Note: Descriptions are shown in the official language in which they were submitted.
CA 02302994 2000-03-24
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,
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CA 02302994 2000-03-24
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],
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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.
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CA 02302994 2000-03-24
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
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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.
CA 02302994 2000-03-24
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
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CA 02302994 2000-03-24
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
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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
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CA 02302994 2000-03-24
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,
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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).
0
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
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CA 02302994 2000-03-24
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
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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
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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:
13
CA 02302994 2000-03-24
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
210-3R
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
14
CA 02302994 2000-03-24
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.
CA 02302994 2000-03-24
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