Differential interference contrast microscopy as a polarimetric instrument

Differential interference contrast microscopy as a polarimetric instrument Andrew Resnick

Differential interference contrast 共DIC兲 microscopy is shown to be equivalent to an incomplete Stokes polarimeter capable of probing optical properties of materials on microscopic-length scales. The Mueller matrix for a DIC microscope is calculated for various types of samples, and the polarimetric properties for DIC component parts of a spaceflight microscope are spectrally measured. As a practical application, a measurement of the index mismatch between colloidal particles and a nearly index-matched fluid bath was performed. © 2002 Optical Society of America OCIS codes: 120.2130, 120.5410, 180.0180, 230.5440.

1. Introduction

Differential interference contrast 共DIC兲 microscopy is a technique used to visualize spatial changes in the index of refraction across an 共ideally兲 phaseonly object.1,2 Because inhomogeneities in the index of refraction of weakly scattering samples are equivalent to inhomogeneities in the retardation, DIC is shown in this paper to be capable of highly accurate quantitative polarimetric measurements of microscopic objects. In addition, the narrow depth of focus provided by DIC allows depth section-type measurements similar to a confocal microscope.3 Thus DIC could prove to be a useful quantitative probe of optical properties of bulk materials. The application that is the primary concern of this research is in the study of colloidal sols and gels in a microgravity environment,4 the optical properties of which can be usefully engineered provided that appropriate optical properties can be measured. Because some of the flight samples are projected to be highly birefringent, exhibit novel optical properties, or even be optically active, use of polarimetry as a probe could prove to be highly interesting and useful. It is hoped that this method proves to be useful in the general field of ellipsometry as a probe of microscopic-scale phe-

A. Resnick 共[email protected]兲 is with Logicon Federal Data, 2001 Aerospace Parkway, Brook Park, Ohio 441422460. Received 25 June 2001; revised manuscript received 6 September 2001. 0003-6935兾02兾010038-08$15.00兾0 © 2002 Optical Society of America 38

APPLIED OPTICS 兾 Vol. 41, No. 1 兾 1 January 2002

nomena.5,6 In addition, use of a microscope allows correlation between structural and polarimetric properties, for example, by a comparison of DIC and bright-field images. As an example of an application of this method, a measurement of the relative index of refraction between polymethylmethacrylate 共PMMA兲 spheres and a nearly index-matching bath of decalin and tetralin was performed. This particular study was motivated by a series of spaceflight experiments that used a microscope to study the statics and dynamics of colloidal systems during nucleation and growth, as well as real-space characterizations of structure 共as opposed to lightscattering-type measurements兲.7,8 The microscope model chosen for spaceflight was a Leica DM RXA, an upright motorized microscope. The lens used was a 100⫻ numerical aperature 共NA兲 1.4 PlanApochromat with an internal iris. Because it is anticipated that colloidal engineering will produce structures with novel optical behavior across the visible and near-infrared portions of the spectrum 共e.g., photonic bandgap materials兲,9 it was desired to spectrally characterize the polarimetric capabilities of the DIC components across a wide spectrum. Following a brief description of DIC, detailed Mueller matrix calculations are given. Results of spectropolarimetric measurements of DIC components and the polarimetric application of DIC microscopy are presented, finishing with a measurement of a sample of interest. An error analysis of this measurement method is performed, and the results of DIC applied to spectropolarimetry are discussed.

2. Description of Differential Interference Contrast Microscopy

A brief description of how DIC works is presented here as background material. For detailed information, the reader is directed to the large body of research on the method, with a good overview presented in Ref. 2. DIC was introduced into microscopy by Nomarski in the 1950s and is a variant of wave-front shearing interferometry. When a wave front is interfered with a laterally shifted copy of itself, the resulting interference pattern appears as though a plane wave were interfering with two identical phase deformations, one positive and one negative. It is the positive and negative phase deformations that give rise to DIC’s relief appearance. DIC uses a lateral shear much less than the resolution limit of the microscope 共differential shear兲, which is also much less than the lateral dimensions of the object of interest. Thus only changes in the phase deformation in the object plane are recorded. The Leica microscope generates a sheared wave front using a linear polarizer and a Wollaston prism. As shown in this paper, the prism acts as a spatially varying linear retarder with a fast axis oriented 45° from the pass axis of the polarizer. A matched prism and de Se´ narmont compensator are located on the objective side to provide destructive interference with the plane-wave component, leaving only the phase deformations. The compensator consists of a linear polarizer and a linear retarder in series. The compensator can encode phase deformations in either intensity or color, depending on orientation. Proper implementation of DIC microscopy also requires that the various components be located at Fouriertransform planes 共Ko¨ hler illumination兲. This implies that the first Wollaston prism be located at the entrance pupil of the condenser lens, whereas the sample is located at the rear focal plane of the condenser lens. In the Mueller formalism, a DIC system takes the form

components are nondepolarizing. If we were to keep the system Mueller matrix as general as possible, the final matrix would be so complex as to be unusable. For the compensator oriented such that light passes first through the polarizer, then through the retardation plate with linear retardance ␦, the system matrix simplifies to

冤

1 ⫺cos ⌿ 1 ⫺cos ␦ cos ⌿ cos ␦ M sys ⫽ 0 0 4 ⫺sin ␦ cos ⌿ sin ␦

冤

1 0 Mw ⫽ 0 0

冤

1 1 1 1 1 Mp ⫽ 2 0 0 0 0

0 0 0 0

冥

0 0 . 0 0

(2)

An approximation that is justified in the visible range of the microscope is shown in Section 3. In addition, depolarizing effects from lenses of high NA10 are neglected. It is also assumed that the individual DIC

冥

0 0 , 0 0

(3)

冥

0 0 0 1 0 0 , 0 cos ⌿ sin ⌿ 0 ⫺sin ⌿ cos ⌿

(4)

and the second Wollaston prism is oriented in an opposite sense to the first. Note that the retardation ␺ is a spatially varying quantity. Justification for use of a linear retarder Mueller matrix for the Wollaston prism is presented in Subsection 3.C. Linear spatial offsets between the prisms effectively change the retardation of the Wollaston prism to ␺⬘ ⫽ ␺ ⫹ ␨. If the Mueller matrix of the sample is a constant, and there is no offset, the system matrix is identically zero. Furthermore, following Chipman,11 we can set the Mueller matrix of any homogeneous, nondepolarizing sample to

冤

冥

0 a b c a 0 ⫺d ⫺e ⫹I Ms ⫽ b d 0 ⫺f c e f 0

(1)

where Mp is the matrix for a linear polarizer, R共␪兲 is a rotation of angle ␪, MW共␺兲 is a Wollaston prism with retardation ␺, Ms is the sample, and Mc is the compensator plate. To clearly see the function of DIC, some simplifications to Eq. 共1兲 are made. For example, it is assumed that the polarizer matrix can be written as

0 0 0 0

冥

where the Mueller matrix for a Wollaston prism is given as

M sys ⫽ R共⫺90兲 M c R共90兲 R共⫺45兲 M w共⫺⌿兲 ⫻ R共45兲 M s R共⫺45兲 M w共⌿兲 R共45兲 M p,

冤

1 1 cos ⌿ cos ⌿ ⫻ Ms 0 0 sin ⌿ sin ⌿

0 ⫺sin ⌿ 0 sin ⌿ cos ␦ 0 0 0 sin ⌿ sin ␦

(5)

in which nonzero elements of the Mueller matrix indicate the presence of spatially homogeneous diattenuation and retardance. The terms have a physical interpretation as follows: • • • • • •

a, linear diattenuation oriented at 0° or 90°; b, linear diattenuation oriented at 45° or 135°; c, circular diattenuation; d, linear retardance oriented at 0° or 90°; e, linear retardance oriented at 45° or 135°; and f, circular retardance.

In this case, the system matrix is still identically zero. Only if the symmetry 共antisymmetry兲 of a, b, and c 共d, e, and f 兲 is violated, as is the case for spatially inhomogeneous materials, or if the system depolarizes incident light is the system Mueller ma1 January 2002 兾 Vol. 41, No. 1 兾 APPLIED OPTICS

39

trix nonzero. Let us represent this residual matrix as

冤

冥

0 ⌬a ⌬b ⌬c 0 0 ⌬d ⌬e , Ms ⫽ 0 0 0 ⌬f 0 0 0 0

(6)

which represents an inhomogeneous nondepolarizing diattenuator and retarder. Then the system Mueller matrix is

冤

z 1 ⫺z cos ␦ ⫺z M sys ⫽ 0 4 ⫺z sin ␦ ⫺z

冥

Fig. 1. Schematic of spectropolarimetric measurement.

z 0 0 cos ␦ 0 0 , 0 0 0 sin ␦ 0 0

z ⫽ ⌬c sin共⌿兲 ⫹ cos共⌿兲关⌬a ⫺ ⌬e sin共⌿兲兴.

(7)

If we allow the system to have a variable depolarization,

冤

冥

1 ⌬a ⌬b ⌬c 0 p ⌬d ⌬e . Ms ⫽ 0 0 q ⌬f 0 0 0 r

(8)

Then the system Mueller matrix is

冤

z 1 ⫺z cos ␦ ⫺z M sys ⫽ 0 4 ⫺z sin ␦ ⫺z

冥

z 0 0 cos ␦ 0 0 , 0 0 0 sin ␦ 0 0

z ⫽ 1 ⫹ cos共⌿兲关⌬a ⫺ p cos共⌿兲兴 ⫹ sin共⌿兲 ⫻ 关⌬c ⫺ ⌬e cos共⌿兲 ⫺ r sin共␺兲兴.

(9)

For the systems studied here, index-matched spheres and fluids, the diattenuation terms are approximately zero, the sample is nondepolarizing, and the only components of the sample residual matrix are those associated with retardance, that is,

冤

0 0 Ms ⫽ 0 0

冥

0 0 0 0 ⌬d ⌬e . 0 0 ⌬f 0 0 0

(10)

3. Spectropolarimetric Measurements

The system Mueller matrix is then

冤

⫺z 1 z cos ␦ z M sys ⫽ 0 4 z sin ␦ z

冥

⫺z 0 0 cos ␦ 0 0 , 0 0 0 sin ␦ 0 0

z ⫽ ⌬e sin共⌿兲cos共⌿兲.

(11)

Thus the detected intensity is proportional to only the linear retardance. A few words should be said regarding the terms cos共␺兲 and sin共␺兲. Because the first Wollaston prism is located at the entrance pupil of the condenser lens, rays that traverse the prism at a particular location traverse the sample at a particular angle. Thus the 40

detected intensity is not dependent on the position of the object, but is a function of the NA of the objective lens. Stopping down the microscope objective limits the NA and thus decreases the sensitivity of DIC. Finally, as shown in Subsection 3.B, the retardation angle of the compensator plate is highly wavelength dependent. Thus illumination wavelength could be used to measure different components of the system Mueller matrix, given a polarization-sensitive detector. It is perhaps unfortunate that, although the Mueller calculus is far more appropriate to describe this particular measurement than the Jones calculus because of use of square-law detectors, the Mueller matrix remains more difficult to interpret.11 Given that the detection method is currently polarization insensitive, many of the Mueller matrix components do not contribute to the detected signal, simplifying the analysis. However, because a complete measurement of the Stokes vector is not performed, this method can perform only relative measurements. For example, the extinction ratio can be measured but the maximum and minimum transmissions cannot. Relative retardance can be measured, but absolute retardance cannot. It is emphasized that, for the systems of interest in this paper, relative measurements are sufficient to characterize the samples. Furthermore, the series of flight experiments will be carried out with polarization-insensitive detectors only.

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Various spectropolarimetric measurements were made of the DIC components in a Leica DM RXA microscope. The DIC components were removed and characterized individually with a rotating sample method.12,13 These measurements were made over the useful wave band of a 100-W halogen bulb, approximately 400 nm to 1.2 ␮m. It should again be emphasized that the spaceflight experiments will probe the optical properties of colloidal crystals over as wide a wave band that can be provided by the instrument. The measurements were performed over the entire spectral range with a StellarNet spectrometer coupled to the light path by a light pipe as a detector 共see Fig. 1兲. The light pipe is an armored glass rod approximately 1.5 mm in diameter and was

supplied by Leica as part of a mercury bulb fiber encoupling kit. Use of a bare fiber tip as the detector restricts measurements to a small region of the DIC element, such that it has homogeneous properties, while at the same time it allows one to average over small-scale effects such as manufacturing defects. The source was a Leica halogen lamp, the output of which was first polarized with a Glan–Thompson polarizer from Karl Lambrect. The manufacturer specifies the extinction ratio of the Glan–Thompson prism over 320 –2300 nm to be 1 ⫻ 10⫺6, below the measurement limit of our laboratory equipment. Thus the various components were illuminated by essentially a perfectly linearly polarized input. When the sample was rotated and the transmitted intensity was measured, information regarding the transmissivity, the diattenuation, and the retardance values could be measured. Normalizing the data to the illumination light 共no sample present兲 eliminates polarization effects in the spectrometer that are due to the presence of a grating. Using the Mueller calculus, we can describe the measurement system by I d ⫽ M共␪兲I 0,

(12)

where Id is the detected intensity Stokes vector, M共␪兲 is the Mueller matrix of a particular DIC component oriented at an angle ␪ relative to the incident polarization, and I0 is the incident intensity Stokes vector. Because the light is initially perfectly linearly polarized along what we take as a principal direction, and the intensity measurement is polarization insensitive, the vector equation above reduces to

冢冣 冤

Id m共␪兲 00 m共␪兲 01 ␦ ␦ ␦ ⫽ ␦ ␦ ␦ ␦ ␦ ␦

␦ ␦ ␦ ␦

␦ ␦ ␦ ␦

冥冢 冣

1 1 , 0 0

(13)

where the components of the Mueller matrix are m共␪兲 00 ⫽ m 00, m共␪兲 01 ⫽ m 01 cos共2␪兲 ⫺ m 02 sin共2␪兲.

(14)

The symbol ␦ represents Stokes vector and Mueller matrix components that were not measured. The extinction ratio is defined by the maximum and minimum transmittance: E⫽

m 00 ⫺ m 01 . m 00 ⫹ m 01

(15)

For the components evaluated here, m02 is much smaller than m01.

冤

A.

Polarizer Measurements

The spectral transmittance and extinction ratio of the polarizer were measured and indicate that the assumption of a perfect linear polarizer breaks down starting at 750 nm. However, replacement of the stock Leica polarizer with, for example, a Glan– Thompson prism may extend the useful spectral range. B.

Compensator Measurements

This measurement is significantly complicated by the fact that the compensator plate has both a retarder and a polarizer sandwiched together. Furthermore, it is not clear what the retardance values are over the spectrum of interest, nor is it indicated how the polarizer pass axis aligns with the fast axis of the retarder. Consequently, several measurements were taken to characterize the compensator. The Mueller matrix for the compensator is given either by M c ⫽ M p R共⫺␥兲 M r R共␥兲

(16)

M c ⫽ R共␥兲 M r R共⫺␥兲 M p,

(17)

or

depending on the orientation of the compensator plate with respect to the optical axis. I assumed that the fast axis of the retarder is rotated an angle ␥ with respect to the pass axis of the polarizer. Note that the rotation angle of the retarder is reversed when the compensator plate is flipped. The term Mp in both Eqs. 共16兲 and 共17兲 should not be confused with the Mueller matrix for the polarizer in Section 2. Equation 共17兲 is simpler, because the retarder is effectively removed from the optical path given a polarization-insensitive measurement. To see this, we evaluate the Mueller matrix for Eq. 共17兲, assuming that the components are linear nondepolarizing components. The detected intensity is I d ⫽ M c00 ⫹ M c01 cos共2␪兲 ⫺ M c02 sin共2␪兲 ⫽ 1⁄ 2 关共␶ max ⫹ ␶ min兲 ⫹ 共␶ max ⫺ ␶ min兲cos共2␪兲兴,

(18)

which is independent of the retardance. Unfortunately, the orientation angle ␥ cannot be measured in this configuration. The Mueller matrix of the compensator in Eq. 共16兲 is

冥

a b共cos2 2␥ ⫹ cos ␦ sin2 2␥兲 b sin 2␥ cos 2␥共1 ⫺ cos ␦兲 ⫺b sin ␦ sin 2␥ 1 b a共cos2 2␥ ⫹ cos ␦ sin2 2␥兲 a sin 2␥ cos 2␥共1 ⫺ cos ␦兲 ⫺a sin ␦ sin 2␥ Mc ⫽ . 2 2 c sin ␦ cos 2␥ 2 0 c sin 2␥ cos 2␥共1 ⫺ cos ␦兲 c共cos 2␥ cos ␦ ⫹ sin 2␥兲 0 c sin ␦ sin 2␥ ⫺c sin ␦ cos 2␥ c cos ␦ 1 January 2002 兾 Vol. 41, No. 1 兾 APPLIED OPTICS

(19)

41

Fig. 2. Measured extinction values for the compensator plate.

Thus, as the compensator rotates, the detected intensity is given by I d ⫽ M c00 ⫹ M c01 cos共2␪兲 ⫺ M c02 sin共2␪兲 ⫻ 共␶ max ⫺ ␶ min兲cos 2␪ ⫺ cos 2␥ sin 2␥共1 ⫺ cos ␦兲 (20)

For proper DIC microscopy, the de Se´ narmont compensator encodes phase gradients as modulated intensity levels. This method would be most sensitive if the retarder is oriented at 45° to the pass axis of the polarizer. Leica has confirmed that the retarder is mounted in that fashion. Given a orientation angle ␥ near 45°, the modulation of the detected signal will be decreased for retardance values less than 90°. Id ⫽ 1⁄2 关共␶max ⫹ ␶ min兲 ⫹ cos ␦共␶ max ⫺ ␶ min兲cos 2␪兴.

(21)

The measured extinction ratio for the compensator in Eq. 共17兲 is shown in Fig. 2. This shows that the compensator plate would no longer act as a perfect polarizer past 750 nm as well. It is unclear at this time how this component could be modified to extend the spectral range of the microscope. Figures 3 and 4 present measured retardation values for Eq. 共16兲. Note that Figure 3 presents the absolute value of the retardation. In fact, the peaks at 390 and 800 nm are out of phase with the peak at 540 nm, shown clearly by the transmitted intensity curve in Fig. 4. Note that the peak at 1050 nm is in

Fig. 3. Absolute value of the retardation angle of the compensator plate. 42

APPLIED OPTICS 兾 Vol. 41, No. 1 兾 1 January 2002

phase with the 540-nm peak. This property allows us to map between color and positive or negative retardation gradients in the sample. C.

⫽ 1⁄ 2 关共␶ max ⫹ ␶ min兲 ⫹ 共cos2 2␥ ⫹ cos ␦ sin2 2␥兲 ⫻ 共␶ max ⫺ ␶ min兲sin 2␪兴.

Fig. 4. Spectral transmission relative to 540 nm at 90° compensator orientation.

Wollaston Prism Measurements

The Wollaston prisms in the DIC path introduce an angular shear between the orthogonal polarization components. This angular shear is in general wavelength dependent. In the microscope, the Wollaston prism is located at the entrance pupil to the condenser lens, which translates to a lateral shear between the two components at the sample. The entire concept of DIC rests on the fact that this lateral shear is below the resolution limit of the microscope. The angular shear introduced by either Wollaston prism was observed to be below the detection limit in the laboratory, which is approximately 1.5 ⫻ 10⫺2 deg. Alternatively, the Wollaston prism functions equivalently as a spatially varying retardation plate. Essentially, rays traversing the prism at various locations undergo various amounts of retardation, the result being a linearly varying retardation just after the prism. The Mueller matrix for a linear retarder located at 45° between crossed polarizers is

冤

1 ⫺ cos ⌿ 1 ⫺ cos ⌿ 1 ⫺1 ⫹ cos ⌿ ⫺1 ⫹ cos ⌿ M⫽ 0 0 4 0 0

0 0 0 0

冥

0 0 , 0 0

(22)

where the retardation ␺ is a one-dimensional function across the prism. Thus, by imaging the Wollaston prism between crossed polarizers, we can measure the retardation as a function of location. This is most conveniently done in the microscope rather than on the benchtop. An image of the Wollaston prism located above the objective lens between crossed polarizers, with a reticle located at the back pupil plane of the condenser lens, is shown in Fig. 5. This image was taken by the Bertrand lens, which images the Fourier-transform plane of the sample plane 共conoscopic view兲. Thus the Bertrand lens images the entrance pupil of the condenser lens, the aperture stop plane, and an intermediate source plane, at which is located a diffuser plate. The diffuser plate will most likely be removed from the flight instrument. Note that the camera field of view is

Fig. 6. Geometry of a sphere in fluid.

where h共 y兲 is Fig. 5. Image of a Wollaston prism between crossed polarizers.

h共 y兲 ⫽ 共r 2 ⫺ y 2兲 1兾2,

(24)

and thus the retardation gradient is slightly overfilled by the image and also that the contrast of the image was changed for this paper. We can see that the retardation varies by a one-half period 共⫹90° to ⫺90°兲 over the size of the pupil. This is shown in Fig. 5 by the inset graph of intensity values taken on a line across the prism. It is also clear how, when the NA is limited, the sensitivity of DIC is decreased because the illumination field is restricted to a smaller portion of the retardation. The uneven illumination levels are due to the uneven output of the halogen lamp filament, located at 45° relative to the Wollaston prism axis. Now it is clear how the various components work together in terms of polarization. A polarizer first orients the illumination polarization 45° relative to a Wollaston prism. The first Wollaston prism encodes a linear retardation at the entrance pupil of the condenser lens, whereas the second prism located above the objective lens removes the linear retardation. Offsets between the prisms change the dc retardation level, controlling the background illumination level. Inhomogeneities in the retardation of the sample can be color coded by the compensator plate, with positive retardations tending toward blue and negative retardations tending toward red.

y ⳵␾ y , ⫽ 2␦n 2 ⳵y 共r ⫺ y 2兲 1兾2

(25)

which is proportional to the intensity for DIC microscopy. The camera used to take the images is a three-chip CCD 共Optronics兲 with all postprocessing 共contrast, sharpen兲 turned off and the exposure time set to manual. The exposure time was fixed by the calibration images and not changed for the test images. The response of the camera 共gamma兲 was set to unity for linear operation. Only the luminance signal of the camera was used in the analysis. A sample image is shown in Fig. 7. Defects in the spheres are clearly visible; note how the shading within the defect is reversed from the normal shading. Two images were taken, one of 2-␮m mean diameter borosilicate glass microspheres 共Duke Scientific Corp.兲 in a bath of microscope immersion oil 共Cargille type DF兲 and another with glass microspheres in a bath of Cargille type FF immersion oil. The samples were illuminated with quasi-monochromatic light 共␭ ⫽ 546 ⫾ 5 nm兲 through use of a narrow-bandpass filter 共Chroma兲. The indices of refraction, as provided by the manufacturers, are n ⫽ 1.5179 共DF兲, n ⫽

4. Analysis of Differential Interference Contrast Images

As an application of this method, images of glass microspheres in various immersion oils and PMMA spheres in an index-matched mixture of decalin and tetralin were taken. The sizes of the spheres were known, and in the first case the indices of refraction were known as well. The geometry is shown in Fig. 6. The phase retardation experienced by a ray traveling in z is ␾ y ⫽ 2关nr ⫹ ␦nh共 y兲兴,

(23)

Fig. 7. DIC image of glass microspheres in Cargille type FF immersion oil. 1 January 2002 兾 Vol. 41, No. 1 兾 APPLIED OPTICS

43

Table 1. Indices of Cargille Fluids at Various Wavelengths

Wavelength 共nm兲

Fluid 1

Fluid 2

Fluid 3

450 546 620

1.569 1.554 1.548

1.524 1.513 1.508

1.469 1.462 1.459

sphere, rather than a more shape-dependent quantity. 5. Error Analysis of Method Fig. 8. Calibration data.

1.4810 共FF兲, and n ⫽ 1.56 共glass兲. Note that the index values of the immersion oils are at 546 nm, whereas the index of the glass was measured by Duke Scientific using index fluids at 589 nm. A graph of intensity versus ⳵␾y兾⳵y gives the constant of proportionality. The graph for both fluids is shown in Fig. 8. With this system calibration, images of PMMA spheres in a nearly index-matched bath of Decalin and tetralin were then analyzed. This material is representative of samples that will be flown in space. The system is nearly index-matched to minimize the sphere–sphere interactions in the fluid, which is crucial to perform a correct experiment. We can calculate the unknown index mismatch by graphing the intensity versus 2k 关 y兾共r2 ⫺ y2兲1兾2兴, where k is the constant of proportionality calculated above. This graph is shown in Fig. 9. From the interpolated line, the index mismatch is calculated to be ⌬n ⫽ 4.0 ⫻ 10⫺3. It should be noted that the diameter of a particular sphere does not affect the quality of this measurement through use of the above normalization. Thus, although the absolute intensity gradient differs from large and small spheres, the scaled intensity gradients all lie on the same curve. In fact, the nonsphericity of the particles does not affect the measurement either—the specification of the glass beads was 80% sphericity. This is because the scaling of the data occurs along a line through the center of the micro-

In an effort to determine the efficacy of the method across a wide spectral range, as well as perform an error analysis, the glass spheres were immersed in a series of index fluids 共Cargille兲 and images were taken at three wavelengths, corresponding to various amounts of retardation of the compensator plate. The index of refraction of the fluids is given in Table 1. The index at 546 nm was provided by the manufacturer, as was the Cauchy equation that was used to calculate the index at the other wavelengths. The data were used to determine the index of refraction of the glass spheres at different wavelengths. This was done by use of the slope of the fit line at various wavelengths, given in Table 2. Each fit line represents an average of four lines taken through different spheres of different sizes. There was no correlation between sphere diameter and slope of the fit line. What is immediately apparent is the role of the retardation angle in the generation of image contrast. At 546 nm, the image contrast is much larger than for 450 or 620 nm. The index of refraction for the glass beads is found by ratios:

冉

冊

slope1 slope1 ⫺ 1 n glass ⫽ n fluid2 ⫺ n fluid1. slope2 slope2

(26)

The values obtained for the refractive index are listed in Table 3. These values agree well with the data given by Duke Scientific, tending to overestimate the index by at most a few percent. At 546 nm, the method agrees with the manufacturer by better than 1%. In addition, the precision of this method is extremely good. Thus DIC microscopy has been shown to be a fairly accurate and precise quantitative diagnostic tool. The primary deficiency at this point is the lack of temperature control. Although the laboratory has nominal climate control, the temperature varies on a day-to-day basis. However, the measurements preTable 2. Slopes of the Fit Line for Glass Microspheres in Cargille Fluids

Fig. 9. Index mismatch measurement. 44

APPLIED OPTICS 兾 Vol. 41, No. 1 兾 1 January 2002

Wavelength 共nm兲

Fluid 1

Fluid 2

Fluid 3

450 546 620

1.62 3.4 2.58

8.75 13.2 7.1

14.5 23.5 12.3

Table 3. Calculated Index of Refraction of Glass Microspheres

Wavelength 共nm兲

nglass

450 546 620

1.589 ⫾ 0.016 1.572 ⫾ 0.014 1.572 ⫾ 0.004

sented here occurred over a short time period, when the temperature can be considered constant. 6. Summary and Conclusions

DIC microscopy has been shown to provide a sensitive method to quantitatively measure spatially varying retardations in microscopic samples. A direct application is the measurement of small differences in the index of refraction between a colloidal particle and the fluid bath. The advantage of this method over use of a series of index fluids is that, at most, few liquids need be used, saving time and equipment. Mueller matrix calculations show that DIC could also be used to probe samples that are diattenuators, but that DIC is not sensitive to circular retardance 共optical activity兲. In addition, materials with spectrally varying polarimetric properties, such as photonic bandgap materials, could be probed with a microscope. The advantages of one using a microscope to perform polarimetric measurements are that small quantities of samples are required, and physical structure can often be directly visualized. It is hoped that this method proves to be a highly useful probe of optical properties of materials. The author thanks John Eustace for helpful discussions and Eric Weeks for providing the Physics of Colloids on Space 2 共PCS-II兲 samples.

References and Notes 1. S. Inoue and K. R. Spring, Video Microscopy: The Fundamentals 共Plenum, New York, 1997兲. 2. M. Pluta, Specialized Methods, Vol. 2 of Advanced Light Microscopy 共Elsevier, New York, 1989兲, Chap. 7, pp. 146 –196. 3. S. Inoue´ , “Ultrathin optical sectioning and dynamic volume investigation with conventional light microscopy,” in ThreeDimensional Confocal Microscopy: Volume Investigation of Biological Specimens J. K. Stevens, L. R. Mills, and J. E. Trogades, eds. 共Academic, London, 1994兲, Chap. 17, pp. 397– 419. 4. The experiments are Physics of Colloids in Space 2 共PCS-II兲, Physics of Hard Spheres Experiment 2 共PHASE-II兲, and Low Volume Fraction Entropically Driven Colloidal Assembly 共L␾CA兲. 5. Z. G. Yu, X. Song, and D. Chandler, “Polarizability fluctuations in dielectric materials with quenched disorder,” Phys. Rev. E 62, 4698 – 4701 共2000兲. 6. B. van Tiggelen and H. Stark, “Nematic liquid crystals as a new challenge for radiative transfer,” Rev. Mod. Phys. 72, 1017–1039 共2000兲. 7. See, for example, B. Berne and R. Pecora, Dynamic Light Scattering 共Krieger, Malabar, Fla., 1990兲. 8. See, for example, B. Dahneke, Measurement of Suspended Particles by Quasi-Elastic Light Scattering 共Wiley, New York, 1983兲. 9. See, for example, J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light 共Princeton U. Press, Princeton, N.J., 1995兲. 10. P. To¨ ro¨ k, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300 –315 共1998兲. 11. R. Chipman, Polarimetry, in Handbook of Optics, M. Bass, ed. 共McGraw-Hill, N.Y., 1995兲, Vol. 2, Chap. 22. 12. See, for example, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light 共Elsevier Science, Amsterdam, 1996兲. 13. D. B. Chenault, “Infrared spectropolarimetry,” Ph.D. dissertation 共University of Alabama, Huntsville, Ala., 1992兲.

1 January 2002 兾 Vol. 41, No. 1 兾 APPLIED OPTICS

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