Superresolution along extended depth of focus with ... - OSA Publishing

Liu et al.

Vol. 25, No. 8 / August 2008 / J. Opt. Soc. Am. A

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Superresolution along extended depth of focus with binary-phase filters for the Gaussian beam Linbo Liu,1 Frédéric Diaz,2 Liang Wang,3 Brigitte Loiseaux,2 Jean-Pierre Huignard,2 C. J. R. Sheppard,1,4 and Nanguang Chen1,4,5,* 1

Graduate Programme in Bioengineering, National University of Singapore 117576, Singapore 2 Thales Research and Technology, RD 128, 91767 Palaiseau Cedex, France 3 Graduate University of Chinese Academy of Science, 100049, China 4 Division of Bioengineering, National University of Singapore 117576, Singapore 5 Department of Electrical and Computer Engineering, National University of Singapore 117576, Singapore *Corresponding author: [email protected] Received April 15, 2008; revised June 16, 2008; accepted June 19, 2008; posted June 20, 2008 (Doc. ID 94988); published July 24, 2008 In the paraxial Debye regime, simple and power-efficient pupil filters are designed to break the diffraction limit along a large depth of focus (DOF) for the Gaussian beam. Dependences of the superresolution factor, DOF gain, Strehl ratio, sidelobe strength, and axial intensity nonuniformity on the Gaussian profile in the pupil plane are characterized using the numerical method. Optimal filter designs are proposed for either highresolution or ultra-large-DOF applications followed by experimental verifications. © 2008 Optical Society of America OCIS codes: 100.6640, 170.1630, 170.4500, 170.2150.

1. INTRODUCTION Focusing to a smaller spot along a large DOF is one of the most important topics in optical microscopy and imaging, especially when ultrahigh-axial-resolution optical coherence tomography (OCT) becomes practical by the use of broad-bandwidth femtosecond laser technology [1–3]. In recent years, there has been a trend toward realizing endoscopic optical biopsy by the use of OCT, and in vivo experiments have been performed successfully with the spatial resolution near the level of histopathology [4–11]; however, it is believed that the ability to image at the cellular level could improve the effectiveness of quantitative macrophage determination [5] and many surgical and microsurgical procedures [4]. Superresolution along an extended DOF is the key to cellular-level imaging with endoscopic OCT for reasons of both theoretical limitations and practical implementations. Conventional focusing optics provides a diffractionlimited spot size of ␭ / 2nNA [full width at half-maximum (FWHM)] and a DOF of 1.4n␭ / NA2 (FWHM) [12] or ␭ / 4n兵1 − 关1 − 共NA/ n兲2兴1/2其 (full width at 0.9 maximum) [13] when a uniform circular aperture is assumed, where ␭ is the vacuum wavelength, n is the refractive index, and NA is the numerical aperture. It then becomes a challenging task to improve transverse resolving power of the endoscope without compromising the DOF, since use of a lens with larger NA results in a much smaller DOF. In addition, a lens of large aperture size or small working distance is not practical for miniaturized probes working in internal tubular organs, such as gastrointestinal tracts, respiratory tracts, and blood vessels. Focusing schemes that can produce superresolution along an extended DOF are essential for miniaturized probes that can accommodate an objective lens of only relative low aperture size 1084-7529/08/082095-7/$15.00

(and therefore relative low NA to maintain the required working distance). To achieve superresolution along extended DOF, many designs of pupil filters have been presented, including amplitude filters, pure-phase filters and complex filters. Amplitude filters and complex filters are not energy efficient since a certain part of the pupil is obstructed. Continuously varying phase filters are difficult to implement in miniaturized probes. The pure-phase binary filter consisting of arrays of concentric rings has been demonstrated to be an effective and energy-efficient axial apodizer [14]; more important, such a filter can be easily fabricated and miniaturized, which makes it the best fit for in vivo imaging applications. Assuming a uniform transverse beam profile, H. Wang et al.demonstrated both numerically and experimentally that smaller spot size and extended DOF can be achieved with binary-phase-only masks in the paraxial Debye regime [15–17]; under the same assumption,a subwavelength and superresolution nondiffraction beam was demonstrated numerically with a binary-phase-only element in the nonparaxial focusing system [18]. However, such an assumption is not valid for most practical applications, where beam shaping becomes difficult and unnecessary. For instance, in a single-mode-fiber based scanning endoscope the transverse profile in the pupil plane of the objective lens is always a Gaussian distribution cut at a certain power level. Little has been written concerning the effects of the transverse beam profile on the performance of the filters, and few experiments have been reported to validate the behaviors of such an important class of filter. In this paper, binary-phase spatial filters (BPSF) are classified into two subclasses according to their axial and transverse performance; dependences of the filter perfor© 2008 Optical Society of America

J. Opt. Soc. Am. A / Vol. 25, No. 8 / August 2008

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mances on the transverse beam profile are presented numerically. Optimized designs are proposed for either high-resolution or ultra-large-DOF applications, and experimental results are shown to verify the theoretically predicted extended DOF and superresolution. A BPSF is composed of an array of concentric rings, and the radii of the rings are r1, r2, r3, r4, r5 , r6, and 1 normalized to the pupil radius [18]. The phase of each ring ⌽ is set to be 0 or ␲ with respect to the center wavelength ␭. The analytical model for such filters in the paraxial Debye regime has been established [14,15,17]. Properties of such a filter include the superresolution factor, DOF (FWHM) gain, Strehl ratio, axial intensity nonuniformity and sidelobe strength: Axial intensity nonuniformity is defined here as the ratio of the largest intensity fluctuation between two intensity maxima to the peak intensity; the sidelobe strength is defined here as the peak power ratio of the first sidelobe to the main lobe in the focal plane. The first three properties can be expressed in terms of the moments of the pupil [19,20]. However, the last two properties can be retrieved only by calculating the diffraction field. The relative importance of these factors depends on the application. The superresolution factor might be an important property for cellular-resolution applications, while filters with a very large DOF gain might be ideal for optical biopsy in vivo near the level of histopathology.

2. DEPTH-INVARIANT SUPERRESOLVING FILTERS To investigate the dependence of the filter performance on the different Gaussian beam intensity profiles, 13 discrete power levels, which define the pupil radius, are chosen to represent the whole power range of the Gaussian beam. For each transverse beam profile cut by the pupil radius at a power level, a filter design is obtained via exhaustive searching in a multidimensional space (r1, r2, r3, r4, r5, r6) for the axial intensity distribution with the best uniformity within an axial range 3 times as large as the conven-

0.7

r2

r3

r4

0.10 0.10 0.10 0.07 0.07 0

0.28 0.28 0.29 0.27 0.26 0

0.47 0.49 0.49 0.48 0.48 33

0.67 0.69 0.69 0.68 0.68 61

0.08

0.23

0.44

0.65

0.09 0.18 0.14 0.09 0.09 0.10 0.15

0.28 0.37 0.31 0.28 0.27 0.17 0.19

0.49 0.53 0.48 0.48 0.47 0.31 0.21

0.68 0.70 0.67 0.67 0.66 0.53 0.40

0.6 0.55

e

0.5

-2

0.4 0.35 100

80 60 40

20 10

1

0.1

Power level at the pupil radius[%] Fig. 1. (Color online) Spot size in the focal plane for conventional and BPSF optimized focusing.

tional DOF. Because of the high computational cost, we mainly focus on filters with ring number no more than 5. In Table 1, filter designs as a result of the search are provided for different transverse beam profiles cut by the pupil radius at power levels ranging from 0.1% to 100%. Diffraction fields of these filters are featured with depthinvariant superresolution, namely, depth-invariant superresolving filters. In Fig. 1, it is expected that the diffraction-limited spot size (solid curve) increases monotonically with the decrease in the power levels at the pupil radius; however the spot size achieved with the depthinvariant superresolving filters (circles) is almost constant around 0.4␭ / NA for power levels above 10%. In Fig. 2, the Strehl ratio (diamonds) fluctuates between 0.14 and 0.15 before 50% power level and goes to its minimum at 1% power level. In Fig. 3, the sidelobe

depth-invariant superresolving filters ultra-large-DOF filters

0.25

0.2

Strehl ratio

100 90 80 70 60 50 (Filter 1) 50 (Filter 2) 40 30 20 13.5 10 1 0.1

r1

0.65

0.45

Table 1. Dimension Parameters of the DepthInvariant Superresolving Filters Power level (%)

diffraction-limited depth-invariant superresolving filters ultra-large-DOF filters

0.75

Spot size [/NA]

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0.15

0.1

0.05

e 0 100

80 60 40

-2

20 10

1

0.1

Power level at the pupil radius[%] Fig. 2. (Color online) Strehl ratio for conventional and BPSF optimized focusing.

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uniformity and/or Strehl ratio for a larger DOF gain within this limit. Shown in Figs. 5(a)–5(c), with the pupil radius defined by 50% power level and NA= 0.01, the intensity pointspread functions (PSFs) are presented for conventional focusing, focusing with a three-ring filter (Filter 1 in Table 1), and focusing with a five-ring filter (Filter 2 in Table 1) respectively. The coordinate z stands for the axial position with the origin at the focal point and r denotes transverse radius. The performance of both filters is listed in Table 2, from which we can see that the five-ring filter is superior in every respect except Strehl ratio. However, both filters demonstrate depth-invariant spot size with a superresolving factor of 0.75.

0.35 depth-invariant superresolving filters ultra-large-DOF filters

0.3

Power ratio

0.25 0.2 0.15 0.1

e

0.05 0 100

80 60 40

2097

-2

20 10

1

0.1

Power level at the pupil radius[%] Fig. 3. (Color online) Sidelobe strength for conventional and BPSF optimized focusing.

Axial intensity nonuniformity [dB]

strength (squares) is roughly the same for all power levels, while the axial intensity nonuniformity (hexagrams in Fig. 4) of power levels above 50% is below −25 dB. These facts indicate that the performance of such a subclass of filters reaches its optimum when the pupil radius is located at around 50% power level of the Gaussian beam. The DOF gain of such filters is at most 4 − 4.5 for the Gaussian beam profile. The DOF gain for filters in Table 1 is around 3.7. One can always trade axial intensity non-

depth-invariant superresolving filters ultra-large-DOF filters

0 -5 -10 -15 -20 -25 -30 -35

e -40 100

80 60 40

-2

20 10

1

0.1

Power level at the pupil radius[%] Fig. 4. (Color online) Axial intensity nonuniformity for conventional and BPSF optimized focusing.

Fig. 5. Intensity distribution of (a) conventional focusing and focusing with (b) Filter 1 and (c) Filter 2 when the pupil radius isdefined by 50% power level; intensity distribution of (d) conventional focusing andfocusing with (e) Filter 3 and (f) Filter 4 when the pupil radius is defined by 1% power level; intensity distribution of the (g) conventional focusing and focusing with (h) Filter 5 and (i) Filter 6 when the pupil radius is defined by 0.1% power level.

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Table 2. Performance Parameters of Optimized Filters for Pupil Radius Defined by 50% and 1% Power Level Pupil Radius Defined by 50% Intensity Level Filter Performance Spot size (␭/NA) Gain in DOF Strehl ratio Power ratio (the first peak vs. the central peak) Axial intensity nonuniformity

Diffraction-Limited

Filter 1

Filter 2

Diffraction-Limited

Filter 3

Filter 4

0.53 1 1 0.013 −⬁

0.41 3.27 0.200 0.119 −20 dB

0.40 3.71 0.156 0.149 −26 dB

0.65 1 1 0.004 −⬁

0.43 6.46 0.072 0.107 −15 dB

0.43 6.08 0.073 0.106 −16 dB

3. ULTRA-LARGE-DOF FILTERS The limitation of DOF gain holds true only for the depthinvariant superresolving filters. Here we present another subclass of filters that have much larger DOF gain. Ultralarge-DOF filter designs are obtained via exhaustive searching within an axial range around 6 times as large as the conventional DOF. The results of the search are provided in Table 3. Corresponding spot size in focal plane, Strehl ratio, sidelobe strength, and axial intensity nonuniformity are shown as crosses in Fig. 1, triangles in Fig. 2, pentagrams in Fig. 3, and dots in Fig. 4. Compared with depth-invariant superresolving filters, ultra-largeDOF filters have higher superresolving factor and lower sidelobe strength; but high axial intensity nonuniformity for power levels above 1% precludes them from highquality imaging applications. Therefore, two filters below −15 dB are chosen as optimized filters. Shown in Figs. 5(d)–5(f), with the pupil radius defined by 1% power level and NA= 0.0116, the intensity PSFs are presented for conventional focusing, focusing with the five-ring filter (Filter 3 in Table 3), and the seven-ring filter (Filter 4 in Table 3), respectively. A dumbbell-shaped intensity PSF indicates that the spot size is depth dependent; nevertheless, the filters are still superresolving Table 3. Optimized Parameters of the Ultra-LargeDOF Filters Power Level (%) 100 90 80 70 60 50 40 30 20 13.5 10 1 (Filter 3) 1 (Filter 4) 0.1 (Filter 5) 0.1 (Filter 6)

r1

r2

0.08 0.34 0.03 0.32 0 0.30 0.16 0.36 0.16 0.36 0.14 0.34 0.07 0.22 0.07 0.20 0.11 0.22 0.13 0.23 0.28 0.39 0.20 0.32 0.05 0.07 r5 = 0.44 0 0.04 0.15 0.19

r3

Pupil Radius Defined by 1% Intensity Level

r4

0.55 0.73 0.54 0.72 0.53 0.71 0.55 0.73 0.55 0.72 0.54 0.72 0.45 0.67 0.43 0.65 0.41 0.64 0.41 0.63 0.42 0.60 0.42 0.59 0.24 0.37 r6 = 0.58 18 40 0.21 0.40

along the full range of the DOF under the Rayleigh range criterion. The performance of the two filters is listed in Table 2, from which we can see that both filters have DOF gain above 6 and a superresolving factor of 0.66. It should be noted that for pupil radius defined by a very low power level, filters with much larger DOF gain and Strehl ratio can be found. A filter similar to Filter 5 has been use for extending focus in real-time swept source OCT [14]. Shown in Figs. 5(g)–5(i), with pupil radius defined by 0.1% power level and NA= 0.0143, the intensity PSFs in the sample space are presented for conventional focusing, focusing with a four-ring filter, and focusing with a five-ring filter (Filter 5 and Filter 6 in Table 3), respectively. For both filters, the achieved superresolution factor, DOF gain, Strehl ratio, and axial intensity nonuniformity are 0.62, 8.9, 0.13, and −16 dB, respectively, while Filter 5 has larger sidelobe strength of 0.4 than the 0.36 achieved by Filter 6. However, both filters are superresolving only in a limited region around the conventional focus.

4. EXPERIMENTAL RESULTS An optically addressed light valve (Thales Research & Technology, France) is used to produce the phase patterns [21]. The transverse beam intensity and phase profile of a Nd:YAG CW laser at 1064 nm is calibrated with a fourwave shearing interferometry wavefront sensor (SID-4, Phasics Corp.). The transverse intensity diffraction patterns are captured using a CCD with a pitch size of 6.7 ␮m to verify the theoretical results. With the pupil radius defined by 50% power level and NA= 0.01, Figs. 6(a)–6(i) show calculated transverse diffraction patterns at different axial positions for the conventional system, system with Filter 1, and system with Filter 2, respectively; Figs. 6共a⬘兲–6共i⬘兲 show the corresponding measured results. The size of each image is 0.6 mm⫻ 0.6 mm. For each profile in the conventional focusing case the intensity is normalized to the local maximum, while for each profile in the case of focusing with filters, the intensity is normalized globally along the optical axis. In Figs. 6(a)–6(c), and Figs. 6共a⬘兲–6共c⬘兲, the spot size for the focusing with filters is smaller than that of conventional focusing. Shown in Figs. 6(d)–6(f) and 6共d⬘兲–6共f⬘兲, this transverse superresolution resolution extends to the axial position of z = 18 mm, where the conventional focus goes far beyond the focal depth and spreads to a very large transverse field. In Figs. 6(g)–6(i) and, even

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Normalized intensity

1

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conventional Filter 1 Filter 2

0.8 0.6 0.4 0.2 0

-100

-50

0

50

100

r [] Fig. 7. (Color online) Measured transverse intensity profiles 共z = 0兲 of conventional focusing (dashed curve, red online) and focusing with Filter 1 (dotted curve) and Filter 2 (solid curve, blue online).

Fig. 6. Calculated transverse intensity distributions in the focal plane of (a) conventional focus and focuses obtained with (b) Filter 1 and (c) Filter 2; transverse intensity distribution of (d) conventional focus and focuses obtained with (e) Filter 1 and (f) Filter 2 at z = 18 mm; transverse intensity distributions of (g) conventional focus and focuses obtained with (h) Filter 1 and (i) Filter 2 at z = 22 mm. 共a⬘兲–共i⬘兲 corresponding measured intensity. Image size: 0.6 mm⫻ 0.6 mm. (g⬘)–(i⬘): (Media 1).

in the case of z = 22 mm, the spot size with Filter 2 is still better than the diffraction-limited one. The complete measured data can be viewed in the associated video file (Media 1). From left to right, transverse diffraction patterns of the conventional system, system with Filter 1, and system with Filter 2 are demonstrated, with the axial position indicated at the top. In Fig. 7, the transverse intensity profiles of the conventional focusing (long-dashed curve), focusing with Filter 1 (short-dashed curve) and Filter 2 (solid curve) are plotted, and the measured superresolution factors are 0.77 and 0.78, respectively. The experimental results agree well with our theoretical predictions. With the pupil radius defined by 1% power level and NA= 0.0116, Figs. 8(a)–8(i) show calculated transverse diffraction patterns at different axial positions for the conventional system, system with Filter 3, and system

with Filter 4, respectively; 8共a⬘兲–8共i⬘兲 illustrate the corresponding measured results. Shown in Figs. 8(a)–8(c) and 8共a⬘兲–8共c⬘兲, the spot size for the focusing with filters is much smaller than that of the conventional focusing in the focal plane. Shown in Figs. 8(d)–8(f) and 8共d⬘兲–8共f⬘兲, this transverse superresolution resolution extends to the axial position of z = 16 mm, where the conventional focus goes far beyond the focal depth and spread to a very large transverse field. Shown in Figs. 8(g)–8(i) and 8共g⬘兲–8共i⬘兲, at z = 36 mm the spot size of focusing with the filters becomes larger than that in the conventional focus. The complete measured data can be viewed in the associated video file (Media 2) . From left to right, transverse diffraction patterns of the conventional system, system with Filter 3, and system with Filter 4 are demonstrated, with the axial position indicated at the top. Figure 9 plots the measured transverse intensity profiles of the focusing with Filter 3 at different axial positions in the sample space. Compared with the diffraction-limited transverse intensity profile (solid curve), Filter 3 is superresolving within z = ± 16 mm. Even in the case of z = 32 mm, the FWHM of the transverse intensity profile of the focusing with Filter 3 (dashed-dotted curve) is only 17% larger than the diffraction-limited one in focus. Again, the experimental results agree very well with our theoretical predictions.

5. DISCUSSION AND CONCLUSION To optimize filter performance, compromises are often made between filter properties. There could be numerous other designs available with changes in the searching criterion. For example, if axial intensity nonuniformity larger than −15 dB is acceptable, one would find many designs demonstrating DOF gains above 6 with the pupil radius defined by any power level. However, in our calculation and experiment, we choose filters with the best axial intensity nonuniformity to ensure a high-quality axial response for biomedical imaging applications.


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Normalized intensity

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conv. Filter 3 z = 0 mm Filter 3 z = 8 mm Filter 3 z = 16 mm Filter 3 z = 32 mm

1 0.8 0.6 0.4 0.2 0

-100

-50

0

50

100

r [] Fig. 9. (Color online) Measured transverse intensity profiles of the conventional (conv.) focus at z = 0 (solid curve, red online) and focus obtained with Filter 3 at z = 0 (dotted curve), z = 8 mm (dashed curve, orange online), z = 16 mm (dash-dot curve, blue online) and z = 32 mm (crosses, green online).

In the paraxial Debye regime, the dependences of filter performances on the Gaussian distribution in the pupil plane are characterized. Optimal filter designs are proposed for either high-resolution or ultra-large-DOF applications followed by experimental verifications, which should facilitate the development of high-performance systems for in vivo biomedical imaging applications.

ACKNOWLEDGMENTS

Fig. 8. Calculated transverse intensity distributions in the focal plane of (a) conventional focus and focuses obtained with (b) Filter 3 and (c) Filter 4; transverse intensity distribution of (d) conventional focus and focuses obtained with (e) Filter 3 and (f) Filter 4 at z = 16 mm; transverse intensity distributions of (g) conventional focus and focuses obtained with (h) Filter 3 and (i) Filter 4 at z = 36 mm. 共a⬘兲–共i⬘兲 corresponding measured intensity. Image size: 0.6 mm⫻ 0.6 mm. (g⬘)–(i⬘): (Media 2).

It can be noticed that there are some differences between calculated diffraction pattern and experimental results, such as spot size, strength, and symmetry of sidelobe. These discrepancies might be explained by aberrations brought about by the light valve. First of all, the nonuniformity of the light valve phase response affects the calibration accuracy of the device and thus the measured results, especially sidelobe intensity of the measured data larger than that of the calculated data. Second, the effective image size of the addressing video projector is about 200⫻ 200, so that the normalized pixel size of the phase mask is limited to 0.01. This limitation may be one of the reasons for the above-mentioned differences. In conclusion, a simple method to optimize the focusing condition using binary-phase spatial filters is presented.

The authors are grateful to Barbara Petit, Anne Delboulbe, and Mane-Si-Laure Lee for their assistance with experiments. This project is supported in part by the Academic Research Grants (R-397-000-024-112 and R-397-000-615-712) from National University of Singapore and benefits from Merlion support no. 2.06.07 from the French Embassy in Singapore.

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