A static Fourier-transform spectrometer has been designed based on .... Redistribution subject to AIP copyright, see http://ojps.aip.org/rsio/rsicpyrts.html.
A static Fourier-transform
spectrometer
based on Wollaston
prisms
M. J. Padgett and A. R. Harvey
Department of Physics and Astronomy, University of St. Andrews, St. Andrews, Fife, KY16 9SS, Utzited Kingdom
(Received 2 September 1994; accepted for publication 21 November 1994) A static Fourier-transform spectrometer has been designed based on Wollaston prisms. It produces an interferogram in the spatial domain which is recorded using a 2D camera. A combination of two Wollaston prisms localizes the interference fringes coincident with the plane of the detector, thereby facilitating an extremely compact design. A personal computer is used to calculate the wavelength spectrum of the input light from the Fourier transform of the interferogram. In addition to its use as a compact and efficient spectrometer, the instrument’s use as a laser wavemeter has also been demonstrated. 0 1995 American Institute of Physics.
1. INTRODUCTION Fourier-transform (FT) spectrometers based on Michelson interferometers are frequently used in both industrial and scientific applications as an alternative to grating-based spectrometers. If the output from a Michelson interferometer is recorded as a function of the path difference between the two arms, then an interferogram is obtained that is the Fourier transform of the spectral distribution of the input light. The principal advantage of FT spectrometers over grating-based instruments is a much larger optical throughput or &endue, making possible the recording of high-resolution spectra at low light levels. This is the Jacquinot advantage. For equivalent resolutions, the &endue of a FT spectrometer based on a Michelson interferometer is typically 190X greater than that of a dispersive spectr0meter.l FT spectrometers based on Michelson interferometers suffer from two disadvantages. First the finite time required to scan the interferometei over the full range of path differences means that the use of J?Tspectrometers with pulsed or rapidly varying sources is problematic. Second, a low-noise interferogram requires the scanning mechanism within the interferometer to be of an exceptionally high quality and well isolated from external perturbations. A number of designs have been proposed that produce interferograms in the spatial rather than the temporal domain. These include Michelson interferometers with mirrors that are tilted so that the path difference varies across the output aperture and the spatially dispersed interferogram has been recorded photographically” or with a detector array.3 Such interferometers have also been configured so that the light in the two arms counterpropagates around an optical ring to give common mode rejection of mechanical perturbations and air currents.4 Recently, we reported an alternative design of FT spectrometer based on a single Wollaston prism.5V6When a Wollaston prism is illuminated with white light and viewed between crossed polarizers, straight-line interference fringes are formed parallel to the edges of the prism.7 For spatially incoherent light the fringes are localized to a plane within the prism and a lens can be used to form an image of these fringes on a detector. The spatial extent and localization of the fringes mean that aberrations in the imaging system can result in a loss of fringe contrast and a distortion in apparent
fringe period. Careful optical design and the use of high quality components are required to limit these effects and the need for low aberration imaging sets a minimum size on the optical system. In this paper, we describe a FT spectrometer in which the lens has been replaced by a second Wollaston prism. The second Wollaston prism localizes the fringes to the plane of the detector without introducing the aberrations associated with the lens based imaging. The use of a second Wollaston prism in place of the lens from our previous design has resulted in a fivefold reduction in the length of the instrument and the absence of aberrations leads to an undisorted fringe pattern. We present spectra calculated from a Fourier transformation of the interferogram, and report preliminary results pertaining to the use of the instrument as a compact laser wavemeter. II. PRlNCDPLE OF OPERATION Light transmitted through a Wollaston prism is split into two orthogonal polarizations which diverge with a half angle a given by’ cu=2(n,--n,)tan
6,
Cl!
where n, and n, are the extraordinary and ordinary refractive indices of the prism, and 6 the angle of the prism wedge interface. The key to operation is a path difference A introduced between the two polarizations upon transmission through the Wollaston prism, given by7 A=cxd,
@I
where d is the lateral displacement from the center of the prism. By placing the prism between two polarizers aligned at 24.5” to the optic axes of the prism material, the common components of the two orthogonally polarized beams interfere with a path difference that is proportional to the lateral displacement from the center of the prism. The period x0 of the fringes produced by light with a wavelength A is given by A
X”=Z(n,--n,)tan
6 ’
The projection of the localized fringe plane to a position outside of the prism pair can be understood with reference to Fig. 1. The first prism splits the incoming light into its two
1995 American Institute of Physics Rev. Sci. Instrum. 66 (4), April 0034-674~/95/66(4)/2807/5/$6.00 2607 Downloaded 31 Oct 2000 1995 to 130.209.6.40. Redistribution subject to AIP copyright,0see http://ojps.aip.org/rsio/rsicpyrts.html.
Orientation of optic Axis
WollastonP+.m
IsotropicMedia
Wollaston Prism Localised
I
I
M---a-b---w I
I
I I Birefringent media
FIG. 1. Localizationof the fringe plane to a position outsidethe Wollaston prisms.
orthogonally polarized components with an angular deviation of al; the second prism is oriented to reverse the splitting introduced by the first prism. Providing that +=Y+ after transmission through the second prism, the two component rays will converge, the crossing point being the position to which the fringes are localized. With reference to Fig. 1, the approximate location of the fringe plane can be calculated from the deviation angles of the Wollaston prisms and their separation, and is given by
h-nd(b+&-;)-a,( .+;-t),
pends on the number of pixels across the array, N, and the short-wavelength operating limit Amin,and is given by 4 AoEwHM=blia * Therefore, for a static FT spectrometer utilizing a CCD camera, oriented with 767 pixels along the interferogram and assuming a X,, of 300 nm, the resolution is predicted to be of the order of 180 cm-‘. Ill. REFRACTION OF THE E-RAY IN A BIREFRINGENT IMEDFA
where a is the center-to-center separation of the prisms, b the distance between the second prism and the localized fringe plane, t the thickness of the prisms, ii the mean refractive indices of the prism material, and ‘y1 and CQ the splitting angles of the first and second prisms, respectively. Exact’calculation of the position of the localized fringe plane needs to take into consideration the internal angle of the prisms and the non-Snell’s law refraction of an e-ray entering or leaving a birefiingent medium. The path’difference A introduced between the two orthogonally polarized beams by the prism pair is given by the sum of the path difference introduced by each individual prism. For small a; this is approximately given by A=d(a,-aI).
FIG. 2. Huygen’sconstructionfor a ray enteringa birefringent medium.
As discussed above, to calculate the exact position of the localized fringe plane it is necessary to take into account the internal angle of the prisms and the nonSnell’s law refiaction of the e-ray. However, most commerically available ray tracing packages are unable to deal with refraction of the e-ray in biretiingent media. To calculate the position of the localized fringe plane for a range of angles of incidence, we Wollastonprism,aA .3”
Wollaston prism, &.I0 /
4
1
Localire
plane
(9
The resolution of a FT spectrometer depends on the maximum path difference Amaxbetween the two arms of the spectrometer. The resolution in wave numbers (full width is approximately given by half maximum), Am-, 1 The exact relationship depends on the choice of apodization window applied to the interferogram prior to the Fourier transform. However, when using a detector array, the maximum path difference is effectively limited by the Nyquist criterion that requires at least two data points per fringe period. Hence, if the interferogram is’ symmetrically recorded about zero path difference, the highest resolution attainable de-
too{
Orientation of optic axes
lOCUll
FIG. 3. Ray tracing trough a pair of Wollastonprisms.
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2808
have ray-traced through the Wollaston prism pair using equations derived from Huygen’s construction which are the equivalent of Snell’s law for a birefringent medium.* A Huygen’s construction for an e-ray entering a birefringent medium is shown in Fig. 2;’ the direction of the refracted ray is given by the line joining the center of the wave velocity
surface to the tangent made between the wave velocity surface and the wave front of the refracted ray. For the simplest case of refraction in the plane containing both the surface normal and the optic axis of the crystal, the geometrical solution for the refracted angle of the e-ray entering a birefringent material is a quadratic in terms of tan r, namely
I
i
n; cos2 p+n;
sin2 p-
(n%cos2p+n; sin2p)2 3 n; sin” i
+nz sin2 p+nz cos2 pnx)sin
i
tan”
r+2(nz-nT)sin
/? cos p l-
n: cos2 p+n:
(
sin2 p
nf sin’ i
1
tan r
P cosPl”=,~ nf sin’ i
(8)
This can be inverted to give the angle of an extraordinary ray leaving a birefringent material; the solution is [(ni cos2 p+nz sin2 p)sin i+(ni--2)sin p cos p cos il2 9 sin2 r=n~[n;(cos p sin ifsin p cos i)2+n~(cos p cos i-sin p sin ij2] ’
(9)
I
where p is the angle between the surface and the optic axis, i the angle of incidence, r the angle of refraction, and ni and n, the refractive indices of the incident and the refracted rays in the isotropic media, respectively.
IV. INSTRUMENT DESIGN The two, 4 mm thick, Wollaston prisms that form the spectrometer are fabricated from calcite. The splitting angles are 2.1” and 1.3”, giving an effective splitting angle of 0.8”. Figure 3 is a ray trace through the prism pair and shows that the fringe plane is localized approximately 3 mm behind the exit face of the second prism and inclined by approximately 4”. The depth of focus over which high contrast fringes can be observed depends on the spatial coherence of the incident light. Successive ray traces show that the fringe plane does not move by more than 100 E.L~for incident angles in the range 22.5” (see Fig. 4). For typical extended sources such as a tungsten lamp, the movement of the fringe plane does not result in a significant loss of fringe contrast. The two polarizers are a simple dichroic sheet located on either side of the prism pair, giving an extremely compact and robust spectrometer measuring 40X40X100 mm. The detector array is a 512X767 element CCD video camera with an active area of 5X7 mm. its output is read via a frame grabber into a personal computer. The nominal 0.8” splitting angle of the prism pair gives a maximum path difference over the 767 pixel width of array of 250 pm, which, from EQ. (6), gives a predicted resolution of 200 cm-’ and a short-wavelength Nyquist limit of 260 nm. (This equates to a resolving power of 80 at 632.8 nm.) To prevent aliasing, it is necessary to attenuate light at wavelengths shorter than the Nyquist limit. This is achieved by the short-wavelength cutoff of the CCD array at 400 nm. Clearly, the spectrometer could be redesigned using Wollaston prisms with greater splitting angles, hence yielding a larger path difference. By matching the Nyquist limit to the short-wavelength cutoff of
the CCD, the resolution could be improved to 130 cm-’ without reducing the spectral coverage of the spectrometer. The data processing is eased by rotating the CCD array so that the fringes lie perpendicular to the lines of the camera image. Averaging the pixel columns of the image generates a single 767 element array containing the fringe information. The data is zero padded to 1024 elements to enable the use of a fast Fourier-transform algorithm from which the spectral power distribution of the input light is calculated. K DISCUSSION OF THE OPTICAL THROUGHPUT OF THE WOLLASTON PRISM BASED SPECTROMETER As discussed, the principle advantage of FT spectroscopy over a dispersive spectrometer is the high &endue in conjunction with high resolution. The &endue E of an interferometer is defined as E=f-hA,
(10) 5
j
2.5
B g
.-............_.. -2.5"
0
i 0 "a .Y n
-----
2.5"
-2.5
-5 2.25
2.50
2.75
3.00
3.25
3.50
Distance behind exit face of secondprism (mm) FIG. 4. The calculatedposition of the localized fringe plane as a function of incident angle.
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where L? is the acceptance solid angle (field of view) of the spectrometer and A is the area of its aperture. The standard derivation of the &endue for a Michelson interferometer considers the change in the path difference between the two arms of the interferometer as the angle of incidence is increased. Once the additional path difference between the onaxis and oblique rays exceeds X/2, the oblique rays are subtracting power from the interferogram. Taking the X/2 condition as defining the maximum angle of incidence, the half angle acceptance iMichelsonfor a FT spectrometer based on a Michelson interferometer is given byY 0
’
I
I-
For a FT spectrometer based on a Michelson interferometer, with a resolving power of 80 at 632.8 run, the full angle acceptance is 6.4”. By contrast, using a similar criteria of a A/2 path difference between the on-axis and the oblique ray, the half angle acceptance of a Wollaston prism, iWoUastonris given by7 (12) where t is the thickness of the Wollaston prism. If the Wollaston prism is illuminated with light that includes ray from outside its angular acceptance, the power in the interferogram is reduced. Using Eq. (12), a FT spectrometer based on a pair of 4 mm thick Wollaston prisms fabricated from calcite has a predicted full angle acceptance of 2.6” at 632.8 nm. In the case of a Michelson interferometer, half of the light is reflected back toward the source and so it has a maximum optical efficiency of 50%. The spectrometer described here may be considered to be a polarizing Mach Zehnder and hence has a maximum efficiency of 25% for unpolarized light and 50% for a linearly polarized source. Combining the field of view and optical efficiency considerations, the etendue for the Wollaston prism based spectrometer is approximately one order of magnitude lower than one based on a Michelson interferometer. Although this is still an order of magnitude higher than the &endue of a spectrometer based on a diffraction grating, it is useful to consider methods by which the &endue may be increased to approach that of a conventional FT spectrometer. The angular acceptance of the Wollaston prism is the limiting factor on the &endue of the spectrometer. From Eq. (12), it follows that the angular acceptance is increased by using thin Wollaston prisms fabricated from a high refractive index material with a low birefringence. However, for a given path difference, a Iow birefringence requires an increase in the internal angle of the Wollaston prism which in turn increases its minimum thickness. For materials with high birefringence the thickness is limited not by the internal angle but by the ability to cut and polish thin wedges of fragile material. These points taken together suggest that materials other than calcite may represent a better alternative. For example, Wollaston prisms can be fabricated from lithium niobate that has refractive indices of 2.4 and 2.5 and
200
400
600
size of detector array
800
;
1000
I
FIG. 5. The apodizedinterferogramof a metal halide street lamp, obtained using the compact static spectrometer.
can be readily polished into thin wedges. Assuming the same maximum path difference as the present instrument, a modified instrument based on 3 mm thick Wollaston prisms fabricated from lithium niobate is predicted to have a half angle accpetance of 6.3”. This figure is comparable to that obtained for a conventional FT spectrometer and suggests that a FT spectrometer based on Wollaston prisms could be designed to have similar &endue to one based on a Michelson interferometer. VI. EXPERIMENTAL RESULTS To illustrate the performance of the Wollaston prism based FT spectrometer, we present a number of results obtained using the instrument. Rotation of either polarizers within the system by 90” reverses the phase of the fringes within the interferogram. By recording both interferograms and calculating their difference divided by their sum, an interferogram is obtained which is normalized with respect to illumination profile and fixed pattern noise on the detector. Alternatively, both phases of the interferogram could be recorded simultaneously by dividing the input aperture into two regions with individual, crossed polarizers and processing the two halves of the CCD array as two separate channels. The latter approach would give an interferogram normalized with respect to the illumination profile but not with respect to the fixed pattern noise. Figure 5 shows the normalized and subsequently apodized interferogram of a metal halide lamp obtained using the interferometer. The half angle subtended by the source was approximately 2”. Figure 6 is the computed spectrum of the lamp obtained by Fourier-transforming the data in Fig. 5 and displayed in terms of wave number. Figure 7 shows the normalized interferogram of a helium neon laser. The coherence of the laser source usually results in speckle within the interferogram. A rotating ground glass plate is used to give a time varying speckle pattern that averages to a uniform illumination over the intergration period of the detector. A Fourier transform of the apodized interferogram enables the wavelength scale of the spectrum to be calibrated. The measured width of the spectral peak corre-
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10000
12000
16000
14000
18000
20000
Wavenumber
22000
0.1845
0.1865
0.1885
(cm-‘)
0.1905
0.1925
0.1945
l/Pixel period
FIG. 6. The measuredspectrumof a metal halide street lamp, obtainedby taking the Fourier transform of the data in Fig. 5. sponds to a resolving power of 80 at 632.8 nm, which agrees closely with that predicted in the previous section. For use as a laser wavemeter, the important criterion is the repeatability of the calculated wavelength. The exact wavelength could be determined by fitting an appropriate function to the dominant peak within the Fouriertransformed spectrum. Alternatively, the center wavelength can be deduced by finding the maximum value in the power correlation function between the interference pattern and a perfect sinusoid (see Fig. 8). This correlation function is equivalent to the Fourier transform but can be calculated for sine waves with an arbitrarily small increment in period. The width of the central peak corresponds to the resolving power of the instrument as determined by the maximum path dif-
FIG. 8. The power correlation of the interferogramwith a perfect sinusoid.
ference, but the high signal-to-noise figure enables the center frequency to be determined to a greater precision. Our preliminary measurements of the same HeNe laser source show a short term reproducibility of approximately 2 in 10’. Over a longer time scale, the reproducibility is limited by the temperature dependence of the birefringence, which for calcite is approximately 1 in lo4 per degree. The fact that the entire interferogram is produced simultaneously means that it could be used for both continuous-wave and pulsed laser sources, In conjunction with temperature stabilized Wollaston prisms or a reference wavelength, it will be possible to increase the long term reproducibility beyond that reported in this work. ACKNOWLEDGMENTS One author (M.J.P.) is a Research Fellow of the Royal Society of Edinburgh, and one (A.R.H.) gratefully acknowledges the support of DIL4, Malvern, UK. ’ R. J. Bell, Introductory Fourier TransformSpectroscopy (Academic,New
: -0.4
0
:
200
i
i 400
: 600
800
1000
FIG. 7. The recordedinterferogramfrom a HeNe laser obtained using the compact static spectrometer.
York, 1972), Chap. 2, p. 22. ‘G. W. Stroke and A. T. Funkhouser,Phys. L&t. 16, 272 (1965). “T. Barns, Appl. Opt. 24, 3702 (1985). 4T. Okamoto, S. Kawata, and S. Minami, Appl. Opt. 23, 269 (1984). ‘M. J. Padgett,A. R. Harvey,A. J. Duncan, and W. Sibbett,Appl. Opt. 33, 6035 (1994). ‘A. R. Harvey,M. Begbie, and M. J. Padgett,Am. J. Phys. 62,1033 (1994). 7M. Fraqon and S. Mallick, Polarization Interferometers (Wiley, New York, 1971), Chap. 2, pp. 26-29. ‘For example, see R. S. Longhurst, Geometrical and Physical Optics (Longmans,New York, 1968), Chap. 22, pp. 484-487. ’ R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic,New York, 1972), Chap. 11, p. 149.
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