High-sensitive and broad-dynamic-range quantitative phase imaging with spectral domain phase microscopy Yangzhi Yan, Zhihua Ding,* Yi Shen, Zhiyan Chen, Chen Zhao, and Yang Ni State Key Lab of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China * [email protected]
Abstract: Spectral domain phase microscopy for high-sensitive and broaddynamic-range quantitative phase imaging is presented. The phase retrieval is realized in the depth domain to maintain a high sensitivity, while the phase information obtained in the spectral domain is exploited to extend the dynamic range of optical path difference. Sensitivity advantage of phase retrieved in the depth domain over that in the spectral domain is thoroughly investigated. The performance of the proposed depth domain phase based approach is illustrated by phase imaging of a resolution target and an onion skin. ©2013 Optical Society of America OCIS codes: (100.5070) Phase retrieval; (170.4500) Optical coherence tomography; (180.3170) Interference microscopy.
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#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25734
18. M. T. Rinehart, N. T. Shaked, N. J. Jenness, R. L. Clark, and A. Wax, “Simultaneous two-wavelength transmission quantitative phase microscopy with a color camera,” Opt. Lett. 35(15), 2612–2614 (2010). 19. J. Zhang, B. Rao, L. Yu, and Z. Chen, “High-dynamic-range quantitative phase imaging with spectral domain phase microscopy,” Opt. Lett. 34(21), 3442–3444 (2009). 20. E. D. Moore and R. R. McLeod, “Phase-sensitive swept-source interferometry for absolute ranging with application to measurements of group refractive index and thickness,” Opt. Express 19(9), 8117–8126 (2011). 21. Y. Zhu, N. T. Shaked, L. L. Satterwhite, and A. Wax, “Spectral-domain differential interference contrast microscopy,” Opt. Lett. 36(4), 430–432 (2011). 22. C. Wang, Z. H. Ding, S. T. Mei, H. Yu, W. Hong, Y. Z. Yan, and W. D. Shen, “Ultralong-range phase imaging with orthogonal dispersive spectral-domain optical coherence tomography,” Opt. Lett. 37(21), 4555–4557 (2012). 23. Y. Z. Yan, Z. H. Ding, L. Wang, C. Wang, and Y. Shen, “High-sensitive quantitative phase imaging with averaged spectral domain phase microscopy,” Opt. Commun. 303, 21–24 (2013). 24. K. Wang, Z. H. Ding, T. Wu, C. Wang, J. Meng, M. H. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express 17(14), 12121–12131 (2009). 25. J. Meng, Z. H. Ding, J. Li, K. Wang, and T. Wu, “Transit-time analysis based on delay-encoded beam shape for velocity vector quantification by spectral-domain Doppler optical coherence tomography,” Opt. Express 18(2), 1261–1270 (2010).
1. Introduction Spectral domain optical coherence tomography (SD-OCT) [1–4] provides high-sensitive cross-sectional images of scattering tissues with axial resolution on the scale of several to tens of microns. It detects spectral interferograms as a function of optical frequency and produces A-scan data by applying Fourier transformation to the acquired interference spectrum. Spectral domain phase microscopy (SDPM) [5–9] is a functional extension of SD-OCT. By employing common-path configuration and phase-sensitive measurement, the sensitivity of the axial displacement measurement by SDPM greatly exceeds the axial resolution of the SDOCT system, allowing for various applications such as material inspection [10], cellular imaging [11, 12], and Doppler flow measurements [13, 14]. In conventional SDPM, the phase due to optical path difference (OPD) between the reference arm and sample arm is extracted in the depth domain. The range of OPD to be measured is restricted to less than half of the source center wavelength due to the well-known 2π ambiguity. To extend the OPD range of SDPM, different phase unwrapping algorithms have been proposed. Conventional phase unwrapping algorithm [15] requires the corresponding OPD along neighboring sampling points to vary gradually, and fails if multiple wrapping of phase occurs. Synthetic wavelength phase unwrapping techniques [16–18] use two or more wavelengths to increase the OPD range without phase unwrapping at the price of increased phase noises. Instead of phase retrieval in the depth domain, phase retrieval in the spectral domain is proposed to extend the OPD range [19–22]. However, the phase sensitivity is degraded since the relevant signals are distributed over a broadband of wavenumbers in the spectral domain in contrast to localization in depth domain, which leads to a reduced signal-to-noise ratio (SNR). Although by averaging spectral phases over multiple wavenumbers can significantly enhance the phase sensitivity [23], the complicated spectral phase retrieval algorithm is time-consuming and error prone. In this paper, we introduce a high-sensitive and broad-dynamic-range quantitative phase imaging method for SDPM. The phase retrieval is still realized in the depth domain to maintain a high sensitivity, while the phase information obtained in the spectral domain is further used for removing the phase ambiguity to extend the dynamic range of SDPM. The phase sensitivity in the spectral domain and in the depth domain will be compared theoretically and experimentally. Phase imaging of a resolution target and an onion skin are presented to evaluate the performance of the proposed phase imaging method. 2. Theory 2.1 Conventional depth domain phase based SDPM Excluding autocorrelation and DC terms, the detected interference spectra of the return lights from a sample reflector and a reference reflector can be described as [2, 19, 23]
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25735
I ( ki ) = 2 S ( ki − k0 ) RR RS cos(2 z0 ki ) + α ( ki ).
(1)
Here ki is the discrete wavenumber ( i = 1 ~ N ) in the spectral domain; k0 is the central wavenumber; S ( ki − k0 ) represents power spectral density function of the light source; RR and RS are reflectivities of the sample reflector and the reference reflector, respectively; z0 denotes the OPD between the reference reflector and the sample reflector; α ( ki ) is the additive Gaussian white noise with a standard deviation of σ k and a mean value of zero due to excluding of the DC term [1, 2]. Conventional SDPM applies the fast Fourier transform (FFT) on Eq. (1) to get the depth signal as I ( zi ) = RR RS Γ( zi − z0 ) exp[ − j 2k0 ( zi − z0 )] + α ( zi ),
(2)
here the coherent ghost image is omitted for simplicity; j is the basic imaginary unit sqrt(−1); zi is the discretely sampled depth position; Γ( zi − z0 ) = [ S ( ki ) ] ⊗ δ ( zi − z0 ) with [•] denotes Fourier transformation; α ( zi ) is the Fourier Transform Spectrum of α ( ki ) , it
is the complex noise in the depth domain corresponding to α ( ki ) in the spectral domain. The standard deviation of α ( zi ) is σ z . The peak value in the depth domain will appear at zi = z0 if assuming a symmetry distribution of S ( ki − k0 ) around its central wavenumber k0 . However, the spectral density of the light source is usually not of an ideal Gaussian form but with asymmetry. The discretely sampled peak value in the depth domain might be at a position inconsistent with z0 . Assuming a deviation of δ z from z0 , the nearest peaked value at sampling point zi = z0 − δ z in the depth domain is then given by I ( z0 − δ z ) = RR RS Γ( −δ z ) exp( j 2k0δ z ) + α ( z0 − δ z ).
(3)
The phase term ϕ ( z0 − δ z ) of I ( z0 − δ z ) is 2k0δ z + Δϕ zi , here Δϕ zi denotes the equivalent phase uncertainty caused by the noise. Obviously, ϕ ( z0 − δ z ) is a linear function of δ z . Subresolution changes in OPD between the reference reflector and the sample reflector can be sensitively detected by measurement of this phase term. Unfortunately, the achieved phase from Eq. (3) is limited to the range of −π to +π and can be written by
ϕ w ( z0 − δ z ) = ϕ ( z0 − δ z ) − 2π floor (
ϕ ( z0 − δ z ) + π ), 2π
(4)
here floor means round toward negative infinity. A problem known as 2π ambiguity occurs when phase change between consecutive measurement falls outside the range [ −π , +π ] , meaning a limitation of the dynamic OPD range less than half a central wavelength. To extend the dynamic range of conventional depth domain phase based SDPM, synthetic wavelength based method is introduced. By windowing the signal spectrum into multiple spectra before applying the Fourier transformation, phase information at multiple center wavelengths are obtained and used for phase retrieval corresponding to a longer synthetic central wavelength. The dynamic range is thus extended but the sensitivity is degraded [16– 18].
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25736
2.2 Spectral domain phase based SDPM
To measure OPD longer than half a central wavelength, spectral domain phase based SDPM [19, 20] is proposed. By Hilbert transformation of Eq. (1), the complex spectral signal is given by I ( ki ) = S ( ki − k0 ) RR RS exp( j 2ki z0 ) + α ( ki ),
(5)
where α ( ki ) is the complex spectral noise. The standard deviation of α ( ki ) is σ k . The phase term ϕ ( k ) of I ( k ) is 2k z + Δϕ , here Δϕ is the equivalent phase uncertainty due to the i
i
i 0
ki
ki
spectral noise. The wrapped phase ϕ w ( ki ) can be obtained from the complex spectral signal described by Eq. (5) as [23]
ϕ w ( ki )=ϕ ( ki ) − 2π floor (
ϕ ( ki )+π ). 2π
(6)
By phase unwrapping and least-square fitting in the spectral domain [19, 20], an estimated phase of ϕ ( ki ) is then obtained by
ϕ ′(ki )=2ki z0′ = 2ki z0 + Δϕ k′ . i
(7)
Here z0′ denotes the fitted slope of the unwrapped phase versus wavenumber, Δϕ k′i denotes the phase errors in ϕ ′(ki ) . Since ϕ ′(ki ) suffers from an amplified phase error [19], i.e. Δϕ k′i > Δϕ ki , the estimated phase is only used for removal of 2π ambiguity of the phase in the spectral domain as [23]
ϕ (ki )=ϕ w (ki )+2π floor (
ϕ ′(ki )+π ). 2π
(8)
Unlike the algorithm presented in reference [19], addition of π is made here for synchronizing of phase jumps with opposite direction in both terms of the right side of Eq. (8) [23]. This spectral domain phase based approach is capable of broad-dynamic-range phase imaging of phase specimen. However, its phase sensitivity is degraded in compared with that achieved in the depth domain and this will be thoroughly investigated in the following section. 2.3 A comparison of phase sensitivity
Phase sensitivity in SDPM is determined by SNR. A higher SNR means a higher phase sensitivity and hence a lower phase uncertainty. The phase uncertainty in SDPM can be expressed by [5] Δϕ = arc tan(
1 )≈ SNR
|σ | 1 = noise , SNR | I sample |
(9)
where σ noise denotes the noise component; I sample denotes the signal. To compare the phase sensitivity in the depth domain and that in the spectral domain, the SNR in both domains should be investigated. To achieve the highest phase sensitivity, the phase information should be retrieved at their 4 ln 2( ki − k0 ) 2 peak values in both domains. Assume a Gaussian distribution S ( ki ) = exp( − ) Δk 2 and FWHM bandwidth of Δk for the light source, the peak values of signal in the depth
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25737
domain and spectral domain can be obtained from Eqs. (2) and (5) to be Δk π RR RS I max ( zi ) = and Imax ( ki ) = 2 RR RS , respectively. Here δ k is the spectral 2δ k ln 2 resolution of the spectrometer in the SDPM. The peak signal ratio between depth domain and spectral domain is given by | I max ( zi ) | π Δk . = | I max ( ki ) | 4 ln 2δ k
(10)
Fig. 1. Simulation results in both domains of SDPM corresponding to a normalized interference signal arising from two reflectors at an OPD of 210 μm. Signal distribution in the depth domain (a) and in the spectral domain (b) corresponding to an ideal normalized interference signal. Depth distribution (c) and spectral distribution (d) from an additive white Gaussian noise under SNR of 70 dB. SNR distribution in the depth domain (e) and in the spectral domain (f).
According to Parseval's theorem of discrete Fourier transformation, the noise variance ratio between depth domain and spectral domain is given by
σ z2 = N. σk2
(11)
N is the number of discrete wavenumbers in the spectral domain. The ratio of SNR between the depth domain and the spectral domain is thus deduced to be SNRz π 1 Δk 2 = ( ). SNRk 16 ln 2 N δ k
(12)
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25738
Using Eq. (9), the ratio of phase uncertainty Δϕ z Δϕ k between the depth domain and the spectral domain is determined by Δϕ z 4 ln 2 = Δϕ k π
N δ k 4 ln 2 N δ k = . Δk π N Δk
(13)
Obviously, Δϕ z Δϕ k is inversely proportional to square root of N since N δ k is comparable to Δk , and the phase sensitivity in the depth domain should be much better than that in the spectral domain as N >> 1 . Simulation is further conducted to investigate the phase sensitivity in both domains. The parameters used for simulation are consistent with the experimental settings described later. A Gaussian light source with a central wavelength of 835nm and FWHM bandwidth of 45nm is adopted. The spectral resolution is set to be 0.0674 nm. The number of discretely sampling points N in both domains is 2048. The OPD between the sample reflector and the reference reflector is set to be 210 μm. Figures 1(a) and 1(b) demonstrate the calculated depth distribution and spectral distribution from an ideal normalized interference signal based on Eqs. (2) and (5), respectively. We can see that the signal distribution in the depth domain is densely concentrated at the location corresponding to OPD while the signal distribution in the spectral domain is broadly distributed over wavenumbers determined by the power spectral density of the light source. Similarly, the calculated depth distribution and spectral distribution from an additive white Gaussian noise for SNR of 70 dB in related to the normalized interference signal are presented in Figs. 1(c) and 1(d), respectively. The calculated SNR distributions in the depth domain and in the spectral domain are shown in Figs. 1(e) and 1(f), respectively. The peak SNR in the depth domain is determined to be 87.50 dB while the peak SNR in the spectral domain is 70 dB. An enhancement of 17.50 dB is resulted in the depth domain, leading to 7.50 fold higher phase sensitivity based on Eq. (9). This simulated result is coincident with the theoretical value of 7.74 determined by Eq. (13). 2.4 The proposed depth domain phase based SDPM As discussed previously, the conventional depth domain phase based SDPM has the advantage in phase sensitivity and the disadvantage in limited dynamic range. To achieve high dynamic range while keeping high sensitivity, we propose a new depth domain phase retrieval method, in which phases in both domains are fully exploited. As discussed in section 2.3, to achieve the highest phase sensitivity, the phase information should be retrieved at the peak values in depth domain. Due to asymmetry of spectral density distribution of the light source and discrete sampling, the peak value of the response profile in depth domain usually cannot be sampled. We take the sampling point zi = z0 − δ z with a maximum value as the estimated peak point. At the estimated peak point, the corresponding wrapped phase is ϕ w ( z0 − δ z ) . Now we introduce how to retrieve ϕ (z0 − δ z ) based on the spectral phase ϕ (ki ) and the wrapped phase ϕ w ( z0 − δ z ) . The spectral phase at the central wavenumber ϕ (k0 ) can be determined by Eq. (8), which is then used to quantify an estimated phase of ϕ ( z0 − δ z ) in the depth domain as
ϕ ′(z0 − δ z )=ϕ (k0 ) − 2k0 ( z0 − δ z ) = 2k0δ z ± Δϕ k . 0
(14)
Since ϕ ′(z0 − δ z ) suffers from an amplified phase error, it is only used for removal of 2π ambiguity of the phase in the depth domain as
ϕ (z0 − δ z )=ϕ w ( z0 − δ z )+2π floor (
ϕ ′(z0 − δ z )+π ). 2π
(15)
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25739
With the location information of the sample point in the depth domain where the phase is retrieved, the absolute OPD between the reference reflector and the sample reflector is thus determined by OPD=z0 − δ z +
ϕ (z 0 − δ z ) 2k0
.
(16)
Compared with the conventional depth domain phase based SDPM, this proposed depth domain phase retrieval method is immune of phase ambiguity. 3. Experiment
As shown in Fig. 2, the proposed phase retrieval method is experimentally evaluated by the SDPM based on an existing SD-OCT system described in our previous papers [24, 25]. The light source is a super luminescent diode with central wavelength of 835 nm and FWHM bandwidth of 45nm. To realize a common-path configuration for the SDPM, the reference arm of the SD-OCT system is disconnected and the light in the sample arm is focused by the objective lens with a focal length of 30mm on the top surface of a coverslip. Back reflected light from the bottom surface of the coverslip acts as the reference light. The interference signal from the SDPM is detected by a high-speed 2048 pixel spectrometer with a resolution of 0.0674 nanometer running at the highest A-scan rate of 29,000 lines/s. Data processing is performed by program written in MATLAB. The computer CPU is an Intel Core(TM) i72600k.
Fig. 2. Schematic of the spectral domain phase microscopy system.
3.1 Coverslip To illustrate the enhanced phase sensitivity of the proposed depth domain phase based SDPM over that achieved by the spectral domain phase based SDPM, 1024 A-scans of a stationary coverslip with nominal optical thickness of 210 ± 35 µm are performed. The OPDs obtained by the spectral domain phase based SDPM and the proposed depth domain phase based SDPM are shown in Fig. 3(a). Evidently, the OPD measured by the proposed depth domain
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25740
phase based SDPM is much more stable than that measured by the spectral domain phase based SDPM. The probability distribution of the measured OPD by spectral domain phase based SDPM and by the proposed depth domain phase based SDPM are shown in Figs. 3(b) and 3(d), respectively. The standard variation (STD) of the former one is 161.81 pm while the STD of the latter one is 21.41 pm. The phase sensitivity of the proposed depth domain phase based SDPM is 7.56 fold higher than that of the spectral domain phase based SDPM, which is in good agreement with the theoretical expectation according to Eq. (13). The phase sensitivity in the spectral domain is degraded since the relevant signals are distributed over a broadband of wavenumbers in contrast to localization in depth domain, which leads to a reduced SNR. On the other hand, the averaging of spectral phases over multiple wavenumbers of our previous reported multiple spectral phases averaged SDPM [23] can again enhance the SNR. To compare the proposed depth domain phase based SDPM with our previous reported multiple spectral phases averaged SDPM, the OPDs obtained from the same experimental data by the multiple spectral phases averaged SDPM is also shown in Fig. 3(a). The probability distribution of the measured OPD by multiple spectral phases averaged SDPM is presented in Fig. 3(c). The STD of the measured OPD is 20.73 pm, which is comparable to that obtained by the proposed depth domain phase based SDPM. Therefore, phase sensitivities in multiple spectral phases averaged SDPM and the proposed depth domain phase based SDPM are similar. However, the previously reported multiple spectral phases averaged SDPM must retrieve and choose the spectral phases deliberately before averaging. Spectral phases corresponding to 343 wavenumbers are selected for the averaging procedure here, which is time-consuming. The processing time taken by the multiple spectral phases averaged SDPM is 31.64 s while that by the proposed depth domain phase based SDPM is only 0.47 s. Hence the proposed method is much more efficient than the previously reported multiple spectral phases averaged approach.
Fig. 3. (a) OPDs corresponding to 1024 repeated A-scans: by the spectral domain phase based SDPM (blue circles); by multiple spectral phases averaged SDPM (green diamonds); by the proposed depth domain phase based SDPM (red crosses). (b) The probability distribution of the measured OPD by spectral domain phase based SDPM. (c) The probability distribution of the measured OPD by multiple spectral phases averaged SDPM. (d) The probability distribution of the measured OPD by the proposed depth domain phase based SDPM.
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25741
3.2 Resolution target To validate the ability of the proposed depth domain phase based SDPM for broad-dynamicrange OPD measurement, a resolution target (1951 USAF, Newport) is adopted as the sample. The sample configuration is shown in Fig. 4(a) where an optical fiber with diameter of 245 ± 5 µm is placed at one end between the coverslip and the resolution target to form a wedge realizing a large dynamic range of OPDs. Interference spectra of the return lights from the top surface of the coverslip and the topological surface of the resolution target are collected and used to measure the OPDs between these two surfaces. Figure 4(b) demonstrates the quantitative phase obtained by conventional depth domain phase based SDPM, where the measurement range is limited by half of the central wavenumber and the patterns on the resolution target are corrupted by phase ambiguity. Figure 4(c) shows the quantitative phase image obtained by the proposed depth domain phase based SDPM, where the wedge-shaped OPD-variations are well reconstructed. The patterns on the resolution target can be clearly resolved from the phase image. The height of the pattern is measured to be 120 nm, which is consistent with the reported value in the literatures [11, 12]. The quantitative phase image obtained by the spectral domain phase based SDPM is also shown in Fig. 4(d) for comparison. However, due to the large dynamic range of OPD, it is hard to notice the difference between the phase images demonstrated in Figs. 4(c) and Fig. 4(d).
Fig. 4. Measurement of the OPD between coverslip and resolution target. (a) Sample configuration. (b) Quantitative phase image by conventional depth domain phase based SDPM. (c) Quantitative phase image by the proposed depth domain phase based SDPM. (d) Quantitative phase image by the spectral domain phase based SDPM.
3.3 Single layer onion skin To demonstrate the feasibility of the proposed depth domain phase based SDPM for biological sample, phase imaging of a single layer of onion skin is done. Figure 5(a) is the image recorded by an optical microscope showing the onion skin cells. The reconstructed phase image is presented in Fig. 5(b), where ellipse shape of onion skin cell (about 30 μm to 125 μm in lateral directions and 30 μm in depth direction) is clearly resolved.
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25742
Fig. 5. (a) Image of onion skin cells by optical microscope. The scale bar indicates 30 μm. (b) Reconstructed phase image of onion skin cells by the proposed depth domain phase based SDPM.
4. Conclusion
By fully exploitation of phase information in the depth domain and the spectral domain, SDPM capable of high-sensitive and broad-dynamic-range quantitative phase imaging is presented. Sensitivity advantage of phase retrieved in the depth domain over that in the spectral domain is confirmed both theoretically and experimentally. Phase ambiguity in the depth domain is overcome by the proposed depth domain phase retrieval method, in which the unwrapped phase in the spectral domain and the location of the sample point in the depth domain is used to extend the dynamic range of SDPM. The performance of the proposed depth domain phase based SDPM with extended dynamic range is illustrated by phase imaging of a resolution target and an onion skin. Further application of this technique could be the study of dynamical phenomenon in biological sample, such as optical measurement of nerve activations. Acknowledgments
The authors would like to acknowledge the financial supports from the Chinese Natural Science Foundation (61275196, 61335003, 61327007) and Zhejiang Province Science and Technology Grant (2012C33031).
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25743
Abstract: Spectral domain phase microscopy for high-sensitive and broaddynamic-range quantitative phase imaging is presented. The phase retrieval is realized in the depth domain to maintain a high sensitivity, while the phase information obtained in the spectral domain is exploited to extend the dynamic range of optical path difference. Sensitivity advantage of phase retrieved in the depth domain over that in the spectral domain is thoroughly investigated. The performance of the proposed depth domain phase based approach is illustrated by phase imaging of a resolution target and an onion skin. ©2013 Optical Society of America OCIS codes: (100.5070) Phase retrieval; (170.4500) Optical coherence tomography; (180.3170) Interference microscopy.
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#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25734
18. M. T. Rinehart, N. T. Shaked, N. J. Jenness, R. L. Clark, and A. Wax, “Simultaneous two-wavelength transmission quantitative phase microscopy with a color camera,” Opt. Lett. 35(15), 2612–2614 (2010). 19. J. Zhang, B. Rao, L. Yu, and Z. Chen, “High-dynamic-range quantitative phase imaging with spectral domain phase microscopy,” Opt. Lett. 34(21), 3442–3444 (2009). 20. E. D. Moore and R. R. McLeod, “Phase-sensitive swept-source interferometry for absolute ranging with application to measurements of group refractive index and thickness,” Opt. Express 19(9), 8117–8126 (2011). 21. Y. Zhu, N. T. Shaked, L. L. Satterwhite, and A. Wax, “Spectral-domain differential interference contrast microscopy,” Opt. Lett. 36(4), 430–432 (2011). 22. C. Wang, Z. H. Ding, S. T. Mei, H. Yu, W. Hong, Y. Z. Yan, and W. D. Shen, “Ultralong-range phase imaging with orthogonal dispersive spectral-domain optical coherence tomography,” Opt. Lett. 37(21), 4555–4557 (2012). 23. Y. Z. Yan, Z. H. Ding, L. Wang, C. Wang, and Y. Shen, “High-sensitive quantitative phase imaging with averaged spectral domain phase microscopy,” Opt. Commun. 303, 21–24 (2013). 24. K. Wang, Z. H. Ding, T. Wu, C. Wang, J. Meng, M. H. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express 17(14), 12121–12131 (2009). 25. J. Meng, Z. H. Ding, J. Li, K. Wang, and T. Wu, “Transit-time analysis based on delay-encoded beam shape for velocity vector quantification by spectral-domain Doppler optical coherence tomography,” Opt. Express 18(2), 1261–1270 (2010).
1. Introduction Spectral domain optical coherence tomography (SD-OCT) [1–4] provides high-sensitive cross-sectional images of scattering tissues with axial resolution on the scale of several to tens of microns. It detects spectral interferograms as a function of optical frequency and produces A-scan data by applying Fourier transformation to the acquired interference spectrum. Spectral domain phase microscopy (SDPM) [5–9] is a functional extension of SD-OCT. By employing common-path configuration and phase-sensitive measurement, the sensitivity of the axial displacement measurement by SDPM greatly exceeds the axial resolution of the SDOCT system, allowing for various applications such as material inspection [10], cellular imaging [11, 12], and Doppler flow measurements [13, 14]. In conventional SDPM, the phase due to optical path difference (OPD) between the reference arm and sample arm is extracted in the depth domain. The range of OPD to be measured is restricted to less than half of the source center wavelength due to the well-known 2π ambiguity. To extend the OPD range of SDPM, different phase unwrapping algorithms have been proposed. Conventional phase unwrapping algorithm [15] requires the corresponding OPD along neighboring sampling points to vary gradually, and fails if multiple wrapping of phase occurs. Synthetic wavelength phase unwrapping techniques [16–18] use two or more wavelengths to increase the OPD range without phase unwrapping at the price of increased phase noises. Instead of phase retrieval in the depth domain, phase retrieval in the spectral domain is proposed to extend the OPD range [19–22]. However, the phase sensitivity is degraded since the relevant signals are distributed over a broadband of wavenumbers in the spectral domain in contrast to localization in depth domain, which leads to a reduced signal-to-noise ratio (SNR). Although by averaging spectral phases over multiple wavenumbers can significantly enhance the phase sensitivity [23], the complicated spectral phase retrieval algorithm is time-consuming and error prone. In this paper, we introduce a high-sensitive and broad-dynamic-range quantitative phase imaging method for SDPM. The phase retrieval is still realized in the depth domain to maintain a high sensitivity, while the phase information obtained in the spectral domain is further used for removing the phase ambiguity to extend the dynamic range of SDPM. The phase sensitivity in the spectral domain and in the depth domain will be compared theoretically and experimentally. Phase imaging of a resolution target and an onion skin are presented to evaluate the performance of the proposed phase imaging method. 2. Theory 2.1 Conventional depth domain phase based SDPM Excluding autocorrelation and DC terms, the detected interference spectra of the return lights from a sample reflector and a reference reflector can be described as [2, 19, 23]
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25735
I ( ki ) = 2 S ( ki − k0 ) RR RS cos(2 z0 ki ) + α ( ki ).
(1)
Here ki is the discrete wavenumber ( i = 1 ~ N ) in the spectral domain; k0 is the central wavenumber; S ( ki − k0 ) represents power spectral density function of the light source; RR and RS are reflectivities of the sample reflector and the reference reflector, respectively; z0 denotes the OPD between the reference reflector and the sample reflector; α ( ki ) is the additive Gaussian white noise with a standard deviation of σ k and a mean value of zero due to excluding of the DC term [1, 2]. Conventional SDPM applies the fast Fourier transform (FFT) on Eq. (1) to get the depth signal as I ( zi ) = RR RS Γ( zi − z0 ) exp[ − j 2k0 ( zi − z0 )] + α ( zi ),
(2)
here the coherent ghost image is omitted for simplicity; j is the basic imaginary unit sqrt(−1); zi is the discretely sampled depth position; Γ( zi − z0 ) = [ S ( ki ) ] ⊗ δ ( zi − z0 ) with [•] denotes Fourier transformation; α ( zi ) is the Fourier Transform Spectrum of α ( ki ) , it
is the complex noise in the depth domain corresponding to α ( ki ) in the spectral domain. The standard deviation of α ( zi ) is σ z . The peak value in the depth domain will appear at zi = z0 if assuming a symmetry distribution of S ( ki − k0 ) around its central wavenumber k0 . However, the spectral density of the light source is usually not of an ideal Gaussian form but with asymmetry. The discretely sampled peak value in the depth domain might be at a position inconsistent with z0 . Assuming a deviation of δ z from z0 , the nearest peaked value at sampling point zi = z0 − δ z in the depth domain is then given by I ( z0 − δ z ) = RR RS Γ( −δ z ) exp( j 2k0δ z ) + α ( z0 − δ z ).
(3)
The phase term ϕ ( z0 − δ z ) of I ( z0 − δ z ) is 2k0δ z + Δϕ zi , here Δϕ zi denotes the equivalent phase uncertainty caused by the noise. Obviously, ϕ ( z0 − δ z ) is a linear function of δ z . Subresolution changes in OPD between the reference reflector and the sample reflector can be sensitively detected by measurement of this phase term. Unfortunately, the achieved phase from Eq. (3) is limited to the range of −π to +π and can be written by
ϕ w ( z0 − δ z ) = ϕ ( z0 − δ z ) − 2π floor (
ϕ ( z0 − δ z ) + π ), 2π
(4)
here floor means round toward negative infinity. A problem known as 2π ambiguity occurs when phase change between consecutive measurement falls outside the range [ −π , +π ] , meaning a limitation of the dynamic OPD range less than half a central wavelength. To extend the dynamic range of conventional depth domain phase based SDPM, synthetic wavelength based method is introduced. By windowing the signal spectrum into multiple spectra before applying the Fourier transformation, phase information at multiple center wavelengths are obtained and used for phase retrieval corresponding to a longer synthetic central wavelength. The dynamic range is thus extended but the sensitivity is degraded [16– 18].
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25736
2.2 Spectral domain phase based SDPM
To measure OPD longer than half a central wavelength, spectral domain phase based SDPM [19, 20] is proposed. By Hilbert transformation of Eq. (1), the complex spectral signal is given by I ( ki ) = S ( ki − k0 ) RR RS exp( j 2ki z0 ) + α ( ki ),
(5)
where α ( ki ) is the complex spectral noise. The standard deviation of α ( ki ) is σ k . The phase term ϕ ( k ) of I ( k ) is 2k z + Δϕ , here Δϕ is the equivalent phase uncertainty due to the i
i
i 0
ki
ki
spectral noise. The wrapped phase ϕ w ( ki ) can be obtained from the complex spectral signal described by Eq. (5) as [23]
ϕ w ( ki )=ϕ ( ki ) − 2π floor (
ϕ ( ki )+π ). 2π
(6)
By phase unwrapping and least-square fitting in the spectral domain [19, 20], an estimated phase of ϕ ( ki ) is then obtained by
ϕ ′(ki )=2ki z0′ = 2ki z0 + Δϕ k′ . i
(7)
Here z0′ denotes the fitted slope of the unwrapped phase versus wavenumber, Δϕ k′i denotes the phase errors in ϕ ′(ki ) . Since ϕ ′(ki ) suffers from an amplified phase error [19], i.e. Δϕ k′i > Δϕ ki , the estimated phase is only used for removal of 2π ambiguity of the phase in the spectral domain as [23]
ϕ (ki )=ϕ w (ki )+2π floor (
ϕ ′(ki )+π ). 2π
(8)
Unlike the algorithm presented in reference [19], addition of π is made here for synchronizing of phase jumps with opposite direction in both terms of the right side of Eq. (8) [23]. This spectral domain phase based approach is capable of broad-dynamic-range phase imaging of phase specimen. However, its phase sensitivity is degraded in compared with that achieved in the depth domain and this will be thoroughly investigated in the following section. 2.3 A comparison of phase sensitivity
Phase sensitivity in SDPM is determined by SNR. A higher SNR means a higher phase sensitivity and hence a lower phase uncertainty. The phase uncertainty in SDPM can be expressed by [5] Δϕ = arc tan(
1 )≈ SNR
|σ | 1 = noise , SNR | I sample |
(9)
where σ noise denotes the noise component; I sample denotes the signal. To compare the phase sensitivity in the depth domain and that in the spectral domain, the SNR in both domains should be investigated. To achieve the highest phase sensitivity, the phase information should be retrieved at their 4 ln 2( ki − k0 ) 2 peak values in both domains. Assume a Gaussian distribution S ( ki ) = exp( − ) Δk 2 and FWHM bandwidth of Δk for the light source, the peak values of signal in the depth
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25737
domain and spectral domain can be obtained from Eqs. (2) and (5) to be Δk π RR RS I max ( zi ) = and Imax ( ki ) = 2 RR RS , respectively. Here δ k is the spectral 2δ k ln 2 resolution of the spectrometer in the SDPM. The peak signal ratio between depth domain and spectral domain is given by | I max ( zi ) | π Δk . = | I max ( ki ) | 4 ln 2δ k
(10)
Fig. 1. Simulation results in both domains of SDPM corresponding to a normalized interference signal arising from two reflectors at an OPD of 210 μm. Signal distribution in the depth domain (a) and in the spectral domain (b) corresponding to an ideal normalized interference signal. Depth distribution (c) and spectral distribution (d) from an additive white Gaussian noise under SNR of 70 dB. SNR distribution in the depth domain (e) and in the spectral domain (f).
According to Parseval's theorem of discrete Fourier transformation, the noise variance ratio between depth domain and spectral domain is given by
σ z2 = N. σk2
(11)
N is the number of discrete wavenumbers in the spectral domain. The ratio of SNR between the depth domain and the spectral domain is thus deduced to be SNRz π 1 Δk 2 = ( ). SNRk 16 ln 2 N δ k
(12)
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25738
Using Eq. (9), the ratio of phase uncertainty Δϕ z Δϕ k between the depth domain and the spectral domain is determined by Δϕ z 4 ln 2 = Δϕ k π
N δ k 4 ln 2 N δ k = . Δk π N Δk
(13)
Obviously, Δϕ z Δϕ k is inversely proportional to square root of N since N δ k is comparable to Δk , and the phase sensitivity in the depth domain should be much better than that in the spectral domain as N >> 1 . Simulation is further conducted to investigate the phase sensitivity in both domains. The parameters used for simulation are consistent with the experimental settings described later. A Gaussian light source with a central wavelength of 835nm and FWHM bandwidth of 45nm is adopted. The spectral resolution is set to be 0.0674 nm. The number of discretely sampling points N in both domains is 2048. The OPD between the sample reflector and the reference reflector is set to be 210 μm. Figures 1(a) and 1(b) demonstrate the calculated depth distribution and spectral distribution from an ideal normalized interference signal based on Eqs. (2) and (5), respectively. We can see that the signal distribution in the depth domain is densely concentrated at the location corresponding to OPD while the signal distribution in the spectral domain is broadly distributed over wavenumbers determined by the power spectral density of the light source. Similarly, the calculated depth distribution and spectral distribution from an additive white Gaussian noise for SNR of 70 dB in related to the normalized interference signal are presented in Figs. 1(c) and 1(d), respectively. The calculated SNR distributions in the depth domain and in the spectral domain are shown in Figs. 1(e) and 1(f), respectively. The peak SNR in the depth domain is determined to be 87.50 dB while the peak SNR in the spectral domain is 70 dB. An enhancement of 17.50 dB is resulted in the depth domain, leading to 7.50 fold higher phase sensitivity based on Eq. (9). This simulated result is coincident with the theoretical value of 7.74 determined by Eq. (13). 2.4 The proposed depth domain phase based SDPM As discussed previously, the conventional depth domain phase based SDPM has the advantage in phase sensitivity and the disadvantage in limited dynamic range. To achieve high dynamic range while keeping high sensitivity, we propose a new depth domain phase retrieval method, in which phases in both domains are fully exploited. As discussed in section 2.3, to achieve the highest phase sensitivity, the phase information should be retrieved at the peak values in depth domain. Due to asymmetry of spectral density distribution of the light source and discrete sampling, the peak value of the response profile in depth domain usually cannot be sampled. We take the sampling point zi = z0 − δ z with a maximum value as the estimated peak point. At the estimated peak point, the corresponding wrapped phase is ϕ w ( z0 − δ z ) . Now we introduce how to retrieve ϕ (z0 − δ z ) based on the spectral phase ϕ (ki ) and the wrapped phase ϕ w ( z0 − δ z ) . The spectral phase at the central wavenumber ϕ (k0 ) can be determined by Eq. (8), which is then used to quantify an estimated phase of ϕ ( z0 − δ z ) in the depth domain as
ϕ ′(z0 − δ z )=ϕ (k0 ) − 2k0 ( z0 − δ z ) = 2k0δ z ± Δϕ k . 0
(14)
Since ϕ ′(z0 − δ z ) suffers from an amplified phase error, it is only used for removal of 2π ambiguity of the phase in the depth domain as
ϕ (z0 − δ z )=ϕ w ( z0 − δ z )+2π floor (
ϕ ′(z0 − δ z )+π ). 2π
(15)
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25739
With the location information of the sample point in the depth domain where the phase is retrieved, the absolute OPD between the reference reflector and the sample reflector is thus determined by OPD=z0 − δ z +
ϕ (z 0 − δ z ) 2k0
.
(16)
Compared with the conventional depth domain phase based SDPM, this proposed depth domain phase retrieval method is immune of phase ambiguity. 3. Experiment
As shown in Fig. 2, the proposed phase retrieval method is experimentally evaluated by the SDPM based on an existing SD-OCT system described in our previous papers [24, 25]. The light source is a super luminescent diode with central wavelength of 835 nm and FWHM bandwidth of 45nm. To realize a common-path configuration for the SDPM, the reference arm of the SD-OCT system is disconnected and the light in the sample arm is focused by the objective lens with a focal length of 30mm on the top surface of a coverslip. Back reflected light from the bottom surface of the coverslip acts as the reference light. The interference signal from the SDPM is detected by a high-speed 2048 pixel spectrometer with a resolution of 0.0674 nanometer running at the highest A-scan rate of 29,000 lines/s. Data processing is performed by program written in MATLAB. The computer CPU is an Intel Core(TM) i72600k.
Fig. 2. Schematic of the spectral domain phase microscopy system.
3.1 Coverslip To illustrate the enhanced phase sensitivity of the proposed depth domain phase based SDPM over that achieved by the spectral domain phase based SDPM, 1024 A-scans of a stationary coverslip with nominal optical thickness of 210 ± 35 µm are performed. The OPDs obtained by the spectral domain phase based SDPM and the proposed depth domain phase based SDPM are shown in Fig. 3(a). Evidently, the OPD measured by the proposed depth domain
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25740
phase based SDPM is much more stable than that measured by the spectral domain phase based SDPM. The probability distribution of the measured OPD by spectral domain phase based SDPM and by the proposed depth domain phase based SDPM are shown in Figs. 3(b) and 3(d), respectively. The standard variation (STD) of the former one is 161.81 pm while the STD of the latter one is 21.41 pm. The phase sensitivity of the proposed depth domain phase based SDPM is 7.56 fold higher than that of the spectral domain phase based SDPM, which is in good agreement with the theoretical expectation according to Eq. (13). The phase sensitivity in the spectral domain is degraded since the relevant signals are distributed over a broadband of wavenumbers in contrast to localization in depth domain, which leads to a reduced SNR. On the other hand, the averaging of spectral phases over multiple wavenumbers of our previous reported multiple spectral phases averaged SDPM [23] can again enhance the SNR. To compare the proposed depth domain phase based SDPM with our previous reported multiple spectral phases averaged SDPM, the OPDs obtained from the same experimental data by the multiple spectral phases averaged SDPM is also shown in Fig. 3(a). The probability distribution of the measured OPD by multiple spectral phases averaged SDPM is presented in Fig. 3(c). The STD of the measured OPD is 20.73 pm, which is comparable to that obtained by the proposed depth domain phase based SDPM. Therefore, phase sensitivities in multiple spectral phases averaged SDPM and the proposed depth domain phase based SDPM are similar. However, the previously reported multiple spectral phases averaged SDPM must retrieve and choose the spectral phases deliberately before averaging. Spectral phases corresponding to 343 wavenumbers are selected for the averaging procedure here, which is time-consuming. The processing time taken by the multiple spectral phases averaged SDPM is 31.64 s while that by the proposed depth domain phase based SDPM is only 0.47 s. Hence the proposed method is much more efficient than the previously reported multiple spectral phases averaged approach.
Fig. 3. (a) OPDs corresponding to 1024 repeated A-scans: by the spectral domain phase based SDPM (blue circles); by multiple spectral phases averaged SDPM (green diamonds); by the proposed depth domain phase based SDPM (red crosses). (b) The probability distribution of the measured OPD by spectral domain phase based SDPM. (c) The probability distribution of the measured OPD by multiple spectral phases averaged SDPM. (d) The probability distribution of the measured OPD by the proposed depth domain phase based SDPM.
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25741
3.2 Resolution target To validate the ability of the proposed depth domain phase based SDPM for broad-dynamicrange OPD measurement, a resolution target (1951 USAF, Newport) is adopted as the sample. The sample configuration is shown in Fig. 4(a) where an optical fiber with diameter of 245 ± 5 µm is placed at one end between the coverslip and the resolution target to form a wedge realizing a large dynamic range of OPDs. Interference spectra of the return lights from the top surface of the coverslip and the topological surface of the resolution target are collected and used to measure the OPDs between these two surfaces. Figure 4(b) demonstrates the quantitative phase obtained by conventional depth domain phase based SDPM, where the measurement range is limited by half of the central wavenumber and the patterns on the resolution target are corrupted by phase ambiguity. Figure 4(c) shows the quantitative phase image obtained by the proposed depth domain phase based SDPM, where the wedge-shaped OPD-variations are well reconstructed. The patterns on the resolution target can be clearly resolved from the phase image. The height of the pattern is measured to be 120 nm, which is consistent with the reported value in the literatures [11, 12]. The quantitative phase image obtained by the spectral domain phase based SDPM is also shown in Fig. 4(d) for comparison. However, due to the large dynamic range of OPD, it is hard to notice the difference between the phase images demonstrated in Figs. 4(c) and Fig. 4(d).
Fig. 4. Measurement of the OPD between coverslip and resolution target. (a) Sample configuration. (b) Quantitative phase image by conventional depth domain phase based SDPM. (c) Quantitative phase image by the proposed depth domain phase based SDPM. (d) Quantitative phase image by the spectral domain phase based SDPM.
3.3 Single layer onion skin To demonstrate the feasibility of the proposed depth domain phase based SDPM for biological sample, phase imaging of a single layer of onion skin is done. Figure 5(a) is the image recorded by an optical microscope showing the onion skin cells. The reconstructed phase image is presented in Fig. 5(b), where ellipse shape of onion skin cell (about 30 μm to 125 μm in lateral directions and 30 μm in depth direction) is clearly resolved.
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25742
Fig. 5. (a) Image of onion skin cells by optical microscope. The scale bar indicates 30 μm. (b) Reconstructed phase image of onion skin cells by the proposed depth domain phase based SDPM.
4. Conclusion
By fully exploitation of phase information in the depth domain and the spectral domain, SDPM capable of high-sensitive and broad-dynamic-range quantitative phase imaging is presented. Sensitivity advantage of phase retrieved in the depth domain over that in the spectral domain is confirmed both theoretically and experimentally. Phase ambiguity in the depth domain is overcome by the proposed depth domain phase retrieval method, in which the unwrapped phase in the spectral domain and the location of the sample point in the depth domain is used to extend the dynamic range of SDPM. The performance of the proposed depth domain phase based SDPM with extended dynamic range is illustrated by phase imaging of a resolution target and an onion skin. Further application of this technique could be the study of dynamical phenomenon in biological sample, such as optical measurement of nerve activations. Acknowledgments
The authors would like to acknowledge the financial supports from the Chinese Natural Science Foundation (61275196, 61335003, 61327007) and Zhejiang Province Science and Technology Grant (2012C33031).
#196955 - $15.00 USD Received 3 Sep 2013; revised 11 Oct 2013; accepted 11 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025734 | OPTICS EXPRESS 25743
