1. Introduction

Most biological specimens are hardly absorptive, lacking sufficient contrast for standard bright-field microscopy. Several widely used phase imaging techniques are Zernike phase contrast (ZPC) microscopy, differential interference contrast (DIC) microscopy and Hoffman modulation contrast (HMC) microscopy. The Zernike phase contrast technique converts changes in phase into corresponding changes in amplitude by using a phase plate. ZPC uses common path interferometry, where a partially coherent light beam passes through the specimen. The light that is diffracted due to phase variations of the specimen and the light that passes without diffraction are then focused together to form a phase contrast image of the specimen. The DIC technique converts changes in the refractive index of the sample into amplitude changes by splitting a polarized beam into two perpendicularly polarized beams less than a micrometer apart. As the index of refraction changes, the phase of the two beams changes after passing through the sample. The two beams are then recombined using a Normaski prism and the recombined beam is sent through an analyzer to form an image of the specimen. The HMC microscopy converts optical phase gradients into amplitude using a spatial filter (or, “modulator”), which has three light-passing zones, a slit plate and a circular polarizer. Different specimen thicknesses deflect the light into different modulator zones thus controlling the contrast of the image. These techniques have well known draw backs such as halo effect (ZPC), shadow effect (DIC), and non-linear phase to amplitude conversion. Because of this non-linearity, phase imaging using these techniques is only qualitative. Quantitative phase imaging is important because it provides information of morphology and refractive indices of specimens. Recently a number of quantitative phase imaging techniques have been developed; Barone-Nugent and team have demonstrated a quantitative phase imaging microscope that separates phase information from amplitude information and produces pure phase images [1], several digital holography techniques have been used for quantitative phase imaging [2, 3], and diffraction phase and fluorescence (DPF) microscopy has been used for simultaneous quantitative phase imaging and epi-fluorescence imaging of living cells [4].

1.1 Phase unwrapping

When the optical depth of an object is greater than the wavelength, the phase image contains 2π ambiguities. Therefore, phase data has to be unwrapped before one can obtain an unambiguous optical thickness profile. Software algorithms that exist for detecting and removing 2π ambiguities are largely computation-intensive and prone to errors when the phase profile is noisy [5]. While the principle of multi-wave phase unwrapping has been known in interferometry [6, 7], known applications have been confined to optical profilers with raster-scanned pointwise interferometry. Other than our recent digital holography experiments [8, 9], multi-wave phase unwrapping has not yet been applied to full-frame phase images. The basic principle of two-wavelength optical phase unwrapping in the context of digital holography was presented by J. Gass, A. Dakoff and M.K. Kim [8]. In this paper, we extend the technique of multi-wavelength optical phase unwrapping to phase-shifting interference microscopy using light emitting diodes at three different wavelengths.

Light emitting diodes (LED) have been used as in interferometric light sources in order to reduce the speckle noise inherent to lasers [10-12]. Interference of coherent waves produces speckle noise, which limits the phase map’s information. Since a LED’s coherence length is in the micron range, speckle noise is greatly reduced. LEDs also cost much less than lasers, are easy to use and replace and can reduce overall apparatus dimensions. The LEDs used in this experiment are Luxeon^™ Emitter diodes from Lumileds Lighting LLC. All of the LEDs used herein have a Lambertian (high dome) radiation pattern. And their spectra are shown in Fig. 1. These spectra were taken with Ocean Optics SD-1000 fiber optics spectrometer.

 

Fig. 1. Spectra for red, red-orange, amber and green LEDs.

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The peak wavelengths, luminous flux, calculated and measured coherence lengths for the red, amber and green LEDs used in this experiment are shown in Table 1. The calculated coherence length of a light source is given by l_C=(2ln 2/π)($\overline{\lambda }$ ²/Δλ), where $\overline{\lambda }$ is the mean wavelength and Δλ is the full width half maximum (FWHM) of Gaussian spectrum [13]. The coherence length was directly measured here by counting the number of fringes in the interference of the tilted mirror object.

Table 1. Characteristics of LEDs. Luminous flux values are at 350 mA, Junction Temperature T_J=5°C [14]

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2. Methodology

The experimental setup is shown in Fig. 2. A set of LEDs illuminate a Michelson interferometer. In a fairly standard arrangement, a combination of a polarizing beam-splitter and two quarter-wave plates in the object and reference arms allow all of the reflected light from the two arms to reach the camera. A polarizer-analyzer pair also allows continuous variation of the relative intensity between the two arms. In order to acquire images with high resolution, two 20X microscope objectives are placed in front of the object and the reference mirror. Images acquired by the CCD are sent to an image acquisition board (National Instruments IMAQ PCI^™-1407) installed in the computer.

 

Fig. 2. Experimental setup. L1, collimating lens; L2,L3, microscope objectives; P, polarizer; QW1,QW2, quarter wave plates; A, analyzer; REF, reference mirror, OBJ, object; PZT, piezo transducer; CCD, charged-coupled device camera.

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The Intensity I(x,y) of the light captured by CCD can be written as;

Here I_O(x, y) is the part of the beam reflected by the object that is coherent with respect to I_R(x,y), the intensity of the beam reflected by the reference mirror. I_B(x, y) is part of reflection from the object that is incoherent with respect to the reference – i.e. outside the coherence length. ϕ(x, y) is the relative phase between the object and the reference mirror and ϕ_i is the phase shift introduced by moving the reference mirror by quarter wavelength intervals. Four images are acquired at ϕ_i=0, π/2, π and 3π/2. The phase map of the object is then given by

3. Multi-wavelength optical phase unwrapping

When an object is imaged by using a wavelength smaller than the object’s height, the phase image contains 2π ambiguities, as shown in Fig. 3.

 

Fig. 3. Phase vs. distance. 2π ambiguities occur when the distance is a multiple of the wavelength.

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From Fig. 3, it is clear that there are many distance values for a given phase value. In order to obtain an unambiguous optical thickness profile, we need to have only one z distance for a given phase. These 2π ambiguities are eliminated by using multi-wavelength optical phase unwrapping method. For the m^th wavelength λ_m, the surface profile Z_m of an object is related to the phase difference ϕ_m as follows;

It is apparent that an unambiguous range of Z can be increased by using a longer λ.

Consider two single wavelength phase maps ϕ₁ and ϕ₂. The beat wavelength Λ₁₂ for λ₁ and λ₂ is given by Λ₁₂=λ₁ λ₂/|λ₁-λ₂|. The Λ₁₂ can be increased by choosing closer values of λ₁ and λ₂. The phase map for Λ₁₂ is obtained by subtracting one single wavelength phase map from the other and then adding 2π whenever the resultant value is less than zero. This phase map is called “coarse map” ϕ₁₂. The surface profile for the coarse map φ₁₂ is given by Z ₁₂=Λ₁₂ ϕ ₁₂/2π. Note that however, the phase noise in each single wavelength phase map is magnified by the same factor as the magnification of the wavelengths. In the two-wavelength optical phase unwrapping method introduced by J. Gass, et.al. [8], the phase noise is reduced by using the following steps.

1. The surface profile Z₁₂ is divided into integer multiples of a single wavelength, say λ₁.

2. The result is added to the single wavelength surface profile Z₁. This significantly reduces the phase noise in the coarse map. However, at the boundaries of wavelength intervals λ₁, the noise of the single wavelength phase map appears as spikes.

3. Remove the spikes by comparing the result with the coarse map surface profile Z₁₂. If the difference is more than half of λ₁, addition or subtraction of one λ₁ depending on the sign of the difference removes the spikes.

The resultant ‘fine map’ has a noise level equal to that of single wavelength surface profile. If a single wavelength phase map ϕ_m contains a phase noise of 2πε_m , the two-wavelength phase unwrapping method works properly for ε_m<λ_m/4Λ₁₂ [8]. For λ₁=653.83 nm and λ₂=550.18 nm, the maximum noise limit is ε_m ~ 4.7 %. Using a larger beat wavelength therefore reduces the maximum noise limit.

3.1 Three-wavelength optical phase unwrapping

The advantage of this three wavelength phase unwrapping method is that the beat wavelength can be increased without reducing the maximum noise limit. Suppose the three chosen wavelengths are λ₁=653.83 nm, λ₂=603.48 nm, and λ₃=550.18 nm. The first two wavelengths give a beat wavelength of Λ₁₂=10.53 µm. Instead of using the surface profile Z₁₂ of Λ₁₂=7.84 µm, which has a high noise, an identical surface profile can be produced by using surface profiles Z₁₃ and Z₂₃ with beat wavelengths of Λ₁₃=3.47 µm and Λ₂₃=6.2 µm, respectively. The resultant “coarse map of coarse maps” ϕ_13–23 with surface profile Z_13–23 also has the same beat wavelength Λ_13–23=Λ₁₃ Λ₂₃/| Λ₁₃-Λ₂₃|=7.84 µm.

The following steps are followed to reduce the noise in Z_13–23.

1. The surface profile Z_13–23 is divided into integer multiples of one of the wavelengths, say Λ₁₃.

2. The result is added to the surface profile Z₁₃.

3. The resultant map is then compared with Z_13–23. If the difference is more than half of Λ₁₃, one Λ₁₃ is added or subtracted depending on the sign of the difference.

This approach reduces the phase noise significantly. Any remaining spikes are due to the noise in ϕ₁₃. We call this result the “intermediate fine map” Z′_13–23. To reduce the noise in intermediate fine map, we repeat the procedure by dividing Z′_13–23 into integer multiples of λ₁ and adding the result to Z₁. Then the resultant map is compared with Z′_13–23. If the difference is more than half of λ₁, one λ₁ is added or subtracted depending on the sign of the difference. Noise in the final map is equal to the noise in single wavelength phase map Z₁. The three wavelength phase unwrapping method works when the maximum noise level ε_m in the single wavelength phase map is the smaller of Λ₁₃/4 Λ₁₂~11.1% or λ₁/4Λ₁₃~4.7%. Therefore, the three wavelength phase unwrapping method increases the beat wavelength without magnifying the noise in the final phase map.

4. Experimental results

4.1 Results for two-wavelength optical phase unwrapping

 

Fig. 4. Results of two-wavelength optical phase unwrapping: (a) single wavelength phase map, (b) two-wavelength coarse map, (c) two-wavelength fine map with reduced noise.

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The object here in Fig. 4 is a micro-electrode array biosensor. It consists of 16 gold electrodes on a Pyrex glass substrate. The center is a 125µm diameter circle with an approximate thickness of 2µm. The center of the device was imaged and the experimental results for two-wavelength optical phase unwrapping are shown in Fig. 4. Red (λ₁=653.83 nm) and green (λ₂=550.18 nm) LEDs are used as the two wavelengths. The beat wavelength Λ₁₂=3.47 µm. Images are of a 184 µm×184 µm area. Fig. 4(a) shows a single wavelength phase map ϕ₁ with λ₁=653.83 nm. The coarse map ϕ₁₂ with Λ₁₂=3.47 µm is shown in Fig. 4(b) and the final phase map with reduced noise is shown in Fig. 4(c). Figure 5 shows cross sections of phase maps along the lines shown in Fig. 4 and the phase noise in the chosen regions. Figure 5(a) is a cross section of single wavelength phase map with λ₁=653.83 nm and Fig. 5(b) is a cross section of coarse map with Λ₁₂=3.47 µm. A cross section of the fine map with reduced noise is shown in Fig. 5(c). For maps (a), (b) and (c), the vertical axis is 4 µm. The root mean square (rms) noise of the coarse map is 43.27 nm. This is shown in Fig. 5(d). Figure 5(e) shows the reduced noise in fine phase map. Since the center of the MEMS device has a curvature, a paraboloid is fitted to the data. The red dotted line is the best-fit parabolic curve. After subtracting the curvature from the data, the Fig. 5(f) shows the corrected phase noise of 10.29 nm.

 

Fig. 5. Surface profiles for two-wavelength phase unwrapping. (a) single wavelength surface profile, (b) surface profile of coarse map, (c) surface profile of final unwrapped phase map with reduced noise, (d) noise of the coarse map in the region between the two markers in plot (b). Rms noise is 43.27 nm, (e) noise of final unwrapped phase map in the area shown in (c). Red dotted line is the best fit parabolic curvature and black solid line is data, (f) corrected phase noise of the unwrapped phase map, after subtracting the curvature of the object. Rms noise is 10.29 nm.

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4.2 Results for three-wavelength optical phase unwrapping

The experimental results for three-wavelength optical phase unwrapping are shown in Fig. 6. Red (λ₁=653.83 nm), amber ((λ₂=603.48 nm) and green ((λ₃=550.18 nm) LEDs are used as the three wavelengths. The beat wavelength Λ=7.84 µm. The object is the center of the same micro-electrode array bio-sensor used in section (4.1). Images are of a 184 µm×184 µm area. Figure 6(a) is the single wavelength phase map with λ₁=653.83 nm. The three wavelength coarse map is shown in Fig. 6(b) with an equivalent wavelength Λ₁₂=7.84 µm. The final fine map with reduced noise is shown in Fig. 6(c).

 

Fig. 6. Results of three-wavelength optical phase unwrapping: (a) single wavelength phase map, (b) three-wavelength coarse map, (c) three- wavelength fine map with reduced noise.

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Cross section of each phase map is taken along the lines shown in Fig. 6. These cross sections and phase noise of coarse and fine maps are shown in Fig. 7. Figures 7(a)-7(c) show surface profiles of single wavelength phase map, coarse map and fine map respectively. Vertical axis for each map is 11 µm. Figure 7(d) shows 105.79 nm rms noise of the coarse map. Because of the curvature of the object surface, a paraboloid is fitted with the final fine map data. This is shown in Fig. 7(e). The black line is data and the red dotted line shows the best- fit parabolic curve. Corrected phase noise in the fine map is 4.78 nm, which is shown in Fig. 7(f).

 

Fig. 7. Surface profiles for three-wavelength phase unwrapping. (a) single wavelength surface profile, (b) surface profile of coarse map, (c) surface profile of final unwrapped phase map with reduced noise. (d) noise of coarse map in the area shown in (b). rms noise is 105.79 nm, (e) noise of final unwrapped phase map in the area shown in (c). Red dotted line is the best fit parabolic curvature and black solid line is data, (f) Final noise of the unwrapped phase map, after subtracting the curvature of the object. Rms noise is 4.78 nm.

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The comparison of the two-wavelength phase unwrapping method to the three-wavelength phase unwrapping method shows that the three-wavelength phase unwrapping method increases the axial range of the object, without increasing phase noise. In our results, the two-wavelength phase unwrapping method produced a 3.47 µm unambiguous range with 10.29 nm phase noise, while the three-wavelength phase unwrapping method produced much larger 7.84 µm unambiguous range with smaller 4.78 nm phase noise.

Multi-wavelength optical phase unwrapping method can also be used for biological cells. Figure 8 shows a single wavelength phase map, a coarse map and a fine map of onion cells using red (653.83nm), amber (603.48nm) and green (550.18 nm) wavelengths. The beat wavelength is 7.84 µm. Image size is 184 µm×184 µm. The Final fine map clearly shows the cell walls by eliminating the 2π ambiguities that would exist in a single wave length phase image.

 

Fig. 8. Results of three-wavelength optical phase unwrapping of onion cells: (a) single wavelength phase map, (b) three-wavelength coarse map, (c) three-wavelength fine map with reduced noise.

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5. Conclusions

In conclusion, the effectiveness of multi-wavelength phase imaging for unwrapping phase images with 2π ambiguities has been demonstrated. The three wavelength optical phase unwrapping method can be used to extend the axial range of the object being imaged, without increasing the noise in the phase map. Broadband light sources provide considerably less coherent noise and have added advantages such as lower cost and reduced apparatus dimensions. The multi-wavelength optical phase unwrapping method presented in this paper can be used to check the quality of surface profiles of microscopic objects such as sensors, integrated circuits and many MEMS devices. It can also be used to determine optical thickness profiles of various optical components and biological samples.