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Claims  |
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We claim:
1. Apparatus for analyzing light having spectral components periodic in
frequency, comprising:
a. light conditioning means for collecting, collimating and transmitting
said light;
b. first interferometric means adapted to receive said light, selectively
separate a preselected periodic spectrum therefrom and transmit said
spectrum in the form of a fringe, said first interferometric means having
interference-producing means for providing a plurality of transmission
windows regularly spaced in frequency, the frequency spacing between
adjacent windows being adjusted to equal substantially the frequency
difference between spectral components of the same periodic spectrum and
to cause the transmission peaks for adjacent orders to coincide with the
spectral lines of the components, whereby said fringe has an intensity
derived from and substantially equal to the combined intensities of said
spectral lines; and
c. second interferometric means in series with said first interferometric
means and adapted to receive said fringe and transmit spectra thereof in
the form of a modified fringe providing a detectable signal, said second
interferometric means having interference-producing means for providing a
plurality of transmission windows regularly spaced in frequency, the
frequency spacing between adjacent windows being adjusted so that (1) the
ratio of the frequency spacing between adjacent windows of the first
interferometric means to the corresponding frequency spacing between
adjacent windows of the second interferometric means is an odd integer, n,
greater than one and (2) the transmission peaks for adjacent nth orders
coincide with the spectral lines of the components, whereby the proportion
of intensity of said modified fringe derived from the combined intensities
of said spectral lines of said spectrum is greater than that of said
fringe.
2. Apparatus for analyzing light having spectral components periodic in
frequency, comprising:
a. light conditioning means for collecting, collimating and transmitting
said light;
b. first interferometric means adapted to receive said light and transmit a
preselected periodic spectrum in the form of a fringe, said first
interferometric means having interference-producing means for providing a
plurality of transmission windows regularly spaced in frequency, the
frequency spacing between adjacent windows being adjusted to equal
substantially an odd integral submultiple, n.sub.1, of the frequency
difference between spectral components of the same periodic spectrum and
to cause the transmission peaks for adjacent n.sub.1 th orders to coincide
with the spectral lines of the components, whereby said fringe has an
intensity derived in part from the combined intensities of said spectral
lines; and
c. second interferometric means in series with said first interferometric
means and adapted to receive said fringe and transmit said spectrum
thereof in the form of a modified fringe providing a detectable signal,
said second interferometric means having interference-producing means for
providing a plurality of transmission windows regularly spaced in
frequency, the frequency spacing between adjacent windows being adjusted
to equal substantially an odd integral submultiple, n.sub.2, of the
frequency difference between spectral components of said periodic
spectrum, the ratio of the frequency spacing between adjacent windows of
the first interferometric means to the corresponding frequency spacing
between adjacent windows of the second interferometric means being a ratio
of odd integers, r = (n.sub.2 /n.sub.1), and the transmission peaks for
each n.sub.2 th interval of orders coinciding with the spectral lines of
the components, whereby the proportion of intensity of said modified
fringe derived from the combined intensities of said spectral lines of
said spectrum is greater than that of said fringe.
3. Apparatus as recited in claim 2, including light source means for
generating monochromatic light and projecting means for directing said
monochromatic light through gaseous material to produce scattered light
having said spectral components periodic in frequency.
4. Apparatus as recited in claim 2, including detecting means for
indicating the intensity of said signal.
5. Apparatus as recited in claim 4, including signal conditioning means
comprising modulating means for modulating the phase difference between
interfering rays of said light so as to vary the intensity of the modified
fringe, the modulating range being no greater than the frequency spacing
between adjacent orders of said modified fringe, and synchronous detection
means for detecting the intensity variation of said modified fringe,
whereby said modified fringe can be identified.
6. Apparatus as recited in claim 5 wherein said modulating means has a
modulating range of about one half the frequency width of said modified
fringe.
7. Apparatus as recited in claim 4, including indicating and recording
means for displaying said signal.
8. Apparatus as recited in claim 5 wherein said modulating means is a
piezoelectric cylinder and said synchronous detection means is a phase
sensitive detection system.
9. Apparatus as recited in claim 3 wherein said light source means is
provided with means for projecting light having a line width and frequency
stability about equal to or less than the instrumental width of said first
interferometric means.
10. Apparatus as recited in claim 3 wherein said light source means is a
pulsed laser.
11. Apparatus as recited in claim 10 wherein said laser is associated with
a time gated electronic detection system having (1) means for measuring
the time interval required to send a pulse from said laser into a sample
of said gaseous material and receive a return signal caused by light
scattered therein and (2) means for measuring the amplitude of said return
signal.
12. Apparatus as recited in claim 2 wherein said preselected spectrum is
the spectrum produced by Raman scattering of a minor constituent of a
gaseous material.
13. Apparatus as recited in claim 12 wherein said gaseous material is air.
14. Apparatus as recited in claim 3 wherein said light source means is
adapted to project plane polarized light and said apparatus includes
polarizing means for receiving said scattered light and transmitting the
light to said first interferometric means.
15. Apparatus as recited in claim 14 wherein said polarizing means is
adapted to reduce the intensity of polarized Rayleigh scattered light to a
degree described by the depolarization ratio thereof and to reduce the
intensity of the Raman scattered light to a substantially smaller degree.
16. Apparatus as recited in claim 2 wherein at least one of said first and
second interferometric means is a solid etalon having temperature control
means associated therewith for adjusting the optical path length thereof.
17. Apparatus as recited in claim 8 including means for applying to said
cylinder a voltage having a square wave form, the limits of said voltage
being adjusted so that the intensity of said modified fringe alternates
between its maximum and minimum values, means for determining for each
cycle of said voltage the difference in photon count between said maximum
and minimum values of said modified fringe to produce a signal count, and
means for accumulating said signal count for a preselected period of time
over a preselected number of cycles of said square wave.
18. Apparatus as recited in claim 17 including means for varying the
preselected time period and the preselected number of cycles inversely
with the intensity of said modified fringe.
19. Apparatus as recited in claim 8, wherein said phase sensitive detection
system is a lock-in amplifier.
20. Apparatus as recited in claim 5 wherein said modulating means is a
piezoelectric cylinder and said synchronous detection means is a photon
counting system.
21. Apparatus as recited in claim 3, wherein said light source means
includes signal conditioning means for varying the frequency of said
monochromatic light to modulate the phase difference between interfering
rays of said scattered light and vary the intensity of said modified
fringe.
22. Apparatus as recited in claim 21, wherein said light source means is a
tunable dye laser.
23. Apparatus as recited in claim 21, wherein said light source means is an
optical parametric oscillator.
24. A method of spectroscopically analyzing light having spectral
components periodic in frequency, comprising the steps of:
a. collecting, collimating and transmitting said light;
b. interferometrically separating a preselected periodic spectrum from said
light and transmitting said spectrum in the form of a fringe having an
intensity derived from and substantially equal to the combined intensities
of said spectral components by directing said light through a first
plurality of transmission windows regularly spaced in frequency, the
frequency spacing between adjacent windows, or first spectral range, being
equal substantially to the frequency difference between adjacent spectral
components of the same periodic spectrum and the transmission peaks for
adjacent orders coinciding with the spectral lines of the components;
c. interferometrically separating said periodic spectrum from said fringe
and transmitting said separated spectrum in the form of a modified fringe
providing a detectable signal by directing said spectrum of said fringe
through a second plurality of transmission windows regularly spaced in
frequency, the frequency spacing between adjacent windows, or second
spectral range, being such that (1) the ratio of the first spectral range
to the second spectral range is an odd integer, n, greater than one and
(2) the transmission peak for each nth interval of orders coincide with
the spectral lines of the components, whereby the proportion of intensity
of said modified fringe derived from the combined intensities of said
spectral lines is greater than that of said fringe.
25. A method of spectroscopically analyzing light having spectral
components periodic in frequency, comprising the steps of:
a. collecting, collimating and transmitting said light;
b. interferometrically transmitting a preselected periodic spectrum in the
form of a fringe having an intensity derived from and substantially equal
to the combined intensities of said spectral components by directing said
light through a first plurality of transmission windows regularly spaced
in frequency, the frequency spacing between adjacent windows, or first
spectral range, being equal substantially to an odd integral submultiple,
n.sub.1, of the frequency difference between adjacent spectral components
of the same periodic spectrum and the transmission peaks for adjacent
orders coinciding with the spectral lines of the components;
c. interferometrically separating said periodic spectrum from said fringe
and transmitting said separated spectrum in the form of a modified fringe
providing a detectable signal by directing said spectrum of said fringe
through a second plurality of transmission windows regularly spaced in
frequency, the frequency spacing between adjacent windows, or second
spectral range, being equal substantially to an odd integral submultiple,
n.sub.2, of the frequency difference between spectral components of said
periodic spectrum, the ratio of the first spectral range to the second
spectral range being a ratio of odd integers, r = (n.sub.2 /n.sub. 1), and
the transmission peak for each n.sub.2 th interval of orders coinciding
with the spectral lines of the components, whereby the proportion of
intensity of said modified fringe derived from the combined intensities of
said spectral lines of said spectrum is greater than that of said fringe.
26. Apparatus for analyzing light having spectral components periodic in
frequency, comprising:
a. light conditioning means for collecting, collimating and transmitting
said light;
b. second interferometric means adapted to receive said light and transmit
spectra thereof in the form of a fringe;
c. first interferometric means in series with said second interferometric
means and adapted to receive said fringe, selectively separate a
preselected periodic spectrum therefrom and transmit said spectrum in the
form of a modified fringe, said first interferometric means having
interference producing means for providing a plurality of transmission
windows regularly spaced in frequency, the frequency spacing between
adjacent windows being adjusted to equal substantially the frequency
difference between spectral components of the same periodic spectrum and
to cause the transmission peaks for adjacent orders to coincide with the
spectral lines of the components, whereby said fringe has an intensity
derived from and substantially equal to the combined intensities of said
spectral lines; and
d. said second interferometric means having interference-producing means
for providing a plurality of transmission windows regularly spaced in
frequency, the frequency spacing between adjacent windows being adjusted
so that (1) the ratio of the frequency spacing between adjacent windows of
the first interferometric means to the corresponding frequency spacing
between adjacent windows of the second interferometric means is an odd
integer, n, greater than one and (2) the transmission peaks for adjacent
nth orders coincide with the spectral lines of the components, whereby the
proportion of intensity of said modified fringe derived from the combined
intensities of said spectral lines of said spectrum is greater that that
of said fringe. |
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Claims |
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Description |
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BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to the field of spectroscopy and more particularly
to apparatus for detecting and quantitatively measuring gaseous
constituents through simultaneous transmission of their periodic spectra.
2. Description of the Prior Art
In apparatus used for spectroscopic gas analysis, light produced by
scattering in gas is collected and transmitted to an interferometer
scanned to transmit selectively and simultaneously the rotational Raman
spectra of a preselected constituent of the gas. The output of the
interferometer is converted to a detectable signal and displayed.
One of the major problems with such apparatus is the difficulty of
analyzing quantities of gaseous constituents present in the low parts per
million range. The output signal from the interferometer represents a
relatively low intensity signal that is frequently altered or obscured by
spectral interference between rotational Raman spectra of the gaseous
constituent being analyzed and spectra of coexistent interfering gases.
The problem is particularly troublesome when the gaseous constituent being
analyzed is located at a point distant from the apparatus. To alleviate
such problems it has been necessary to provide the apparatus with highly
sensitive forms and combinations of detectors, sources, filteres, control
systems and the like, which are relatively expensive.
SUMMARY OF THE INVENTION
The present invention provides apparatus having increased sensitivity for
spectroscopically analyzing light having spectral components periodic in
frequency. Such apparatus has light conditioning means for collecting,
collimating and transmitting the light. A first interferometric means
adapted to receive the light selectively separates a preselected periodic
spectrum therefrom and transmits the spectrum in the form of a fringe.
Such first interferometric means has interference-producing means for
providing a plurality of transmission windows regularly spaced in
frequency. The frequency spacing between adjacent windows, or spectral
range of the first interferometric means is adjusted to equal
substantially the frequency difference between spectral components of the
same periodic spectrum and to cause the transmission peaks for adjacent
orders to coincide with the spectral lines of the components, whereby the
fringe has an intensity derived from and substantially equal to the
combined intensities of the spectral lines. A second interferometric means
in series with the first interferometric means is adapted to receive the
fringe and transmit the spectrum in the form of a modified fringe
providing a detectable signal. Such second interferometric means has
interference-producing means for providing a plurality of transmission
windows regularly spaced in frequency. The frequency spacing between
adjacent windows, or spectral range, of the second interferometric means
is adjusted so that (1) the ratio of the spectral range of the first
interferometric means to the spectral range of the second interferometric
means is an odd integer, n, greater than one and (2) the transmission
peaks for adjacent nth orders coincide with the spectral lines of the
components, whereby the proportion of intensity of the modified fringe
derived from the combined intensities of the spectral lines of the
spectrum is greater than that of the fringe.
Further, the invention provides a method for spectroscopically analyzing
light having spectral components periodic in frequency, comprising the
steps of collecting, collimating and transmitting the light;
interferometrically separating a preselected periodic spectrum from said
light and transmitting said spectrum in the form of a fringe having an
intensity derived from and substantially equal to the combined intensities
of said spectral lines by directing said light through a first plurality
of transmission windows regularly spaced in frequency, the frequency
spacing between adjacent windows, or first spectral range, being equal
substantially to the frequency difference between adjacent spectral
components of the same periodic spectrum and the transmission peaks for
adjacent orders coinciding with the spectral lines of the components;
interferometrically separating said periodic spectrum from said fringe and
transmitting said separated spectrum in the form of a modified fringe
providing a detectable signal by directing said spectra of said fringe
through a second plurality of transmission windows regularly spaced in
frequency, the frequency spacing between adjacent windows, or second
spectral range, being such that (1) the ratio of the first spectral range
to the second spectral range is an odd integer, n, greater than one and
(2) the transmission peaks for adjacent nth orders coincide with the
spectral lines of the components, whereby the proportion of intensity of
said modified fringe derived from the combined intensities of said
spectral lines of the spectrum is greater than that of said fringe.
Although the light which is sujected to analysis can be received from an
external source, it is usually produced by the apparatus. Thus, the
apparatus preferably has light source means for generating monochromatic
light. A projecting means associated with the light source means directs
the monochromatic light through the gaseous material to produce scattered
light having spectral components periodic in frequency. Light conditioning
means are provided for collecting, collimating and transmitting the
scattered light to interferometric means of the type described.
Several known interferometric means may be adapted for use with the above
apparatus. Preferably, each of the interferometric means is a Fabry-Perot
interferometer (FPI), the first interferometric means having a mirror
separation, d.sub.1, adjusted to transmit substantially all rotational
lines of a preselected molecular species, or constituent, of the gaseous
material and the second interferometric means having a mirror separation,
d.sub.2, adjusted to transmit substantially all of the aforesaid
rotational lines and reject interfering rotational Raman lines in the
vicinity of the rotational lines of the preselected species. This
condition obtains when
##EQU1##
where d.sub.1 is the mirror separation of the first FPI, d.sub.2 is the
mirror separation of the second FPI, n.sub.1 and n.sub.2 are odd integers,
.mu. is the index of refraction of the medium between the mirrors, and B
is the molecular rotational constant of the species. For a given molecular
species, the rotational constant B is a unique quantity. Thus,
identification of the species emitting a particular rotational Raman
spectrum is made positively by determining the mirror separation of the
FPI at which all the rotational Raman lines of the species are
simultaneously transmitted. Advantageously, combination of the two FPI's
in series produces a larger finesse, that is, much narrower transmission
windows for a given spectral range. Moreover, with two FPI's in series the
contrast factor is increased. Spectral interference is minimized, the
sensitivity of the apparatus is increased and highly sensitive forms and
combinations of light sources, filters and control systems are
unnecessary. Accordingly, the method and apparatus of this invention
permits gaseous constituents to be detected and measured more accurately
and at less expense than systems wherein the spectra are transmitted
through a single interferometric means.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be more fully understood and further advantages will
become apparent when reference is made to the following detailed
description of the preferred embodiments of the invention and the
accompanying drawings in which:
FIG. 1 is a block diagram showing means for increasing the accuracy of
apparatus for the simultaneous transmission of periodic spectra;
FIG. 2 is a schematic diagram of the means of FIG. 1, including,
additionally, means for producing light carrying such spectra;
FIG. 3 is a side view, partially cut away, showing means for modulating the
second interferometric means of FIGS. 1 and 2; and
FIG. 4 is a schematic representation of transmission profiles produced by
the apparatus of FIG. 1 during analysis of light carrying periodic spectra
.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
Light having spectral components periodic in frequency is found in each of
the visible, infrared and ultraviolet frequency regions, at intensities
sufficient to permit detection of the components. As a consequence, the
invention will function with light having a relatively wide range of
frequencies. For illustrative purposes, the invention is described in
connection with apparatus for measuring rotational Raman spectra of
gaseous material scattered by light from the visible frequency region.
When applied in this manner, the invention is particularly suited to
detect and to measure quantitatively minor constituents of a gaseous
material such as air. It will be readily appreciated that the invention
can be practiced using light from any of the foregoing frequency regions,
and that it can be employed for similar and yet diversified uses, such as
the analysis of vibration-rotation spectra, the determination of molecular
gas constants and the like.
Referring to FIG. 1 of the drawings, there is shown preferred apparatus for
the simultaneous transmission of preselected periodic spectra. In the
basic apparatus, shown generally at 10, light conditioning means 12
collect, collimate and transmit light 14 having spectral components
periodic in frequency. First interferometric means 15 receives the light
14, selectively separates a preselected periodic spectrum therefrom and
transmit the spectrum to second interferometric means 16. The latter
further separates the spectrum from the light and transmits the spectrum
in the form of a detectable signal 18.
More specifically, first interferometric means 15 is connected in series
with and between light conditioning means 12 and second interferometric
means 16. First interferometric means 15 receives the light 14,
selectively separates therefrom a preselected periodic spectrum, and sends
the spectrum in the form of a fringe to second interferometric means 16.
The latter receives the fringe and transmits spectrum thereof in the form
of a modified fringe which provides a detectable signal 18. Detecting
means 22 is adapted to receive the signal 18 and determine the intensity
thereof. Signal conditioning means 20 is associated with second
interferometric means 16 and detecting means 22. The signal conditioning
means 20 has modulating means for modulating the phase difference between
interfering rays of light 14 transmitted by the second interferometric
means 16 so as to vary the intensity of the fringe transmitted thereby.
Signal conditioning means 20 also has synchronous (e.g. phase sensitive)
detecting means for detecting the intensity variation of the modified
fringe, whereby the modified fringe can be identified by detecting means
22.
FIG. 2 schematically shows the apparatus of FIG. 1, including,
additionally, means for producing light carrying periodic spectra. As
previously noted, the light 14 which is subjected to analysis can be
received from an external source. Generally, however, the light 14 is
produced by the apparatus 10. Hence, the apparatus 10 has light source
means 36, such as a conventional argon ion laser, a frequency doubled,
pulsed ruby laser or the like, for generating a highly monochromatic,
coherent, collimated beam of radiation. The resolving power of the
interferometric means 16 is best utilized when the light source means 36
is provided with means for projecting light having a line width and
frequency stability about equal to or less than the instrumental width,
described hereinafter in greater detail, of the first and second
interferometric means 15, 16.
The use of a pulsed laser as the light source means 36 together with a time
gated electronic detection system permits determination of pollutant
concentration and location of a sample of gaseous material remote from the
apparatus 10. For example, by providing the apparatus 10 with (1) means
for measuring the time interval required to send a laser pulse into the
sample and receive a return signal caused by light scattered therein and
(2) means for measuring the amplitude of the return signal, the distance
of the sample from the apparatus 10 as well as the pollutant concentration
thereof is readily obtained. A pulsed laser adapted to determine pollutant
concentration and location in the above manner preferably has means for
projecting light having a line width and frequency stability about equal
to or less than the instrumental width of each of the first and second
interferometric means associated therewith. Such means typically includes
a mode selecting etalon disposed in the laser cavity.
A projecting means associated with the light source means 36, introduces
the radiation, schematically represented by ray path 38, into gaseous
material in sample compartment 40 in one direction, which will be
considered to be substantially vertical for convenience in referencing
directions, but may, of course, be in any direction desired. Raman
scattered radiation hereinafter referred to as light 14, from the gaseous
material in sample compartment 40 is collected, collimated and transmitted
to the first interferometric means 15 by light conditioning means 12,
which may be a lens, or other suitable optical system. As long as the
gaseous material contains (1) molecules which are of the linear or
symmetric top variety or (3) slightly asymmetric top molecules which have
nearly periodic spectra, the light 14 will exhibit spectral components
periodic in frequency.
The first interferometric means 15 has interference-producing means for
providing a plurality of transmission windows regularly spaced in
frequency. In addition, the first interferometric means 15 can be provided
with scanning means for variably controlling the frequency of each other.
The interference-producing means is adjusted so that the frequency spacing
between adjacent windows substantially equals the frequency difference
between adjacent spectral components of a preselected periodic spectrum.
Generally such preselected periodic spectrum is that produced by
scattering of a minor constituent of the gaseous material as, for example,
the rotational Raman spectra of carbon dioxide or carbon monoxide in a
sample of air. The scannning means is then adjusted so that the
transmission peaks for adjacent orders coincide with the spectral lines of
such components. When the first interferometric means 15 is adjusted in
the above manner, the preselected spectrum is transmitted therefrom in the
form of a fringe and is thereby selectively separated from the light 14.
The separated spectrum is transmitted by first interferometric means 15 to
second interferometric means 16.
The second interferometric means 16 has interference-producing means for
providing a plurality of transmission windows regularly spaced in
frequency, and can be provided with scanning means for variably
controlling the frequency of each order. Second interferometric means 16
is disposed in series with first interferometric means 15 in the path of
the spectrum transmitted thereby. Its interference-producing means is
adjusted so that (1) the ratio of the frequency spacing between adjacent
windows (spectral range) of the first interferometric means 15 to the
corresponding frequency spacing between adjacent windows (spectral range)
of the second interferometric means 16 is an odd integer, n, greater than
one. The scanning means of the second interferometric means is then
adjusted so that the transmission peaks for adjacent nth orders coincide
with the spectral lines of the components. The separated spectrum is
transmitted by the second interferometric means 16 in the form of a
modified fringe which provides a detectable signal 18.
The first interferometric means transmits simultaneously all of the
spectral lines of the spectrum for the preselected species. Hence, the
fringe transmitted by the first interferometric means 15 is derived from a
plurality of spectral lines and has an intensity substantially equal to
their sum. The second interferometric means 16 also effects simultaneous
transmission of all spectral lines of the spectrum for the preselected
species and thereby produces a signal having the additive feature produced
by the first interferometric means 15. Surprisingly, however, the
combination of interferences produced by the first and second
interferometric means 15, 16 causes the latter to transmit a modified
fringe having a proportion of intensity (derived from the combined
intensities of all spectral lines of said spectrum for the preselected
species) which is as much as three times greater than that of the fringe
transmitted by the first interferometric means 15.
Before describing how the apparatus of FIG. 2 can be used to determine the
intensity of signal 18, it would be helpful to explain the construction
and operation of the first and second interferometric means 15 and 16. The
transmission function of an FPI (I.sub.t /I.sub.0) is given by the Airy
formula
##EQU2##
where T + R + A = 1, I.sub.0 is the intensity of the incident light and
the phase difference .phi. is expressed as .phi. = 4.pi..mu..omega.d for
rays normal to the FPI mirrors. The symbols A, R and T represent,
respectively, the absorbance, reflectance and transmittance of the FPI
mirrors; .mu. is the refractive index of the medium between the FPI
mirrors; d is the FPI mirror separation; .omega. is the frequency of the
incident light expressed in wavenumbers. The parameter F in equation (1)
is defined by the equation.
##EQU3##
The transmission maxima for I.sub.t occur when the condition for
constructive interference is satisfied, i.e., the phase difference .phi.
must be an integral multiple of 2.pi., viz.,
.phi. = 2.pi.m (3)
or
m = 2.mu..omega.d (4)
where m is the order of interference. The transmission maxima for I.sub.t
are referred to in the specification and claims as transmission windows.
For a specific value of the mirror separation, d, the FPI provides a
plurality of transmission windows regularly spaced in frequency. The
frequency spacing, .DELTA..omega., between adjacent windows (or spectral
range) of the FPI is
.DELTA..omega. = (2.mu.d).sup.-.sup.1. (5)
By varying the mirror spacing, d, of the FPI, .DELTA..omega. can be
adjusted to equal substantially the frequency difference between adjacent
spectral components of a specific periodic spectrum. The finesse, N, of
the FPI is equal to the ratio of the spectral range (.DELTA..omega.) to
the full width of the transmission window at half transmission points. The
finesse is equal to
##EQU4##
Consider the case of two Fabry-Perot interferometers in series, one of the
interferometers having a small mirror spacing d.sub.1 and the other having
a large mirror spacing d.sub.2 such that
d.sub.1 /d.sub.2 = n.sub.1 /n.sub.2 (7)
where n.sub.1 and n.sub.2 are odd integers with n.sub.1 <n.sub.2. For
simplicity, it is assumed that n.sub.1 = 1 and n.sub.2 = n. Equation (7)
may be written as
d.sub.2 = nd.sub.1 (8)
where n is an odd integer. The transmission of a rotational Raman spectrum
by two series FPI's with n = 3 is shown schematically in FIG. 4. The
transmission of the first and second interferometers and the combined
transmission of the interferometers in series is represented in FIG. 4 by
transmission profiles A, B and A + B, respectively. Using equation (1) and
neglecting the absorption losses of the Fabry-Perot Mirrors, the combined
transmission, T.sub.c, for the two series connected FPI's may be written
as
##EQU5##
where, for generality, it is assumed that the finesse of the
interferometers is not the same. The parameter F is given by equation (2)
and the phase difference .phi. is defined by .phi. = 4.pi..mu..omega.d.
Using equation (7) and letting .theta. = .phi./2, equation (9) may be
written as
T.sub.c = [1 + F.sub.1 sin.sup.2 .theta..sub.1 + F.sub.2 sin.sup.2
n.theta..sub.1 + F.sub.1 F.sub.2 sin.sup.2 .theta..sub.1 sin.sup.2
n.theta..sub.1 ].sup.-.sup.1. (10)
The maximum values of T.sub.c occur for values of .theta. = m.pi. where m
is an integer. For these values of .theta..sub.1, T.sub.c (max) is unity.
The frequency separation (.DELTA..omega.)between two adjacent maximum
values of T.sub.c may be determined as follows:
(m + 1).pi. - m.pi. = 2.pi..mu..omega.'d.sub.1 - 2.pi..mu..omega."d.sub.1 =
2.pi..mu.d.sub.1 (.DELTA..omega..sub.1) (11)
where .DELTA..omega..sub.1 = .omega.' - .omega." is the spectral range of
the first interferometer with mirror spacing d.sub.1, i.e.,
.DELTA..omega..sub.1 = (2.mu.d.sub.1).sup.-.sup.1. (12)
Therefore the use of the two interferometers in series results in a
spectral range equal to the spectral range of the interferometer with the
smaller mirror spacing. The minimum transmission value of T.sub.c
[equation (10)] occurs for values of .theta..sub.1 = m.pi./2, where m is
an integer. Hence equation (10) yields
T.sub.c (min) = [1 + F.sub.1 + F.sub.2 + F.sub.1 F.sub.2 ].sup.-.sup.1.
(13)
the contrast, C, of the series interferometers is defined by the equation
##EQU6##
In the immediate region of a transmission maximum, the approximation sin
.theta. .perspectiveto. .theta. is valid and equation (10) becomes
T.sub.c = [1 + F.sub.1 .theta..sub.1.sup.2 + n.sup.2 F.sub.2
.theta..sub.1.sup.2 + n.sup.2 F.sub.1 F.sub.2 .theta..sub.1.sup.4
].sup.-.sup.1. (15)
if we let .beta. = .theta..sup.2, then equation (15) may be written as
T.sub.c [1 + (F.sub.1 + n.sup.2 F.sub.2).beta. + n.sup.2 F.sub.1 F.sub.2
.beta..sup.2 ] = 1. (16)
solving equation (16) for .beta. yields
##EQU7##
Since .beta. = .phi..sub.1.sup.2 /4, equation (17) is an expression
describing the behavior of the phase difference in the immediate region of
a transmission window of the two series interferometers. At the
transmission peak, .beta. = 0 and the value of T.sub.c derived from
equation (17) is unity.
The combined finesse, N.sub.c, of the two series interferometers is
determined as follows: At the half transmission points, T.sub.c = 0.5, and
equation (17) yields
##EQU8##
To further simplify this analysis, it is assumed that the finesse of each
interferometer is the same, i.e., F.sub.1 = F.sub.2 = F. Under such
conditions,
##EQU9##
Expressing equation (19) in terms of the phase difference .phi..sub.1/2,
produces the equation
##EQU10##
Since
##EQU11##
where .delta..sub.c is the full width of the transmission window at half
transmission points for the series interferometers, by definition,
##EQU12##
Hence the combined finesse, N.sub.c, is
##EQU13##
For a mirror reflectivity of 0.95, the parameter F [equation (2)] has a
value of 1520. From equation (6), the finesse of each individual
interferometer is equal to 61.2. For n = 3, the overall finesse [equation
(23)] for the two series interferometers is 201.5. The contrast factor for
each individual interferometer may be obtained from equation (1) with the
result that
C = 1 + F. (24)
in the above example, F = 1520 and hence a contrast factor of 1521 is
obtained. From equation (14), with F.sub.1 = F.sub.2 = F, the contrast for
the series interferometers is equal to (1 + 2F + F.sup.2) and for F =
1520, C = 2,313,441. This large contrast is useful for detecting very low
intensity spectral components in the presence of large intensity spectral
components. Therefore the combination of two interferometers in series
results in a high finesse, high resolution instrument with a large
contrast factor.
A computer program was used to estimate the effectiveness of the
series-connected interferometers for rejecting unwanted spectra. The
calculation was performed in connection with detection of rotational Raman
scattering from a small amount (300 ppm) of carbon monoxide in air. For a
given molecular species with a rotational constant B and a centrifugal
distortion constant D, the frequencies of the rotational Raman lines are
.omega..sub.R = .omega..sub.0 .-+. (4B - 6D) (J + 3/2) .+-. 8D(J +
3/2).sup.3 (25)
where .omega..sub.0 is the laser exciting frequency, J is the rotational
quantum number and the upper and lower signs refer to Stokes and
anti-Stokes scattering, respectively. The peak intensity for
Stokes-shifted rotational Raman lines is
##EQU14##
with a similar expression for anti-Stokes shifted lines. The calculated
Raman intensities were multiplied by factors proportional to the
scattering cross-section and concentration for oxygen, nitrogen and carbon
monoxide.
For each rotational Raman line, the intensity transmitted by each
interferometer was calculated using equation (1) and the intensity
transmitted by the series interferometers was determined by using equation
(9). The total transmitted intensity, in each case, was found by adding
the transmitted intensities for all the individual Raman lines. This
calculation is only approximate since the peak intensities of the
individual Raman lines were used rather than the broadened line profiles.
The first interferometer was specified so that its spectral range was
approximately equal to 4B for CO with transmission windows exactly located
at the most intense (J=6) Stokes and anti-Stokes lines for CO. This
occurred for an optical path (.mu.d) equal to 0.0650402 cm. The second
interferometer was specified so that its spectral range had a value equal
to one fifth the spectral range of the first interferometer and its
optical path (.mu.d) was equal to 0.325201 cm. The values of the
transmission functions of equations (1) and (9) will change as the phase
difference .phi. is varied. Since .phi. = 4.pi..mu..omega.d, the
transmission value will change when either the refractive index .mu., the
light frequency .omega. or the mirror separation d is varied. For the
purpose of calculation, the scanning was achieved by keeping the optical
paths (.mu.d) of both interferometers fixed and varying the laser
frequency .omega..sub.0. Experimentally this condition could be realized
by using a tunable dye laser as the excitation source along with fixed
spacing solid Fabry-Perot etalons. The laser frequency for transmitting
the CO signal was calculated to be 19430.29 cm.sup.-.sup.1. The
calculations were performed for two other frequencies on both sides of
.omega..sub.0, viz., .omega..sub.- = .omega..sub.0 - 0.25 cm.sup.-.sup.1
and .omega..sub.+ = .omega..sub.0 + 0.25 cm.sup.-.sup.1. The results are
shown below in Table I.
TABLE I
__________________________________________________________________________
Summary of Calculations of Relative Signal in Detection
of 300 ppm of CO in Air by Using a Single Etalon and
Two Etalons in Series.
__________________________________________________________________________
Excitation
CO Air Ratio of CO to
Frequency
Signal Signal
Air Signals
__________________________________________________________________________
Etalon I .omega..sub.0
0.775 13.31
0.058
only(Spectral
.omega..sub.-
0.093 21.57
4.31.times.10.sup..sup.-3
Range .perspectiveto. 4B)
.omega..sub.+
0.0742 22.16
3.35.times.10.sup..sup.-3
Etalon II .omega..sub.0
0.502 83.0 6.05.times.10.sup..sup.-3
only(Spectral
.omega..sub.-
8.75.times.10.sup..sup.-3
72.8 1.20.times.10.sup..sup.-4
Range .perspectiveto. 4B/5)
.omega..sub.+
5.85.times.10.sup..sup.-3
75.3 7.77.times.10.sup..sup.-5
Etalon I .omega..sub.0
0.452 0.0977
4.63
+ .omega..sub.-
4.62.times.10.sup..sup.-3
0.1471
0.031
Etalon II .omega..sub.+
2.53.times.10.sup..sup.-3
0.1495
0.017
__________________________________________________________________________
For etalon I, the CO-to-air signal ratio is 0.058 at the exciting frequency
.omega..sub.0, whereas the corresponding ratios for etalon II and the
series combination of etalons I and II are 6.05 .times. 10.sup.-.sup.3 and
4.63, respectively. Therefore the series etalon combination offers
improvements in the CO-to-air signal ratio by factors of 79.8 and 761.5 in
comparison to the use of etalon I or etalon II alone.
In the preceding example for carbon monoxide, the relationship between the
mirror spacings for the series connected interferometers is given by
equation (8) with n = 5. In general, however, the mirror spacings for the
series connected interferometers are related according to equation (7),
viz., d.sub.1 /d.sub.2 = n.sub.1 /n.sub.2 where n.sub.1 and n.sub.2 are
odd integers with n.sub.2 >n.sub.1. For this general case, the combined
transmissions, T.sub.c, for the series connected interferometers is given
by equation (9) with the phase differences .phi..sub.1 and .phi..sub.2
obeying the equation
##EQU15##
Since n.sub.2 >n.sub.1, an upper limit on the value of n.sub.2 (relative
to n.sub.1) may be established by requiring that the spectral range
(.DELTA..omega..sub.2) of the second interferometer be greater than the
full width at half transmission points (.delta..sub.1) of the first
interferometer. The finesse of the firsit interferometer is (by
definition) equal to
##EQU16##
and using equation (6), N.sub.1 is also equal to
##EQU17##
From equations (28) and (29), the transmission width, .delta..sub.1, of
the first interferometer is
##EQU18##
since .DELTA..omega..sub.1 = (2.mu.d.sub.1).sup.-.sup.1, equation (30)
becomes
##EQU19##
The spectral range for the second interferometer (.DELTA..omega..sub.2) is
given by the equation
##EQU20##
The upper limit on the value of n.sub.2, namely (n.sub.2)max, may be
obtained by equating equations (31) and (32). The result is
##EQU21##
Therefore, the upper limit on the value of n.sub.2 is equal to the product
of n.sub.1 and the finesse of the first interferometer, N.sub.1.
In the event that light projected by light source means 36 in ray path 38
is plane polarized, a further reduction of the background caused by
Rayleigh scattering can be achieved by positioning polarizing means, such
as a polarizing element, in light path 14 so as to minimize the
transmission of Rayleigh scattered light. With this configuration of the
polarizing means, the Rayleigh scattered light passed through the
polarizing means is reduced to a degree described by the depolarization
ratio thereof. Since the Rayleigh depolarization ratio of light produced
by scattering in gases such as oxygen and nitrogen is about 1 percent, the
polarizing means operates to reduce the intensity of the Rayleigh light
transmitted to the interferometric means by a factor of about 100. The
pure rotational Raman scattered light from sample 40 is depolarized and
hence passes through the polarizing means with a substantially smaller
reduction in intensity, as in the order of about a factor of 2.
As previously noted, modulating means 42 is associated with second
interferometric means 16 for modulating the phase difference, .phi., so as
to vary the intensity of the transmitted fringe. The modulating means 42
can, alternatively, be associated with the first interferometric me | | |
