Throughput of tilted interferometers

Jerome Genest, Pierre Tremblay, and Andre Villemaire

The throughput of a tilted Fourier-transform spectrometer ~FTS! with collimation is calculated. It isshown that the maximum off-axis angle that is acceptable in the interferometer is inversely proportionalto the distance between the detector and the location where the tilt is applied to the wave fronts and isalso inversely proportional to the tilt angle. This effect leads to tilt sensitivity in a scanning FTS andto the loss of the throughput advantage in a FTS with no moving part in which a tilt between twocollimated beams is used to disperse the interferogram spatially. Experimental verification confirms thethroughput condition with tilt angle. © 1998 Optical Society of America

OCIS codes: 120.0120, 120.3180, 200.6300, 230.6120.

1. Introduction

A Fourier-transform spectrometer ~FTS! with nomoving parts spatially disperses the interferogramand detects all the desired path differences simulta-neously by use of a detector array or matrix.1–3 Suchinstruments are often called spatially modulated in-terferometers ~SMI’s! to differentiate them frommore-conventional time-scanning instruments.

Because SMI’s are FTS’s, it is usually assumed,2often without formal proof, that they preserve thewell-known4–6 throughput advantage. This israrely the case. Good SMI designs such those thatuse the image optical path difference ~OPD! superpo-sition3 and source-doubling techniques,1,7–10 will ben-efit from a higher throughput than time-scanningFTS’s or any other spectrometers. It has indeedbeen shown1,7 that source-doubling SMI’s are not lim-ited in throughput by interferometric considerations.Horton3 tentatively demonstrated the same phenom-enon for image OPD superposition SMI’s, althoughthe throughput limit in this case must come from thedepth of focus.

On the other hand, one of the more intuitive waysto produce a spatial interferogram is to tilt one mirrorof a collimated interferometer. Because of their sim-plicity, such beam-tilting SMI’s have been widely

J. Genest and P. Tremblay are with the Departement de GenieElectrique et de Genie Informatique, Universite Laval, QuebecG1K 7P4, Canada. A. Villemaire is with Bomem, Inc., 450 Ave-nue Saint-Jean-Baptiste, Quebec G2E 5S5, Canada.

Received 25 September 1997; revised manuscript received 17March 1998.

0003-6935y98y214819-04$15.00y0© 1998 Optical Society of America

used.2,11,12 We show here that beam-tilting SMI’sare severely limited in throughput and are thereforepoorer solutions for acceptance of wide angle beamsthan well-designed SMI’s.

In beam-tilting SMI’s ~Fig. 1! the collimated beamsthat come out of the interferometer are tilted withrespect to each other and the OPD varies linearlyalong one direction of space, say, y. With a detectorarray one can then simultaneously record N OPDpoints of a single point source. If a detector matrixis to be used, the second dimension can be used toimage a line of the field of view. This can be donewith a cylindrical lens ~in Fig. 1 the imaging directionis perpendicular to the plane of the sheet!.

This operating mode is not limited to tilted Mich-elson designs; it includes all interferometers that ac-cept a quasi-collimated beam from an extendedsource, split this beam into two, and apply a tilt be-tween the two parts before recombining them. Thisis the case for several designs of tilted Sagnac andMach–Zehnder as well as wave-front-dividing inter-ferometers.13,14

The formal derivation of the maximum off-axis an-gle that is acceptable in a tilted interferometer withcollimation in the direction of interferogram disper-sion is presented here. Knowledge of this angle isuseful not only for evaluation of the throughputachievable with common beam-tilting SMI’s but alsobecause this angle could add tilt sensibility in a time-scanning FTS. The calculations in this paper arerelated and complementary to the research done byJunttila13 and by Juntilla et al.15

2. Calculations

We made the calculations in this paper by assuminga pair of tilted mirrors placed side by side ~Fig. 2!; the

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only purpose of the mirrors is of course to permit twotilted wave fronts to be obtained from one originalcollimated beam. This procedure will permitthroughput evaluation for tilted interferometers withcollimation, such as beam-tilting SMI’s. Inasmuchas all the calculations in this paper involve idealplane waves, the mirrors must be mentally extendedto take infinite dimensions. They should be consid-ered as if the virtual image of one mirror were super-imposed upon the other mirror, as would be the casein an interferometer with a beam splitter. Thephysical size of the mirror would then be importantonly for defining the region in space where the twotilted beams overlap and thus interfere.

A plane wave incident upon a pair of mirrors at anangle u with respect to the optical axis ~Fig. 2! will bereflected at angles 2a 1 u and 2a 2 u by the twomirrors, where a is the tilt angle of one mirror. Thetwo interfering waves will therefore be

E1 5 Eo1 exp$2jk@cos~2a 1 u!z 2 sin~2a 1 u!y#%,

E2 5 Eo2 exp$2jk@cos~2a 2 u!z 1 sin~2a 2 u!y#%, (1)

where the phasor representation of the electric field isused.

Fig. 1. Tilted Michelson interferometer: FOV, field of view.

Fig. 2. Beam incident at an angle upon a tilted mirror pair.

4820 APPLIED OPTICS y Vol. 37, No. 21 y 20 July 1998

Let us suppose, for simplicity, that Eo1 5 Eo2 5 Eo.The sum of the waves is therefore

E1 1 E2 5 Eo@exp$2jk@cos~2a 1 u!z 2 sin~2a 1 u!y#%

1 exp$2jk@cos~2a 2 u!z 1 sin~2a 2 u!y#%#.(2)

The quantity measured by a detector is, however, P~u!5 1y2 uE1 1 E2u2. One readily obtains

P~u! 5 2Eo2 cos2HkFScos~2a 2 u! 2 cos~2a 1 u!

2 Dz

1 Ssin~2a 2 u! 1 sin~2a 1 u!

2 DyGJ. (3)

In Eq. ~3! the y term will give rise to the desiredspatial interference and the z term is unwanted.For instance, if one sets u 5 0, one has

P~u 5 0! 5 2Eo2 cos2@k sin~2a!y#, (4)

which is the ideal spatial interferogram for a wavefront impinging upon the optical axis of the mirrorpair, with its scale depending on the tilt angle a.

Now Eq. ~3! must be integrated over the finite an-gle range corresponding to the range of angles admit-ted in the interferometer, say, from umin to umax or bysymmetry and for simplicity from 2umax to umax. Toperform the integration it is convenient to use asmall-angle approximation for a and u, that is, sin~A!' A and cos~A! ' 1. The integral therefore reducesto

P 5 *2umax

umax

2Eo2 cos2@k2a~uz 1 y!#du. (5)

The result obtained after integration is

P 5 2Eo2umaxF1 1 cos~4aky!

sin~4akumaxz!

4akumaxzG . (6)

It is obvious from Eq. ~6! that the spatial interfero-gram in cos~4aky! is multiplied by a cardinal sine~sinc! term that will reduce the fringe visibility for allpath differences. To preserve the fringe contrast wemust keep the sinc argument as small as possible.

To permit the formulation of a condition for themaximal angle in a beam-tilting SMI, let us specifythat the sinc argument must be smaller than py2.We then easily obtain

umax ,p

8akz. (7)

To realize fully the effect of this dependency of themodulation on z and to understand the reason whythe coordinate system is attached to the plane pass-ing by the line that is common to both mirrors ~Fig. 2!one must return to the basis of the problem @Eq. ~1!#.Two tilted plane waves are traveling in free space.In our construction they are in phase at the origin,e.g., at y 5 0 and z 5 0. From Fig. 2 it is clear that

Fig. 3. Schematic of the experimental setup.

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the two waves are in phase on the line common toboth mirrors because both reflected waves originatefrom that plane, without delay, from the same ray ofthe incoming wave front. The origin of the coordi-nate system is therefore located somewhere on thetop of the roof formed by the two mirrors. The re-maining degree of freedom corresponds to the x di-rection and is irrelevant in our treatment.

Inequality ~7! thus shows explicitly that the max-imum allowable angle in a beam-tilting SMI is in-versely proportional to the distance z between thedetector and the separation of wave fronts and to thetilt angle a. Equation ~7! also shows that thethroughput of beam-tilting SMI’s is not subject to thesame limitation as standard FTS’s. Because z isnormally quite large, usually larger than the aper-ture size, the throughput of beam-tilting SMI’s ismuch smaller than the etendue of conventional Mich-elson instruments. Beam-tilting SMI’s do not there-fore benefit to its full extent from the throughput~Jacquinot! advantage of standard FTS’s.

3. Tilt Sensitivity of Time-Scanning Interferometers

As it is the case for beam-tilting SMI’s, the modu-lation loss with tilt angle may also be observed intime-scanning interferometers. Suppose that thedetector array of a SMI is replaced by a single detec-tor and one mirror is scanned to produce a time-varying interferogram. One would then make surethat the tilt is small enough that the detector seesonly a fraction of a single fringe of the ~now unwant-ed! spatial interferogram. The residual tilt will,however, still reduce the fringe contrast according toEq. ~6!. This is a supplementary tilt sensitivity ef-fect, and if z is big ~which could be the case becausecurrent time-scanning interferometers are not de-signed with this fact in mind!, the maximum off-axisangle can be severely limited or, equivalently, themodulation loss with tilt angle can be important.

4. Experimental Verification

To verify the expression of fringe visibility loss withtilt angle derived in this paper, we made measure-ments by using a tilted Michelson setup ~Fig. 3!.

According to Eq. ~6!, the argument of the sinc termdepends on tilt angle a, distance z, and the maximumoff-axis angle in the interferometer, umax. Oneshould therefore be able to measure a sinc variationof the visibility if two of the parameters are keptconstant while the third is changed.

For the data presented here ~Fig. 4! the fringe vis-ibility was measured as a function of tilt angle a.The values used for the other parameters of the sincargument in the theoretical prediction are those ofthe experimental setup and are listed in Table 1.

The experimental measurements are compared inFig. 4 with the theoretical prediction. One can see

Fig. 4. Visibility as a function of tilt angle.

Table 1. Values of the Parameters Needed To Predict Visibility

Parameter Value

z 3 cmumax 5 mradWave number 10,000 cm21

the good agreement between the theoretical and mea-sured values when the amplitude of the theoreticalprediction is globally adjusted. This adjustment hasto be made to take into account the fact that the beamsplitter does not have perfect efficiency. The visibil-ity at zero tilt angle should be 1, but it is near 0.58because a UV beam splitter that was not optimizedfor the visible range was used in the setup.

5. Conclusion

We have derived the criteria for the maximum off-axisangle that is acceptable in a tilted Fourier-transformspectrometer with collimation. This effect limitsthe throughput of beam-tilting spatially modulatedinterferometers and adds sensitivity to tilt in time-scanning collimated interferometers. An experi-mental verification of this condition was presented.

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