coding phase mask used if low coded phase mask was used and the ratio of EDoF also decrease which is proportional to the period of the error structure. For a phase mask which was manufactured in the study, because of the two direction error structure existed, the ratio of EDoF will approach to zero.
References and links
1. S. C. Tucker, W. T. Cathey, and E. R. Dowski, “Extended depth of field and aberration control for inexpensive digital microscope systems,” Opt. Express 4, 467–474 (1999) [CrossRef] [PubMed]
2. G. E. Johnson, E. R. Dowski, and W. T. Cathey, “Passive ranging through wave-front coding: information and application,” Appl. Opt. 39, 1700–1710 (2000). [CrossRef]
3. E. R. Dowski Jr, “Wavefront coding for mobile phone imaging,” Photon. Spectra 40, 56–58 (2006).
4. T. Hellmuth, “Spatial imaging with wavefront coding and optical coherence tomography,” Adv. Atomic Molec. Opt. Phys. 53, 105–138 (2006)
5. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995) [CrossRef] [PubMed]
6. E. J. Tremblay, J. Rutkowski, I. Tamayo, P. E. X. Silveira, R. A. Stack, R. L. Morrison, M. A. Neifeld, Y. Fainman, and J. E. Ford, “Relaxing the alignment and fabrication tolerances of thin annular folded imaging systems using wavefront coding,” Appl. Opt. 46, 6751–6758 (2007). [CrossRef] [PubMed]
7. W. Z. Zhang, Z. Ye, T. Y. Zhao, Y. P. Chen, and F. H. Yu, “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15, 1543–1552 (2007). [CrossRef] [PubMed]
8. W. Z. Zhang, Z. Ye, Y. P. Chen, T. Y. Zhao, and F. H. Yu, “Ray aberrations analysis for phase plates illuminated by off-axis collimated beams,” Opt. Express 15, 3031–3046 (2007). [CrossRef] [PubMed]
9. T. Y. Zhao, Z. Ye, W. Z. Zhang, Y. P. Chen, and F. H. Yu, “Wide viewing angle skewed effect of the point spread function in a wavefront coding system,” Opt. Lett. 32, 1220–1222 (2007). [CrossRef] [PubMed]
10. F. Yan, L. G. Zheng, and X. J. Zhang, “Image restoration of an off-axis three-mirror anastigmatic optical system with wavefront coding technology,” Opt. Eng. 47, 8 (2008).
11. H. Dagnall, Exploring Surface Texture. (Rank Taylor Hobson, 1980), Chap. 3.
12. J. W. Goodman, Introduction of Fourier Optics, (Stanford University, 1968), Chap. 4.
