Optimization of wavefront-coded infinity-corrected microscope systems with extended depth of field

Tingyu Zhao; Thomas Mauger; Guoqiang Li

doi:10.1364/BOE.4.001464

1. Introduction

Applications of traditional light microscope systems are limited by small depth of field. Thick samples cannot be observed at one time. Usually mechanical scanning is used to achieve 3D volume information. In this way, it is impossible to use a traditional microscope system to observe the samples with dynamic changes.

Wavefront coding technology [1–3] is one of widely used methods to extend the depth of field in recent years. By applying a specially designed phase mask at the pupil plane, the whole system is insensitive to object distance. Although the image captured at the image plane is blurred, they can be restored by simple digital process because of the insensibility to the defocus. In this way, this two-step imaging system can extend the focal depth to 10 times of the traditional one. Notice that wavefront-coded technology uses phase mask instead of amplitude mask. Therefore, it does not sacrifice resolution or illumination compared to other methods such as optical pupil apodization.

In this paper, wavefront coding technology is employed to extend the depth of field of an infinity-corrected microscope system. Instead of modifying the structure of the objective [4], a specially designed phase mask is added between the infinity-corrected objective and the tube lens. In this way, the optical and mechanical structure will remain the same except adding the phase mask. It is cost effective and more flexible in comparison with the previous method which alters the structure of the objective.

Optimization of the phase mask is one of the key tasks in the design of the wavefront-coded microscope. However, most of the previous research focused on optimization of the phase mask for ideal optical systems [5–9]. In those designs, optical aberrations are ignored in order to reduce the complexity of the optimization, and that methodology needs to be greatly improved for designing practical optical systems. Here aberrations are taken into account for the design of the phase mask so that it fits a practical microscope system. For demonstration, the commercial optical design software Zemax is used to get the real analysis data of the optical system. The optical optimization methods for both non-symmetric and symmetric phase masks are proposed. Polynomial phase mask and rational phase mask [9] are used as examples for non-symmetric and symmetric phase masks respectively.

This paper is organized as follows. The structure of the wavefront-coded microscope system is introduced in Section 2. In section 3, the optimization methods for the design of the phase masks are presented and examples are given. Conclusions are given in the last section.

2. A wavefront-coded infinity-corrected microscope system

An infinity-corrected microscope system is composed of the infinity-corrected objective and the tube lens. The specimen locates at the front focal plane of the objective, while the detector locates at the back focal plane of the tube lens. As shown in Fig. 1(a), a phase mask is placed between the objective and the tube lens. Here, we use an infinity-corrected objective with magnification ratio $M$ = 32 × and numerical aperture NA = 0.6 as an example. An achromatic doublet AC254-200-A (Thorlabs Inc.) with focal length $f_{2} = 200 mm$ is used as the tube lens. Obviously, the focal length of the objective is $f_{1} = f_{2} / M = 6. 25 mm$ . The optical structure of the objective is shown in Fig. 1(b). A CCD with a dimension of 3.6mm*4.8mm is placed in the image plane.

Fig. 1 (a) Schematic drawing of a wavefront-coded infinity-corrected microscope system; (b) The structure of an infinity-corrected objective lens.

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Without any phase mask, the depth of field (DoF) of the traditional optical system is described as [10]: $D o F = λ / (2 N . A .^{2}) \approx 0.75 μ m,$ where $λ$ is the wavelength and for the visible spectral range, the central wavelength is assumed to be 550 nm. Therefore the DoF of this objective is too small for observing certain volume of a thick specimen. In this way, a specially designed phase mask is needed to extend the depth of field so that the three-dimensional (3D) volume information can be acquired at one time.

3. Optimization of the wavefront-coded microscope system

Design of the phase mask is the key to the optimization of the wavefront-coded microscope system with large DoF. Different from previous research which focused on ideal optical systems, here Zemax is used as an example of optical design software to design and analyze the practical microscope system where aberrations are taken into consideration. The optical design process is divided into 4 steps:

Step 1. Setup the structure of the traditional microscope system, including infinity-corrected objective and tube lens.

Step 2. Insert a phase mask between the objective and the tube lens. The initial structure of the phase mask is usually a parallel plane. Notice that sometimes the type of the surface of the phase mask is not included in the surface database of the software. In this case, the surface type needs to be defined by the user according to the expression of the phase mask.

Step 3. User-define the merit function which is suitable for the wavefront-coded microscope system with large DoF. Notice that for the system with extremely large DoF, the default merit function [11] RMS cannot be used for the design. We will discuss this in details in the following sections.

Step 4. Optimize the optical design and analyze the results.

Phase masks can be classified into two categories. One is the non-symmetric phase mask and the other is the symmetric phase mask. Here polynomial phase mask is used as an example of non-symmetric phase mask, while rational phase mask is used as an example of symmetric phase mask.

3.1 Non-symmetric phase mask

In general, optical encoding and digital decoding are essential for a wavefront-coded system when non-symmetric phase mask is used. That means that it is important to use a defocus-insensitive point spread function (PSF) as the deconvolution kernel. Notice the corresponding relationship between the defocus and the object distance, as well as that between the PSF in space domain and the modulation transfer function (MTF) in frequency domain. Therefore, the consistency of the MTF at different object distances is one of the targets for optimization. Here we use the standard deviation of the MTF at certain frequencies to represent the consistency of the MTF and define it as the merit function $E :$

E = \frac{\sum_{j = 1}^{N_{v}} \sum_{i = 1}^{N_{u}} w (i, j) \sqrt{\sum_{n = 1}^{N_{F}} \sum_{m = 1}^{N_{o}} {[M T F (D_{o} (m), F (n), u (i), v (j)) - \bar{M T F} (u (i), v (j))]}^{2}}}{N_{F} N_{o} \sum_{j = 1}^{N_{v}} \sum_{i = 1}^{N_{u}} w (i, j)},

where

D_{o} (m)

is the object distance at the m^th configuration and

N_{o}

is the total number of configurations;

F (n)

is the n^th field and

N_{F}

is the total number of fields;

u (i)

and

v (j)

are frequencies at the i^th and the j^th positions respectively;

w (i, j)

is the weight at the frequencies

u (i)

and

v (j)

;

M T F (\cdot)

represents the MTF value;

N_{u}

and

N_{v}

are the number of frequencies

u (i)

and

v (j)

respectively. The average value of the MTF,

\bar{M T F} (u (i), v (j)),

is described as

\bar{M T F} (u (i), v (j)) = \frac{1}{N_{F} N_{o}} \sum_{n = 1}^{N_{F}} \sum_{m = 1}^{N_{o}} M T F (D_{o} (m), F (n), u (i), v (j)) .

In order to make the MTF curves at different object distances and different fields be consistent with each other, the merit function in Eq. (1) should be as small as possible. However, the MTF values will be reduced to zero if no penalty factors are taken into consideration. For this reason, a threshold method is used as a penalty factor. Let

P (m, n) = {\begin{cases} P_{0} \min_{i, j} M T F (D_{o} (m), F (n), u (i), v (j)) \leq T \\ 0 o t h e r w i s e \end{cases},

where

T

is the threshold of the MTF;

P_{0}

is a large constant defined by user. Let

w_{p}

be the weight of the penalty factor. Then the modified merit function can be represented as

E' = E + w_{p} \sum_{n}^{N_{F}} \sum_{m}^{N_{o}} P (m, n) .

Using the merit function defined by Eq. (4), we design a non-symmetric phase mask to extend the field depth of the microscope system described in section 2. As we know, cubic phase mask is regarded as one of classic phase masks in wavefront-coded system [1], and the general cubic phase mask has been presented as a better phase mask because of its good performance for large field [6, 12]. Both of them can be regarded as special cases of the polynomial phase mask, which is described as

z (x, y) = \sum_{n = 2}^{k} \sum_{m = 0}^{n} C_{m (n - m)} x^{m} y^{n - m},

where

x, y

are the coordinates in the pupil plane;

m, n

are positive integrals;

k

is the highest power of

x, y

. Considering the computation efficiency, we take

k

= 5 in this paper. For simplicity, only sagittal and tangential MTF values are taken into consideration in optimization. The MTF values calculated during the procedure are within frequency 20 lp/mm with interval 2 lp/mm. Finally, we obtain the optimized coefficients for the polynomial phase mask as shown in Table 1. The MTF curves at different object distance are shown in Fig. 2. The graphs in Figs. 2(a), 2(b), 2(c), and 2(d) correspond to object distances of 0.355 mm, 0.358 mm, 0.362 mm, and 0.365 mm, respectively, with a variation of the object distance being equal to 10 μm. It can be seen that the on-axis MTF curves are similar to each other and do not have zero values within the frequency of 20 lp/mm. Although the performance of off-axis MTFs is a little worse than the on-axis ones, it can be ignored since the on-axis performance is more important than that of the off-axis in the microscope system. Therefore, by using the specially designed phase mask, the DoF is increased to 10 μm, which is 13 times more than the DoF of the traditional microscope.

Table 1. Coefficients of the optimized polynomial phase mask

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Fig. 2 MTF curves of the wavefront-coded microscope system with the specially designed polynomial phase mask at different object distances: (a) 0.355mm; (b) 0.358mm; (c) 0.362mm; (d) 0.365mm.

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Such a non-symmetric phase mask can be fabricated using a diamond turning machine, and more axes are needed for operation of the machine in comparison with making a rotationally symmetric phase mask. Furthermore, it has a stringent requirement on the alignment of the non-symmetric phase mask in the optical system because the optical performance is sensitive to the centering and the orientation of the mask. To fabricate and use a symmetric phase mask can be much easier in these two aspects.

3.2 Symmetric phase mask

When the symmetric phase mask is used, digital process is not necessary for the wavefront-coded system [9] and the consistence of the MTF or PSF is not maintained as well as the case when the non-symmetric phase mask is used. Different merit function should be used for optimization. Considering most optical design software has its own default merit function for the traditional optical system, we will modify the default merit function so that it is suitable for the design of the symmetric optical system. In Zemax, the default merit function is calculated by tracing certain rays. Then we can get the displacement $Δ (D_{o} (m), W (n), r (i))$ , which is introduced by these rays relative to the chief ray at the image plane. Then the standard deviation of the displacement at the m^th object distance is described as

ξ (m) = \frac{1}{N_{w} N_{r}} \sum_{n}^{N_{w}} \sum_{i}^{N_{r}} {[Δ (D_{o} (m), W (n), r (i)) - \bar{Δ} (D_{o} (m))]}^{2},

where

W (n)

is the n^th wavelength;

r (i)

is the i^th radial position in the pupil;

N_{w}

and

N_{r}

are the number of wavelengths and radial positions respectively, and

\bar{Δ} (D_{o} (m)) = \frac{1}{N_{w} N_{r}} \sum_{n}^{N_{w}} \sum_{i}^{N_{r}} Δ (D_{o} (m), W (n), r (i)) .

In order to allow the system to have a large depth of field, we need to make the following, i.e., to minimize the value E defined by the following expression:

E = \frac{1}{N_{o}} \sum_{m}^{N_{o}} {[ξ (m) - \bar{ξ}]}^{2},

where

\bar{ξ}

is the average of

ξ (m)

. However, using Eq. (8) as the merit function may lead to very large RMS. Therefore another penalty factor is needed to avoid this situation. Here we use the ratio of the diffraction-limited MTF to that of the practical system as the penalty factor:

P = \frac{\sum_{j = 1}^{N_{j}} \sum_{i = 1}^{N_{i}} M T F_{0} (u (i), v (j))}{\frac{1}{N_{F} N_{o}} \sum_{j = 1}^{N_{j}} \sum_{i = 1}^{N_{i}} \sum_{n = 1}^{N_{F}} \sum_{m = 1}^{N_{o}} M T F (D_{o} (m), F (n), u (i), v (j))},

where MTF₀ means the diffraction-limited MTF. Let

w_{p}

be the weight of the penalty factor. Then the modified merit function is defined as

E' = E + w_{p} P .

Using the merit function defined by Eq. (10), we design a rational phase mask for the new wavefront-coded microscope system. The expression of the rational phase mask is taken as

z (r) = \frac{\sum_{i = 0}^{N} p_{i} r^{i}}{\sum_{i = 0}^{N} q_{i} r^{i}},

where

r

is the radial coordinate in the pupil plane;

p_{i}

and

q_{i}

are the coefficients of the phase mask to be determined by optimization. Considering the computation efficiency, here N is chosen as 12. Default settings for ray tracing in Zemax are used to calculate our merit function. After optimization, we obtain the coefficients for the rational phase mask as shown in Table 2. Then we get the performance of the wavefront-coded microscope system. The MTF curves for different object distances are shown in Fig. 3. The graphs in Figs. 3(a), 3(b), 3(c), and 3(d) correspond to object distances of 0.358 mm, 0.361 mm, 0.364 mm, and 0.367 mm, respectively. The on-axis performance is much better than the off-axis one. Firstly, there are no zero values for on-axis MTF within the frequency of 20 lp/mm. Secondly, the value of the on-axis MTF at high frequency is much higher than the others. Although the performance of the off-axis MTF is not as good as the on-axis one, it is still acceptable since the center field in microscope is more important than the other fields. The depth of field can reach 9 μm and is 12 times more than the DoF of the traditional microscope at a little sacrifice of off-axis performance.

Table 2. Coefficients of the optimized rational phase mask

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Fig. 3 MTF curves of the wavefront-coded microscope system with rational phase mask at different object distances: (a) 0.358mm; (b) 0.361mm; (c) 0.364mm; (d) 0.367mm.

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4. Conclusions

The depth of field of an infinity-corrected microscope system is extended by applying a specially designed phase mask between the objective and the tube lens. Different from the previous research, this paper takes every optical aberration into account and focuses on the optimization method based on optical design software. Two merit functions are proposed to design non-symmetric and symmetric phase masks respectively. A polynomial phase mask and rational phase mask are used as examples. Simulation results show that the optimization methods work well for a 32 × and 0.6 NA infinity-corrected microscope system. The depth of field can be extended to about 13 times of the traditional one. The method can be used for other type of phase masks, such as sinusoidal phase mask and logarithmic phase mask. The microscope with extended depth of focus can be used in many circumstances where visualization of the three-dimensional volume information is important, and such applications may include industrial inspection, cell biology, pathology, and surgical operation. We believe that this method is not limited to the wavefront-coded system for extension of the depth of field. It can be used for the design of other special optical systems such as a range finder by modification of the merit function. Experimental results will be reported in the near future.

Acknowledgments

G. Li would like to thank the support from the Ophthalmology Department fund, the ElectroScience Laboratory CERF fund, National Institutes of Health National Eye Institute (through grant R01 EY020641), National Institute of Biomedical Imaging and Bioengineering (through grant R21 EB008857), National Institute of General Medical Sciences (through grant R21 RR026254/R21 GM103439), and Wallace H. Coulter Foundation Career Award (through grant WCF0086TN).

References and links

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4. T. Zhao, Z. Ye, W. Zhang, W. Huang, and F. Yu, “Design of objective lenses to extend the depth of field based on wavefront coding,” Proc. SPIE 6834, 683414, 683414-8 (2007). [CrossRef]

5. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004). [CrossRef]

6. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003). [CrossRef]

7. Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272(1), 56–66 (2007). [CrossRef]

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11. Zemax 12 Optical design program user’s manual (Radiant Zemax, 2012).

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$C_{20}$	$C_{11}$	$C_{02}$	$C_{30}$	$C_{21}$	$C_{12}$
$3.910 \times 10^{- 6}$	$1.946 \times 10^{- 6}$	$8.498 \times 10^{- 6}$	$- 2.905 \times 10^{- 5}$	$- 1.548 \times 10^{- 6}$	$- 7.287 \times 10^{- 6}$
$C_{03}$	$C_{40}$	$C_{31}$	$C_{22}$	$C_{13}$	$C_{04}$
$- 1.988 \times 10^{- 5}$	$5.079 \times 10^{- 7}$	$- 1.655 \times 10^{- 7}$	$3.410 \times 10^{- 7}$	$- 1.897 \times 10^{- 7}$	$- 1.128 \times 10^{- 7}$
$C_{50}$	$C_{41}$	$C_{32}$	$C_{23}$	$C_{14}$	$C_{05}$
$- 2.944 \times 10^{- 7}$	$1.712 \times 10^{- 7}$	$2.296 \times 10^{- 7}$	$- 1.408 \times 10^{- 6}$	$6.299 \times 10^{- 7}$	$- 1.408 \times 10^{- 6}$

$p_{0}$	$p_{1}$	$p_{2}$	$p_{3}$	$p_{4}$	$p_{5}$
$1387$	$- 2.884 \times 10^{4}$	$- 1.768 \times 10^{4}$	$1.327 \times 10^{4}$	$9574$	$- 3715$
$p_{6}$	$p_{7}$	$p_{8}$	$p_{9}$	$p_{10}$	$p_{11}$
$- 4375$	$- 165.0$	$36.40$	$2.606$	$2.430$	$- 1.206$
$p_{12}$	$q_{0}$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$
$0.846$	$- 3.424 \times 10^{7}$	$3.234 \times 10^{8}$	$4.473 \times 10^{7}$	$- 1.399 \times 10^{7}$	$- 8.233 \times 10^{5}$
$q_{5}$	$q_{6}$	$q_{7}$	$q_{8}$	$q_{9}$	$q_{10}$
$- 1.827 \times 10^{6}$	$7.973 \times 10^{5}$	$1.226 \times 10^{5}$	$4.535 \times 10^{4}$	$1.120 \times 10^{5}$	$- 8894$
$q_{11}$	$q_{12}$
$- 3974$	$- 3.605$

$C_{20}$	$C_{11}$	$C_{02}$	$C_{30}$	$C_{21}$	$C_{12}$
$3.910 \times 10^{- 6}$	$1.946 \times 10^{- 6}$	$8.498 \times 10^{- 6}$	$- 2.905 \times 10^{- 5}$	$- 1.548 \times 10^{- 6}$	$- 7.287 \times 10^{- 6}$
$C_{03}$	$C_{40}$	$C_{31}$	$C_{22}$	$C_{13}$	$C_{04}$
$- 1.988 \times 10^{- 5}$	$5.079 \times 10^{- 7}$	$- 1.655 \times 10^{- 7}$	$3.410 \times 10^{- 7}$	$- 1.897 \times 10^{- 7}$	$- 1.128 \times 10^{- 7}$
$C_{50}$	$C_{41}$	$C_{32}$	$C_{23}$	$C_{14}$	$C_{05}$
$- 2.944 \times 10^{- 7}$	$1.712 \times 10^{- 7}$	$2.296 \times 10^{- 7}$	$- 1.408 \times 10^{- 6}$	$6.299 \times 10^{- 7}$	$- 1.408 \times 10^{- 6}$

$p_{0}$	$p_{1}$	$p_{2}$	$p_{3}$	$p_{4}$	$p_{5}$
$1387$	$- 2.884 \times 10^{4}$	$- 1.768 \times 10^{4}$	$1.327 \times 10^{4}$	$9574$	$- 3715$
$p_{6}$	$p_{7}$	$p_{8}$	$p_{9}$	$p_{10}$	$p_{11}$
$- 4375$	$- 165.0$	$36.40$	$2.606$	$2.430$	$- 1.206$
$p_{12}$	$q_{0}$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$
$0.846$	$- 3.424 \times 10^{7}$	$3.234 \times 10^{8}$	$4.473 \times 10^{7}$	$- 1.399 \times 10^{7}$	$- 8.233 \times 10^{5}$
$q_{5}$	$q_{6}$	$q_{7}$	$q_{8}$	$q_{9}$	$q_{10}$
$- 1.827 \times 10^{6}$	$7.973 \times 10^{5}$	$1.226 \times 10^{5}$	$4.535 \times 10^{4}$	$1.120 \times 10^{5}$	$- 8894$
$q_{11}$	$q_{12}$
$- 3974$	$- 3.605$

Optimization of wavefront-coded infinity-corrected microscope systems with extended depth of field

Abstract

1. Introduction

2. A wavefront-coded infinity-corrected microscope system

3. Optimization of the wavefront-coded microscope system

3.1 Non-symmetric phase mask

3.2 Symmetric phase mask

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Tables (2)

Equations (11)

Biomedical Optics Express