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The problem of nonlinear phase errors introduced by misalignment of a Michelson interferometer
THE
PROBLEM OF NONLINEAR PHASE ERRORS INTRODUCED BY MISALIGNMENT OF A MICHELSON INTERFEROMETER B. SCHROEDER and R. GEICK Physikalisches Institut. RGntgenring 8, D-8700 Wiirzburg. Germany
Abstract--ln Fourier transform spectroscopy, a slight angular misalignment or canting of one of the mirrors in the Michelson interferometer causes not only a reduction of the oscillatory part of the interferogram but may aho fead to an asymmetry. In order to elucidate the reasons for this ef?ect. we performed calcufations for a Michelson interferometer incfuding the e&cts originating from a source of Finite area and from a finite aperture. Our treatment also takes into account the effect of a thin film beam-splitter in the parallel beam with a relatively large angle of incidence. The beam-splitter may be considered a dispersive medium in the sense that its efficiency varies strongly with frequency. The essential results of our calculations are that nonlinear phase errors arise under the considered conditians depending on the product of the angle of mirror misalignment and the solid angle subtended by the source from the collimator mirror. Experimentally. we studied the occurrence of asymmetric interferograms and nonlinear phase errors due to misalignment for three different types uf interferometers. In the case of lamellar grating interferometer, a misalignment of one of the cam systems caused only a symmetric reduction of the oscillatory part of the interferogram. A similar symmetric reduction with rto asymmetry was observed fur a two-beam interferometer with the beam-splitter placed at a focus. For a Michelson interferometer with a thin film beam-splitter in the paraftel beam {Polytec FIR 3Oj_however, a mirror misa~i~ment of a few hundredths of a degree caused asymmetries in the jnt~rf~f~gram of as much as 30%, in satisfactory quantitative agreement with our theoretical approach.
INTRODUCTXON
is well known in Fourier spectroscopy that a slight angular misalignment or canting of one of the mirrors in a Michelson interferometer causes not only a reduction of the oscillatory part of the interferogram but may also lead to an asymmetry. In two cases, an asymmetric interferogram is obtained. If the two arms of the Michelson interferometer are not optically identical (e.g. a sample in one arm) such an asymmetry will be observed. If, however, the interferometer is optically symmetrical. an asymmetry of the interferogram wilI.be the result of a nonlinear phase error. ffere, we are considering only slow-scan Fourier transform spectrometers for the far-infrared, and not rapid-scan instruments where a symmetric interferogram may be recorded asymmetrically due to the frequency dependence of the response function of the detector and the electronic system. In our discussion we will concentrate on the nonlinear phase error producing an asymmetric interferogram and on the question of how the nonlinear phase error depends on the mirror misalignment. Our investigation is performed in two ways. For several types of interferometers, we have studied the symmetric reduction of the oscillatory part of the interferogram and the occurrence of an asymmetry experimentally. Then, we have calculated the effects of mirror misalignment theoretically. For this purpose, we consider a Michelson interferometer with collimator mirrors and paraflet beam at the thin film beam-splitter. The angle of incidence of the radiation at the beam-splitter is about 45”. In our treatment, we take into account the finite aperture of the optical system and the finite area of the source. The effect of mirror misalignment has been studied by Williams,“’ taking into account only the finite aperture, and by Kunz and Goorvitch”’ far finite aperture and converging beam. It
THEORETICAL CONSIDERATIONS Our theoretical calculations have been performed for a Michelson interferometer of the type shown in Fig. 1. Apart from the thin film beam-spIitter, from the two plane 595
596
H. SCHR~DLR
and R. GEKK
-7 Y
X
Fig.
I. Schematic
diagram
of the
Michelson interferometer treatment.
considered
for
our
theoretical
mirrors (fixed and movable one) and from the collimator mirrors, the path of the radiation is shown which is emitted at a given point of the source of finite area and which is reflected at a certain point of the first collimator mirror. If there were no mirror misalignment, the radiation would reach the same point of the detector giving rise there to the well-known interference effects. In this case, the path difference in the interferometer is given for the particular ray in Fig. 1 by’3-5’ s(1 -- ;-n?)
(1)
where s is the geometrical path difference and H the angle of the parallel beam and the optical axis. since the emitting point of the source of finite area is generally not located on the optical axis. We assume 0 to be a small quantity and have approximated cos 0 by (1 - $9’). For our treatment. one of the plane mirrors is tilted by an angle x as indicated in Fig. 1. Then, the radiation emitted from one point of the source arrives at two different points of the detector. For an ideal optical system and for a detector with rather small correlation between neighbouring points of the detector area, the interference between the two parallel beams (oscillatory part of the interferogram) will be go\‘erned by the overlap of the diffraction patterns which originate from the finite sire of the mirrors (finite aperture). In our case, this overlap is given by J ,(4n%R)
(2)
(47&R) where J1 is the first Bessel function. it the wave radius of the circular mirror limiting the arguments and on a different way, we obtain for (6 = 0, the source one point only) the same result introduced by the beam-splitter, the path difference tilted by an angle a is as follows:
number of the radiation. and R thu aperture. With different physical a finite aperture and a point source as Williams.“’ Except for the effects A in the interferometer with a mirror
A = s(l - $0’) + L(&cos~
+ x2)
(3)
where Lis twice the path travelled by the light from the tilted mirror to the detector and /? is the azimuthal angle of the considered point on the source (p = 0 is in the plane of drawing of Fig. 1). Here again, 0 and z are considered to be small quantities (0 < 1, x 6 1) and only terms up to second order in these quantities are kept.
The problem
of nonlinear
The effect of the beam-splitter is taken into account reflection and transmission coefficients, respectively : I’)’ = - G t,, =
(1 -
,-~~
R,)
,,go ~g
(e-2hr
i
_
e-2iCjl+
597
phase errors
IT)
=
by considering
1 _
RGe-if2K+*)r
=
x=0
e21f
1 _
&
_
its amplitude
Roemzif
(1 - R,)e-” 1 _ K. eezLr-
(4)
The summation of the partial waves in Eqn 4 means taking into account the interference effects in the beam-splitter which cause the wavelength dependence of its efficiency. In Eqn 4, R, is the power reflectivity for the beam-splitter material (dx). For radiation polarized perpendicular and parallel to the plane of incidence, R,, and R,,, are for an angle of incidence of 45”, respectively.
R
01
=
_ n2 2
(
J2n2 JP?-12
1 + 1 1
R
011
,,I T---T 2n = ( V’mm
+
(5)
,2 1
In Eqn 4, P is the phase shift of the light originating from one pass through the beamto the splitter. For the radiation coming from the source and reflected or transmitted plane mirrors, we have
r = and for the radiation
2xi;d
!
Jn’
v’
from the untilted I- = 2Gd
ocosp
- i mirror
jn’-+
n -
I,
1
(6)
to the detector
+ v
! and for that from the tilted mirror
,2-
I9cos p , ,,,2 _ +
(7)
to the detector (8)
For R, (see Eqn 5) we have neglected all influences of the small quantities r and j% and in the phase shift we need take into account only terms up to first order in r and 0 since the thickness d of the beam-splitter is about 5 orders of magnitude smaller than the length L in Eqn 3. It should be noted, however, that these linear terms in c1 and 0 appear in Eqns 6 and 7 only for the case of non-normal incidence. Generally, these terms are proportional to sin 26, where 6 is the angle of incidence (sin 26 = 1 for 6 = 45’). Under ideal conditions, the radiation power converted by the detector to an electrical signal is proportional to: (a) the spectral intensity I(?) emitted by the source: (b) the solid angle 0, subtended by the collimator from the source; (c) the source area A, = nt-’ (assuming a circular source with radius I’). In the following, we consider only the oscillatory part of the interferogram from which the phase errors due to the tilting of the mirror can be extracted. The constant terms of the interferogram are disregarded. For the oscillatory part of the interferogram, i.e. power at the detector, we obtain
P(s) =
I,’ d$ s,’pdpIo2^
- 2(K - ,D + p -
djI4l(;)n,
T)$
COS /3 +
[4R0( 1 - R012 (sin PO + Jr7 sin
‘$
(IT
+
zr,)]
(9)
1)2X)‘] I
B. SCHR~DER
598
and R. GEICK
In Eqn 9, the integrations mean summations of all wavelengths (d;) and over the area of the source (p .dp ‘d/I). Further, the following abbreviations are used :
2ni;A = ~(1 - $7’) with CT= 27~;s and b = 2niL. Integration over B yields a zeroth
+ b(aO cos /I + z2)
Bessel function
according
to
2n cos (A
+
B cos p)
d/j’ = 2n cos A J,(B)
(10)
s0 and integration
over p = f0 leads then to an infinite
series of Bessel functions:
J,(DS)udp=nr’expi[C-~]
z~~~~cos(c--$)x
x i,ih
(i0$r
:x”,“-‘{{)~
+ conjugate
(11)
complex.
Since G/f-’ 2 10m4 6 1 we need only take into account the first two terms of the series with /n = 0 and HI = 1. Along these lines. the final result for the oscillatory part of the interferogram is P(s) =
- Ro)’ (sin’ r. + ry sin 2r0)]
[’ d\;81(,)LI,/4,J1(u) ---[4Ro(l
-_
or2
J2(GKppr)
f'
(12)
($,,,,,) sln[o[l
-
G)
+
where *
KllOT
= bar/f
+ 2(p - ti - p + r)y(r/f):
c&,~ = ba2 + 2(ti - p - p + r)P, A, = m2 (=area
of the source).
In the limit r--+0 and with y(r/‘) t 0. the summations out. and we obtain P(s) = 2
’ I(V)R,A
'
- $sin[ojl
+ (22 + 1)ccy:
4R0(l - R,) sin’ P, [(l - Ro)2 + 4R, sin’ ToI
in Eqn
{cos[+
12 can easily
be carried
- $)]
- $)I}.
Disregarding terms of higher order than linear in or’/f 2. Eqn 13 agrees with standard result for finite source area’3-5) * sin(0r2,i4f’) 4R,,( 1 - R,) sin* To cos[a(i - &:)I. I( ;)n, ‘4, P(s) = 2 ar2i4f2 [(l - Ro)2 + 4Ro sin2 ro]’
the
(14)
In Eqn 14, the factor 4Ro(l - Ro)sin2ro “‘~“~* “’ “‘* C(l _ R,)2 + 4Ro sin2 p,]Z describes
(13)
the beam-splitter
efficiency.
The problem
Next, we derive the interferogram forms of P(s):
of nonlinear
phase
errors
599
the phase error introduced by the tilting of one of the mirrors from in Eqn 12. For this purpose, we have to consider the Fourier trans+z P,(G) =
s -1
s
P(s) cos (27~;s) ds
(15)
+X
P*(F) =
P(s) sin (2Cs)
ds
--*
and the phase error Q(i) is given by :
PA(G)
(16)
Q(C) = arctg ~ Ps( C)
For performing the Fourier transform, we neglect the frequency shift introduced by the finite source area {CT[l - (r’/2f’)] = CJ- 2 7cGsi. The factor 0 = 2C.s in front of J~($;,,,) requires a differentiation i;(a/S) in the Fourier transformed function. For this differentiation, I@) and Jr(a)/a are considered slowly varying functions of Yj and their derivatives are neglected. Also aysin 2r0 is neglected in comparison to sin2F, in the following. In this way, we have obtained P,(iJ and PA(~) analytically from Eqn 12. The expressions are too lengthy to be reproduced here. The expressions can be simplified considerably by developing the functions of II/,,,, and $J,,,, for small yr/f and cry, respectively. and keeping only terms of first order in y/f and 2-y. Then, the summations ZhlIJIT can be carried out. Finally. we obtain for the phase error according to Eqn 16:
@(iq = -hx’
- arctg
R, - cos2f, f 2 tioJ*Wo) 4Rocry - t& (1 - R,)’ - 4R, sin’ r. - Y’ J2(ll/o)
2ro ctgro
1
8RoTosin2Fo (1 -R,)’
f” Ii/o J,(Ic/o) - r’ J2(tio)
2FClctgfo
+ 4R,sin”F,
1
(17)
where $o = but-/f = 27GLcrrJf. The first term ba2 = 27rCLa2 in Eqn 17 is the linear phase error which does not give rise to an asymmetry in the interferogram. The second term represents the nonlinear phase error Q&. For realistic values of the parameters, the argument of the arctg function is rather small (10-3-10-5), except for the zeros of J,(I/I~) which occur approximately at It/o = nrr + (7r/4). In the neighbourhood of these zeros of Jr, the denominator and the numerator of the second term in Eqn 17 become zero at slightly different values of tie. Within an extremely small range of tie, tgQNL increases from positive values near zero to infinity, changes sign, increases further (decrease of absolute value), goes through zero and has small positive values again. This means for QuL a step of 7c in these ranges near the zeros of .J,(tio) (see Fig. 2). For our further considerations, we have approximated GNL by @NL =O;F(S)
= ei%
= 1;O 5 i; s
(PN,. = nn; F(C) = eiaNL = (- 1)“;
5
8Larl.f (18)
4n + 1 < ~ < 4iI + 5
8L~r/f
-
- 8Larlf’
To be able to judge the effect of this nonlinear phase error on an otherwise symmetrical interferogram lo(s) corresponding to a spectrum I,@), we have to consider the Fourier transform F(s) of F(i). Using some formulae which are compiled in Ref. 6 and with the help of some physical arguments, we have performed this Fourier transform and obtain : F(s) = 2 sm nz;LYr
_ 4 i m=O
sin~;;,r++li;~4
[
6 (2,,r + l)LZ ; - s
1
(19)
600
Fig. 2.
Nonlinear
This function
form of Eqn phase @u, (according to the approximate vs wawsumhers nnd reduced quantity $,,.
F(s) has the following
IX) ;ind F(V) = P’\l
properties:
(a) It is an odd function of the tilt angle 2, and thus a resulting asymmetry of the interferogram should be antisymmetric with respect to x. (b) For r--+0. we obtain F(s)-+~(s). That means the interferogram is the original undistorted one. (c) While the first term of Eyn 19 is sytnmetric with respect to .s, the sum over m in the second term will indeed cause asymmetries of the interferogram. (d) The effect of the nonlinear phase error depends on Lxr:f’ (remember I/I* = 3tCLxr;:f’). i.e. on the combined effect of the tilting of one of the mirrors and of the finite size of the source. (e) The nonlinear phase error does not depend on the finite aperture. FLIrtherin~rc. we believe that in actual detector systems the spatial correlation is much grcatcr than the overlap of the d&-action pattern. Therefore. it seems to t>c more realistic to drop the factor J,(-ln~i~R)/(4nrCK) in Eqns 9 and 17. The distorted
+, s
interferogram
lcJit,(S)=
is the convolution
I”(Y) E‘(s -
s’) ds’
I
of F(s) with I,(.s):
(‘0)
-7
For sufficiently small values of Lar;,j’. the first term F(s) in Eqn 19 may he approximated by B(s). Then. I,,,,,(a) becomes a rather simple expression:
The problem
By means of Eqn using for I,(s)‘5’:
21, we have
of nonlinear
performed
phase
comparisons
IIJ(S)= 7tyof ~COS(2niblsi 0
601
errors
to our
experimental
- 40)e-“;n’
findings
(22)
where 4.
= arc sin yO 25,
and
;; = &
- ;t-r:.
Apart from the asymmetry of the interferogram due to the nonlinear phase error 4jN,,, there is a reduction of the oscillatory part of the interferogram which can well be approximated by the Bessel functions with index 1 in Eqn 12:
where the influence of the finite aperture has been dropped in the second line according to the argument given above, Then, the reduction of the oscillatory part of the interferogram depends also on Lctrif. But, of course, it is symmetric with respect to CC.Equation 23 also has been used for a comparison with the experimental results. COMPARISON
WITH
EXPERIMENTAL
DATA
For the comparison of our theoretical approach with experimental data, we employed three different types of Fourier spectrometers. With a lamellar grating spectrometer as well as with a double beam interferometer of the type described by Genzel et al.,“’ a misalignment led to a strong decrease of the oscillatory part of the interferogram, but practically no asymmetry could be generated. Therefore, for the further investigations we employed a Polytec FIR 30 interferometer. The setup is almost identical with that shown in Fig. 1. The tilt angle of the fixed plane parallel mirror was varied continuously around the horizontal axis (Q) or the vertical axis (av) with respect to the plane of the interferometer (cf. Fig. 1). The instrument could not be evacuated as we measured the tilt angle by the deflection of an He-Ne laser beam. For the choice of the optimum beam-splitter some additional facts have to be considered. Typical sizes for our instrument are L 2 130cm and r/f = 0.05. Therefore, an angle of 0.001” will cause a shift of the focus of about 1 mm on the detector area. The Golay cell window diameter is about 5 mm. This indicates that even such low values of the tilt angle--together with the finite aperture-may cause additional shadowing effects which are not contained in our theory. Therefore. the angles should be restricted to lower values. Even then, a comparison of the dependence of the symmetrical reduction of the interferogram (see Eqn 23) on the tilt angle is problematic and such a comparison, therefore, was not considered. The beam-splitter thickness n enters together with the tilt angle a.* If a thicker beam-splitter is employed, the tilt angle has to be increased to obtain the same effects. This increases the additional effects of shadowing and makes a thinner beam splitter favourable. A thinner beam-splitter, on the other hand, means very low values of tilt angle to be measured. Also, the influence of microphony increases, because the instrument has to be operated at atmospheric pressures as mentioned before. For our investigations, we found a 15 pm thick mylar beam-splitter was the optimum compromise between the indicated contrary factors. In Fig. 3, we plotted three typical interferograms, obtained at angles xH = 0 and various values of s(v (zv with respect to the vertical axis). Besides a reduction of the * In the approximative form of the final Eqn 19. the beam-splitter thickness does not enter explicitly. But it affects the behaviour of the nonlinear phase error as given in the more exact form of Eqn 17.
602
B. SCHROEDER and R. GEICK
:b)
Fig. 3. Experimentally observed interferograms (IS /cm beam-splitter) (a) for optimum adjustment (1, = 0). (b) fixed mirror tilted at about 2, = -6 x IO-” with respect to the vertical axis (see text). (c) fixed mirror tilted at about 3, = - 14 x 10~“. For our treatment, the asymmetry A is defined as A = Al;I.
oscillatory part of the interferogram. an increasing asymmetry angle. As a convenient parameter to measure this asymmetry.
is evident with increasing we define
A = Al/I. See Fig. 3 for explanation of I and AI. The negative sign of the angle in Fig. 3 is a convention and shall not ue considered in detail. In Fig. 4, we plotted the dependence of the measured asymmetry on the tilt angle is not contained in our xH around the horizontal axis for s[v = 0. This dependence theory. but in the range of experimental error we find an asymmetry which is an even function of c(~. This is reasonable, as the setup of the spectrometer is symmetrical with respect to the horizontal plane. From Fig. 4, we also estimate that the error in measuring the tilt angle was well below 10-4, which is necessary for our investigations. also not for No asymmetry is caused for a tilt angle of rH = 0, but surprisingly XH I +5 x lo-“. This can be seen also in Fig. 5. Here, we plotted the observed asymmetry A as a function of the tilt angle rv (with respect to the vertical axis) for rH = 0 and rH = 5 x 10e4. At rv = 0. both curves pass A = 0 with good accuracy. The observed asymmetry values are an odd function of rv which is identical with the angle z in the theoretical part. It is obvious that for a fixed and not too large interferograms. For such value of rH there exist three positions CC\,with symmetrical an additional value xv # 0, we plotted an interferogram in Fig. 3 (curve b). The asymmetry is negligible for the first minima. However. the intensity of the first side maxima are not equal. This, together with the lower modulation, enables the experimentor to identify such a wrong adjustment of the interferometer.
The problem
of nonlinear
phase
errors
603
20 -
o-
o-
-10 ,
I -0001
I
0 001
0
=H Fig. 4. Measured
asymmetry
as defined in Fig. 3 vs tilt angle q, around (with respect to the interferometer plane).
the horizontal
axis
For the comparison of these data with our theory, we employed an interferogram of the form of Eqn 22 with values off = 1. y0 = 50 cm -I, b0 = 100 cm - *. This yields a spectrum with a maximum intensity at i = lOOcm_’ and decreasing intensity for C+ cc and C-+0. This spectrum is somewhat similar to the undisturbed spectrum obtained with our instrument with the 15 pm beam-splitter. However, the absorption due to water lines is disregarded in our theoretical calculations as we are interested in the fundamental effects around the central maximum at s = 0. This means that we are dealing with low resolution and would not be able to resolve any narrow line. The calculated values for the asymmetry are plotted in Fig. 6 vs the tilt angle rv. A comparison with the experimental value in Fig. 5 shows a surprisingly good agreement with respect to the numerous simplifications made. Especially, the sign change of the
c ‘0 -
o-
-10 -
d
.
IO -
$jLJL#& . 0
/
x)L
/r o/
I 5x10-4
1 0
I 5x10-*
="
Fig. 5. Measured asymmetry A vs tilt angle r, with respect to the vertical axis of the fixed mirror. Filled circles: rH = 5 x 10e4, open circles: q, = 0 (see also Figs 2 and 3).
f i i
-:
Fig.
6. Calculated
asymmetry
A (see text) vs tilt an& X, of the fixed mirror the wrtical axis for x5, = 0.
Fig. 7. Calcuiatcd intcrfero~r~ms (see text1 for scvcrai till intcrfcrogram): b: x, = -3 x IO-‘: C: 1, = -2.X x IO-‘: comparison the dccrcase of the modulation
with
respect to
an&s x,. a: x, = 0 (UndjstLlrbed d: X, L -3.X x 10~’ For hcttcr wks disregarded.
The problem
of nonlinear
phase errors
605
asymmetry is described correctly. The absolute calculated values are somewhat larger than the corresponding experimental data. However, the order of magnitude is correct. In Fig. 7. we plotted typical calculated interferograms. Again, the agreement with the experimental data in Fig. 3 is good. CONCLUSION
Our experimental and theoretical studies confirm that there arises a error in the interferograms. This error originates from a mirror tilt interferometer of the standard type. Our theoretical considerations yield tory explanation of this nonlinear phase error. Our studies show on that there exist other types of two beam interferometers not showing of the interferogram due to a mirror tilt.
nonlinear phase in a Michelson a very satisfacthe other hand an asymmetry
REFERENCES I. 2. 3. 4. 5. 6.
7.
WILLIAMS,
c.
S..
Appl.
Opf.
5.
1084 (1966). KUNZ. L. W. & D. V. GOORVITH. Appl. Opt. 13. 1077 (1974). CHANTRY. G. W.. Suhn~illiwwrrr Spwrroscop)~. Academic Press. MBLLER. K. D. & G. ROTHSCHILIX Fur I~$wrd Specrroscopy. GHCK. R. IR Fouricv Truwform Sp~rroscopy. Topics iu Curre~tt berg (1975). OBERHETTINGER, F. Fourier E.xpunsion.r. Academic Press, New GENZEL, L.. H. R. CHANDRASEKHAR & J. KUHL. Opr. Commun.
London
(197 I).
Wiley Interscience. Clwmistry.
New York (1971). Vol. 58. p. 73. Springer. Hcldcl-
York (1973). 18, 381 (1976).
PROBLEM OF NONLINEAR PHASE ERRORS INTRODUCED BY MISALIGNMENT OF A MICHELSON INTERFEROMETER B. SCHROEDER and R. GEICK Physikalisches Institut. RGntgenring 8, D-8700 Wiirzburg. Germany
Abstract--ln Fourier transform spectroscopy, a slight angular misalignment or canting of one of the mirrors in the Michelson interferometer causes not only a reduction of the oscillatory part of the interferogram but may aho fead to an asymmetry. In order to elucidate the reasons for this ef?ect. we performed calcufations for a Michelson interferometer incfuding the e&cts originating from a source of Finite area and from a finite aperture. Our treatment also takes into account the effect of a thin film beam-splitter in the parallel beam with a relatively large angle of incidence. The beam-splitter may be considered a dispersive medium in the sense that its efficiency varies strongly with frequency. The essential results of our calculations are that nonlinear phase errors arise under the considered conditians depending on the product of the angle of mirror misalignment and the solid angle subtended by the source from the collimator mirror. Experimentally. we studied the occurrence of asymmetric interferograms and nonlinear phase errors due to misalignment for three different types uf interferometers. In the case of lamellar grating interferometer, a misalignment of one of the cam systems caused only a symmetric reduction of the oscillatory part of the interferogram. A similar symmetric reduction with rto asymmetry was observed fur a two-beam interferometer with the beam-splitter placed at a focus. For a Michelson interferometer with a thin film beam-splitter in the paraftel beam {Polytec FIR 3Oj_however, a mirror misa~i~ment of a few hundredths of a degree caused asymmetries in the jnt~rf~f~gram of as much as 30%, in satisfactory quantitative agreement with our theoretical approach.
INTRODUCTXON
is well known in Fourier spectroscopy that a slight angular misalignment or canting of one of the mirrors in a Michelson interferometer causes not only a reduction of the oscillatory part of the interferogram but may also lead to an asymmetry. In two cases, an asymmetric interferogram is obtained. If the two arms of the Michelson interferometer are not optically identical (e.g. a sample in one arm) such an asymmetry will be observed. If, however, the interferometer is optically symmetrical. an asymmetry of the interferogram wilI.be the result of a nonlinear phase error. ffere, we are considering only slow-scan Fourier transform spectrometers for the far-infrared, and not rapid-scan instruments where a symmetric interferogram may be recorded asymmetrically due to the frequency dependence of the response function of the detector and the electronic system. In our discussion we will concentrate on the nonlinear phase error producing an asymmetric interferogram and on the question of how the nonlinear phase error depends on the mirror misalignment. Our investigation is performed in two ways. For several types of interferometers, we have studied the symmetric reduction of the oscillatory part of the interferogram and the occurrence of an asymmetry experimentally. Then, we have calculated the effects of mirror misalignment theoretically. For this purpose, we consider a Michelson interferometer with collimator mirrors and paraflet beam at the thin film beam-splitter. The angle of incidence of the radiation at the beam-splitter is about 45”. In our treatment, we take into account the finite aperture of the optical system and the finite area of the source. The effect of mirror misalignment has been studied by Williams,“’ taking into account only the finite aperture, and by Kunz and Goorvitch”’ far finite aperture and converging beam. It
THEORETICAL CONSIDERATIONS Our theoretical calculations have been performed for a Michelson interferometer of the type shown in Fig. 1. Apart from the thin film beam-spIitter, from the two plane 595
596
H. SCHR~DLR
and R. GEKK
-7 Y
X
Fig.
I. Schematic
diagram
of the
Michelson interferometer treatment.
considered
for
our
theoretical
mirrors (fixed and movable one) and from the collimator mirrors, the path of the radiation is shown which is emitted at a given point of the source of finite area and which is reflected at a certain point of the first collimator mirror. If there were no mirror misalignment, the radiation would reach the same point of the detector giving rise there to the well-known interference effects. In this case, the path difference in the interferometer is given for the particular ray in Fig. 1 by’3-5’ s(1 -- ;-n?)
(1)
where s is the geometrical path difference and H the angle of the parallel beam and the optical axis. since the emitting point of the source of finite area is generally not located on the optical axis. We assume 0 to be a small quantity and have approximated cos 0 by (1 - $9’). For our treatment. one of the plane mirrors is tilted by an angle x as indicated in Fig. 1. Then, the radiation emitted from one point of the source arrives at two different points of the detector. For an ideal optical system and for a detector with rather small correlation between neighbouring points of the detector area, the interference between the two parallel beams (oscillatory part of the interferogram) will be go\‘erned by the overlap of the diffraction patterns which originate from the finite sire of the mirrors (finite aperture). In our case, this overlap is given by J ,(4n%R)
(2)
(47&R) where J1 is the first Bessel function. it the wave radius of the circular mirror limiting the arguments and on a different way, we obtain for (6 = 0, the source one point only) the same result introduced by the beam-splitter, the path difference tilted by an angle a is as follows:
number of the radiation. and R thu aperture. With different physical a finite aperture and a point source as Williams.“’ Except for the effects A in the interferometer with a mirror
A = s(l - $0’) + L(&cos~
+ x2)
(3)
where Lis twice the path travelled by the light from the tilted mirror to the detector and /? is the azimuthal angle of the considered point on the source (p = 0 is in the plane of drawing of Fig. 1). Here again, 0 and z are considered to be small quantities (0 < 1, x 6 1) and only terms up to second order in these quantities are kept.
The problem
of nonlinear
The effect of the beam-splitter is taken into account reflection and transmission coefficients, respectively : I’)’ = - G t,, =
(1 -
,-~~
R,)
,,go ~g
(e-2hr
i
_
e-2iCjl+
597
phase errors
IT)
=
by considering
1 _
RGe-if2K+*)r
=
x=0
e21f
1 _
&
_
its amplitude
Roemzif
(1 - R,)e-” 1 _ K. eezLr-
(4)
The summation of the partial waves in Eqn 4 means taking into account the interference effects in the beam-splitter which cause the wavelength dependence of its efficiency. In Eqn 4, R, is the power reflectivity for the beam-splitter material (dx). For radiation polarized perpendicular and parallel to the plane of incidence, R,, and R,,, are for an angle of incidence of 45”, respectively.
R
01
=
_ n2 2
(
J2n2 JP?-12
1 + 1 1
R
011
,,I T---T 2n = ( V’mm
+
(5)
,2 1
In Eqn 4, P is the phase shift of the light originating from one pass through the beamto the splitter. For the radiation coming from the source and reflected or transmitted plane mirrors, we have
r = and for the radiation
2xi;d
!
Jn’
v’
from the untilted I- = 2Gd
ocosp
- i mirror
jn’-+
n -
I,
1
(6)
to the detector
+ v
! and for that from the tilted mirror
,2-
I9cos p , ,,,2 _ +
(7)
to the detector (8)
For R, (see Eqn 5) we have neglected all influences of the small quantities r and j% and in the phase shift we need take into account only terms up to first order in r and 0 since the thickness d of the beam-splitter is about 5 orders of magnitude smaller than the length L in Eqn 3. It should be noted, however, that these linear terms in c1 and 0 appear in Eqns 6 and 7 only for the case of non-normal incidence. Generally, these terms are proportional to sin 26, where 6 is the angle of incidence (sin 26 = 1 for 6 = 45’). Under ideal conditions, the radiation power converted by the detector to an electrical signal is proportional to: (a) the spectral intensity I(?) emitted by the source: (b) the solid angle 0, subtended by the collimator from the source; (c) the source area A, = nt-’ (assuming a circular source with radius I’). In the following, we consider only the oscillatory part of the interferogram from which the phase errors due to the tilting of the mirror can be extracted. The constant terms of the interferogram are disregarded. For the oscillatory part of the interferogram, i.e. power at the detector, we obtain
P(s) =
I,’ d$ s,’pdpIo2^
- 2(K - ,D + p -
djI4l(;)n,
T)$
COS /3 +
[4R0( 1 - R012 (sin PO + Jr7 sin
‘$
(IT
+
zr,)]
(9)
1)2X)‘] I
B. SCHR~DER
598
and R. GEICK
In Eqn 9, the integrations mean summations of all wavelengths (d;) and over the area of the source (p .dp ‘d/I). Further, the following abbreviations are used :
2ni;A = ~(1 - $7’) with CT= 27~;s and b = 2niL. Integration over B yields a zeroth
+ b(aO cos /I + z2)
Bessel function
according
to
2n cos (A
+
B cos p)
d/j’ = 2n cos A J,(B)
(10)
s0 and integration
over p = f0 leads then to an infinite
series of Bessel functions:
J,(DS)udp=nr’expi[C-~]
z~~~~cos(c--$)x
x i,ih
(i0$r
:x”,“-‘{{)~
+ conjugate
(11)
complex.
Since G/f-’ 2 10m4 6 1 we need only take into account the first two terms of the series with /n = 0 and HI = 1. Along these lines. the final result for the oscillatory part of the interferogram is P(s) =
- Ro)’ (sin’ r. + ry sin 2r0)]
[’ d\;81(,)LI,/4,J1(u) ---[4Ro(l
-_
or2
J2(GKppr)
f'
(12)
($,,,,,) sln[o[l
-
G)
+
where *
KllOT
= bar/f
+ 2(p - ti - p + r)y(r/f):
c&,~ = ba2 + 2(ti - p - p + r)P, A, = m2 (=area
of the source).
In the limit r--+0 and with y(r/‘) t 0. the summations out. and we obtain P(s) = 2
’ I(V)R,A
'
- $sin[ojl
+ (22 + 1)ccy:
4R0(l - R,) sin’ P, [(l - Ro)2 + 4R, sin’ ToI
in Eqn
{cos[+
12 can easily
be carried
- $)]
- $)I}.
Disregarding terms of higher order than linear in or’/f 2. Eqn 13 agrees with standard result for finite source area’3-5) * sin(0r2,i4f’) 4R,,( 1 - R,) sin* To cos[a(i - &:)I. I( ;)n, ‘4, P(s) = 2 ar2i4f2 [(l - Ro)2 + 4Ro sin2 ro]’
the
(14)
In Eqn 14, the factor 4Ro(l - Ro)sin2ro “‘~“~* “’ “‘* C(l _ R,)2 + 4Ro sin2 p,]Z describes
(13)
the beam-splitter
efficiency.
The problem
Next, we derive the interferogram forms of P(s):
of nonlinear
phase
errors
599
the phase error introduced by the tilting of one of the mirrors from in Eqn 12. For this purpose, we have to consider the Fourier trans+z P,(G) =
s -1
s
P(s) cos (27~;s) ds
(15)
+X
P*(F) =
P(s) sin (2Cs)
ds
--*
and the phase error Q(i) is given by :
PA(G)
(16)
Q(C) = arctg ~ Ps( C)
For performing the Fourier transform, we neglect the frequency shift introduced by the finite source area {CT[l - (r’/2f’)] = CJ- 2 7cGsi. The factor 0 = 2C.s in front of J~($;,,,) requires a differentiation i;(a/S) in the Fourier transformed function. For this differentiation, I@) and Jr(a)/a are considered slowly varying functions of Yj and their derivatives are neglected. Also aysin 2r0 is neglected in comparison to sin2F, in the following. In this way, we have obtained P,(iJ and PA(~) analytically from Eqn 12. The expressions are too lengthy to be reproduced here. The expressions can be simplified considerably by developing the functions of II/,,,, and $J,,,, for small yr/f and cry, respectively. and keeping only terms of first order in y/f and 2-y. Then, the summations ZhlIJIT can be carried out. Finally. we obtain for the phase error according to Eqn 16:
@(iq = -hx’
- arctg
R, - cos2f, f 2 tioJ*Wo) 4Rocry - t& (1 - R,)’ - 4R, sin’ r. - Y’ J2(ll/o)
2ro ctgro
1
8RoTosin2Fo (1 -R,)’
f” Ii/o J,(Ic/o) - r’ J2(tio)
2FClctgfo
+ 4R,sin”F,
1
(17)
where $o = but-/f = 27GLcrrJf. The first term ba2 = 27rCLa2 in Eqn 17 is the linear phase error which does not give rise to an asymmetry in the interferogram. The second term represents the nonlinear phase error Q&. For realistic values of the parameters, the argument of the arctg function is rather small (10-3-10-5), except for the zeros of J,(I/I~) which occur approximately at It/o = nrr + (7r/4). In the neighbourhood of these zeros of Jr, the denominator and the numerator of the second term in Eqn 17 become zero at slightly different values of tie. Within an extremely small range of tie, tgQNL increases from positive values near zero to infinity, changes sign, increases further (decrease of absolute value), goes through zero and has small positive values again. This means for QuL a step of 7c in these ranges near the zeros of .J,(tio) (see Fig. 2). For our further considerations, we have approximated GNL by @NL =O;F(S)
= ei%
= 1;O 5 i; s
(PN,. = nn; F(C) = eiaNL = (- 1)“;
5
8Larl.f (18)
4n + 1 < ~ < 4iI + 5
8L~r/f
-
- 8Larlf’
To be able to judge the effect of this nonlinear phase error on an otherwise symmetrical interferogram lo(s) corresponding to a spectrum I,@), we have to consider the Fourier transform F(s) of F(i). Using some formulae which are compiled in Ref. 6 and with the help of some physical arguments, we have performed this Fourier transform and obtain : F(s) = 2 sm nz;LYr
_ 4 i m=O
sin~;;,r++li;~4
[
6 (2,,r + l)LZ ; - s
1
(19)
600
Fig. 2.
Nonlinear
This function
form of Eqn phase @u, (according to the approximate vs wawsumhers nnd reduced quantity $,,.
F(s) has the following
IX) ;ind F(V) = P’\l
properties:
(a) It is an odd function of the tilt angle 2, and thus a resulting asymmetry of the interferogram should be antisymmetric with respect to x. (b) For r--+0. we obtain F(s)-+~(s). That means the interferogram is the original undistorted one. (c) While the first term of Eyn 19 is sytnmetric with respect to .s, the sum over m in the second term will indeed cause asymmetries of the interferogram. (d) The effect of the nonlinear phase error depends on Lxr:f’ (remember I/I* = 3tCLxr;:f’). i.e. on the combined effect of the tilting of one of the mirrors and of the finite size of the source. (e) The nonlinear phase error does not depend on the finite aperture. FLIrtherin~rc. we believe that in actual detector systems the spatial correlation is much grcatcr than the overlap of the d&-action pattern. Therefore. it seems to t>c more realistic to drop the factor J,(-ln~i~R)/(4nrCK) in Eqns 9 and 17. The distorted
+, s
interferogram
lcJit,(S)=
is the convolution
I”(Y) E‘(s -
s’) ds’
I
of F(s) with I,(.s):
(‘0)
-7
For sufficiently small values of Lar;,j’. the first term F(s) in Eqn 19 may he approximated by B(s). Then. I,,,,,(a) becomes a rather simple expression:
The problem
By means of Eqn using for I,(s)‘5’:
21, we have
of nonlinear
performed
phase
comparisons
IIJ(S)= 7tyof ~COS(2niblsi 0
601
errors
to our
experimental
- 40)e-“;n’
findings
(22)
where 4.
= arc sin yO 25,
and
;; = &
- ;t-r:.
Apart from the asymmetry of the interferogram due to the nonlinear phase error 4jN,,, there is a reduction of the oscillatory part of the interferogram which can well be approximated by the Bessel functions with index 1 in Eqn 12:
where the influence of the finite aperture has been dropped in the second line according to the argument given above, Then, the reduction of the oscillatory part of the interferogram depends also on Lctrif. But, of course, it is symmetric with respect to CC.Equation 23 also has been used for a comparison with the experimental results. COMPARISON
WITH
EXPERIMENTAL
DATA
For the comparison of our theoretical approach with experimental data, we employed three different types of Fourier spectrometers. With a lamellar grating spectrometer as well as with a double beam interferometer of the type described by Genzel et al.,“’ a misalignment led to a strong decrease of the oscillatory part of the interferogram, but practically no asymmetry could be generated. Therefore, for the further investigations we employed a Polytec FIR 30 interferometer. The setup is almost identical with that shown in Fig. 1. The tilt angle of the fixed plane parallel mirror was varied continuously around the horizontal axis (Q) or the vertical axis (av) with respect to the plane of the interferometer (cf. Fig. 1). The instrument could not be evacuated as we measured the tilt angle by the deflection of an He-Ne laser beam. For the choice of the optimum beam-splitter some additional facts have to be considered. Typical sizes for our instrument are L 2 130cm and r/f = 0.05. Therefore, an angle of 0.001” will cause a shift of the focus of about 1 mm on the detector area. The Golay cell window diameter is about 5 mm. This indicates that even such low values of the tilt angle--together with the finite aperture-may cause additional shadowing effects which are not contained in our theory. Therefore. the angles should be restricted to lower values. Even then, a comparison of the dependence of the symmetrical reduction of the interferogram (see Eqn 23) on the tilt angle is problematic and such a comparison, therefore, was not considered. The beam-splitter thickness n enters together with the tilt angle a.* If a thicker beam-splitter is employed, the tilt angle has to be increased to obtain the same effects. This increases the additional effects of shadowing and makes a thinner beam splitter favourable. A thinner beam-splitter, on the other hand, means very low values of tilt angle to be measured. Also, the influence of microphony increases, because the instrument has to be operated at atmospheric pressures as mentioned before. For our investigations, we found a 15 pm thick mylar beam-splitter was the optimum compromise between the indicated contrary factors. In Fig. 3, we plotted three typical interferograms, obtained at angles xH = 0 and various values of s(v (zv with respect to the vertical axis). Besides a reduction of the * In the approximative form of the final Eqn 19. the beam-splitter thickness does not enter explicitly. But it affects the behaviour of the nonlinear phase error as given in the more exact form of Eqn 17.
602
B. SCHROEDER and R. GEICK
:b)
Fig. 3. Experimentally observed interferograms (IS /cm beam-splitter) (a) for optimum adjustment (1, = 0). (b) fixed mirror tilted at about 2, = -6 x IO-” with respect to the vertical axis (see text). (c) fixed mirror tilted at about 3, = - 14 x 10~“. For our treatment, the asymmetry A is defined as A = Al;I.
oscillatory part of the interferogram. an increasing asymmetry angle. As a convenient parameter to measure this asymmetry.
is evident with increasing we define
A = Al/I. See Fig. 3 for explanation of I and AI. The negative sign of the angle in Fig. 3 is a convention and shall not ue considered in detail. In Fig. 4, we plotted the dependence of the measured asymmetry on the tilt angle is not contained in our xH around the horizontal axis for s[v = 0. This dependence theory. but in the range of experimental error we find an asymmetry which is an even function of c(~. This is reasonable, as the setup of the spectrometer is symmetrical with respect to the horizontal plane. From Fig. 4, we also estimate that the error in measuring the tilt angle was well below 10-4, which is necessary for our investigations. also not for No asymmetry is caused for a tilt angle of rH = 0, but surprisingly XH I +5 x lo-“. This can be seen also in Fig. 5. Here, we plotted the observed asymmetry A as a function of the tilt angle rv (with respect to the vertical axis) for rH = 0 and rH = 5 x 10e4. At rv = 0. both curves pass A = 0 with good accuracy. The observed asymmetry values are an odd function of rv which is identical with the angle z in the theoretical part. It is obvious that for a fixed and not too large interferograms. For such value of rH there exist three positions CC\,with symmetrical an additional value xv # 0, we plotted an interferogram in Fig. 3 (curve b). The asymmetry is negligible for the first minima. However. the intensity of the first side maxima are not equal. This, together with the lower modulation, enables the experimentor to identify such a wrong adjustment of the interferometer.
The problem
of nonlinear
phase
errors
603
20 -
o-
o-
-10 ,
I -0001
I
0 001
0
=H Fig. 4. Measured
asymmetry
as defined in Fig. 3 vs tilt angle q, around (with respect to the interferometer plane).
the horizontal
axis
For the comparison of these data with our theory, we employed an interferogram of the form of Eqn 22 with values off = 1. y0 = 50 cm -I, b0 = 100 cm - *. This yields a spectrum with a maximum intensity at i = lOOcm_’ and decreasing intensity for C+ cc and C-+0. This spectrum is somewhat similar to the undisturbed spectrum obtained with our instrument with the 15 pm beam-splitter. However, the absorption due to water lines is disregarded in our theoretical calculations as we are interested in the fundamental effects around the central maximum at s = 0. This means that we are dealing with low resolution and would not be able to resolve any narrow line. The calculated values for the asymmetry are plotted in Fig. 6 vs the tilt angle rv. A comparison with the experimental value in Fig. 5 shows a surprisingly good agreement with respect to the numerous simplifications made. Especially, the sign change of the
c ‘0 -
o-
-10 -
d
.
IO -
$jLJL#& . 0
/
x)L
/r o/
I 5x10-4
1 0
I 5x10-*
="
Fig. 5. Measured asymmetry A vs tilt angle r, with respect to the vertical axis of the fixed mirror. Filled circles: rH = 5 x 10e4, open circles: q, = 0 (see also Figs 2 and 3).
f i i
-:
Fig.
6. Calculated
asymmetry
A (see text) vs tilt an& X, of the fixed mirror the wrtical axis for x5, = 0.
Fig. 7. Calcuiatcd intcrfero~r~ms (see text1 for scvcrai till intcrfcrogram): b: x, = -3 x IO-‘: C: 1, = -2.X x IO-‘: comparison the dccrcase of the modulation
with
respect to
an&s x,. a: x, = 0 (UndjstLlrbed d: X, L -3.X x 10~’ For hcttcr wks disregarded.
The problem
of nonlinear
phase errors
605
asymmetry is described correctly. The absolute calculated values are somewhat larger than the corresponding experimental data. However, the order of magnitude is correct. In Fig. 7. we plotted typical calculated interferograms. Again, the agreement with the experimental data in Fig. 3 is good. CONCLUSION
Our experimental and theoretical studies confirm that there arises a error in the interferograms. This error originates from a mirror tilt interferometer of the standard type. Our theoretical considerations yield tory explanation of this nonlinear phase error. Our studies show on that there exist other types of two beam interferometers not showing of the interferogram due to a mirror tilt.
nonlinear phase in a Michelson a very satisfacthe other hand an asymmetry
REFERENCES I. 2. 3. 4. 5. 6.
7.
WILLIAMS,
c.
S..
Appl.
Opf.
5.
1084 (1966). KUNZ. L. W. & D. V. GOORVITH. Appl. Opt. 13. 1077 (1974). CHANTRY. G. W.. Suhn~illiwwrrr Spwrroscop)~. Academic Press. MBLLER. K. D. & G. ROTHSCHILIX Fur I~$wrd Specrroscopy. GHCK. R. IR Fouricv Truwform Sp~rroscopy. Topics iu Curre~tt berg (1975). OBERHETTINGER, F. Fourier E.xpunsion.r. Academic Press, New GENZEL, L.. H. R. CHANDRASEKHAR & J. KUHL. Opr. Commun.
London
(197 I).
Wiley Interscience. Clwmistry.
New York (1971). Vol. 58. p. 73. Springer. Hcldcl-
York (1973). 18, 381 (1976).