The combined effects on performance of shear between the two arms, defocus of die detector, and difference in wavefront between the two arms of a Fourier transform spectrometer using cube corner retroreflectors were investigated. Performance was characterized by the amplitude of the fringe signals coming from a detector as the path-length difference was scanned. A closed-form expression was found for the combined effects of shear and defocus, and it was found that defocus had no effect in the absence of shear. The effect of wavefront error was modeled numerically and assumed to be independent of shear and defocus. Results were compared with measurements made on the breadboard and engineering model of the Composite Infrared Spectrometer for the Cassini mission to Saturn, and good agreement was found

Hagopian, John G.

Martino, Antonio J.

English

NASA Technical Reports Server

_. 5,._ _,,.' /.
Effects of shear, defocus, and wavefront error on the theoretical
performance of the composite infrared spectrometer for Cassini
Anthony J. MartinC alld John G. Hagopian h
_Laser/Electro-Optics Branch, Code 554, NASA Goddard Space Flight Center, Greenbelt, MD 2()77 l
bOptics Branch, Code 551, NASA Goddard Space Flight Center, Greenbelt, MD 20771
ABSTRACT
The combined effects on performance of shear between the two arms, defocus of the detector, ,and difference in wavefront
between the two arms of a Fourier transform spectrometer using cube comer retroretlectors were investigated. Performance
was characterized by the amplitude of the fringe signals coming from a detector as the path-length difference was scanned. A
closed-form expression was tbund for the combined effects of shear ,'uuldefocus, and it was found that defocus had no effect in
the absence of shear. The effect of wavefront error w_ mtxleled ntanerically _aul a.ssumed to be independent of shear and
defocus. Results were compared with measurements made on the breadboard and engineering model of the Composite Infrared
Spectrometer for the Cassini mission to Saturn, and good agreement w,a.s found.
Keywords: Cassini, CIRS, Michelson interferometer, Fourier tr:msform spectrometer, reflectance, shear, defocus,
wavefront, alignment, modulation
1. INTRODUCTION
The Composite Infrared Spectrometer _'2 (CIRS) for the Cassini mission to Saturn is a Fourier transtbrm infrared spectrometer
that contains three Michelson-type interferometers. One is used for spectral measurements in the mid-infrared (MIR) band of
7 to 17 I.tm; another is used in the far-infrared (FIR) b,'md of 17 to 1000 gin. The moving elements of these two
intefferometers are mounted on a common scan mech:mism x. The thh'd is a reference interferometer a that me,'tsures the
position of the scan mechanism.
During the development of the MIR interferometer, it bcc:une necess_u'y to detennine the theoretical perfonnance of the real,
as opposed to ideal, interfe_ometer. Effects of all known departures from ideal were to be calculated and compared with the
measured performance of the actual hardware. Effects that were included in the c,'dculations were reflectances of ,all surfaces in
the beamsplitter/compensator stuck, wavefront aberrations due to the figures of these surfaces and the retroretlectors, and two
forms of misalignment: shear and defocus. Performance of the intefferometer was measured in terms of the observed
modulation of the output signal.
2. CONCEPTS AND DEFINITIONS
A conceptual sketch of the MIR interferometer is showti in Figure 1. A source is nomin:llly located at the foc:d plane of the
input lens. An image is formed, nominally at the foc:d plzme of the output lens. A detector, the area of which defines the
field of view, is also lcx:ated nominally at the output focal plane. [n the subsequent atl:dysis, defocus will refer to axial
displacement of the source or detector from the input or output foc:d plane.
At the center of the interferometer is a be:unsplittcr/ct_ml3Cns:_tor p:dr. The subst,'at¢ matcri:d is potassium bromide (KBr).
The beamsplitter coating, which is deposited on the sccolld suflztcC ot the bc:unsplittcr, is I_t_mitl:tlly 50% rellective :aid 50%
transmissive for both l'Xfl:u-izatiotls across the spcctr:d b:mdwidth eft' iltlercst.
https://ntrs.nasa.gov/search.jsp?R=19990014595 2020-06-18T00:44:59+00:00Z
Tilebe,'unstnms£nittedmidtel]cotedtrt,nlthebc;unq_liucr_,urfaccpropagal¢toap;tirofcul',c-etHncrretrorctlectors.These
retrorelleetorstermitlate tile ;runs of the Michcls(m intofcro,ncter, mht2 be:tins th'dt rettJrll lt+t)tll the rcutwclleelors ;.Ire
recombirmd at the bemnsplitter, then focused by the output lens t>nt() the detector. ,,'ks the moving retrorcllcctor trzmslatcs
_dong the optical axis, the path lcagdt dilfcrcncc hetweeu the two arms clumgcs. This gives rise to at v:u-ymg signal recorded
by the detector. This sign,'d can be characterized by its modulation, given by
M=S=_, -- Smin
S,.= + S,.,.
where S is the signal recorded by the detector.
(1)
f i xed
r et ro
inp ut
lens
I,o0u,z A oo+oo tor
Ifooal
I0,+.I
r output
lens
_) movingret ro
The actmd rellect,'mce of the beamsplitter
smI"ace deviates from ideal perfc, rm,ance.
Also, there is no ,'mtiretlection coating on
the first surface of the be,'unsplitter or either
surface of the compensator. The reflectance
of uncoated KBr at the relevant angle of
incidence (37") is high enough that a
significant ,'unount of light goes into stray
be,'uns that ,are reflected at surfaces other than
the nominM be_unsplitter surface. These
surfaces ,are slightly tilted with respect to the
bemnsplitter surface, so the stray be,'u-ns do
not contribute to the modulation of the
interferognun recorded at the detector.
However, they do contribute ,'m offset term.
output focal plane
Figure 1. Conceptual sketch of the MIR interferometer.
input lens. In this case, rays in the fixed and moving re'ms that come from a single ray incident on the bemnsplitter meet
again when the beams ,are recombined.
Nomimdly, the apexes of the cube comers
m-e located on tim optic;d axis, which is
detined by the tkx:,d point mid center of the
If one or both of the cube comers is displaced laterally from its nomin:d position, the rays returning from the two ,arms are
displaced. This displacement of the two returning be:uns is cMled shear. Alone, mid in combination with def(x:us, it affects
the modulation of the detected signal.
Errors in the surface figures of the be,'unsplitter, compensator, ,and retroreflectors cause a v_uiatiou of ph:L';e difference across
the aperture of the interferometer. This ;dso contributes to a loss of modulation.
Except for shem" and dehx:us, these effects are, It) a good approxhnation, independent tit"each odler. Consequently, dmy c:m be
treated separately and then combined to yield a timd result.
3. SURFACE REFLECTANCE
It is well known s that, for an ideal Michelsoq intcrfc,ometcr, the modulation is given by
M = 4RT , (2)
where R is the retlect;u_ce ;rod T is the trausmitt:mee t)t" fl_c bczunsplitter. When there m'e additional reflective surfaces
associated with the be,'unsplitter, addition:d stray borons mc ere:lied di:lt contribute to the tot:d power received by the detector
but not to the interferogr:un.
The zero order beam of the interferometer consists t>f rays tllat split ;.uld rcct,mbittc tit tile bc:unspliltcr stJrf;.tces, ;uld dr) not
reflect at any other be:.unsplitter t)r compensator surface. The zcru order power tllat reaches die detector (the m;.dn bc;.un) from
the moving and fixed arms is given by
e.. =
P,,: = P_ T +T."R T _3)UG Ul Ce t'l
where Pm is the input power; tile subscripts "'c" and "u" murat "coaled" _md "uncoated'" widl tile bc_un._plittcr coating; m_d "i"
and "e" mean intel_.al and external rellection. Then the mnplitude of the fringes is
a = 24P.j,P . =24T_+'T_,SRT,,R,T, C4)
A similar expression can be found for zero order power that is returned to die source. In die MIR interferometer, the fraction
of the input power that reaches the detector in a zero-order bemn varies between I 1% and 37%, depending on wavelength and
polarization. This leads to modulation of the main begun that varies between 89% and 97%. The complementary main beam,
which returns to the source, accounts for another 32% to 78% of the input power.
Stray beams can be classified as first order, second order, etc.. ba._d on the number of rellections made at the front
beamsplitter surface and the compensator surfaces. Together, these higher-order be,'uns account fi_r 10% to 35% of the input
power. If we (very roughly) approxhnate that the higher-order lx_wer is split between the detector and the source according to
the same ratio as for the zero-order power, and c,'dculate modulation as
A
M = . (5)
/'..,
where P_ower reaching, the detector from all bemns, we obtain the results shown in Figure 2.
0.75
0.74
0.73
_ o.72
Wavelengt h (tam)
Figure 2. Modulation of the output light due to bemnsplitter and
compensator surface reflecu_nce, averaged between s and p polmizations.
4. WAVEFRONT ERROR
The electric field at a point (x,y) in the aperture, asstuning eqmd IX)wers m flat: two arms, is
1 Eo[e/_tx., , + ei,¢,<x.y,+k=+lx¢,,x.y,, ] (6)E(x, y) = _-
where _l is the phase at (x,y) from one interferomctcr mm in radians. A_, is the phzt,_e difference between the two arms due
to wavefront errors, ,'rod z is the mean path length difference.
The power at the interference pl:me is
P(x, y) = E(x, y)E'(x, y)
1 2
= _ E o [1 + cos(kz + A¢,(x, y))]
= e,[l + cos(kz + A¢.(x, y))]
0
Figure 3. Phase difference in waves (at 9.1 I.tm) between
the fixed and moving arms of the CIRS MIR interferometer
due to surface figures. Horizontal units ,are fractitms of
pupil radius.
(7)
0.9
0.8
0.7
0.6
0.5
20
wavelengt h (#m)
25
Figure 4. Modulation of the CIRS MIR interferometer due
to the wavefront difference shown in Figure 3.
The interferogram is prtxluced by focusing the tot,'d power
in the aperture to a spot, then capturing the spot on a detector:
l(z) = _ /',[1+ cos(kz + A¢.(x, y))]dxdy
A
= P,A + Po_ cos(kz + A¢,(x, y))dxdy
&
= P,A + P, cos kz_ cos A¢, (x, y)dxdy
A
Let
H_
n 2
Then
= _ cos A¢, (x, y)dxdy
A
= f sin A_. (x, y)dxdy
A
- Po sin kz_ sin A¢. (x, y)dxdy
A
l(z) = AP, + Poll t cos kz - Poll2 sin kz
= AP o + Po._Ht 2 + Hz2(cos(kz + _))
H,
where tan • = .-7 ' Then dm me:m of the intcrfcrogr_un is APo and tim modulation is
(8)
(9)
(10)
M
H12 + H22
A (11)
For some wavefront shapes, He and H, could bc calculated ,m;dytically. [n practice, it is necessary to compute them
numerically from measured dam. To model the CIRS MIR iatcrferomctcr, surface figures or w:tvel'ront distortions were
measured for each component using a Zygo interferometer. Waver'fonts from the lixed and moving arms, ,'rod wavefront
difference, were computed using these mea.sured data. Figure 3 is a plot of computed wavefront difference at 7.1 I.tm
wavelength, and Figure 4 is a plot of the resulting modulation It,; a function of wavenumber.
5. SHEAR AND DEFOCUS
A treatment of the effect of shear in a cube-corner Michelson interterometer has been published by Murty 6. For a cube-corner
Michelson interferometer with a circuh'u" field of view, no defi)cus, at zero path difference, the modulation is given by
M = J_(2o-s_)
where o" is the wavenumber, s is the wavefmnt shear (twice the cube corner displacement), and f_ is the solid angle
subtended by the field of view. Kauppinen and Sam'inen v have extended this theory to predict line shape distortions in the
spectral domain.
image detector output lens
_lane plane rear principal plane
si
o_,0) (fl,7) polar (p,r)
(x,y) (u,v) rect angular (l,m)
S_mrinen and Kauppinen s have ex,'unined the effect of a
defi)cused point source on the interferogr,'un ,and in the
specmd domain, expanding on work by Guelachvili 9.
Kunz and Goorvitch t° have ex_unined the combined effects
of source defocus and mirror misaligiunent in a Michelson
interferometer that uses plane mirrors. Here we present a
treaunent of the combined effects of defocus and shear on
the modulation at ZPD of a cube-comer Michelson
interferometer looking at an object that fills a small but
nonzero field of view.
The geometry is shown in Figure 5. Each point in the
source (not shown) is imaged to a point in the image
plane (nomin_dly the focal plzme of the output lens) at a
distance .v, from the rear princip_d pl_me of the output lens.
A pkuuu" detector is located at some distm_ce 8 from the
hnage plane. For each point (u,v) in the detector pl,'me
mid (l.m) in the pupil, there is a con'esponding point (x,y)
in file image plane.
Figure 5. Geometry fiw delocus/shear theory, showing The pupil of tile optic:tl system is ch'cular, with radius Rp.
significant pl,'mes and poh_r _md rectangular coordinates used In general, the source is not at the tbcad plane of the input
therein, lens, so tile wavet'ront incident on the output lens is
spherical with radius R,,. In this zmalysis, R, is assumed
to be much greater th,'m the maxhnum path length difference between the two rems of tile interfer¢)meter, st) the wavefronts
from the two arms can be treated as having the s_une radius t)f curvature.
Consider a point on the y-axis in the image pkme. The wavcli'tmt d_:tt is focused t)mo it is dcscril_cd p;tr, Lxi;tlly in the pupil
plane ,'ks
12 +m 2 /(I - (y / s, ):)
z(l, m) = m y +
s, 2R
A wavelYont that is sheared with respect to this ot_c by s is
l 2 +(m +s) 2/(1-
z(l,m) = (m + s) y +
s_ 2R+
2(y/s )
(13)
(14)
Then the phase difference between the two waves is
2sin + s2
AC,(O,y,l,m) = 2rr(s y + )z))Yt s, 2R(1 - (y / s,
(15)
Since the image height is much smaller them the image distance, we can eliminate terms of order 2 or higher in y/s, yielding
2sin + s2
AO(0, y, l, m) = --2_r (s --y + )
s+ 2R+
It can be shown that for arbitrary field points, under the szune conditions, the s,'une result holds. Then
(16)
ys
P(x, y, l, m, z) = Po + Po cosk(z +-- +
Si
The power that strikes a point (u,v) on the detector is
e(u, v, z) = pdp drP(o:, 0, p, "r) ,
0 0
S 2 snl
+--)
2R R
(17)
(18)
where (u.v). (ct.0), and (p.r) ,are ,all coilinear. Substituting (17) into (18) .'rod using some geometry to express image pl,'me
coordinates in terms of pupil and detector plm_e ct×_rdinate._
f 2xP(u, v, z) = 7r-R,2Po + Po pdp_ dz cos k
0 0
we obtain
Z +
R s_(L - _ psin'_
S_V SV S 2
+ +--+--
si(s i - 63 s i 2R
(19)
which evaluates to
P (u , v, z) = _Tr-R,2Po
s ll(E'v s211COS k s, + -- + Z •R+ s,(s, - 63 - ¢5 2R+ (20)
In the case of a rectanguh'u detector with dhnensions A.,,t_md/iv. the power integrated across the detector is
P(z) = 7rRp2 PoAuAv
J_(q_,) s, - S (sin--
+2xPoRp_Au
q_ ks (+ COS-
where
s,(s, - o3
Then the modulation is
M(s, 6, Av) =
where
ksAv%=
2(s, - b-')
2 J' (_) sin(_U2 )
). 0006
-0.0005 _ ] 3.0004
shear (m)
. _ o I/
0002
0 ,0005
defocus (m) o
Figure 6. Effect of shear and defocus on modulation of the
CIRS MIR interferometer at I0 lira wavelength.
sin _,. _ _) L2R + z
cos kSV,,._ / sin k[ £: + ]
s, - 8) L2R. zJ
(21)
(22)
(23)
(24)
An interesting feature of this result is that, while shear affects
the modulation by itself, defocus at the output affects
mtxlulation only in the presence of shear. With shear present,
mtxlulation is maximized by placing the detector not at the
fi)cal plane of the output lens, but at the image of the source.
(The unage of the source coincides with the focal plane of the
output lens if there is no defocus at the input.)
Figure 6 illustr4tes the effect of shear and output defocus on
the modulation of the CIRS MIR interferometer. The detector
in this institute is ,an ,array of pixels, each of which can be
regarded as a 200/.tin square.
6. COMPARISON OF THE COMBINED EFFECTS WITH EXPERIMENT
All of these effects were combined to produce prediction.s of the pet'fiwm;mce of the CIRS MIR interferometer. The
predictions were comp_u'ed with measurements made ou the engineering :rod flight m(_lels of the instrument. The details of
the experimental mea.surements are described in ;ulother papez "j_.
In Figure 7, we show predictions of mea.,_ured spectra using :t 50(Y C blackbody source under v_u-ious conditions. The
calculations include the effects described above :rod the relative spccu':d responsivity of ;.tHgCdTe detector. Figure 8 is ,'m
ex,'unple of actu,'d data measured under correspoudit_g conditions in a CIRS brcadbo:u'd, using a detector simil,'u but not
identical to the one whose responsivity was used ia the calculation.
shear 0
/--i
t/
shear 250 lam
1 000 _-1 2000 1000 (_-I
shear 500 lam shear 1000 jam
2OOO
1000 (f-11000 _-! 2000 2000
Figure 7. Predicted performance of the CIRS MIR interferometer measuring a 773 K blackbody spectrum.
Effects of beamsplitter/compensator reflect:race, sudace figures, shear, output detbcus, ,'rod detector
responsivity are included. Each plot shows detector reslxmse in _ubilxury units as a function of wavenumber
with defocus (from top) 0, +250 pro, +500 l.tm, and +1000 p,n.
• • • I • • •
6 800 1 000__11 200 1400 1600
Figure 8. Spectrum of a 773 K blackbody, measured with
(from top) 200 p.m she,'u" ,'rod no det'ocus, 10(X) pm t_t"shc:tr
and 0, 200 Ilm, 300 [am, 400 pm, and 600 pin of output
de focus.
With no shear, the theoretical plot is simply the blackbody
spectrum multiplied by the relative responsivity of the
detector. In die absence of defocus, the effect of shear is
noticeable but mild, being most pronounced at higher
wavcnmnbers (shorter wavelengths).
As shear in increased, the effects of detbcus become more and
lnore pronounced. Large degra&itions appear fin'st at higher
wavenumbcrs, then propagate to lower wavenumbers as the
she:u- or detbcus increases. Note that there is little difference
between the effects of lx)sitive _md negative detbcus.
The expcrimcnt:d results agree well with the theory. As
ex[',cctcd, 200 pm of .she:u" with no detocus shows a neurly flat
spectrum. With 100() p.m of shear, the perflmnance at higher
wavcnumbcrs in degraded m:u'kedly even with no dehx:us.
Pcrtonn:mcc :it tile lower wavenumbers is still ",.dfected only
slightly. An dclbcus increases, perhmnance drops quickly at
high w:lvcnumbcrs :rod more slowly at low wavenumbers. At
the largest defocus (lO00 pin), the performance actu:dly recovers slightly at the hight.st w:wenumbers.
The theory presented in this paper was used in the di;kgnosis :rod ch:trztctcriz:tlitm of Lhc CIRS instrument. With this
inh_rmation, it was possible to determine that ',dlcr :digmttcnt _t :ttl d_c optic:,, Ihc in_,trumcnt pcrl_wm:ulce w:t.s as good as
could be obtained with die fabricalcd h,trdw:uc.
7. REFERENCES
1. V.Kundeet. al., "Ca.'_sini infrared Fourier spcctrosct}pic investiguticm." Pr_)c. SPIE 28113. pp. 162-177, 1996.
2. P. W. Maymon et al., "Optical design of tile composite inlY_trcd spcctr_unctcr (CIRS) for the Cassiui mission," Proc.
SPIE 1945, pp. 100-111, 1993.
3. C. E. Hakun and K. A. Blumensttx:k, "A cryogenic scan mechanism for use in Fourier trzmsfl}rm spectrometers," in
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5. R.J. Bell, Introductory Fourier Transfi_rm Spectroscopy, p. 112, Academic Press, New York, 1972.
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7. J. Kauppinen and P. Sa,.arinen, "Line-shape distortions in mis:digned cube cota_er interferometers," Appl. Opt. 31, pp. 69-
74, 1992.
8. P. Saarinen and J. Kauppinen, "Spectral line-shape distortions in Michelson interferometers due to oft-h_cus radiation
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Effects of Shear, Defocus, and Wavefront Error on the Theoretical Performance of the Composite Infrared Spectrometer for Cassini

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