CA2302994A1

CA2302994A1 - Static fourier transform spectrometer with enhanced resolving power

Info

Publication number: CA2302994A1
Application number: CA002302994A
Authority: CA
Inventors: Evgeny V. Ivanov
Original assignee: Individual
Current assignee: Individual
Priority date: 2000-03-24
Filing date: 2000-03-24
Publication date: 2001-09-24

Abstract

Static Fourier transform spectrometers, as well as multichannel dispersive spectrometers, usually have a relatively low value of the product of resolving power and relative free spectral range. This value is limited by the number of pixels in a line of the detector array used in the spectrometer. In this invention, a substantial increase of the SR-factor over the prior art static Fourier spectrometers is provided by means of introducing a stepped retardation in a static double beam Michelson or Mach-Zender interferometer with two-dimensional detector array.
The stepped retardation can be introduced into the interferometers by using a stepped-profile reflective (10) or refractive element into one of the interferometer arms. A
two-dimensional interference pattern that contains folded interferograms is formed on the detector plane, captured by the detector array, and digitized by the analog-to-digital converter in the signal processing unit. Then the interferogram, as a function of the light intensity vs.
linearly increasing retardation, can be reconstructed by merging the folded interferograms corresponding to adjacent lines of the interference pattern together, and the original spectrum of the analyzed radiation can be retrieved by applying the Fourier transform to the reconstructed interferogram.

Description

STATIC FOURIER TRANSFORM SPECTROMETER
WITH ENHANCED RESOLVING POWER.
AUTHOR: Evgeny Ivanov FIELD OF THE INVENTION
This invention pertains generally to the field of optical instruments and particularly to static (stationary) Fourier transform spectrometers.
BACKGROUND OF THE INVENTION
Fourier transform spectroscopy is a well-recognized method for making spectroscopic measurements in the ultraviolet, visible and infrared regions. It offers distinct advantages over the dispersive spectrometers, such as throughput and multiplex advantages which give superior signal-to-noise performance and allows to accommodate for much larger source solid angles or smaller sample size, large wavenumber range per scan, high wavelength accuracy, large resolving power achievable with smaller size and lower weight of the interferometer than dispersive spectrometer, nonsensitivity to stray light. Thanks to these advantages, Fourier transform spectrometers currently dominate in infrared spectroscopy. However, the disadvantages of conventional Fourier transform spectrometers, such as high accuracy required for mechanical mirror drive, and high flatness required for the interferometer mirrors and beamsplitter surfaces, make Fourier transform spectrometers expensive and complex instruments, especially in the UV and visible range.
In order to overcome these disadvantages, an alternative way of stationary Fourier transform spectroscopy was proposed in [Stroke G.W., Funkhouser A.T., "Fourier Transform Spectroscopy Using Holographic Imaging without Computing and with Stationary Interferometers", Physics Letters, V.16, N.3, 1965, p.272] and has been the subject of intensive studies in the last few decades. In this method, the optical parts of the interferometer are fixed in their positions, interferogram is produced in space rather than in time and recorded by a detector array or photographic plate. The applications of various types of interferometers, such as:
- Michelson [Breckinridge J.B., O'Callaghan F.G., "Integrated Optics in an Electrically Scanned Imaging Fourier Transform Spectrometer"" US Patent 4,523,846, 1985], - triangle common path [Okamoto T., Kawata S., Minami S., "Fourier Transform Spectrometer with a Self Scanning Photodiode Array", Applied Optics, V.23, 1984, p.269], [Rafert B.J., Sellar G.R., Blatt J.H., "Monolithic Fourier Transform Imaging Spectrometer", Applied Optica~, v. 34, 1995, p.7228], [Dierking M.P., "Solid Stationary Interferometer Fourier Transform Spectrometer", US Patent 5,541,728, 1996], - Sagnac [Barnes T.H., "Photodiode Array Fourier Transform Spectrometer with Improved Dynamic Range", Applied Optics, v.24, 1985, p.3702], - Lloyds mirror [Bliss M., Craig R.A., Anheier N.C., "Demonstration of a static Fourier transform spectrometer", Fiber Optic and Laser Sensors and Applications;
Including Distributed and Multiplexed Fiber Optic Sensors VII, Proc. SPIE Vol. 3541, 1999, p. 103], - Mach-Zender [Simeoni D., "New Concept for a High-Compact Imaging Fourier Transform Spectrometer", Proceedings of SPIE Symposium on OElAerospace Sensing, SPIE
Vol. 1479, 1991, p.127], [Simeoni D, Cerutti-Maori G., "Interferometer Devices, Especially for Scanning Type Multiplex Fourier Transform Spectrometry", US Patent 5,223,910, 1993], [Juntti:la M.-L., Kauppinen J., Ikonen E., "Performance Limits of Stationary Fourier Spectrometers", Journal of the Optical Society of America A, v.8, No 9, 1991, p.1457], [Horton R.F., "High Etendue Imaging Fourier Transform Spectrometer", US Patent 5,777,736, 1998], - polarization [Hashimoto M., Kawata S., "Multichannel Fourier Transform Infrared Spectrometer", Applied Optics, v.31, 1992, p.6096], [Smith W.H., Hammer P.D., "Digitally Scanned Interferometer: Sensors and Results", Applied Optics., v.35, 1996, p.2902], [Courtial J., et.al., "Design of a Static Fourier Transform Spectrometer with Increased Field of View", Applied Optics, v.35, 1996, p.6698], [Pagett M.J., et.al., "Single Pulse, Fourier Transform Spectrometer Having No Moving Parts", Applied Optics, v.33, 1994, p.6035], - folded mirror [Junttila M.-L., "Stationary Fourier Transform Spectrometer", Applied Optics, v.31, 1992, p. 4106], in static Fourier transform spectrometers were reported and reviewed in [Egorova L.V., Ermakov D.S., Kuvalkin D.G. and Taganov O.K., "Static-type Fourier spectrometers", Soviet Journal of Optical Technology, v. 59, 1992, p.65], and their performance limits were studied in [Junttila M.-L., Kauppinen J., Ikonen E., "Performance Limits of Stationary Fourier Spectrometers", Journal of the Optical Society of America A, v.8, No 9, 1991, p.1457] and [Sellar G.R., Rafert J.B., "Effects of Aberrations on Spatially Modulated Fourier Transform Spectrometers", Optical Engineering, v.33, No 9, 1994, p.3087]. In addition to common advantages of the Fourier spectroscopy mentioned above, static Fourier transform spectrometers have the following advantages:
- They have no moving parts and consequently avoid many of the mechanical problems associated with moving-mirror interferometers.
- They can be embodied in a simple, rugged and inexpensive design.
- Since the detector array always samples the interferogram at the same points, the need for a referencing laser is eliminated.
- As opposed to scanning Fourier transform spectrometers, static Fourier transform spectrometers obtain the full interferogram at once and therefore are not sensitive to flicker (fluctuation) noise or signal change during the measurement cycle. This makes them potentially useful for flame emission spectroscopy and time-resolved spectroscopy.
- Certain configurations of static Fourier transform spectrometers have relaxed requirements to the surface flatness of the optical components, as opposed to conventional scanning Fourier spectrometers. This becomes possible due to the fact that in the static Fourier transform spectrometers the entire aperture wavefront is sensed by the detector array, as opposed to scanning Fourier transform spectrometers, where the wavefront is usually optically focused on the single element detector. The latter technique requires high level of phase uniformity over the aperture in order to avoid degradation of the interferogram contrast, while the former technique can tolerate much higher phase nonuniformity and further correct it in the computer data processing using a reference interferogram obtained by a monochromatic source.
The mentioned feature, however, does not apply to the field-widened interferometers, such as Sagnac and triangle common path interferometers, which usually require much higher quality beamsplitter and mirrors and low-aberration imaging optics.

The mentioned advantages of static Fourier transform spectrometry make it an attractive alternative to the dispersive spectrometers in the ultraviolet and visible range, in spite of the fact that the multiplex advantage of Fourier transform spectroscopy in the ultraviolet and visible range is lost.
However, all of the described static Fourier transform spectrometers have one common disadvantage: they all have relatively low resolving power (normally less than a thousand) due to the fact that, according to the Nyquist sampling theorem, the number of resolution elements in Fourier transform spectrometers is limited by half of the number of the interferogram samples, which in the case of static Fourier transform spectrometers is equal to half of the number of pixels N in the detector array along the direction of the interferogram (or the retardation gradient in the detector plane) y"'"" y""" < N/2, where v~"~ and v",~ are the higher and lower Ov extremes of the wavenumber range, and Ov is the instrument resolution in wavenumbers. A
method of heterodined static Fourier transform spectrometry with enhanced resolving power was described in [Barnes T.H., Eiju T., Matsuda K., "Heterodyne Photodiode Array Fourier Transform Spectrometer", Applied Optics, v. 25, 1986, p.1864] and [Roesler, F.L., Harlander J., "Spatial Heterodyne Spectrometer and Method", US Patent 5,059,027, 1991]. In these spectrometers, however, the SR-factor, which is equal to the product of the relative free spectral range S = y'"~" y""" and the resolving power R = y"'a" , is still limited by half of the number of v",;~ 0V
pixels in the detector array SR= y"'~" y"'~" S N / 2 , so that the enhanced resolving power in the Ov heterodined spectrometers is achieved to the expense of narrowing the free spectral range.
It is important to note that in the dispersive spectrometers with detector arrays the SR-factor is limited by a similar rule SR = y"'~'Qv min ~ N ~ where N is the number of pixels in the detector array along the direction of dispersion. Therefore, the dispersive spectrometers with detector arrays are subject to the same limitation. Due to this limitation, simple and inexpensive dispersive spectrometers with detector arrays and without grating positioning mechanics are available only with relatively low resolution (typically worse than 0.5-1 nm).
On the other hand, high resolution dispersive spectrometers are usually equipped with precision grating positioning mechanics to overcome the limitation on the free spectral range, which makes them more complex and expensive.
In principle, the limitation on the SR-factor in the mentioned static Fourier transform spectrometers is not dictated by the total number of pixels in the detector arrays, since the number of pixels in two-dimensional detector arrays can be as many as several millions.
However, the static Fourier transform spectrometers described above have the interferometer retardation linearly or monotonically increasing along a certain direction in the detector plane, therefore their SR-factor is limited by the number of pixels in the array line along the retardation gradient in the detector plane.
The possibility of using an increased number of samples provided by two-dimensional detector arrays for the enhancement of the SR-factor of static Fourier transform spectrometers has been demonstrated in [Ebizuka N, et.al., "Development of a Multichannel Fourier Transform Spectrometer", Applied Optics, v.34, 1995, p.7899]. Ebizuka and co-authors developed a polarization static Fourier transform spectrometer in which the interferogram is folded within the two-dimensional detector by means of producing four folded interferograms which can be further connected with each other to form a single interferogram with extended retardation range. The folding was introduced by means of phase retarding birefringent plates. The possibility of increasing the resolving power by a factor of 2 using this method was shown.
However, the proposed design based on Lithium Niobate and Calcite birefringent crystals was relatively expensive, and the possibility of achieving higher values of resolving power has not been proven.
It is hardly possible to achieve high values of resolving power in the proposed arrangement of polarization interferometers, since it requires a large number of relatively long and thin retardation birefringent plates. For example, in order to provide the resolving power of 104, twenty Calcite retardation plates with a thickness of 1 mm and a length up to 28 mm are required.
As a result, static Fourier transform spectrometers with SR-factor more than 1000-1500 have never been reported. If the static Fourier transform spectrometers had an enhanced SR-factor, they would be potentially useful for a variety of spectroscopic applications requiring simultaneously high resolution and broad free spectral range, and would be able to replace the conventional dispersive spectrometers in the mentioned applications due to their simple and inexpensive design with no moving parts.

SUMMARY OF THE INVENTION.
In this invention a substantial increase of the SR-factor over the prior art static Fourier transform spectrometers is provided by means of introducing a stepped retardation in a double beam Michelson or Mach-Zender interferometer with two-dimensional detector array. Static Fourier transform spectrometers based on this invention can have increased resolving power up to 104 - 105, a broad spectral range limited only by the detector response, a throughput advantage with respect to the dispersive spectrometers that can reach a value of tens or even hundreds, a simple, compact, and rugged design with no moving parts, and relaxed requirements to the surface flatness of the optical components. Due to these properties, they can be an attractive and more affordable alternative to high-resolution dispersive spectrometers in the ultraviolet, visible and near-infrared range.
The spectrometer of the invention utilizes a double beam static Michelson or Mach-Zender interferometer which includes an input collimator, beamsplitter, two reflective elements, two-dimensional detector array, and may include another beamsplitter (required in Mach-Zender interferometer) and output imaging optics before the detector array, in which a linearly or monotonically increasing retardation between two interfering beams is introduced in one direction and a retardation increasing in a stepped manner is introduced in another direction. The monotonically increasing retardation can be introduced into the interferometer either by tilting one of the reflective elements (mirrors in one of the interferometer arms with respect to the reflective element in the other arm, or beamsplitters), or by introducing a wedged refractive element in one of the interferometer arms, or by a combination of both methods. The stepped retardation can be introduced into the interferometer either by using a stepped-profile reflective element (a stepped profile mirror instead of a plane mirror in one of the interferometer arms, or a stepped profile beamsplitter instead of a plane beamsplitter), or by introducing a stepped profile refractive element in one of the interferometer arms, or by a combination of both methods. As a result, a two-dimensional interference pattern that contains folded interferograms is formed on the detector plane. This interference pattern is captured by the detector array and digitized by the analog-to-digital converter in the signal processing unit, then the interferogram as a function of the light intensity vs. linearly increasing retardation can be reconstructed by merging together the folded interferograms corresponding to adjacent lines of the interference pattern, and the original spectrum of the analyzed radiation can be retrieved by applying the Fourier transform onto the reconstructed interferogram. For the folded interferograms to be connected to each other, the interferometer is adjusted so that the retardation difference between two adjacent folded interferogram lines is slightly less than the retardation difference between two opposite extreme points along each individual folded interferogram line. The positions of optical components of the spectrometer are fixed during a single period of integration of the interference pattern. A
variety of optical configurations of the interferometer with stepped retardation can be utilized.
The exact nature and advantages of this invention will be apparent from the following detailed description when taken in conjunction with the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS.
FIG. 1 is a schematic view of a generalized static Fourier transform spectrometer based on the Michelson or Mach-Zender interferometer with stepped retardation which illustrates the principles of this invention.
FIG. 2 is a schematic view of the static Michelson interferometer with stepped retardation in which the stepped retardation is introduced by means of using a stepped-profile reflective element in one of the interferometer arms.
FIG. 3 is a schematic view of the static Michelson interferometer with stepped retardation in which the stepped retardation is introduced by means of using a stepped-profile refractive element in one of the interferometer arms.
FIG. 4 is a schematic view of the static Michelson interferometer with stepped retardation in which the stepped retardation is introduced by means of using both stepped profile refractive element and stepped profile reflective element in one of the interferometer arms.
FIG. 5 is a schematic view of the static Mach-Zender interferometer with stepped retardation in which the stepped retardation is introduced by means of using a stepped-profile reflective element in one of the interferometer arms.
FIG. 6 is a schematic view of the static Michelson interferometer with stepped retardation in which the stepped retardation is introduced by means of using a stepped-profile reflective element in one of the interferometer arms, and the interference pattern is imaged onto the detector by means of imaging optics.
FIG. 7 is a piece of experimentally obtained interference pattern produced in the interferometer shown on FIG. 6 illuminated by a monochromatic light and sensed by a two-dimensional CCD array.
FIG. 8 and FIG. 9 show the spectrum of the Mercury discharge lamp obtained with the experimental prototype of static Fourier transform spectrometer based on the Michelson interferometer with stepped retardation shown on FIG. 6.
DETAILED DESCRIPTION OF THE INVENTION
A schematic view of a generalized static Fourier transform spectrometer based on the Michelson or Mach-Zender interferometer with stepped retardation in accordance with this invention is shown on FIG.1. The incoming electromagnetic radiation is received through an entrance field diaphragm 1 and collimated into a parallel beam of aperture A
by collimator 2. The wavelength range of the incoming radiation can lie in the ultraviolet, visible or infrared region.
Collimator 2 can be made as a refractive, reflective, or any combination of refractive and reflective collimating elements, such as spherical or aspherical mirrors or lenses. The collimated radiation is then received into a double beam Michelson or Mach-Zender interferometer with stepped retardation 3 which has the characteristic of splitting the incoming beam into two coherent beams, introducing a phase retardation between these coherent beams, and then recombining them together in order to generate an interference pattern. The retardation 0(x,y) between the wavefronts 4 and 7 of these beams introduced by the interferometer is a monotonically increasing function of a coordinate x along one arbitrary direction in the plane of wavefront 7 and is a stepped function of a coordinate y along another arbitrary direction in the plane of wavefront 7 (y direction is not coincident with x). In a general case, a stepped function F(z) of a variable z can be defined as a sum of Heaviside step functions 6(z) (6(z)=0 for z<0 and 6(z)=1 for z>0) K
F(z) _ ~ hk ~ 6 (z - k ~ ak ) , X=o so that the retardation function of the present invention 0(x,y) in a general case can be expressed as follows:
K
0(x~Y) _ ~, hk (x) ' 6(Y - k ' ak (x)) k=0 where the partial derivative a~a~' Y) is either positive everywhere in the x y plane, or negative everywhere in the x y plane. In order to simplify the optical design and data processing, directions x and y can be made orthogonal to each other, the monotonic functional dependence of the retardation vs. coordinate x can be made linear, and the height h and width a of the retardation "steps" can be kept constant over the beam aperture, so that the function 0(x,y) can be expressed as:
K K
O(x,y)=8+x~tan(a)+~h~o'(y-ak)=8+xa+~h~6(y-ak), (I) k=0 k=0 where 8 is the initial retardation at x=y=0, and oc is the tilt angle between two wavefronts. The recombined wavefronts 4 and 7 form a two-dimensional interference pattern on the plane of the two-dimensional detector array 5. This interference pattern is captured by detector array 5, digitized by analog-to-digital converter, and processed in the electronic unit and computer analyzer 6 according to the algorithm described further. A typical interference pattern produced on the detector array by illuminating the spectrometer of FIG. 1 with a monochromatic source is shown on FIG. 7, and contains a series of folded partial interferograms.
Assuming that the amplitudes of the two interfering beams are equal to each other, the interference pattern as a function of coordinates x and y in the detector plane can be expressed as:
~m;~
~(x~Y) = J I°2~)~l+exp{2~o(x,Y)}~Zd~ = j I°2~) [I+cos{2~cvo(x,Y)~~d~ ~ (2>
~m;~ ~~u~
where I°(v)is the intensity of the incoming radiation as a function of the wavenumber v=1/~, of the radiation, ~, is the wavelength of the radiation. The interferometer is adjusted so that the retardation difference between two adjacent folded interferogram lines h is slightly less than the retardation difference between two opposite extreme points along each individual folded interferogram line ocA, where A is the aperture of the beam. In this case the computer analyzer can connect the adjacent partial interferograms with each other at the points of equal retardation, thereby reconstructing the full interferogram as a function of the light intensity vs. linearly increasing retardation O
J'(0) = f to (v) eos~2~cv0~dv . (3) ~m;~
The constant interferogram offset was removed in the expression (3) by subtracting the moving average from the original interferogram J(~). The original spectrum of the analyzed radiation can be retrieved by applying the Fourier transform to the reconstructed interferogram o,~x I'(v ) = J J'(0) cos f 2~cv0~d0 . (4).

The resolution Ov of the obtained spectrum I~(v) is limited both by the maximum retardation D,nax [Griffiths P.R., de Haseth J.A.., Fourier Transform Infrared Spectroscopy, Wiley, NY, 1986, p.ll] and half of the total number of pixels in the detector array N:
Ov > (0"~ )-' and 0v > 2 ~~' . The number of "steps" K in the retardation function (1) must be less than the number of pixels in the row of the detector array NroW. In reality, however, the resolution is limited by aberrations and image blur in the optical system, which will be considered further in this description.
The basic configurations of the Michelson interferometer with stepped retardation are shown on FIG. 2, FIG. 3 and FIG 4. All of them have beamsplitter 9 splitting the incoming collimated beam 8 into two coherent beams. These two beams are reflected back by means of stepped profile reflective element 10 and plane mirror 11 on FIG. 2 and FIG.
4, or plane mirrors 12 and 11 on FIG. 3, and then are recombined together at beamsplitter 9. The interference pattern, formed by interference between these recombined beams, is captured by the two-dimensional detector array 5. The stepped retardation function ( 1 ) can be introduced by means of using a reflective element with stepped profile 10, similar to a reflective diffraction grating, in one of the arms of the interferometer, and simultaneously tilting plane mirror 11, beamsplitter 9 or the stepped profile mirror 10 in the direction along the "grooves" of the stepped profile mirror by the angle a, as shown on FIG. 2 and FIG. 4, or by introducing a refractive element with a stepped profile 14 similar to a transmissive diffraction grating and a wedged refractive element into any of the interferometer arms, as shown on FIG. 3, or by using both reflective and 1o refractive stepped profile elements and wedged refractive element, and tilting any of the reflective elements, as shown on FIG. 4. The combination of reflective and refractive stepped profile elements as shown on FIG. 4 can be used to achieve wider field of view acceptance angle and therefore larger throughput, according to the method described in [Ring J., Schofield J.W., "Field Compensated Michelson Spectrometers", Applied Optics, v.l l, 1972, p.507]. In addition to the methods mentioned, the stepped retardation can be introduced into the Michelson interferometer by means of replacing plane beamsplitter 9 with a stepped profile beamsplitter (not shown on figures), which can be made similar to the stepped profile mirror 10 with partially reflective instead of fully reflective coating.
Similarly, the stepped retardation can be introduced into a Mach-Zender interferometer.
FIG. 5 shows the Mach-Zender interferometer in which a stepped retardation is introduced by replacing one of the mirrors by a stepped profile reflective element and tilting the other mirror. A
stepped retardation can be introduced into the Mach-Zender interferometer by a variety of methods similar to those used for Michelson interferometer, such as using a stepped profile mirror or beamsplitter instead of one of the plane mirrors or beamsplitters and tilting any of the reflective elements (plane mirror, stepped profile mirror or any of the beamsplitters), or by using a stepped profile refractive element and a wedged refractive element in any of the interferometer arms, or by using a combination of both refractive and reflective elements.
It is known that in the Fourier transform spectrometers based on scanning Michelson interferometers the maximum field-of view (FOV) angle /3max is determined by the resolving power [Griffiths P.R., de Haseth J.A.., Fourier Transform Infrared Spectroscopy, Wiley, NY, 1986]:
= Ov R Z ~ (5) However, in all of the interferometers shown on FIG. 1 - FIG. 5 high values of the resolving power R = y "'~" can be achieved only to the expense of a decrease in the FOV
angle below the Ov value determined by the expression (5), and, therefore, to the expense of a certain loss in the system throughput. This is caused by the following mechanisms:
1. The image of the groove of the stepped profile element on the detector array is blurred due to the divergence of the input beam caused by finite FOV angle ~3 of the input collimating optics 1 and 2. The blur size is proportional to the product of the FOV angle (3 and the distance between the detector plane and the surface of the stepped profile element. If the blur size exceeds the distance between the images of the adjacent grooves on the detector, the partial interferograms will overlap. In order to have a large density of grooves to achieve high values of the resolving power, the FOV angle in the direction orthogonal to the direction of grooves has to be decreased.
2. The interferogram contrast is deteriorated at high retardation values and finite FOV angle found in the interferometers with the transversal shear between the wavefronts interfering on the detector plane. This mechanism was studied in [Junttila M.-L., Kauppinen J., Ikonen E., "Performance Limits of Stationary Fourier Spectrometers", Journal of the Optical Society of America A, v.8, No 9, 1991, p.1457]). The extent of deterioration is proportional to the product of the FOV angle in the direction along the grooves of the stepped profile element and the distance d between the detector plane and the line of virtual wavefront crossing (i.e. the line of wavefront crossing where there is no transversal shear between two wavefronts, as shown on FIG.1). In order to achieve high values of the resolving power, the FOV angle in the direction parallel to the direction of grooves must be decreased.
A possible way to avoid both problems, and thereby preserve the throughput advantage associated with Fourier transform spectrometers, is to image both the surface of the stepped profile element and the line of the virtual wavefront crossing onto the detector array by means of appropriate imaging optics. Although the surface of the stepped profile element and the line of the virtual wavefront crossing may not coincide, they can be imaged independently on the detector plane by means of two cylindrical imaging optical units orthogonal with respect to each other, since the direction of the image blur of the stepped profile element and the direction of the interferogram blur caused by wavefront shear are orthogonal to each other. In this case the blur of the image of the stepped profile element and the deterioration of the interferogram contrast will be determined by the product of the FOV angle and the longitudinal aberrations of the imaging optics, rather than the product of the FOV angle and the distance between the detector plane and the surface of stepped profile element or the line of the virtual wavefront crossing. In the interferometer shown on FIG. 2 the surface of the virtual wavefront crossing coincide with the surface of the stepped profile mirror. Therefore, in this case, only one imaging element (made as a refractive, reflective, or any combination of refractive and reflective elements, such as spherical or aspherical mirrors and lenses) can be used instead of two cylindrical imaging elements. One of the simplest optical configurations of the reflective Michelson interferometer with stepped retardation that takes advantage of the mentioned imaging solution is shown on FIG. 6, where imaging objective 16 is installed into the reflective interferometer shown on FIG.
2 in order to increase the throughput of the interferometer. However, in order to increase the throughput, imaging optics made of a pair of cylindrical imaging optical units can be introduced in any of the above-mentioned interferometer configurations, both Michelson and Mach-Zender, with stepped retardation introduced by reflective or refractive elements or a combination of both.
Let us estimate the limitations of the resolving power of the interferometer shown on FIG. 6.
Apparently, the detector plane should be tilted to accommodate for the tilted image of the stepped profile mirror. Consider the blur of the image of stepped profile mirror 10 on the detector in the direction orthogonal to the direction of the grooves. This blur is determined by the convolution of the diffraction pattern caused by the diffraction of the beam on system aperture A, and the aberration pattern caused by imaging optics aberrations. The maximum characteristic size of the image blur Da can be approximately characterized as the sum of spot sizes caused by each of these effects ~a = ~ad~~r + U(.LUberr - "'max lF /# ) + ~Lub , where Lab is the maximum value of the longitudinal aberrations of the imaging objective 16, F/#
is the F-number of the imaging objective, 7~",ax is the maximum wavelength of the analyzed radiation. The images of the grooves should not overlap, therefore the condition for the image blur is E = Da l a < 1, where a is the groove's width and ~ is the relative blur of the groove image.
Therefore, the maximum number of grooves will be:
_ A _ _A~ _ A~ ~,~) Km~ amin Da ~",a,~ (F l#) + /3L~b ' and the maximum achievable resolving power will be:
_ Ncol Kmax _ Ncol AE
Rmax 2 2(/Lmax lF l# ) + /3Lub ~ ' where N~o, is the number of pixels in the detector array column along the direction of the stepped element grooves. By substituting /3 = R z , according to (5), and solving the quadratic equation, Rmax can be expressed as:

z R Lub 1 + N~~r A~~,",~ (F l# ) -1 m~ = 2~.,~ (F l#) Lab ~ (8)' For typical values ar=0.25, N~o,=1000, F/# =4 and ~,",~ 1000 nm, and for a real imaging objective with a typical value of longitudinal aberrations Lab=2 mm, the maximum resolving power calculated from (8) will be R"~ = 45000, and the number of grooves per millimeter will be K l mm,~ _ "'~" =10 . The free spectral range of the measurements is limited only by AN,~W
the detector response and transmission of optics. For interferometers with fused silica or reflective optics and thinned back-illuminated CCD detector, the free spectral range can be from 100000 to 500000 cm ~, so that the relative free spectral range is S = y"'~"
ym'" = 0.8 . In this V max case, the SR factor can be as much as 40000 for the example given. The practical limitation on the resolving power is determined by the dimensions of the optical assembly, namely, by the focal length of the imaging objective necessary to provide sufficiently small aberrations required by the expression (8). Therefore, the present invention can provide a substantial increase in the SR-factor and resolving power of static Fourier transform spectrometers as compared with static Fourier transform spectrometers of the prior art.
In addition to high resolving power, the Fourier transform spectrometers based on the present invention have all the mentioned advantages of static Fourier transform spectrometers with respect to the scanning Fourier transform spectrometers and dispersive spectrometers, such as throughput advantage with respect to the dispersive spectrometers, large wavenumber range per scan, compact, rugged and inexpensive design with no moving parts, relaxed requirements to the surface flatness of the optical components, insensitivity to stray light and input light intensity fluctuations.
However, the static Fourier transform spectrometers of the present invention, as well as other static Fourier spectrometers, are relatively sensitive to the light intensity nonuniformity over the beam aperture caused by cosmetic defects and contamination of optics, as compared to the scanning Fourier transform spectrometers and dispersive spectrometers.
Even though the mentioned nonuniformity can be corrected to some extent in the data processing software, certain residual systematic noise will always be left in the interferogram and degrade the signal-to-noise ratio of the final spectrum. Therefore, in order to provide a high dynamic range of the measurements, the static Fourier transform spectrometers usually require optical components with clean and smooth surfaces and minimal cosmetic defects.
The present invention was experimentally verified in the visible spectral range using the interferometer configuration shown on FIG. 6. A regular grade 7v./4 plane mirror, pellicle beamsplitter and achromatic doublet with 40 mm focal length were used for components 11, 9 and 16 respectively, a reflective grating with 4 grooves per millimeter was used as a stepped profile minor 10, and a 512x760 pixels front-illuminated CCD was used as a detector array 5.
FIG. 7 shows a piece of the interference pattern obtained with the experimental prototype of the spectrometer illuminated by a He-Ne laser. FIG. 8 shows the spectrum of a Mercury discharge lamp obtained with the experimental prototype of the spectrometer, and FIG. 9 shows the magnified part of the FIG. 8 spectrum with the resolved Mercury doublet 577-579 nm. In order to demonstrate the highest resolution, the original interferogram was not apodized. This, together with cosmetic defects of the grating and dust on the optical elements, gave rise to small satellite peaks and excessive noise around the emission lines. According to FIG. 9, the resolution of the experimental prototype is 0.2 nm at 577 nm, which corresponds to the resolving power of 4200 at the minimum 400 nm wavelength.
It is understood that the present invention is not confined to the particular embodiments set forth herein as illustrative, but embraces all such modified forms thereof as come within the scope of the following claims.

PUBLICATIONS CITED:
1. Barnes T.H., "Photodiode Array Fourier Transform Spectrometer with Improved Dynamic Range", Applied Optics, v.24, 1985, p.3702.
2. Barnes T.H., Eiju T., Matsuda K., "Heterodyne Photodiode Array Fourier Transform Spectrometer", Applied Optics, v. 25, 1986, p.1864.
3. Bliss M., Craig R.A., Anheier N.C., "Demonstration of a static Fourier transform spectrometer", Fiber Optic and Laser Sensors and Applications; Including Distributed and Multiplexed Fiber Optic Sensors Trll, Proc. SPIE Vol. 3541, 1999, p. 103.
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Claims

1. A Fourier transform spectrometer for analyzing electromagnetic radiation which comprises:
(a) A double beam Michelson interferometer constructed to receive a single input beam and produce an output beam composed of two beams formed from the input beam and recombined so that the retardation between two said beams .DELTA.(x,y), being a function of arbitrary coordinates x and y in the wavefront plane of one of the output beams, is a monotonically increasing function along the direction of coordinate x, and a stepped function along the direction of coordinate y;

(b) input means for receiving the radiation from the source and producing a collimated beam which is provided as the input beam to the said interferometer;
(c) an imaging two-dimensional detector array placed on the way of the output beam from the interferometer; and (d) means for processing the image sensed by said detector array to determine the spectral content of the received radiation, wherein the interferometer comprises at least one beamsplitting element serving as means to split the input beam into two beams and then recombine them together, and two reflective elements serving as means to reflect the said two beams back to the beamsplitter in order for them to be recombined, with the positions of said beamsplitter and mirrors being fixed during one measurement period.

2. A spectrometer of claim 1 wherein the reflecting surfaces of the beamsplitter and one of the reflecting elements are plane, and the reflecting surface of the other reflecting element has a stepped profile.

3. A spectrometer of claim 1 wherein the reflecting surfaces of both reflecting elements are plane, and the reflecting surface of the beamsplitter has a stepped profile.

4. A spectrometer of claim 1 wherein the reflecting surfaces of the beamsplitter and both reflecting elements are plane, and which further includes a refractive element with stepped profile placed in one of the interferometer arms, and a wedged refracting element placed in one of the interferometer arms.

5. A spectrometer of claim 2 which further includes a refractive element with stepped profile placed in one of the interferometer arms, and a wedged refracting element placed in one of the interferometer arms.

6. A spectrometer of claim 1 which further includes a means for imaging the output beam from the interferometer onto the said detector array in order to increase the contrast of the interference pattern on the detector array.

7. A spectrometer of claim 6 wherein said imaging means consist of two imaging optical units, one of which is translationary symmetrical with respect to the direction of coordinate x, as defined in claim 1(a), and the other is translationary symmetrical with respect to the direction of coordinate y, as defined in claim 1(a).

8. A spectrometer of claim 6 wherein said imaging means consist of imaging optical unit axially symmetrical with respect to the direction of the propagation of the interferometer output beam, the reflecting surfaces of the beamsplitter and one of the reflecting elements are plane, and the reflecting surface of the other reflecting element has a stepped profile.

9. A Fourier transform spectrometer for analyzing electromagnetic radiation which comprises:
(a) A double beam Mach-Zender interferometer constructed to receive a single input beam and produce an output beam composed of two beams formed from the input beam and recombined so that the retardation between two said beams .DELTA.(x,y), being a function of arbitrary coordinates x and y in the wavefront plane of one of the output beams, is a monotonically increasing function along the direction of coordinate x, and a stepped function along the direction of coordinate y;
(b) input means for receiving the radiation from the source and producing a collimated beam which is provided as the input beam to the said interferometer;
(c) an imaging two-dimensional detector array placed on the way of the output beam from the interferometer; and (d) means for processing the image sensed by said detector array to determine the spectral content of the received radiation, wherein the interferometer comprises at least one beamsplitting element serving as means to split the input beam into two beams, another beamsplitting element serving as means to recombine said beams together, and two reflective elements serving as means to direct the said two beams to the second recombining beamsplitter in order for them to be recombined, with the positions of said beamsplitters and mirrors being fixed during one measurement period.

10. A spectrometer of claim 9 wherein the reflecting surfaces of the beamsplitters and one of the reflecting elements are plane, and the reflecting surface of the other reflecting element has a stepped profile.

11. A spectrometer of claim 9 wherein the reflecting surfaces of both reflecting elements are plane, and the reflecting surface of one of the beamsplitters has a stepped profile.

12. A spectrometer of claim 9 wherein the reflecting surfaces of both beamsplitters and both reflecting elements are plane, and which further includes a refractive element with stepped profile placed in one of the interferometer arms, and a wedged refracting element placed in one of the interferometer arms.

13. A spectrometer of claim 10 which further includes a refractive element with stepped profile placed in one of the interferometer arms, and a wedged refracting element placed in one of the interferometer arms.

14. A spectrometer of claim 9 which further includes a means for imaging the output beam from the interferometer onto the said detector array in order to increase the contrast of the interference pattern on the detector array.

15. A spectrometer of claim 14 wherein said imaging means consist of two imaging optical units, one of which is translationary symmetrical with respect to the direction of coordinate x, as defined in claim 9(a), and the other is translationary symmetrical with respect to the direction of coordinate y, as defined in claim 9(a).

16. A spectrometer of claim 14 wherein said imaging means consist of imaging optical unit axially symmetrical with respect to the direction of the propagation of the interferometer output beam, the reflecting surfaces of the beamsplitters and one of the reflecting elements are plane, and the reflecting surface of the other reflecting element has a stepped profile.