Characteristics of LEDs. Luminous flux values are at 350 mA, Junction Temperature TJ= 25 C. * LuxeonTM Emitter and Star sample information AB11, 2 (Feb 2002).
1. Introduction
Quantitative phase imaging is a vital technique in many areas of science. Studying properties and characteristics of biological and other microscopic specimens has been facilitated with new quantitative phase imaging microscopy methods. In quantitative phase imaging, phase images are obtained by interfering two light beams – one reflected from, or traversed through, the specimen and the other reflected from a reference mirror. This can be achieved by two methods; holography or phase-shifting interferometry. In holography, one interferogram is used to produce the phase image, while phase-shifting interferometry uses three or more interferograms.
Each fringe in an interferogram represents an area of data ranging from 0 to 2
In the basic phase unwrapping method, the phase image is divided to horizontal lines and these lines are unwrapped separately by scanning pixels and adding an offset to each pixel. At each discontinuity a 2
In 1994 Ghiglia and Romero used a least squares integration method with phase unwrapping. In this method, which is known as least squares integration of phase gradient method, the phase gradient is obtained as wrapped phase differences along two perpendicular directions and the gradient field is least squares integrated to obtain continuous phase. However, this method is not effective for phase maps with high noise. P. G. Charette and I. W. Hunter proposed a robust phase unwrapping method for phase images with high noise content. The basic concept behind this method is to identify contiguous areas that are not on or close to a fringe boundary by locally fitting planes to the phase data. Then these areas are phase shifted with respect to one another by multiples of 2
Software algorithms that exist for detecting and removing 2
In this chapter we will present quantitative phase images of cells and other microscopic samples, using multi-wavelength optical phase unwrapping. Three types of light sources are used in a standard four-frame phase shifting interferometer to obtain phase profiles with larger beat wavelengths, thus removing 2
2. Multi-wavelength optical phase unwrapping
When an object is imaged by a wavelength smaller than the object’s height, phase image of the object contains 2
For years it has been known that a longer wavelength light source produces fewer fringes over a given object than will a short wavelength light source, thus reducing the number of 2
The term ‘two-wavelength interferometry’ was first used by C. Polhemus in a paper where he introduced a two-wavelength technique for interferometric testing. In the method of static interferometry, a fringe pattern obtained with a light source of wavelength
In 1984, two-wavelength phase shifting interferometry was introduced as an optical phase unwrapping method. In this method, phase shifting interferometry and two-wavelength interferometry were combined to extend the phase measurement range of single-wavelength phase shifting interferometry. Two methods were introduced to solve 2
The two-wavelength phase unwrapping method has been extended to multiple wavelengths; enabling measurements of steep surfaces without software phase unwrapping. A hierarchical phase unwrapping algorithm that chooses a minimum number of wavelengths to increase the accuracy of optical unwrapping has been introduced by C. Wagner, W. Osten and S. Seebacher. The basic principle of this method is to start with a larger beat wavelength. Then a systematic reduction of beat wavelengths is used to improve the accuracy of the measurement while the information of the preceding measurements is used to eliminate 2
2.1. Principle of two-wavelength optical phase unwrapping
The basis of multi-wavelength optical phase unwrapping method is the idea of beat wavelength. For two wavelengths
For the
It is apparent that unambiguous range of
Consider two single wavelength phase maps
First, the surface profile
2.2. Principle of three-wavelength optical phase unwrapping
The advantage of 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
The noise reduction is done as follows. In the first step, the quantity of integer multiples of
In the next step, the result is added to the surface profile
The resultant map is then compared with
The resultant surface profile
Then the result is added to the single wavelength surface profile
The resultant map
The method can be applied to phase images obtained with any type of light source, regardless of the coherence length of the source.
3. Multi-wavelength optical phase unwrapping experiments
In this experiment, four step phase shifting algorithm is applied to the interference microscope. Though the minimum number of intensity values needed for phase calculation is three, even a small error in measurements can cause a large phase error. Taking four intensity measurements can reduce this effect. In order to acquire phase images, the Michelson interferometer is used as the experimental setup as shown in Figure (2).
The light is expanded and collimated by the microsocpe objective MO and the lens L1, respectively. The light is then linealry polarized by the polarizer P. The polarized beam splitter (PBS) splits the incoming beam into an S-polarized (polarization plane is perpendicular to polarization axis) ray and a P-polarized (polarization plane is parallel to polarization axis) ray. The S-polarized beam is reflected at the PBS to illuminate the sample object OBJ and the P-polarized beam is transmitted throught the PBS to illuminate the reference mirror REF. When the S-polarized light passes through the quarter wave plate (QW1), the phase changes by 90 and it becomes circulary polarized. After reflcting at the mirror and going through another 90 phase shift at QW1, the light becomes P-polarized. This change from S-polarization to P-polarization avoids light traveling back to the light source and directs all reflected light to the charge-coupled device (CCD). Similarly, P-polarized light illuminating the REF changes to S-polarized light and travels to the CCD. At the analyzer A, the two S-polarized and P-polarized light beams are changed into a common polarization state so that the interference can occur on the CCD plane.
The polarizer-analyzer pair also controls the variation of the realtive intensity between the two arms. The reference mirror is mounted on a piezo-electric transducer (PZT). A function generator supplies a ramp signal to the PZT to dither the reference mirror by a distance of one wavelength. Images are recorded at quarter wavelength intervals.
Images acquired by the CCD are sent to an image acquisition board (National Instruments IMAQ PCI-1407) installed in the computer. The Intensity
The phase map of the object is given by;
Once a phase profile of the specimen is obtained, it can be used to determine the height profile of the specimen.
The optical path difference (OPD) between the object wave and reference wave is given by
4. Multi-wavelength optical phase unwrapping using light emitting diodes
Experimental results of multi-wavelength optical phase unwrapping using light emitting diodes (LED) are presented below. In interferometry, LEDs have been used as light sources in order to reduce the speckle noise inherent to lasers. Since LEDs have coherence lengths in micron range, speckle noise is greatly reduced. All the LEDs used in the experiment are LuxeonTM Emitter diodes from Lumileds Lighting LLC and have a Lambertian (high dome) radiation pattern. The peak wavelength, luminous flux, calculated and measured coherence lengths for red, red-orange, amber and green LEDs used in this experiment are shown in the Table 1. The calculated coherence length of a light source is given by
Colour | Luminous Flux Φ (lm) * |
Peak Wavelength λ (nm) | Spectral Width (nm) | Calculated Coherence Length (μm) | Measured Coherence Length (μm) |
Red | 44 | 653.83±0.07 | 27.24±0.15 | 6.91±0.04 | 9.15±2.45 |
Red-Orange | 55 | 643.42±0.07 | 23.21±0.14 | 7.85±0.05 | 10.29±2.57 |
Amber | 36 | 603.48±0.03 | 17.53±0.05 | 9.14±0.03 | 10.86±2.56 |
Green | 25 | 550.18±0.09 | 38.39±0.19 | 3.42±0.02 | 3.85±1.46 |
4.1. Results for two-wavelength optical phase unwrapping
The object here in the Figure 3 is a micro-electrode array biosensor. It consists of 16 gold electrodes on a Pyrex glass substrate. The center is a 125 μm diameter circle with an approximate thickness of 2 μm. The center of the device was imaged and the experimental results for two wavelength optical phase unwrapping are shown in Figure 3. Red (
4.2. Results for three-wavelength optical phase unwrapping
The experimental results for three-wavelength optical phase unwrapping are presented in Figure 5. Red (
Cross section profile of each phase map is taken along the lines shown in Figure 5. These cross section profiles and phase noise of coarse and fine maps are shown in Figure 6. Figure 6 (a)-6(c) show surface profiles of single wavelength phase map, coarse map and fine map respectively. The vertical axis is 11 μm. Figure 6(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 as shown in Figure 6(e). The black line is data and the red dotted line shows the best-fit parabolic curve. Corrected phase noise in the final fine map is 4.78 nm, which is shown in Figure 6(f).
The comparison of the two-wavelength optical phase unwrapping method to the three-wavelength optical phase unwrapping method shows that the three-wavelength phase unwrapping increases the axial range of the object, without increasing phase noise. These results show that 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 7.48 μm unambiguous range with smaller 4.78 nm phase noise.
Multi-wavelength optical phase unwrapping methods can also be used for biological cells as shown in Figure 7. Here, Figure 7 shows a single wavelength phase map, a coarse map and a fine map of onion cells using red (653.83 nm), amber (603.48 nm) and green (550.18 nm) wavelengths. The beat wavelength is 7.48 μm. Image size is 184 μm × 184 μm. The final fine map clearly shows the cell walls by eliminating the 2
5. Three-wavelength optical phase unwrapping using laser diodes & lasers
In the previous section, incoherent light sources (light emitting diodes) were used to reduce the speckle noise inherent in lasers. However, light emitting diodes are available in only several different wavelengths. Therefore, wavelength combinations that produce large beat wavelengths are limited. Because of small coherence lengths of light emitting diodes, imaging phase profiles of samples with features larger than the coherence range is not possible. In this section, the effectiveness of the three-wavelength optical phase unwrapping method is tested by using laser diodes and a ring dye laser. Laser diodes have been frequently used as a light source in interferometry due to their frequency tunability, smaller size and cost, compared to those of lasers. They also have shorter coherence lengths, typically few centimeters, compared to coherence length of lasers. However, laser diodes also have a limited availability of wavelength choices. Using a ring dye laser, the beat wavelength can be extended to more than a hundred micrometers. In this section, the effectiveness of the optical phase unwrapping method with any type of light source is presented. The results of three-wavelength optical phase unwrapping using laser diodes are shown in Figure 8 and the results obtained with a ring dye laser as the light source are shown in Figure 9, Figure 10 and Figure 11.
Figure 8 shows experimental results of three-wavelength optical phase unwrapping method using laser diodes. The object here is a micro-electrode array biosensor with 16 gold electrodes on a Pyrex glass substrate. The three wavelengths are
In Figure 9, results show phase images of a sample of cheek cells (basal mucosa) illuminated with a ring dye laser. The sample is illuminated by using wavelengths
In Figure 10 and Figure 11, the sample is a piece of 33 1/3 rpm long playing (LP) record. For 33 1/3 rpm records the typical width at the top of the groove ranges from 25.4 μm to 76.2 μm and the groove depth is ~15 μm. The sample is coated with a layer of 200 nm aluminum for better reflectivity. The three wavelengths used for the optical phase unwrapping process is
appears in darker color. The final fine map with reduced noise is shown in Figure 10(c). Figure 10(d) is the 3-D rendering of the final fine map. In the final unwrapped phase map, the width of the top of the groove is measured along the line shown in Figure 4(d). The measured width is 44 μm.
Cross-sections and phase noise of the coarse and fine maps are shown in Figure 11. Figure 11(a) is the unwrapped coarse map and Figure 11(b) is the final fine map with reduced noise. Figure 11(c) is the surface profile of the coarse map along the line shown in Figure 11(a). The RMS noise in the coarse map in the area shown is 2.12 μm and this is shown in Figure 11(d). Figure 11(e) shows the surface profile of fine map along the line shown in Figure 11 (b). The groove depth h = 18 μm.
5. Summary
In summary, this chpater demonstrates the effectiveness of the multi-wavelength optical unwrapping method. To our knowledge this is the first time that three wavelengths have been used in interferometry for phase unwrapping without increasing phase noise. Unlike conventional software phase unwrapping methods that fail when there is high phase noise and when there are irregularities in the object, the multi-wavelength optical phase unwrapping method can be used with any type of object. Software phase unwrapping algorithms can take more than ten minutes to unwrap phase images. This is a disadvantage when one needs to study live samples in real time or near – real time. The multi-wavelength optical unwrapping method is significantly faster than software algorithms and can be effectively used to study live samples in real time. Another advantage is that the optical phase unwrapping method is free of complex algorithms and needs less user intervention.
The method is a useful tool for determining optical thickness profiles of various microscopic samples, biological specimens and optical components. The optical phase unwrapping method can be further improved by adding more wavelengths, thus obtaining beat wavelengths tailored for specific samples.
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