A wavefront coding (WFC) technique provides an extension of the depth of field for a microscopy imaging system with slight loss of image spatial resolution. Through the analysis of the relationship between the incidence angle of light at the phase mask and the system pupil function, a mixing symmetrical cubic phase mask (CPM) applied to 5X40X microobjectives is optimized simultaneously based on pointspread function (PSF) invariance and nonzero mean values of the modulation transfer function (MTF) near the spatial cutoff frequency. Optimization results of the CPM show that the depth of field of these microobjectives is extended 310 times respectively while keeping their resolution. Further imaging simulations also prove its ability in enhancing the defocus imaging.
I. INTRODUCTION
A microobjective plays a decisive role in microscopy imaging systems, determining the resolution and magnification. In order to achieve better image quality and high resolution, a microobjective is designed with relatively large aperture, which means the depth of field is small.
Wavefront coding (WFC) is an opticaldigital hybrid imaging technique developed to extend the depth of field of conventional optical systems
[1

3]
. By introducing a phase mask into the pupil of the system to modulate the wavefront of the incident light, it is possible to make the pointspread function (PSF) nearly invariant over a wide range of defocus. Then sharp images can be restored from both focus and defocus images by the same digital deconvolution filter, extending the depth of field.
In the past several years, many different kinds of WFC phase mask have been designed
[4
,
5]
. Because of relative simplicity in manufacture, the cubic phase mask is the most widely used form in the WFC technique
[6

8]
. Various optimized methods have been used to improve the performance of the cubic phase mask (CPM)
[9

12]
. Zhang and Chen optimized the CPM based on Strehl ratio. Carles presented an analytical approach to determine the optimal CPM strength, and Liu made the stationary phase analysis of CPM. However, most optimized CPMs in microscopy systems are only applied to one power microobjective
[13]
. That means microobjectives with different powers each need a specific CPM, which is not practical and economical.
In this paper, we optimize a mixing symmetrical CPM. Through the analysis of the relationship between the incidence angle of light at the phase mask and system pupil function, we build a merit function (MF) based on the PSF invariance and nonzero mean values of the modulation transfer function (MTF) near the spatial cutoff frequency to optimize the CPM. Then, we analyze the ability of optimized CPM in extending the depth of field for different power microobjectives and its effect on spatial resolution. Finally, we show its enhancement on imaging of 5X40X microobjectives with different defocus distances.
II. THE THEORETICAL ANALYSIS OF WAVEFRONT CODING PHASE MASK OF MICROOBJECTIVES WITH DIFFERENT POWERS
A schematic of a wavefront coding microscopy imaging system is shown in
Fig. 1
.
Schematic of a wavefront coding microscopy system.
A WFC phase mask is located at the system pupil plane after the microobjective. CCD detects an encoded image with the same blur degree at different defocus distances due to the modulation of the phase mask. Then, a sharp image will be restored by the same deconvolution filter, which is built through the system PSF.
Numerical aperture (NA) of a microobjective is closely related to the maximum incidence angle of light in object space, which determines its resolution. That means microobjectives with different powers are not the same. When introducing a WFC phase mask in a microscopy system, the incidence angle of light at the phase mask also changes following the converting of microobjectives. It is shown in
Fig. 2
. In order to extend the depth of field of microobjectives with different powers using one phase mask, the influence of different incidence angle of light at the phase mask is analyzed as follows.
The incidence angle at the phase mask in the optical path of microobjectives with different powers.
We select a mixing symmetrical CPM due to its manufacturing convenience. Its surface shape is represented by
Where
x
and
y
are the normalized coordinates,
α
and
β
are the coefficients of the CPM.
For an aberrationfree optical system with only defocus, its pupil function modulated by CPM is represented by
Where,
ϕ
is the phase of the pupil wavefront,
W
_{020}
is the coefficient of defocus and
λ
is the wavelength. If 2
π
/
λ
is a constant, we analyze that the pupil function is a simple form by ignoring that constant and representing the pupil function by
The difference of incidence angle at CPM could be regarded as a rotation of coordinates, shown in
Fig. 3
, and transformation formulas between two coordinates can be represented by
The incidence angle at phase mask and its coordinate transform.
Where,
x
,
y
,
z
are the original coordinates,
x
′,
y
′,
z
′ are the transformed coordinates,
θ
is the difference of two different incidence angles at CPM.
Substituting Eq. (4) into Eq. (1) and simplifying, the pupil function with different incidence angles is represented by
The coefficients of CPM in pupil function vary with the incidence angle. It is shown in
Fig. 4
. Coefficient
α
is getting larger with incidence angle increasing both at meridian and sagittal directions. It grows faster at the meridian direction. But coefficient
β
remains unchanged at the sagittal direction and is getting smaller at the meridian direction. Two conclusions can be drawn from this, 1) selecting suitable
α
and
β
can restrain the influence of incidence angle; 2) there is a difference between meridian and sagittal directions.
The incidence angle at phase mask and its coordinates transform.
III. THE OPTIMIZATION OF WAVEFRONT CODING PHASE MASK OF MICROOBJECTIVES WITH DIFFERENT POWERS
Based on the invariance of PSF, the merit function (MF) for WFC phase mask is represented by
Where,
Std
(
PSF_{diffdefocus}
)
_{i,j}
is the standard deviation of PSF at the same spatial location of microobjectives with different powers, defined to represent the PSF invariance,
x
and
y
are sampling numbers of meridian and sagittal directions, N is the sum of sampling numbers.
Variation of coefficients of phase mask will introduce more aberration in the microscopy imaging system, which results in decrease of value of MTF near spatial cutoff frequency. That creates a decline of resolution of the optical system.
Thus, optimization of our CPM adds a punishment function (PF) in order to eliminate or decrease this impact, and taking the difference between meridian and sagittal directions into consideration. The PF is represented by
Where,
Mean
(
MTFM_{freq}
) and
Mean
(
MTFM_{freq}
) are mean values of MTF near the spatial cutoff frequency in meridian and sagittal directions, N is the number of sampling frequencies.
Then, the merit function for CPM is represented by
Finally, through an optimization search algorithm, such as a genetic algorithm (GA) or simulated annealing algorithm (SA), we could solve for the optimal coefficients of the CPM.
IV. OPTIMIZATION RESULT AND DISCUSSION
Shape of optimized CPM is shown in
Fig. 5
.
Shape of WFC phase mask.
We combine this phase mask with 5X, 10X, 20X and 40X microobjectives, computing their MTF in optical design software (ZEMAX). These results are shown in
Fig. 6
. The MTFs of all microobjectives with the phase mask have an acceptable value near the cutoff frequency, by which we mean not losing any high spatial frequency, maintaining the resolution of microscopy. In addition, the higher power the microobjective is, the larger is the extension of the ranges of depth of field with the optimized CPM. For 5X objective, its depth of field is extended from 75 µm (without CPM) to about 200 µm (with CPM). It can be seen from
Fig. 6(c)
, the value of MTF for the optical system with CPM at defocus distance 200 μm is already very small. That demonstrates the maximal ability of the CPM for extending the depth of field, because the value of MTF will drop to zero probably when defocus distance is over 200 μm for a practical optical system with aberrations. Similarly, the optimized CPM multiplies the depth of field of 10X, 20X, 40X objectives five to ten times.
The MTFs of 5X, 10X, 20X, 40X microobjectives without (a), (d), (g), (j) and with (b), (e), (h), (k) the CPM. (c), (f), (i), (l) are the partial enlarged views of (b), (e), (h), (k) near the spatial cutoff frequency.
Since the microobjectives with optimized CPM maintain their proper spatial cutoff frequency, the shape images can be restored from the encoded images. Below we present the comparison of microscopy imaging simulation results of microobjectives with different powers at different defocus locations. These results are exhibited in
Tables 1

4
. The target image is a spoke pattern with numbers and marks at corners of which the size is 512 × 512 pixels. The coordinates in all figures in Tables are in pixels.
Imaging simulation results of 5X microobjectives
Imaging simulation results of 5X microobjectives
Imaging simulation results of 10X microobjectives
Imaging simulation results of 10X microobjectives
Imaging simulation results of 20X microobjectives
Imaging simulation results of 20X microobjectives
Imaging simulation results of 40X microobjectives
Imaging simulation results of 40X microobjectives
V. CONCLUSION
Based on the theoretical analysis of the relationship between the incidence angle of light and pupil function, a mixing symmetrical cubic wavefront phase mask has been optimized. The depth of field of 5X40X microobjectives optical system have multiplied 3 to 10 times larger after introducing optimized CPM. In addition, the results of microscopy imaging simulation with different defocus distance show its ability in enhancement for defocus imaging.
Acknowledgements
This work was supported by the Major National Science and Technology Programs in the “Twelfth FiveYear” Plan period, No.2009ZX02205.
Dowski E. R.
,
Cathey T. W.
1995
“Extended depth of field through wavefront coding,”
Appl. Opt
34
1859 
1866
DOI : 10.1364/AO.34.001859
Sherif S. S.
,
Cathey W. T.
,
Dowski E. R.
2004
“Phase plate to extend the depth of field of incoherent hybrid imaging systems,”
Appl. Opt
43
2709 
2721
DOI : 10.1364/AO.43.002709
Zhao H.
,
Li Y.
2010
“Optimized sinusoidal phase mask to extend the depth of field of an incoherent imaging system,”
Opt. Lett.
35
267 
269
DOI : 10.1364/OL.35.000267
Takahashi Y.
,
Komatsu S.
2008
“Optimized freeform phase mask for extension of depth of field in wavefrontcoded imaging,”
Opt. Lett.
33
1515 
1517
DOI : 10.1364/OL.33.001515
Dowski E. R.
,
Johnson G. E.
1999
“Wavefront coding: A modern method of achieving high performance and/or low cost imaging system,”
Proc. SPIE
3779
137 
145
Tucker S. C.
,
Cathey T. W.
,
Dowski E. R.
1999
“Extend depth of field and aberration control for inexpensive digital microscope systems,”
Opt. Express
4
467 
474
DOI : 10.1364/OE.4.000467
Narayanswamy R.
,
Baron A. E.
,
Chumachenko V.
2004
“Applications of wavefront coded imaging,”
Proc. SPIE
5299
163 
174
Zhang W. Z.
,
Chen Y. Y.
,
Zhao T. Y.
2006
“Simple Strehl ratio based on method for pupil mask’s optimization in wavefront coding system,”
Chin. Opt. Lett
4
515 
517
Chen S. Q.
,
Fan Z. G.
,
Chang H.
2011
“Nonaxial Strehl ratio of wavefront coding systems with a cubic phase mask,”
Appl. Opt.
50
3337 
3345
DOI : 10.1364/AO.50.003337
Carles G.
2012
“Analysis of the cubicphase wavefrontcoding function: Physical insight and selection of optimal coding strength,”
Opt. Lase. Eng
50
1377 
1382
DOI : 10.1016/j.optlaseng.2012.05.014
Liu M.
,
Dong L. Q.
,
Zhao Y. J.
,
Hui M.
,
Jia W.
2013
Stationary phase analysis of generalized cubic phase mask wavefront coding,”
Opt. Commun.
298
67 
74
Ortyn W. E.
,
Perry D. J.
,
Venkatachalam V.
,
Liang L. C.
,
Hall B. E.
,
Frost K.
,
Basiji D. A.
2007
“Extend depth of field imaging for high speed cell analysis,”
Cyto. Part A
71
215 
231