We propose a spectral-domain interferometric technique, termed spectral modulation interferometry (SMI), and present its application to high-sensitivity, high-speed, and speckle-free quantitative phase imaging. In SMI, one-dimensional complex field of an object is interferometrically modulated onto a broadband spectrum. Full-field phase and intensity images are obtained by scanning along the orthogonal direction. SMI integrates the high sensitivity of spectral-domain interferometry with the high speed of spectral modulation to quantify fast phase dynamics, and its dispersive and confocal nature eliminates laser speckles. The principle and implementation of SMI are discussed. Its performance is evaluated using static and dynamic objects.
© 2015 Optical Society of America
Quantitative phase imaging (QPI) has emerged as an important tool for studying structures and dynamics of unstained biological specimens . QPI techniques rely on the intrinsic phase distribution of a sample as contrast mechanism, similar to conventional Zernike phase contrast microscopy and Nomarski differential interference contrast microscopy [2, 3]. These classic techniques, however, do not measure phase but rather convert it into intensity variations for direct visualization on a microscope. In contrast, QPI techniques are capable of accurate quantification of both optical phase (or equivalently optical pathlength, OPL) and amplitude, providing a powerful means for studying cellular morphology and surface metrology [4–7]. Commercial developments are on the rise as applications and acceptance in life sciences and other fields continue to grow.
Most QPI systems are based on either interferometric or computational approach, and usually represent a tradeoff between sensitivity, speed and complexity. There have been numerous reports on laser-based interferometric phase imaging, such as digital holography [8–12]. These systems are in general simple and fast, limited only by camera speed. While OPL sensitivity has been reported to reach nanometer or sub-nanometer level, significant improvement is limited by the presence of laser speckles, often visible in the resulted phase images. Broadband light-based systems have also been developed using various types of wavefront engineering schemes to mitigate speckle artifacts and achieve sensitivity gain [13–18]. Among computational QPI techniques, major examples include transport-of-intensity equation-based techniques and recently emerging optical ptychographic microscopy [19–22]. Although simple and robust, they require sequential acquisition of multiple sub-images to compute one output, hence limited in speed.
Compared to these full-field implementation, point-scanning phase microscopy based on spectral-domain low-coherence interferometry (also known as whitelight interferometry) can improve phase sensitivity by more than an order of magnitude (tens of picometers) [23–26]. The drawback is the slow acquisition speed that normally requires several seconds for an image of typical resolution, therefore unable to map fast phase dynamics in two dimensions.
In the following research, we seek to achieve high-sensitivity, speckle-free quantitative phase imaging while maintaining high acquisition speed. We introduce spectral modulation interferometry (SMI), which combines spectral-domain low-coherence interferometry with spectral modulation technique. The spatial phase and amplitude of a specimen are modulated onto the spectrum of a broadband light. We discuss the principle and implementation of SMI-based QPI system and demonstrate its performance using both static reflective objects and dynamic biological specimens.
2. Spectral modulation interferometry
Figure 1 shows a Linnik interferometer-based implementation of SMI for phase imaging. The broadband light from a singlemode superluminescent diode (SLD, Superlum; λ = 837 nm, ∆λ = 54 nm) is passed through a 50/50 fiber coupler and a 1D galvo scanner (x direction), dispersed (y direction) by a transmission grating (Wasatch Photonics; 600 l/mm), and focused by two identical objectives (Carl Zeiss; 40 × , 0.65) on both the sample and reference mirrors to form a wavelength dispersed line illumination. The reference arm can be axially translated to adjust the pathlength difference between the two arms. Interference spectrum between light reflected from both arms is detected by a custom spectrometer (SPM) with a linescan camera (e2v; EM1, 1024 pixel, maximum line rate 78 kHz), which covers 61 nm spectral range with 0.06 nm/pixel spectral resolution. Sample information of all points along the illuminated line is simultaneously captured in the interference spectrum, and full-field images can be obtained by a simple 1D scanning of the galvo. The camera exposure time is fixed at 12 μsec for all scanning speeds. When the galvo is within the linear portion of its forward scan period, a train of trigger pulses initiate camera acquisition, capturing successive lines that form a 2D image. Increasing scan speed (frame rate) reduces forward scan duration as well as the number of lines per image along scan direction. For surface metrology measurement, light directly reflected by sample surface is detected; for thin, relatively transparent samples such as live cells, they can be placed on a reflective surface for double-pass transmission measurement.
The dispersed illumination in SMI is similar to that of spectrally encoded imaging [27–30]. Some of these prior reports were also based on interferometry [28–30]. In these techniques, however, sample information is handled locally in a small neighborhood of each spatial point using processing techniques such as short-time Fourier transform to estimate interference fringe density. In contrast, SMI treats the complex sample field of the entire line as a continuous function and analyzes the entire spectrum globally, thus requiring completely different signal demodulation processes and also providing new insights into signal modulation processes, as mentioned below.
The complex reference and sample fields detected by the system can be written as and , respectively, where Lr and Ls are arm OPL of the Linnik interferometer, and the corresponding phase depends on spatially dispersed wavenumber, k(y), which can be written as with k0 being the staring wavenumber and being the dispersion coefficient. α can be further expressed as , where β is the angular dispersion coefficient of the grating, M is the magnification of the 4f system after the grating, and fobj is the focal length of the objective. The total detected interference intensity hence is
Equation (1) clearly shows this is a modulation technique. The sample amplitude Es(x,y) and phase ϕs(x,y) are modulated onto an interferometrically generated carrier wave with a frequency of , hence the name spectral modulation interferometry. With the carrier wave, the complex sample field is shifted to high frequency and separated from baseband components and noises.
It is important to note the inherent connection between SMI and two other technologies. First, SMI is the same concept as amplitude/frequency/phase modulation (AM/FM/PM) in analog communication, but implemented in optical spectral domain instead of time domain. Secondly, we point out that Eq. (1) is mathematically identical to the interferograms produced by off-axis digital holographic microscopy, which can be viewed as the same modulation concept implemented in spatial domain.
The similarity between SMI and digital holography suggests the two-dimensional interferogram Itot(x,y) in Eq. (1) may be analyzed as a digital hologram with some minor modifications. The processing steps are briefly described as follows. First, due to the uneven wavenumber sampling of the spectrometer, frequently seen in spectral-domain interferometry, resampling into linear wavenumber domain is necessary and is done using sinc interpolation. A 2D Fourier transform is then performed, where the DC term, the carrier-shifted signal and its complex conjugate are separated. The shifted signal is then filtered, down-shifted back to baseband, and inversely Fourier transformed to spatial domain. The intensity of the resulted complex signal represents the brightfield image of the sample. Also, sample phase is extracted and unwrapped using standard Goldstein algorithm , and then converted to OPL using corresponding wavelength at each position. Finally, background variation is removed by Zernike polynomial subtraction .
Compared to digital holographic microscopy, a significant advantage of SMI is that it is free of speckles generated by optics defects, contaminations and dust particles. The primary reason is that for each wavelength the system operates as a confocal microscope, which rejects coherent scattering from those sources. In addition, the dispersive illumination means incoherence between an image point with a specific wavelength and the diffraction field produced by other wavelengths, further suppressing the strength of speckle patterns. The absence of speckles can substantially improve phase sensitivity. At the same time, the line illumination scheme still allows for fast acquisition.
3. Experimental results
3.1 Resolution target
To demonstrate SMI, we first use a 1951 USAF resolution target (Edmund Optics). An 83.9 × 83.9 µm2 area is imaged with 1024 × 1024 pixel resolution. The average y (spectral) direction sampling interval is 0.082 µm, producing sufficient sampling bandwidth to allow complete separation between carrier-shifted signal and DC without causing aliasing. In theory, the maximum sampling interval, or equivalently the minimal sampling frequency, may be closely estimated from Eq. (1) with knowledge of system’s point spread function (PSF), as often done in digital holography. In SMI, however, the confocal PSF depends on both the objective and the input singlemode fiber , and has not been theoretically characterized for this particular system. The sampling frequency was instead experimentally determined by adjusting the magnification of the 4f system before the Linnik interferometer.
The x-direction sampling interval is set to the same as that of y direction. It should be noted that this is not required and the choice of x interval is flexible as long as it satisfies Nyquist rate because no carrier separation is needed along x direction. This permits greater x sampling interval or fewer number of x samples when larger field of view or higher acquisition speed is needed.
Figure 2(a) shows a 2D interferogram of the resolution target, with each column representing a sample-modulated spectrum at the corresponding position. The OPL difference between the two arms is approximately 1.23 mm, producing 4.7 spectral samples per interference fringe. The fringe density determines the frequency of the carrier wave. The 2D FFT of the interpolated interferogram in Fig. 2(b) shows that the modulated signal is shifted to high frequency, similar to that of a typical off-axis digital hologram. Following the signal processing procedure outlined above, the 2D OPL and intensity images of the area are obtained, as shown in Fig. 2(c) and 2(d). The cross-sectional profiles at marked positions in these images are plotted in Fig. 2(e). From the 10%-90% edge response of the OPL curve, the lateral resolution of the system is estimated to be 0.99 µm. The intensity curves are consistent with the spectral shape of the SLD. If needed, they can be normalized to restore the actual intensity profile of the sample.
Further examination of the measured height of the Cr pattern in Fig. 2(e) brings up an interesting and important issue in optical phase measurement. The average measured height by our optical system is 113.50 nm. However, an AFM determines the height as 143.15 nm, confirmed by a Dektak stylus profiler. Both are contact, mechanical measurement devices. To investigate the discrepancy, an optical Zygo surface profilometer was used, measuring the thickness as ~126 nm. The smaller values by non-contact optical devices, as compared to contact measurement, is a result of the complex reflection coefficient at the Cr-air interface, which causes additional phase change in addition to the ideal phase shift of π. In the meantime, the air/glass interface does not experience this additional shift. We then coated the entire resolution target with silver and new measurement resulted in 138.94 nm, in good agreement with AFM results. There is still a 4.2 nm difference, likely because of our non-optimized optics, e.g. Zeiss objective was used without a matching Zeiss tube lens to provide optimal correction. Overall, this experiment signifies the need for cautious interpretation in optical metrology measurement on multi-material surfaces.
Next, we evaluate the temporal OPL sensitivity of the SMI system using a mirror surface as the sample at three galvo-scanning rates, as plotted in Fig. 2(f). Since the camera exposure time and light source power are fixed, this experiment studies how galvo motion affects system stability and sensitivity. At 0 Hz (non-scanning), only the center line of the field of view was imaged and the temporal standard deviation of OPL was calculated for each point along the line, ranging from 0.1 nm to 0.17 nm with an average of 0.12 nm for over 90% of all spectral points except for edges, where source power is low. Without galvo vibration, this represents the sensitivity limit of this setup. Expectedly, this limit depends on signal strength and the curve inversely correlates to the SLD spectrum shown as line 3 in Fig. 2(e). For 66.9Hz scanning rate (1024 × 1024 resolution, 0.082 μm x step), the sensitivity remains the same. The imaging speed is then increased to 120.6 Hz by reducing the number of x steps to 512 while doubling step size to maintain the same field of view. The sensitivity drops moderately to an average of 0.17 nm, which is likely caused by galvo vibration at high scanning rate and can be alleviated by optimizing galvo size and driving pattern, as well as stabilizing the sample stage. Figure 2(h) shows the temporal sensitivity across the entire field of view at 120.6 Hz. As can be seen, the sensitivity curves are all similar to that of the center line. There is a relatively small variation along the scan direction, which is believed to be caused by the system’s power efficiency variation at different scan positions.
Spatial OPL sensitivity is also calculated using the 120.6 Hz image in Fig. 2(g) as example. The standard deviation of the OPL across the full field of view is 0.98 nm. This value is believed to be largely dominated by surface flatness and roughness of the sample and reference mirrors. Such irregularities are clearly visible in the image. A 10 μm × 10μm, relatively flat area was then chosen and its standard deviation is calculated to be 0.66 nm. Better surface quality of the mirrors is expected to further improve the spatial sensitivity.
3.2 Dynamic imaging of peranema
The dynamic performance of SMI on biological specimens is demonstrated using Peranema culture (Carolina Biological Supply). Samples were prepared by sandwiching a drop of the culture between sample mirror surface and a coverslip, and then imaged at 120.6 frames per second (fps) with the same resolution and field of view mentioned above. Figure 3(a) and 3(b) show 2D OPL images from a recorded video (Media 1) in part (c). The high acquisition rate allows the fast moving flagellum to be closely examined in slow motion (30 fps), as shown in Fig. 3(d) (Media 2). Another example is shown in Fig. 3(e) (Media 3) and (f).
In both the resolution target and Peranema experiments, the OPL images confirm that the SMI system produces no observable speckles, and the background texture seen in Fig. 3(c), 3(d) and 3(e) is dominated by surface roughness of the sample and reference mirrors, which can be improved with better optics if needed. Such speckle-free imaging, combined with spectral-domain interferometry, enables the system for high sensitivity phase quantification.
4. Discussion and conclusion
Spectral modulation interferometry is a novel spectral-domain interferometric technique and has been demonstrated for quantitative phase imaging of static and dynamic samples with high OPL sensitivity and high acquisition speed. Compared to traditional holographic techniques, SMI offers similar signal format and processing procedure, but its dispersive and confocal nature eliminates speckle-induced instability and inaccuracy. Current system achieves a temporal OPL sensitivity of 0.12 nm. This performance is likely shot-noise limited, as suggested by preliminary simulated and experimental analysis, and therefore may be further improved by optimizing system signal-to-noise ratio.
The SMI technique also offers high acquisition speed, the limit of which is decided by both the required number of samples along the scanning direction and the speed of the linescan camera. Potentially, such a system could reach hundreds of fps for reduced field of view.
In summary, we have discussed the theoretical principle and experimental validation of SMI for quantitative phase imaging. SMI is well suited for quantitative study of dynamic processes in both biological and non-biological samples requiring exquisite sensitivity.
The authors thank Jung Ki Hong of the Department of Sustainable Biomaterials and Donald Leber of the Department of Electrical and Computer Engineering at Virginia Tech for their help in resolution target characterization. This work was supported in part by a Junior Faculty Collaborative grant from the Institute for Critical Technology and Applied Science (ICTAS) at Virginia Tech.
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