USING STANDARD PRA S

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J. Opt. Soc. Am. A / Vol. 16, No. 9 / September 1999

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Spectrally resolved white-light interferometry for measurement of ocular dispersion Daniel X. Hammer and Ashley J. Welch Department of Electrical and Computer Engineering, University of Texas, Austin, Texas 78712-1084

Gary D. Noojin, Robert J. Thomas, and David J. Stolarski TASC, 4241 Woodcock Drive, Suite B-100, San Antonio, Texas 78228-1330

Benjamin A. Rockwell U.S. Air Force Research Laboratory, Human Effectiveness Directorate, Directed Energy Bioeffects Division, Optical Radiation Branch, 8111 18th Street, Brooks Air Force Base, Texas 78235-5215 Received November 6, 1998; revised manuscript received February 22, 1999; accepted April 26, 1999 Spectrally resolved white-light interferometry was used to measure the wavelength dependence of refractive index (i.e., dispersion) for various ocular components. Verification of the technique’s efficacy was substantiated by accurate measurement of the dispersive properties of water and fused silica, which have both been well-characterized in the past by single-wavelength measurement of the refractive index. The dispersion of bovine and rabbit aqueous and vitreous humors was measured from 400 to 1100 nm. In addition, the dispersion was measured from 400 to 700 nm for aqueous and vitreous humors extracted from goat and rhesus monkey eyes. An unsuccessful attempt was also made to use the technique for dispersion measurement of bovine cornea and lens. The principles of white-light interferometry, including image analysis, measurement accuracy, and limitations of the technique, are discussed. In addition, alternate techniques and previous measurements of ocular dispersion are reviewed. © 1999 Optical Society of America [S0740-3232(99)01409-X] OCIS codes: 260.2030, 290.3030, 170.4460, 120.3180, 170.3660.

1. INTRODUCTION Recently a technique known as spectrally resolved whitelight interferometry (SRWLI) was developed to make highly accurate, real-time measurements of the dispersive properties of transparent media.1–4 White-light interferometry was also used to measure the dispersion of dielectric multilayer structures.5 Dispersion is the physical property of a substance that describes the wavelength dependence of its index of refraction. It is present in all optical elements that use the refraction of light for imaging. Dispersion leads to chromatic aberration. The simplest example is a dispersive glass lens that has a wavelength-dependent focal length. Longitudinal (or axial) chromatic aberration becomes apparent when the lens focuses broadband white light: The focal points of different wavelengths are spread along the optical axis. The importance of refractive-index measurements cannot be underestimated. The complex index is related to the absorptive and scattering properties of a medium.6 Measurement of dispersion can also allow one to infer information about other optical and electrical material properties7 or other physical quantities, such as pressure, density, stress, or concentration. Measurement of the temperature dependence of refractive index can aid in the measurement of thermal phenomena, for example, thermal lensing due to focused laser pulses. Moreover, light propagation models must incorporate dispersion in their calculations to permit accurate determination of the be0740-3232/99/092092-11$15.00

havior of light. This is especially important in predicting the propagation of light through ocular media and the focusing of laser pulses on the retina. Thus analysis of laser safety data depends on an accurate measurement of dispersion for the major components of the eye. The refractive index of transparent liquids or solids can be measured by several different techniques. Some of the methods are based on refractometry, in which the refractive index is determined by Snell’s law from the angle at which a beam of light exits an optical system that includes the sample. With the minimum-deviation method the refractive index is found from the prism apex angle and by precise measurement of the minimum deviation the beam makes upon traversing the prism. A hollow dispersing prism can be constructed with optical windows to measure the refractive index of liquids or gases. The minimum-deviation method is best suited for visible light, since the minimum angle must be located (with a telescope) while the rotation of the prism is adjusted. The Pulfrich refractometer is a variation of this technique by which the deflection angle of light passing through a V-shaped prism with the liquid sample placed inside the inclined faces of the prism is measured.8 Other methods of refractometry include the constant-deviation method9 and the critical-angle method.10 All the above methods operate under a limitation apart from the restriction of source wavelength: Knowledge of prism index (or index of other optical elements in the system) is required. A © 1999 Optical Society of America

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technique involving knife-edge measurement of the displacement of a beam impinging on a cuvette at an oblique angle has been developed to circumvent both restrictions.11 A second approach to measurement of refractive index is interferometry, in which the number of fringes or the phase evolution is detected as the arms of an interferometer are moved precise distances. Several different types of interferometers have been used, including Michelson,12 Fabry–Perot,13 and Mach–Zehnder.14 Recently several novel optical arrangements have been developed for measurement of the absolute refractive index with little prior knowledge of the sample, for example, without knowledge of thickness15 or expansion coefficient.16 Moreover, systems have been constructed with multiple interferometers,17 fiber optics,18 and diffraction gratings.19 Spectrally resolved white-light interferometry has many advantages over the techniques described above. SRWLI eliminates the need to use many lasers or many wavelength lines to make single-point measurements of the refractive index over the entire visible and nearinfrared spectrum. It can be used to make measurements anywhere in the electromagnetic spectrum, provided that the grating and the detector are sensitive, the sample is transparent, and the source emits a broad spectrum in the region of interest. SRWLI makes a single, real-time measurement of the differential index across a band of wavelengths whose range and resolution depend on the grating of the spectrometer. Also, the technique removes the need for complex analyses or nonlinear interpolation and extrapolation to predict the refractive index for wavelengths for which measurements have not been made.20–22 Last, measurement of the refractive index with SRWLI can be accomplished without any knowledge of the sample characteristics other than thickness. For obvious reasons the bulk of previous measurements of refractive index in liquids were made on water.23,24 Querry et al. highlighted the most important measurements of this century.24 The temperature dependence of the refractive index of water has also been measured extensively. Tilton and Taylor used the minimumdeviation method with hollow prisms to determine the refractive index of distilled water in the wavelength range 404.66–706.52 nm and the temperature range 0°–60 °C.25 Relatively few measurements of refractive index have been reported for ocular media across the entire spectrum of visible and near-infrared wavelengths. The reason for this is that even a single index measurement at one wavelength can require considerable instrumentation. Moreover, measurement of refractive index is much more difficult for tissue samples such as lens or cornea than for other solids or liquids. Adding to this difficulty, the lens of the eye is a gradient-index structure, with the index decreasing sharply with increasing distance from the optical axis. Several classical texts on the optics of the eye, when discussing computation of the reduced focal length or positions of the principal points of the eye, list refractive-index values for ocular media.26,27 These values are usually assumed to be at 589.3 nm (Fraunhofer sodium D line). This is also true for papers on comparative ophthalmology, in which different species of animals are contrasted.28,29 Le Grand listed index calculations

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based on Cornu’s formula and a few preliminary measurements near the beginning of this century.30 Sivak and Mandelman, who measured the index31 and computed the chromatic aberration32 for several species including man, compiled the most extensive tabulation of refractive indices to date. Other species and media measured include the four major ocular layers (i.e., cornea, aqueous humor, lens, and vitreous humor) of rat33 and rabbit34 and the lenses of porpoise21 and fish.35 In addition, Palmer and Sivak have measured the dispersion of human, rat, rabbit, and pigeon lens.36 In several studies of the human eye, both the transverse37 and the longitudinal (or axial)38 chromatic aberrations have been measured, although direct measurement of ocular dispersion was not accomplished. The refractive indices of several types of highly scattering tissues, such as human dermis, have also been measured.39,40 Ocular dispersion data are particularly sparse for nearinfrared wavelengths. Most recently Drexler et al. measured the group refractive index of ocular media at 814 and 855 nm.41 However, this study based some of its calculations on the dispersion of water rather than on the actual measurement of ocular dispersion. As can be seen, few studies have been completed that are entirely devoted to the measurement of the refractive index in ocular media. The aim of this research was to remedy that deficiency.

2. EXPERIMENTAL SETUP A. Optical Arrangement White-light interferometry requires an optical layout that includes a broadband white-light source, a traditional Michelson interferometer, and an imaging spectrograph with a CCD camera. The experimental setup used in this study is shown in Fig. 1. For the visible data a 75-W xenon arc lamp was used (Oriel Inc., Model 6263). This lamp produced a relatively intense beam of light. However, in the infrared region, the arc lamp produced several sharp spikes in the emission spectrum. These spikes appeared in the interferogram and prevented measurement of the phase. For the infrared data a 100-W quartz-tungsten-halogen (QTH) lamp was used (Oriel Inc., Model 6333). The QTH lamp was less intense than the xenon arc lamp, but it provided a relatively constant spectral curve. The Michelson interferometer shown in Fig. 1 consisted of two 1-in. aluminum mirrors, M4 and M5, and a beamsplitting cube. The interferometer arms consisted of mirrors mounted on translation stages. A sample cuvette was placed in one arm, and an empty cuvette was placed in the other arm to cancel the effect of the cuvette material dispersion. The cuvettes were made of quartz, which has high transmittance from 400–1100 nm. The fixed arm, A2, consisted of a mirror mounted on a standard micrometer translation stage. Once gross alignment was accomplished and fringes became apparent on the image monitor, this arm was not moved during the measurement. The variable arm, A1, of the interferometer consisted of a mirror mounted on a stage with a microprocessor-based motion controller with 100-nm resolution (Aerotech, Inc.). The cube was used in lieu of a

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Fig. 1. Optical layout for white-light interferometry. A1, variable arm; A2, fixed arm; M1, dielectric-coated (broadband visible) mirror; M2–M5, aluminum mirrors; BS, fused silica beam splitter; I1–I4, iris apertures (I4 was used for initial alignment, with M4 removed); S1 and S2, sample and empty cuvettes; L1, ⫺100-mm focal-length lens; L2, 750-mm-focal-length lens; L3, interferogram lens (several types were used); ND, neutral-density-filter stack; LP, 750-nm long-pass filter for infrared measurements; DMA, direct memory access.

parallel-plate beam splitter and compensating plate. Light traveled through the same thickness of glass (two complete passes through the cube) regardless of which arm it traveled through. A He–Ne laser was aligned collinear to the white light to ensure that the interferometer mirrors were normal to each other (i.e., there was not a wedge of air in one of the paths). Owing to the laser’s long coherence length (⬃10 cm or greater), He–Ne fringes were readily visible even when the optical path lengths of the arms were unequal. A lens, L3, and neutral-density filters, ND, were placed at the output of the interferometer to adjust the intensity and spatial extent of light that entered the spectrograph. A long-pass filter, LP, was placed between the interferometer and spectrograph to block second-order visible wavelength lines produced by the grating when near-infrared spectra were obtained. The filter blocked all wavelengths of light below 750 nm, and its average transmittance from 750 to 2000 nm was 85%. Optics placed after the interferometer had no effect on the dispersion measurement. Polychromatic light incident on the entrance slit of the quarter-meter imaging spectrograph (Chromex, Inc.) was the incoherent superposition of the fringe patterns produced by the interference of each monochromatic component within the spectrum of the white-light source. The spectrograph separated the intensity by wavelength, and the 12-bit, 512 ⫻ 512–array CCD camera (Hamamatsu,

Fig. 2. Portions of images of white-light fringes: (a) nondispersive sample, (b) fused silica, and (c) illustration of stationaryphase position (arrow).

Inc.) recorded an interference image (called an interferogram). Several examples of interferograms are shown in Fig. 2. The image was acquired by use of Hamamatsu software and was displayed on the image monitor. Since the largest sources of vibration on the surface of the optical table were the fans in the camera, the camera was isolated on a small platform attached to the base of the table. An interferogram has spatial dependence (the slit

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length) along the vertical axis (y axis) and frequency (or wavelength) dependence along the horizontal axis (␭ axis). The wavelength of each image was calibrated by use of between 2 and 13 lines from rare-gas spectral calibration lamps (Oriel, Inc., krypton, neon, and mercury lamps). A Nd–YAG laser also provided calibration for some of the infrared images. The imaging spectrograph has three gratings, all of which were used in this study: G1, 600 grooves/mm blazed at 500 nm; G2, 600 grooves/mm blazed at 1 ␮m; and G3, 100 grooves/mm blazed at 450 nm. G1 and G3 were used exclusively for visible data, whereas G2 was used exclusively for infrared data. Only high-resolution gratings were used to determine phase functions. The low-resolution grating was used infrequently to identify and locate the stationary-phase position. However, the two types of grating gave nearly identical results when they were used on the same wavelength region. The measured spectral window width of the 600-grooves/mm gratings was 62 nm (0.12 nm/pixel). The spectral window width of the 100-grooves/mm grating was 364 nm (0.71 nm/pixel). The effect of coma is minimal below a wavelength of 1500 nm for G1 and G2 and 6 ␮m for G3. Moreover, it was found that the response of the gratings was linear with wavelength (i.e., with increasing wavelength, the corresponding pixel number increased linearly) across the spectrum in which they were used.

White-light interferometry allows the calculation of dispersion because the wave number–dependent optical delay with a dispersive sample in one of the arms is a function of the wave number–dependent refractive index. In this case, the relationship between optical delay (also known as optical path-length difference) ⌬ and the index of refraction can be expressed as

B. Sample Preparation Freshly recovered eyes were obtained either from a slaughterhouse or from populations of research animals (see note following Acknowledgments). All measurements were completed within several hours after tissue dissection. In addition to ocular media, measurements were performed on high-purity triply distilled water (18 M⍀/cm) and fused silica optical flats (Reynard Enterprise, Inc., Model 1668), with specified flatness not greater than ␭/10 over 25 mm.

Thus the phase of white-light fringes is dependent on wave number in the same manner that the phase of monochromatic light fringes (produced with a He–Ne laser, for example) is dependent on path-length difference. In other words, the appearance of fringes at the output of the spectrograph occurs because the monochromatic components of the white light interfere with different phase with respect to each other. Equations (3) and (4) also show that if a sample is nondispersive (i.e., the index of refraction is constant with respect to wave number), then ⌬ will be constant and the phase will vary linearly with respect to wave number [see Fig. 2(a)]. We can now derive an expression that will allow phase calculation from the interferogram. Irradiance is proportional to the time average of the magnitude of the electric field intensity squared. The total irradiance at a given point in space that is due to the superposition of the electric field intensity from two sources emitting parallel waves of the same frequency is

C. Image Analysis Once an image was acquired with the setup described in Subsection 2.A, the images were processed by a LabVIEW software virtual instrument (National Instruments, Inc.). The following image analysis algorithm is based on the work of Saínz et al.3 The goal of this research was to obtain the dispersion of ocular media. Dispersion may be expressed as the differential index of refraction: ⌬n ⫽ n 共 ␴ 兲 ⫺ n 共 ␴ 0 兲 ,

(1)

where n is the index of refraction (phase), ␴ is the wave number in a vacuum (wave number is the reciprocal of wavelength, ␴ ⫽ 1/␭), and n( ␴ 0 ) is the index at an arbitrary known wave number. Alternatively, dispersion may be expressed as higher-order terms in a Taylor-series expansion of the refractive index about the center index, n 0 ⫽ n( ␴ 0 ):

n共 ␴ 兲 ⫽ n0 ⫹

dn d␴

共␴ ⫺ ␴0兲 ⫹

1 d2 n 2 d␴

共␴ ⫺ ␴0兲 . 2 2

(2)

⌬共 ␴ 兲 ⫽ d关 n共 ␴ 兲 ⫺ 1 兴 ⫺ S0 ,

(3)

where d is the thickness of the sample and S 0 is the initial path difference before a sample is introduced into the interferometer. S 0 is equivalent to the distance between the stationary-phase position without a sample and the acquired image position (measured by the stagealignment virtual instrument that controls the variable arm, A2, of Fig. 1). When light passes through the spectrograph and is dispersed by the grating, the irradiance of each monochromatic component of the white light is displayed on the CCD camera. Thus the irradiance at the output of the spectrograph is modulated with respect to wave number. The phase function of the modulation is proportional to the optical delay, which in turn depends on the index of refraction. The relationship between the phase of the white-light fringes and the optical delay is

␾ 共 ␴ 兲 ⫽ 4 ␲␴ ⌬ 共 ␴ 兲 ⫽ 4 ␲␴ 兵 d 关 n 共 ␴ 兲 ⫺ 1 兴 ⫺ S 0 其 .

I ⫽ I 1 ⫹ I 2 ⫹ 2 冑I 1 I 2 cos ␾ ,

(4)

(5)

where I 1 and I 2 are the irradiances of the two sources and ␾ is the phase difference resulting from the path difference between the sources.42 The irradiance is maximum when the phase difference between sources is zero (i.e., when cos ␾ ⫽ 1) and the waves completely overlap. The irradiance is minimum when the phase difference is 180° (i.e., when cos ␾ ⫽ ⫺1) and the peaks of one source overlap the troughs of the other source. The total irradiance for these two extremes can be expressed as I max ⫽ I 1 ⫹ I 2 ⫹ 2 冑I 1 I 2 ,

(6)

I min ⫽ I 1 ⫹ I 2 ⫺ 2 冑I 1 I 2 .

(7)

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Equations (5)–(7) can be combined to yield a compact expression for the irradiance: I ⫽ I 0 共 1 ⫹ ␥ cos ␾ 兲 ,

sult with Eq. (12). Alternatively, Eqs. (12) and (4) may both be differentiated with respect to ␴ and compared. This yields

(8) A ⫽ 2␲d

with I0 ⫽

␥⫽

I max ⫹ I min 2 I max ⫺ I min I max ⫹ I min

,

(9)

,

(10)

where ␥ is known as the fringe visibility or modulation. In an interferogram produced by white-light interferometry, the irradiance, and hence the phase, is a function of wave number at every point. The phase function of a single line of an interferogram 关 ␾ img( ␴ ) 兴 can be computed by cosine inversion of Eq. (8):

␾ img共 ␴ 兲 ⫽ cos⫺1

冋

2I 共 ␴ 兲 ⫺ 共 I max ⫹ I min兲 I max ⫺ I min

册

,

(11)

where I max and I min are the irradiance at a local maximum and minimum, respectively. There are two ways to compute the phase function. The first is to acquire three images (I, I 1 , I 2 ) and insert the values into Eq. (5). The second is to acquire one image and with the use of Eq. (11) compute the phase for each segment of points between local minima and maxima. The two methods are equivalent, and as the second technique requires acquisition of only one image, it was the preferred method. Since the phase at a fringe peak is equal to an integer multiple of ␲, the segments between local minima and maxima can be joined with an additive phase correction (i.e., 0, ␲, 2␲, ...) to produce the phase function for an entire interferogram line. The first maximum in the region where the phase is calculated is set to zero. For this reason, and because we have assigned an arbitrary integer value to the multiplicative factor of ␲, the resultant phase function should be thought of as the relative phase and not the true phase. However, the relative phase can be corrected to the true phase. As will be shown, this is accomplished by application of a polynomial fit to the measured phase function and determination of the relationship between fit coefficients and optical delay. Owing to the shape of the phase curve as a function of wavelength for all media reported in this paper, it was found that a third-order polynomial fit achieved accurate results for wavelengths below 550 nm, while a secondorder polynomial fit achieved accurate results for wavelengths greater than 550 nm (as determined by the mean squared error of the fit). For the remainder of the analysis, a third-order polynomial fit will be used. Similar expressions may be derived for a second-order fit. The fit may be expressed as

␾ fit共 ␴ 兲 ⫽ A ␴ 3 ⫹ B ␴ 2 ⫹ C ␴ ⫹ D,

(12)

where A, B, C, and D are the fit coefficients, which are related to the higher-order dispersion terms of Eq. (2). This relation can be found by substitution of the righthand side of Eq. (2) into Eq. (4) and comparison of the re-

B ⫽ 4␲d

d2n d␴ 2

冉冉

dn d␴

,

(13)

⫺␴

C ⫽ 4␲d n ⫺

S0 d

d2n d␴ 2

冊

,

(14)

⫺1⫺␴

dn d␴

⫹

␴ 2 d2n 2 d␴ 2

冊

.

(15)

(This step in the analysis involves an approximation. Either a fixed number of terms in the Taylor-series expansion are used or the higher-order terms, e.g., d3n/d␴ 3 and higher, are discarded in the differentiation.) As mentioned previously, the phase measured from the interferogram must be adjusted to the actual phase before the dispersion can be calculated. This is done by selecting a point in the phase curve that can be corrected to the actual phase and then adjusting the rest of the phase curve accordingly. This point is the center index [n 0 in Eq. (2)] with optical delay ⌬ 0 . (The subscript zero attached to any variable always refers to the value at this center wave number.) Since the phase is known at the minima and maxima of the interferogram, a peak became the logical choice for the center wavelength. The phase at a peak (minimum or maximum) is

␾共 ␴0兲 ⫽ m␲,

m ⫽ 0, 1, 2, ... .

(16)

Therefore when the multiplicative factor m is found, the true phase function can be calculated. The first step is to find the relationship between the variables in the phase equations [Eqs. (4), (12), and (16)]. By use of Eqs. (3), (13), (14), and (15), an expression emerges that relates the fit coefficients to the optical delay at the center wave number ⌬ 0 : A ␴ 0 2 ⫹ B ␴ 0 ⫹ C ⫽ 4 ␲ 共 dn 0 ⫺ d ⫺ S 0 兲 , ⫽ 4␲⌬0 .

(17) (18)

Equating the two expressions for the phase at the center wavelength [Eqs. (4) and (16)], and using Eq. (18) for the optical delay at the center wave number, we obtain the following expression: m ⫽ 4 ␴ 0⌬ 0 ⫽

A ␴ 03 ⫹ B ␴ 02 ⫹ C ␴ 0

␲

.

(19)

After m is calculated it is rounded to the nearest integer (m ⬘ ) to correspond to an irradiance peak, and the optical delay is recalculated: ⌬ 0⬘ ⫽

m⬘ 4␴0

.

(20)

Once the optical delay at the center wave number is known, the phase curve can be adjusted. Two additive corrections must be applied. First, since the center wavelength was chosen to correspond to a peak that will lie somewhere on the phase curve and the phase curve was arbitrarily chosen to begin at zero, the phase calcu-

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lated from the interferogram [Eq. (11)] must be corrected by the center phase, ␾ img( ␴ 0 ). This is equivalent to correcting the fit phase by the intercept term (i.e., D). Second, the phase at the center wave number must be corrected to the actual value of the phase calculated by Eq. (4). The corrected phase thus becomes

␾corr(␴)⫽␾img(␴)⫺␾img共 ␴ 0 兲 ⫹ 4 ␲␴ 0 ⌬ 0⬘ .

(21)

When the sample in the interferometer is nondispersive or when no sample is placed in the interferometer, the optical delay at every wave number is equal to the optical delay at the center wave number. Therefore the phase in the nondispersive case is

␾ nd共 ␴ 兲 ⫽ 4 ␲␴ ⌬ 0⬘ ⫽ m ⬘ ␲

冉冊 ␴

␴0

,

(22)

with Eq. (20) used for the optical delay. Finally, the differential index can be calculated from the corrected phase of Eq. (21) and the nondispersive phase of Eq. (22). Inserting these terms into Eq. (4) and subtracting yields ⌬n ⫽ n ⫺ n 0 ⫽

␾ corr共 ␴ 兲 ⫺ ␴ nd共 ␴ 兲 4 ␲␴ d

.

Fig. 3. Measurement of the refractive index of fused silica. Dashed curve, calculation based on a three-term Sellmeier dispersion formula (Ref. 43); solid curve, compilation of index curves obtained from several interferograms.

(23)

For the majority of ocular media reported in this paper, there were no previous measurements of refractive index with which to compare our measurements. However, it is possible to calculate the absolute refractive index from the interferogram with Eqs. (3) and (20) if the thickness of the sample is known precisely. Thus the absolute refractive index can be calculated as n0 ⫽ 1 ⫹

⌬ 0⬘ ⫹ S 0 d

.

(24)

One final correction had to be applied to the absoluteindex curves. Since there were systematic errors in the measurement of the center index (see Subsection 3.C), an additive correction was applied to the index curves. The final correction was that which minimized the difference between adjacent segments or previous measurements (see Figs. 3–5 below).

3. RESULTS A. Reference Media Since measurements of dispersion and refractive index of fused silica and water with other techniques have been abundantly reported in the literature, we measured these two materials to illustrate the accuracy and precision of the SRWLI technique. The accuracy of differential-index measurement should be distinguished from the accuracy of absolute-refractive-index measurement. The former depends solely on the image obtained and the subsequent analysis. The latter depends on the measurement accuracy of the interferometer mirror positions, among other things. Accuracy and error analysis will be presented more thoroughly in Subsection 3.C. Measurement of the refractive index of fused silica and water for the visible and near-infrared spectral regions is shown in Figs. 3 and 4, respectively.

Fig. 4. Measurement of the refractive index of water. Points represent values obtained from previous measurements.15,24,25,44 Solid curve, compilation of index curves obtained from several interferograms.

Malitson measured the refractive index of opticalquality fused silica for 60 wavelengths from 0.21 to 3.71 ␮m by the minimum-deviation method.43 He fitted the measurements to a three-term Sellmeier dispersion equation. Comparison of the measurements by SRWLI and those made by Malitson is shown in Fig. 3. With the exception of the region greater than 900 nm, the technique appeared to be quite accurate (measured index within 0.0005 of index found by Malitson). The measurements for water were compared with those of four previous studies.15,24,25,44 Again, for wavelengths from 400 to 900 nm, there is excellent agreement between the measurements of Tilton and Taylor,25 Austin and Halikas,44 and Stolarski et al.15 and those made with SRWLI (see Fig. 4). Since the measurements presented by Querry et al.24 were listed only to three decimal places, the deviation between Querry and the others was probably due more to precision than to accuracy. Also, the data of Querry et al. were computed from Kramers– Kronig analysis of the imaginary index 关 k(␭) 兴 found by measurement of the absorption and reflectance spectra of water over an extremely broad range of wavelengths (107 – 10⫺2 ␮ m).

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B. Ocular Media The refractive index of ocular media for several different animals was measured. For reasons to be explained below, interferograms were not obtained for cornea or lens tissue from any species. Figure 5 shows the measurements for bovine aqueous and vitreous humors, along with the previous measurements of Sivak and Mandelman.31 To present the results succinctly for all examined species, we applied a fit based on Cornu’s empirical formula to each dispersion curve obtained.21,30 Cornu’s formula is valid for transparent media in the visible spectrum and may be expressed as n ⫽ n ⬁ ⫹ K/ 共 ␭ ⫹ ⌳ 兲 ,

(25)

where n ⬁ , K, and ⌳ are the parameters of the fit. If the data are extended into the infrared, a more appropriate fit (i.e., one with a theoretical foundation, such as a Sellmeier fit) should be used.45 The parameters of the fits for all data are listed in Table 1. C. Measurement Precision and Accuracy Using SRWLI, Saínz and co-workers obtained a precision between 10⫺6 and 10⫺8 for a differential-index measurement.1–3 They found that the uncertainty in the

Fig. 5. Measurement of refractive indices of (a) bovine aqueous humor and (b) vitreous humor. Points represent values obtained from previous measurements.31 Solid curves, compilation of index curves obtained from several interferograms; dashed curves, index data fitted to Cornu’s formula, Eq. (25).

differential index was bounded by three factors: uncertainty in the phase measurement, uncertainty in spectral resolution, and uncertainty in path length, d ⫻ ⌬n [see Eq. (23)]:

␦ 共 ⌬n 兲 ⌬n

⭐

2

␲␴ d⌬n 冑D

⫹

2␦ 共␴兲 ⌬␴

⫹

␦ 共d兲 d

,

(26)

where ␦ is the uncertainty in a given variable, D is the dynamic range of the detector (I max ⫺ Imin), ␦ (␴) is the spectral resolution, and ⌬␴ is the bandwidth (or window width). For our experiments, D was 2 12 (i.e., a 12-bit CCD camera), d was generally 1 cm, ␦ (d) was not greater than 10 ␮m, ⌬n was approximately 5 ⫻ 10⫺3 , and the spectral resolution was approximately 1 cm⫺1. Therefore the theoretically calculated precision was ⬇10⫺5 . Table 2 lists the mean and maximum final corrections (absolute value) and standard deviations of the dispersion measurements for all the results presented in this paper. The individual index curves, derived from a single interferogram, give a single value of the final correction and standard deviation. The mean and maximum are taken from the set of individual index curves that collectively make up the complete measurement across the visible to near-infrared spectral range. As mentioned above, the final correction is applied to rectify errors in the absolute index by shifting the entire index curve up or down to match previously obtained (and presumably more accurate) results. Since image analysis is performed to yield the mean index of several hundred lines of an interferogram with slightly differing phase curves, the standard deviation of the index curve quantifies the uncertainty in the differential index. Although the standard deviation in Table 2 indicates that there was a large uncertainty in the differentialindex measurement, the largest standard deviation was obtained from interferograms centered below 550 nm and greater than 1000 nm. The reason for the large standard deviation below 550 nm was that the phase curves in this region were fitted to a third-order polynomial, which yielded larger deviations than did a lower-order fit. For longer wavelengths both the lamp intensity and the camera sensitivity decreased. Therefore the fringe visibility varied immensely across an image centered at 1000 or 1040 nm, and accurate measurement of peak locations and phase calculation was compromised. Moreover, the absorption of ocular media increased as wavelength increased from 400 to 1100 nm. This meant that the irradiance of the light in the reference arm was much greater than that in the sample arm of the interferometer. This led to a degradation in the fringe visibility across the entire image. Also, the limit of precision calculated above was increasingly important for the region beyond 1000 nm because in this region the index curve becomes extremely flat (i.e., progressively nondispersive). The mean standard deviation for all media was 0.00183. When only data between 550 and 1000 nm were considered, the mean decreased almost an order of magnitude to 0.00020. Two sources of error left uncorrected in the differentialindex measurement were the dispersion of air and the system spectral response. Although the dependence of

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Table 1. Fit Parameters for Cornu’s Formula, Eq. (25), for All Speciesa Media Fused silica Water Bovine aqueous Bovine vitreous Goat aqueous Goat vitreous Monkey aqueous Monkey vitreous Rabbit aqueous Rabbit vitreous Stolarski et al.b

n⬁

K

⌳

r2

Absolute Residual

1.44330 1.31978 1.32165 1.32144 1.30746 1.31256 1.31618 1.30504 1.32031 1.32147 1.31872

6.53887 5.60903 4.99108 5.17987 21.65739 14.99220 10.36987 26.94137 6.23868 5.07154 5.70056

⫺155.669 ⫺154.857 ⫺168.892 ⫺153.851 228.545 124.747 ⫺4.765 349.150 ⫺125.222 ⫺158.706 171.37

0.99905 0.99282 0.99233 0.99412 0.99960 0.99937 0.99934 0.99922 0.99208 0.99384 0.98965

0.000135 0.000341 0.000339 0.000249 0.000043 0.000048 0.000054 0.000050 0.000317 0.000267 0.001187

a The square of the correlation coefficient and the mean absolute residual (mean of the absolute value of the difference between measured and calculated values) are also listed. b The data of Stolarski et al. (Ref. 15) for water are shown for comparison.

the refractive index of air on wavelength with fixed temperature and pressure is small, it is nonetheless nonzero. The change in the refractive index from 410 to 465 nm computed with the dispersion equation for air at 25 °C and 1 atm (7.60 ⫻ 102 Torr) was ⫺3.82 ⫻ 10⫺8 nm⫺1 (Ref. 46). For the same spectral region, dispersion measured without a sample in the setup shown in Fig. 1 was ⫺2.01 ⫻ 10⫺6 nm⫺1 (or a change in index of 0.00011 from 410 to 465 nm). This value for the dispersion was less than the mean standard deviation for all the data listed in Table 2. The large discrepancy between measured and calculated values was probably due to the precision limitation of SRWLI. The measured dispersion of air was near the limit of precision calculated above. Despite the seemingly large air dispersion compared with calculated values, the effect on ocular media measurements was marginal at best and did not account for the error in differential index. The system spectral response includes source spectral response, transmittance of optics, grating efficiency, CCD detector sensitivity, and spectral output of the lamps. The effects of the transmittance of optics and detector sensitivity were negligible in the visible region, but they could have contributed to the error beyond 1000 nm. The system spectral response affects differential-index measurement by altering the position of the fringe peaks. If a peak is located in a region where the response curve has a sharp slope, its position may be changed (backward for a negative slope and forward for a positive slope). However, when a correction corresponding to the system spectral response was included in the image analysis, the location of the peaks remained unchanged. Thus it was concluded that the slope of the system response curve was never large enough to induce a change in phase. As seen in Table 2, the mean final correction applied to the data was 0.00562. This value represents the error in absolute index measurement. For fused silica at visible wavelengths, the stage positions were not recorded, and thus the mean correction includes only near-infrared data. The source of inaccuracy in the absolute index measurement can best be elucidated by examination of the individual measurements that comprise the calculation of the center index. The center index was calculated

Table 2. Mean and Maximum Added Corrections (Absolute Value) and Standard Deviations for All Data

Media

Mean (Max) Absolute Correction (All Data)

Mean (Max) Standard Deviation (All Data)

Fused silica Water Bovine aqueous Bovine vitreous Goat aqueousb Goat vitreousb Monkey aqueousb Monkey vitreousb Rabbit aqueous Rabbit vitreous Mean

0.00198 (0.00400)a 0.00543 (0.01630) 0.00334 (0.00860) 0.00643 (0.01400) 0.00383 (0.00670) 0.00970 (0.01700) 0.00367 (0.00610) 0.00999 (0.02150) 0.00374 (0.00900) 0.00818 (0.01850) 0.00562

0.00091 (0.00674) 0.00088 (0.00198) 0.00223 (0.00668) 0.00094 (0.00618) 0.00252 (0.00514) 0.00137 (0.00225) 0.00409 (0.00901) 0.00183 (0.00380) 0.00133 (0.00453) 0.00222 (0.01210) 0.00183

a b

Near-infrared data only. Visible data only.

with Eq. (24). The sample thickness d, for all cases except bovine cornea, was measured with calipers that had a resolution of 0.001 inches, or 25.4 ␮m. For fused silica the thickness was simply measured by placing the glass within the calipers. For the liquid samples the thickness of the walls of the cuvette were subtracted from the entire thickness of the cuvette. The thickness of the cuvette probably varied by a much greater amount over its dimensions than did the thickness of the fused silica. S 0 was measured by recording the variable-arm stage position at a known landmark in the interferogram, namely, the stationary-phase position [see Fig. 2(c)]. Although the stage had a resolution of 100 nm, its uncertainty was greater than 1 ␮m. This was particularly true when the stage was moved long distances. A rudimentary test of stage drift was made by measurement of the stationary-phase position without a sample over the course of the day. For the most part the stationaryphase position varied by less than 3 ␮m over several hours. Another factor that affected measurement of the stationary-phase position was the spectrograph grating-

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groove frequency. With a higher groove frequency, greater spectral resolution, and hence greater differential index precision, is achieved. However, the high groove frequency makes it difficult to locate the exact stationaryphase position, which may encompass 50 nm in the spectral domain. There is clearly a trade-off between spectral resolution and location of the stationary-phase position. Since the useful range of the lower-groovefrequency grating was restricted to visible wavelengths and near-infrared measurements were required, the choice of grating to use in the measurement became unequivocal. The last variable that may have affected measurement of the center index was ⌬ 0 , the optical delay at the center wave number of the interferogram. Its calculation depended exclusively on the fit coefficients and the center wave number. In general, for a second-order polynomial fit, the standard deviation for the fit coefficients was two orders of magnitude below their mean. The center wave number depended on the wavelength calibration and the proximity of the peak to the center pixel (256). Since the gratings were quite linear and the wavelength calibration routine accounted for variations in the position of the spectral calibration lines (i.e., if they were tilted from the top to the bottom of the image), the wavelength calibration, and hence measurement of the center wave number, was extremely accurate. What was not exact was the position of the center peak with respect to the center of the image. For example, although the center wavelength of the spectrograph may be set to 440 nm, the closest fringe could be 442 nm. This was particularly true if the interferogram was close to the stationary-phase position (because the fringes surrounding the stationary-phase position were quite wide). However, this was not thought to be a major source of error because the stationary-phase position was extremely wide and the placement of its center exactly on the center of the image was not a highly precise undertaking. As an example of the effect on the measurement of the center index, each of the variables in Eq. (24) will be altered in the calculation of the center index for bovine vitreous humor with a center wavelength of 632 nm. In the original analysis the average values for d, S 0 , and ⌬ 0⬘ were found to be 1.0, 0.3443, and ⫺0.0159 cm, respectively. This yielded a value of 1.3284 for the center index, to which a correction of 0.0040 was added to obtain the final result. If the measured thickness of the 1-cm cuvette was in error (i.e., smaller) by 1%, the value would change by 100 ␮m (or 0.004 inches) and would alter the value for the center index at 632 nm by 0.0033 (to 1.3317). If the value for S 0 was inaccurate by 10 ␮m, the center index would change by 0.0010 (to 1.3294). The original analysis produced the following values for the mean fit coefficients (mean ⫾ standard deviation): A ⫽ 1.45 ⫻ 10⫺5 ⫾ 6.69 ⫻ 10⫺8 , B ⫽ ⫺4.28 ⫻ 10⫺1 ⫾ 2.13 ⫻ 10⫺3 , C ⫽ 3.13 ⫻ 103 ⫾ 1.70 ⫻ 101 . The value for the center wavelength was 630.7 nm (15,855.7 cm⫺1). If the value of the mean fit coefficients was altered by one standard deviation, the change in ⌬ 0⬘

Hammer et al.

would be approximately 2.54 ⫻ 10⫺4 , which in turn would result in only a 0.0003 change in center index. The above discussion suggests that errors in the measurement of sample thickness were most likely responsible for errors in center index, followed by S 0 and then ⌬ 0⬘ . However, for the data below 550 nm, where a third-order polynomial fit was used, the variance of ⌬ 0⬘ will dominate the other terms involved in the calculation of the center index. The mean additive corrective values presented in Table 2 are reasonable, given the accuracy with which the thickness measurement was made and the large variance when higher-order fits were used. It is necessary to mention one final source of error that led to the additive correction of the absolute index. Throughout the analysis it was assumed that the refractive index of the air was that of a vacuum. However, air is known to have a refractive index of ⬃1.0003 at visible wavelengths.42 This will slightly reduce the mean correction shown in Table 2.

4. DISCUSSION Recently there has been interest in an accurate measurement of the refractive index of ocular media to use in propagation models. Even though research has been done at visible wavelengths for some species, there are large gaps to be filled, especially in the near-infrared region. White-light interferometry is a technique that has the potential to fill those gaps. In this preliminary study, dispersion in the visible and the near-infrared regions was measured to establish the feasibility and accuracy of the technique. It is most important to find the dispersion of the cornea, of all the ocular components, for two reasons. First, the change in refractive index is larger between air and cornea than the change between any other two successive layers of the eye (ignoring for now the tear layer). Therefore the majority of the refractive power of the eye is due to the cornea. Second, the difference in index is larger between water and cornea than between water and aqueous or vitreous humor. This means that water cannot be used as a substitute for cornea in the models. There were several problems with the measurement of the differential index of bovine cornea. The excised cornea often became wrinkled and always became more turbid, and this prevented measurement of sharp interferograms. Also, with this technique it was considerably more difficult to measure accurate curves when the sample was thin. Moreover, in contrast to aqueous and vitreous humors, which could be placed in a cuvette of known thickness, the exact thickness of the cornea was not known. This precluded any attempt at measurement of the absolute index. The latter problem may be remedied by use of alternative techniques to make single index measurements (e.g., refractometry) or alternative techniques to make thickness measurements (see the use of partial coherence interferometry in Ref. 41). The opacity may be cleared by heating the tissue to body temperature or by using freshly recovered tissue. Similar problems existed during measurement of the lens, with one additional difficulty. The optical power of the lens caused an intermediate focus between the mirror

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Table 3. Comparison between Bovine and Water Dispersiona Sivak and Mandelman (Ref. 31)

This Study

Tilton and Taylor (Ref. 25)

␭ Range (nm)

Aqueousb

Vitreous

Water

Aqueous

Vitreous

Waterc

440–486 486–590 590–650

⫺4.20 ⫺4.02 ⫺2.92

⫺4.80 ⫺3.38 ⫺2.80

⫺4.59 ⫺3.86 ⫺3.37

⫺4.78 ⫺3.65 ⫺2.67

⫺4.13 ⫺3.37 ⫺2.83

⫺6.14 ⫺4.00 ⫺2.75

All values are 10⫺5 nm⫺1. Used 577 nm instead of 590 nm. c Used 435.8 and 656.3 nm instead of 440 and 650 nm, respectively. a b

and the spectrometer, and hence the beam traveling through the sample arm was of different width than the beam in the reference arm. This problem can be alleviated by use of a convex mirror, with focal length matching that of the lens, in the sample arm of the interferometer. In this study interferograms were not obtained for the lens. The differential refractive index of aqueous and vitreous humors measured with white-light interferometry yielded very reliable results. On average, the difference in index between present and previous measurements was not greater than ⬃0.0005 for the spectral region from 400 to 950 nm. Although very accurate results were achieved for the differential index, it was not possible to measure the absolute index as accurately. The average error in the measurement of absolute index was 0.00562. Several broad comments can be made regarding the dispersion results of ocular media with respect to water. Sivak and Mandelman31 concluded from their seminal study that the humors are slightly less dispersive than water, whereas the cornea is more dispersive at short wavelengths and the lens is significantly more dispersive at all wavelengths. Table 3 compares those measurements with the ones presented in this study for bovine aqueous and vitreous humors for visible wavelengths. As can be seen, the dispersion of aqueous and vitreous humors for all visible regions matched the measurements of Sivak and Mandelman quite closely. The largest divergence between water and the humors occurs at the shortest visible wavelengths. In the region between 486 and 650 nm the largest difference in refractive index between water and aqueous or vitreous humor was 0.00068, a difference too small to be reliably distinguished with SRWLI. In the near-infrared spectral region, Drexler et al. measured the group dispersion of human ocular media.41 Although direct comparison with the results in this paper are not possible, relative differences between ocular media and water can be examined. Drexler found that the group dispersion between 814 and 855 nm for aqueous humor, cornea, lens, and water was ⫺1.43, ⫺8.82, ⫺3.08, and ⫺1.62 (all values ⫻10⫺5 nm⫺1), respectively. Thus aqueous humor was slightly less dispersive and cornea and lens were significantly more dispersive in this region. The phase dispersion of bovine aqueous and vitreous humors in the region from 814 to 855 nm was ⫺0.85 and ⫺0.93 (⫻10⫺5 nm⫺1), compared with the dispersion of water in a similar band (800–853 nm) found to be ⫺0.75

⫻ 10⫺5 nm⫺1 by Stolarski et al.15 Although these measurements differ from those of Drexler et al., the difference in both phase dispersion and group dispersion between water and liquid ocular media in the near-infrared is minimal.

5. CONCLUSION The refractive index of ocular media from 400 to 1100 nm of several species was measured with spectrally resolved white-light interferometry. SRWLI provided accurate, real-time dispersion measurement of transparent media. Although difficulties were encountered in the measurement of cornea and lens, accurate measurement of aqueous and vitreous humors for bovine, monkey, goat, and rabbit was achieved. The dispersion of these components did not significantly vary from the dispersion of water. Although a theoretical precision of less than 10⫺5 may be attainable, the average error in differential and absolute indices for all ocular media was 0.00183 and 0.00562, respectively, for the current measurements. With future refinements of the SRWLI technique, we may finally obtain a better understanding of the propagation of light through the eye.

ACKNOWLEDGMENTS This work was funded by the U.S. Air Force Office of Scientific Research (AFOSR grants 2312A103 and F4962098-1-0199), the AFOSR Summer Graduate Research Program, Armstrong Laboratory (now the Human Effectiveness Division of the U.S. Air Force Research Laboratory), and the University of Texas at Austin. The authors thank Tom Milner for his comments. The animals involved in this study were procured, maintained, and used in accordance with the Federal Animal Welfare Act and the ‘‘Guide for the Care and Use of Laboratory Animals,’’ prepared by the Institute of Laboratory Animal Resources, National Research Council.

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