JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Optimization of the Czerny-Turner Spectrometer*ARTHUR B. SHAFER, LAwRENCE R. MEGILL, AND LEANN DROPPLEMAN

Central Radio Propagation Laboratories, National Bureau of Standards, Boulder, Colorado(Received 12 August 1963)

Geometrical optic techniques are used to analyze and compare the symmetrical spherical mirror to theunsymmetrical spherical-mirror Czerny-Turner spectrometer. The aberration problems due to diffractionfrom the grating are analyzed and methods of partial correction of the aberrations are derived. The flux andresolution advantage of gratings with high blaze angles used in the unsymmetrical spherical-mirror Czerny-Turner is shown. A design and ray tracing routine employing a digital computer is utilized to illustrate thegeometric effects of the diffraction grating and the partial corrective measures. Slit curvature is analyzednumerically and some general results are abstracted from the numerical data. It may be inferred from theresults of theory and numerical calculations that the unsymmetrical Czerny-Turner spectrometer usingtwo spherical mirrors can be made superior to a similar symmetrical Czerny-Turner spectrometer. A com-parison of luminosity is made between the Czerny-Turner spectrometer, utilizing a high blaze grating,and various interferometric and modulating spectrometers and it is shown that the luminosity of the Czerny-Turner spectrometer is comparable or superior.

I. INTRODUCTION

IT was first recognized by Czerny and Turner,' thatthe aberrations introduced by a spherical collimating

mirror M1 can be partially corrected by a symmetrical,but oppositely oriented, spherical condensing mirror M2

M2

M,

\ V B

A

FIG. 1. A two-dimensional schematic of a generalized sphericalmirror Czerny-Turner arrangement. A is the entrance slit, B theexit slit. M1 and M2 are the collimating mirror and condensingmirror, respectively. G is the grating. This is not the type ofinstrument analyzed by M. Czerny and A. F. Turner, since it isasymmetric.

* Sections 1, 4, and portions of 5 were presented at the meetingof the Optical Society of America at Los Angeles, California,October 1961; J. Opt. Soc. Am. 51, 1464 (1961).

1 M. Czerny and A. F. Turner, Z. Physik. 61, 792 (1930).

(Fig. 1). More recently, Fastie 2 independently designeda spectrometer that was first designed and built byEbert.3 Kudo4 has expanded the Hamilton point-characteristic function for the symmetrical Czerny-Turner, Ebert, and Pfund spectrometers and has givendata about slit curvature. Leo5' 6 has shown that becauseof the change in beam width after diffraction, theoriginal symmetrical arrangement of Czerny and Turneris not satisfactory (see Sec. IV) and has analyzed theproblem for the grating5 and the prism.6 In all of thework referred to above, except Leo's, it is assumed thatthe grating has little or no effect upon the quality ofthe image.

This work, essentially, is an extension and generali-zation of the work of Leo5 ; we shall take into account(1) the aberration introduced by the collimatingmirror, (2) the change in beam width due to the tilt ofthe grating, (3) the grating or diffraction inducedchange in the shape of the surface of constant phase,and (4) the amount of, or lack of, correction to thewavefront provided by the second (condensing) mirror.

To obtain analytically the approximate focusingcondition and the relative orientation of the slits,mirrors, and grating, use has been made of Hamilton'scharacteristic V function in two dimensions andFermat's principle. To obtain a more exact relation-ship between slits, mirrors and grating, a three-dimen-sional design and ray trace routine is used, employinga digital computer.

2 W. G. Fastie, J. Opt. Soc. Am. 42, 641, 647 (1952).3 H. Ebert, Wiedemann Ann. 38, 489 (1889).4K. Kudo, Sci. Light (Tokyo) 9, 1 (1960).5 W. Leo, Z. Angew. Phys. 8, 196 (1956).6 W. Leo, Z. Instrumentenk. 66, 240 (1958); 70, 9 (1962).7 W. R. Hamilton, Mathematical Papers, Geometrical Optics (Cam-

bridge University Press, London, 1931); Vol. I, Chap. 17, p. 168;J. L. Synge, Geometrical Optics, Cambridge University Press(1937); J. L. Synge, Hamilton's Metlhod in Geometrical Optics (TheInstitute for Fluid Dynamics and Applied Mathematics, Univer-sity of Maryland, College Park, Maryland, 1951); J. L. Synge,J. Opt. Soc. Am. 27, 75 (1937); M. Herzberger, ibid., p. 133.

879

VOLUME S4, NUMBER 7 JULY 1964

SHAFER, MIEGILL, AND DROPPLEMAN

[I. FLUX AND RESOLUTION

Several authors8 -'3 have written about the fluxand/or resolution advantages of the grating with ahigh-blaze angle (echelle) over a grating with a low-blaze angle (echellette). Some have shown that thisadvantage is proportional to the sine of the incidentor diffraction angle (the angular dispersion) or accruesto the instrument owing to a possible reduction in focallength (possible only if aberrations are ignored).We shall show, on the basis of the excellent work ofJacquinot and Dufour8 and Jacquinot,0 that it is largerthan shown by previous authors.

The equations for the flux due to a spectral line or acontinuum may be written as

tD = kTBxbX (I/F) (2W11 sin y cosx/X),

b= krBxQ (bX)2 (1/F) (2WH sin9 cosx/X)

(1)

(2)

where k is a factor related to diffraction by the exitpupil (grating) and the width of the entrance and exitslits,85' r is the transmittance of the optical assembly;Bx is the source radiance; 6X, the resolution; 1, thelength of the entrance slit; F, the focal length of thecollimating mirror; W and H are the width and heightof the grating; X is the wavelength; sp is the blaze angle;and x is the angle between the incoming central rayand the blaze normal or between the central diffractedrav and blaze normal at the blaze wavelengths.

If we wish to maintain the same focal length, aperture,and resolution, but change from a low-blaze to a high-blaze grating, the variables which need to be consideredare W, a, and k.

The approximate relation between the width ofa high-blaze grating and low-blaze grating is W,,= lWJ(Cos'p1/COS ',) where the subscript It refers to highblaze and I to low blaze. If 'Pi=20' and pD,=760 thencos051/cosqOh=3.9, also sinyjt/sinyz= 2.9. The Rayleighresolving power of the high-blaze grating is greater by afactor of 12 over that of the low-blaze grating; hence,the factor k may increase from =1.2 to =4. Themaximum increase under the above conditions maythen be obtained from Eq. (1) or Eq. (2); this is4'h,/14144; the minimum increase is cP7®= 11.

It is not possible in all cases to obtain a high-blazegrating wide enough to maintain the aperture. If thelow-blaze grating is replaced by a high-blaze gratingof the same width, the variables are then so and k. Underthese conditions the increase in 'p will give a flux

P P. Jacquinot and C. Dufour, J. Rech. Centre, Nat!. Rech. Sci.Lab, Bellevue (Paris) No. 6, 91 (1948).

9P. Jacquinot, J. Opt. Soc. Am. 44, 761 (1954)."G. W. Stroke, J. Opt. Soc. Am. 51, 1321 (1961). Stroke has

stressed with the help of numerical values, the advantages inusing high-bliazed gratings in spectrometers and spectrographs.See also, G. W. Stroke, Progr. Opt. 2, 3 (1963).

1V. . Malyshev and S. G. Rautian, Opt. i Spectroskopiya6, 550 (1959) [English transl.: Opt. Spectry 6, 351 (1959)].

12 R. Chabbal, Rev. Opt. 37, 49, 336, 501 (1958).'3 P. Jacquinot, Rept. Progr. Phys. 23, 267 (1960).

increase of 2.9 (see above) and k will be increased from1.2 to 3. The total increase in flux may then be

==3 to 9.By considering the possible increase in resolution

when using a high-blaze grating, from Eqs. (1) and (2)by keeping the flux constant, t= F,,, we may writefor the line spectra

6Xh/6X1= sinceq cos0o5./sin.oh COSsoi,

and for the continuum

b5X/bX1= (sinsci coswso,/sinsc, cossz)t-

(3)

(4)

Hence the spectral slitwidth, for the same amount offlux, can be reduced.' 0

III. EFFECT OF THE GRATING UPONWAVEFRONT ABERRATION

Before going into the effect of the grating upon theincident wavefront, it may be well to define threecommonly used terms. The image of the entrance slit,formed off-axis by the Czerny-Turner spectrometer, isa structure formed from many types of image aber-rations. The common terms such as coma, sphericalaberration, or astigmatism usually refer to the third-order Seidel aberrations; but for the case of a spectromn-eter where a diffraction grating intervenes betweenthe object and image, the usual meaning of these termsis lost. To circumvent the above, the following termsshall be used: (1) Coma is an asymmetry of the imageof the entrance slit and is orthogonal to the curved exitslit. This will produce an asymmetry of the spread.function of the spectrometer and hence an asymmetryin the contour of the observed line. (2) Astigmatism isan extension of the image of a point of the entrance slit.The extended point image, if properly curved slits areused, is tangential to the curve of the exit slit. Astigma-tism, according to geometrical optics, will have noeffect upon the spread function [see Sec. V (b) forprobable effects of diffraction]. (3) Spherical aberrationis a symmetrical spreading of the image of a point ofthe entrance slit. Spherical aberration will produce asymmetrical broadening of the spread function andhence a widening of the line contour with a possiblereduction in resolution or contrast. These terms havethe defined meaning throughout this work unlessotherwise specified. The effect of the diffraction gratingupon the incident wavefront is twofold; it changes thebeamwidth and transforms the shape of the incidentwavefront. The collimated beamwidth is changed fromTV, before diffraction, to approximately [V cosflo/cosa0 o after diffraction where a00 and j30o refer to theincidence and diffraction angles of the central ray.This forces the rays to depart from the conjugaterelationship between mirror surfaces of the standardCzerny-Turner condition.5

Any aberration in the wavefront produced by thecollimating mirror will be transformed by diffraction.

880 Vol, 54

July 1964 OPTIMIZATION OF CZERNY-TURNER SPECTROMETER

By taking the differential of the diffraction equation

n\/d = cos4' (sina,+ sin8,),

we may obtain, by assuming a small angle variation,

,=2(sinag0/cosjBo) tan4tAV

- (COosa0o/coSl30o)za, (5)

where t/ is the angle between a plane orthogonal to theface of the grating, the grating rulings and any ray;the angular increments Al 0,, zao,, and AV/ are the deriva-tions of any ray from the central ray angles. An exami-nation of Eq. (5) shows that a symmetrically curved

incident wavefront will be diffracted asymmetricallyand that an asymmetric wavefront is made moreasymmetric.

IV. APPROXIMATE IMAGING CONDITIONS

The approximate imaging conditions for the Czerny-Turner arrangement will be derived in two dimensionsby using Hamilton's characteristic V function andapplying Fermat's condition for imaging.

The expansion of the light path function is takendirectly from Beutler'4 and reduced to two dimensionsgiving

w 2 r-cos2 osa\ /cos2 3 cosf\] Fsina Ccos2a cosa sin3cos 2 cosf3\1F= r+r'-w(sina+sinm) + -YL)rtR) - + _ R-J +r R)2 - r RJ \rf R - -r r R rl rf R A

± sin2(aYcos 2 aY cosat sin 2 1 cos2i3 cosiO] W -4 -1 cos) - +I COST) (6)

r2 r R r'2 r' R 8RL2 r R r' R

In Eq. (6), R is the radius of the mirror, r is the object distance, r' is the image distance, and a and A are the anglesof incidence and reflection, respectively.

Applying Fermat's principle to Eq. (6) and assuming the central ray approximation where f= -a, and acollimating condition, r'= co we obtain

aF 'cos2a 2 cosa\ gina cos2 a sina cosa\=WI - J3V -

Ow\ r R r2 rR

Sin2a cos 2a Sin2 a CoSa\ WI /1 2 Cosa\+2W 3 )+-(-- )+..=O. (7)

r3 r2R / 2R2 \r RI

If the first bracket in Eq. (7) is set equal to zero weobtain a central ray solution given by

r= ro= (R/2) cosa. (8)

The path variation for a single surface may becomputed from

'a FzXF=f= -dw, (9)

Jo aw

where wi is a width coordinate of the projection of thegrating upon the collimating mirror. If we assume thatthe aberration is asymmetrical relative to the centralray, i.e.,

Wi OF p' OFI dw==-j -dw,

by uw adWi ( ew

by using (8), (9), and the second term in (7) we obtain

(10)AFj= iE (wi3/R2) sina.

Hence, for the collimating mirror

AFT= ± (u,2/R 12) sinan)

and for the condensing mirror

Wm13

AFm',= 1- sinam,,'. (12)R2 2

On the assumption that the aberration of the extremerays is greater than that of the interior rays, correctionshall be applied to the extreme rays that strike thegrating. The half-width of the second mirror may beexpressed as

WM,3 cos3f3 cos 3ami3 =~ - (13)

8 cosa 0 , cos 3 a'

WT, is the projected width of the grating upon thecollimating mirror, a,, and d, are angles of incidenceand diffraction, am and am' are the off-axis angles ofincidence and reflection of the central ray upon thecollimating and condensing mirror. Finally, the desiredcondition is

AFI+F+ AmF0= . (14)

(11) 14 H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).

881

SHAFER, MIEGILL, AND DROPPLEMAN

A B

X

FIG. 2. The mirror surface M1 is tangent to the OY axis at 0,hence its center of curvature is at C1l 2 on the OX axis. For anEhert system, i.e., a concentric Czerny-Turner spectrometer, themirror su face M2 is tangent to the OY axis at 0 and its centeris at CS, . F or a corrected Czerny-Turner spectrometer by makingR2<R1 such that amsaX,,' a central ray will trace a path similarto the path from A to B; or if one has the condition a,0 <am'and R 1 R2 then a central ray will follow a path similar to theone from A to B'. Under this condition, the mirror surface, M2'has its center-of-curvature at some point C2 '. The central rayshown is parallel to the OX axis; this is not necessary and anylpath of the central ray from A to Ml to the grating is valid.

Substituting Eqs. (11), (12), (13) into Eq. (14) gives

sinam'=R 22 g/R1?(cosa, cosacm'/cos3, cosam)3 sina, (15)

or

R smial /R2= R 12

-sinal ,

( Cosa, cosam a I

cos3,0 cosa,,,

Equations (15) and (16) may be approximated,

sinacm,'= (R22 /R 12) (cos3 a,/cost3,) sina,,,, (17)

andR 2= R, (sina,0 ' cos3l,/sina,,, costma,) i, (18)

by ignoring the third-order am and a,,,' dependence.1 5

Equations (15) through (18) indicate that thesymmetrical arrangement normally used in Czerny-Turner spectrometers does not correct coma and thatan asymmetrical tilt of the second mirror or change inradius of curvature is needed. The difference betweenan Ebert spectrometer and a corrected Czerny-Turnerspectrometer is shown in Fig. 2.10

16 G. Rosendahl, J. Opt. Soc. Am. 52, 412 (1962), has derivedEq. (Io) without the dependence upon the radii of curvature, byutilizing a different geometrical optic technique. W. G. Fastie(private communication and U. S. Patent 3,011,391) has derivedan equation similar to (17) but without the dependence upon theradii of curvature.

16 W. G. Fastie experimentally noted that the effect of comamay lie corrected by tilting the second mirror [Symp. Toter-ferometry, No. 11, 243 (1960), Nati. Phys. Lab., G. Brit.]. Thelarge, high-resolution Ebert spectrometer, which has a focallength of 183cm [W. G. Fastie, H. M. Crosswhite, and P. Gloersen,J. Opt. Soc. Am. 48, 106 (1958)], achieved a resolving power inexcess of 500,000. This was done at an f number >20, i.e., a

The difference between the above analysis and Leo's5

stems from the fact that Leo assumes that r=R/2and we assume the Gaussian off-axis condition r= (R12) CosaOm. We have also, to a rough approximation,taken into account the effect of coma upon the wave-front curvature after diffraction. Leo5 assumes a solutionof the form R 1/R 2 = COSa0 /COSj30 or R 2= RI(cosd 0/cosa 0 ).If we solve Eq. (18) by setting am= am' we obtainR2=R1 (cosfl0 /cos1a0 ), which is similar to Leo's equa-tion, the difference having been stated previously.

The reduction in the radius of R2 or increase of am' isquite adequate when an echellette grating is used;but if an echelle is utilized, R2 is reduced too much anda compromise must be made between increasing am' andreducing R2. The best imaging, as will be shown, iswhen R2 <R1 and am'>a,,,. It is not adequate to letR2 =Ri and a,,,'> am, to correct for coma.

It will be assumed that primary spherical aberrationis the only remaining aberration. If WI is the approxi-mate width of the beam before diffraction we may writefor the maximum beamwidth

/ - 1 /CO~poCOSA1\' IF1 1 cosOfa cosa,0 t -iWI= 64XN ) _ItL RI'1(o'~ cosa,, cosam" !

(19)

where X is the wavelength and N is a number such as0.25 which will give one the Rayleigh criterion. Itshould be noted that as the grating angle increases, theeffect of spherical aberration due to the second mirroris reduced. Equation (19) will give a good approxi-mation for practical design work.

V. RESULTS OF CALCULATIONS

When the wavefront aberration is less than about4X or the spatial frequencies are high (the case formedium and high resolution spectrometers) geometricaloptics and ray tracing will give little information aboutthe structure of the image and hence of the resolutionor contrast at the exit slit; but, since the ray trajectoriesrepresent the mean value of the Poynting vectors it ispossible to state a relative value relation of spot dia-gram A to spot diagram B.

All the instruments analyzed have a collimatingmirror radius of curvature of 200 cm and normalbeamwidth from the collimating mirror of about 14 cm.The off-axis angle of the collimating mirror is 2.580 andthe distance from the center of the entrance slit to theedge of the collimated beam is about 2 cm. The angle

projected grating width <9 cm. A re-examination of the linesrevealed an asymmetry and a coma-like flare (private communi-cation). After a discussion with one of the authors, R. Brower(private communication) of Brower Laboratories, Wellesley Hills,Massachusetts set up a point source, two spherical mirrors, andgrating. By using the grating in the zeroth norder and aligningthe instrument, the effect of coma is eliminated with a symmetricalarrangement; to eliminate the effect of coma for the first orhigher orders of diffraction of an echellette grating, the secondmirror had to be tilted. When R2 is made <R1 , Brower found thatthe image quality is superior to that with R,=R2.

882 Vol. 54

July 1964 OPTIMIZATION OF CZERNY-TURNER

between the blaze normal and the incoming or diffractedcentral ray at the blaze wavelength is about 5.20; thisis larger than usual for a Czerny-Turner monochromatorwith the above aperture. Normally this angle is about30. This has no bearing on the following analysis orcomparison between the symmetrical and unsym-metrical Czerny-Turner spectrometer.

A. Coma Correction

The spot diagram produced by an Ebert system inwhich the grating angle is 23.20 is shown in Fig. 3(a).This shows the coma and broadening due to diffractionfrom the grating. Figure 3(b) shows the spot diagramfrom a Czerny-Turner spectrometer with a gratingangle of 23.20 and the second mirror tilted so that theoff-axis angle is 3.42°. Figure 3(c) shows the spot dia-gram for the case in which the radius of curvature of thesecond mirror is 180 cm and the off-axis angle is 2.620.In Fig. 4(a) the spot diagram from an Ebert arrange-ment is shown with a grating angle of 58.30; the comais obvious. For the unsymmetric Czerny-Turner spec-trometer in Fig. 4(b) the asymmetry through the centerof the pattern is corrected absolutely, but the curvature(coma) of the astigmatic pattern is quite large; theoff-axis angle of the second mirror is 6.250. In 4(c) theoff-axis angle is 5.150; the correction is quite good, butto achieve this the central regions of the pattern are

40 Or

C 0

-60 0 60(a)

1

i ....:. -

....

!; '

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,..'s,,;-.;.-

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''''''I'''

".-',. .

' -

;:.

's,".'-'.N'.

- { I I i I , ,-60 0 60

(b)

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FIG. 3. The above spot diagrams at the exit slit are of a low-blaze Ebert spectrometer (a); a low-blaze Czerny-Turner spec-trometer where R1 =R2= 200 cm and coma correction is made byincreasing the angle am' (b); and a Czerny-Turner spectrometerwhere R1 = 200 cm and R2 = 180 cm (c). The coma shown in (a)is reduced in (b) although there is a slight increase of the astigma-tism. In both patterns a slight curvature of the astigmatic patternis noted. The asymmetric Czerny-Turner (c) shows a reductionof both coma and astigmatism. The numerical scale is in microns.

4400

0

-400

.1650,

0

-60 ' ' 60 -60 6 60

(a) lb)

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I -. '.-

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FIG. 4. A comparison of spot diagrams from instrumentsutilizing high-blaze gratings, hence a steep grating angle; (a) is anEbert spectrometer; (b) a Czerny-Turner spectrometer wherecoma is absolutely corrected through the center of the patternby tilting the second mirror, R1=R2=200 cm; (c) is the sameinstrument as (b) except that the tilt of the second mirror hasbeen reduced, bringing coma to the center of the pattern, butpartially balancing the diagram; (d) is a Czerny-Turner spec-trometer where R1 = 200 cm and R2= 130 cm. The off-axis angleof the second mirror is approximately that of the collimatingmirror. Note the change in vertical scale for (b) and (c). In (b)and (c) the curvature of the astigmatic pattern is more pronouncedthan in the low-blaze case (Fig. 3). The spot diagram (d) indicatesthat simply tilting the second mirror without a reduction in theradius of curvature will not be a satisfactory correction. All tracesoriginate from the slit center. The numerical scale is in microns.

displaced to the left and there is a general broadeningof the pattern. In Fig. 4(d) the radius of the secondmirror is 130 cm and the off-axis angle of the secondmirror is 2.70. The curvature (coma) of the pattern hasbeen reduced.

The curvature of the patterns in Figs. 3(b) and (c)and Figs. 4(b) and (c) is caused by the rays which strikethe upper and lower sections of the tilted second mirrorand by the astigmatism and spherical aberration fromthe collimating mirror being transformed by the grating.Equations (15)-(18) indicate that decreasing the radiusof curvature of the second mirror will have the sameeffect as increasing its off-axis angle. It is here, aspreviously mentioned, that the two dimensionalanalysis fails.

When a high-blaze grating is used, the ray traceresults indicate that a reduction in the radius of thesecond mirror, such that am-am' and R2<R1 , will be abetter imaging condition than simply tilting the secondmirror and letting R1 =R 2 . This may be seen by com-paring the spot diagrams of Figs. 3 and 4. The abovecondition also reduces the astigmatism of the instru-ment; this shall be discussed in the next section. 'Whateffect, if any, reducing the radius of the second mirrorwill have upon the angular scanning range of the gratinghas not yet been determined.

The reason that a spectral line would appear to becoma-corrected in an unsymmetric Czerny-Turner spec-

0� i::..q �.

I .' ;:.,

I ',

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I

- . . A.. ,-60 0 60

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S P E C T R 0 M E T E R

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SHAFER, MEGILL, AND DROPPLEMAN

0

23

(a)

fc.

X

(b)

FIG. 5. In (a), (1) is an approximate prolate ellipse, (2) a circle,and (3) an approximate oblate ellipse. They have a common centerat point P. These curves correspond to the directions of the normalto the blaze step at blaze wavelength shown in (b), A, B, and C.The optical path of S M'f G is for a central ray striking the gratingpositioned at A, B, or C in drawing (b). The central ray is shownparallel to the OX axis as in Fig. 2. But this is not necessary sinceslit curvature is independent of the trajectory of the central rayfrom M1 to G.

trometer when R1 =R2 and am<am' is that the inte-grated effect of a series of asymmetric spot diagrams,such as illustrated in Fig. 4(c), results in an imagewhich contains a symmetrical aberration (sphericalaberration).

B. Astigmatism

It is well known that astigmatism is not correctedin a Czerny-Turner arrangement utilizing two sphericalmirrors. The astigmatism is greater for the correctedCzerny-Turner spectrometer where ,mt'> aO, and thiscan be seen in comparing Figs. 3 and 4. Geometricaloptics show that this has little effect upon resolution butaccording to physical optics this may not be correct.

The work of Nienhuis,"7 Van Kampen,"8 and Nienhuisand Nijborn1 0 shows that for a circular exit pupil,primary astigmatism produces an asteroid that broadensthe diffraction pattern of an astigmatic line. This may beseen in Nienhuis,17 Born and Wolf,20 Cagnet, Franconand Thrierr,2 1 or Wolf.22 Whether this is deleteriousto medium or high spectral resolution in a spectrometercannot be determined by geometrical optics. Astigma-tism due to the collimating mirror will also have anasymmetric effect upon the diffracted wavefront [seeSecs. III and V(a)].

" K. Nienhuis, thesis, University of Groningen, 1948.,8 N. G. Van Kampen, Physica 15, 575 (1949).1° K. Nimnhuis an(d B. R. A. Nijl)orn, Physica 15, 590 (1949).20 M. Born and E. Wolf, Principles of Optics (Pergamon Press

Inc., New York, 1959), p. 477.21 M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical

Phenomena (Springer-Verlag, Berlin, 1963).22 E. Wolf, Rept. Progr. Phys. 14, 95 (1951).

C. Slit Curvature

The slit curvature for an instrument is obtainednumerically by iterative techniques which determinethe radius of curvature of the slit, if circular, measuredfrom the axial line OX [see Figs. 2 and 5 (b)], the majorand minor axes of an ellipse whose center lies on a linethrough the grating and is parallel to the OX axial lineor whose center lies on the OX axial line, and allpertinent data so that a ray trace may proceed fromany point on the calculated slit curve.

The general results of our calculations of slit curva-ture for several instruments are: (a) slit curvature is afunction of the distance from the entrance slit to thecollimating mirror; for best imagery, this distance isnot necessarily equal to the focal length of the mirrorfor any type of Czerny-Turner orientation." Slitcurvature is also a function of orientation of the normalto the blaze step at the blaze wavelength to the axialline OX [Fig. 2 and Fig. 5(b)]; (b) slit curvature maybe any type of second degree curve determined by theabove; (c) under the condition that O<x <7r/4 and thatthe best focusing condition has been chosen, the curveapproximates an ellipse. The approximate ellipse de-generates to a circle when the blaze normal at blazewavelength is parallel to the OX axis. This is the well-known Fastie slit.2

In Fig. 5 (a) the circular slit (No. 2) is shown and the

.350-

0

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(a)

C

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%.d-...:,...

.@,.....

is'':....

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0

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FIG. 6. These spot diagrams all originate from 2.75 cm above thecenter of a circular slit. This would be a slit length of 5.5 cm. Theyare for low-blaze grating instruments; (a) is an Ebert spectrom-eter; (b) is a Czerny-Turner where R, =R 2 =200 cm, and (c) is aCzerny-Turner spectrometer where R1 = 200 cm and R2 180 cm.The comatic shift towards the right, as compared to Fig. 3 isquite pronounced and, as explained in the text and Fig. 7, limitsthe slit length. The numerical scale is in microns.

23 To illustrate the sensitivity of the slit curvature upon thedistance from the entrance slit to the collimating mirror: for oneof the instruments analyzed, a shift of 0.3 mm distorted the slitcurvature from a circle to a smooth curve which deviates from acircle by i=6 A at the slit center to ;0.2 u, 2.75 cm above orbelow the slit center.

884 Vol. 54

I

July 1964 OPTIMIZATION OF CZERNY-TIURNER SPECTROMETER

approximate prolate and oblate elliptical slits (Nos. 1,and 3, respectively) are shown. The elliptical characterof the slit was predicted by Kudo,4 but not generalizedas to grating position.

It makes no difference where the grating is locatedalong the path M1 to G as illustrated in Fig. 5(b).The positions of the center of the grating A, B, and Chave no effect upon slit curvature; the orientation ofthe blaze normal at the blaze wavelength numbers 3, 2,and 1 is critical. These numbers correspond to the curvenumbers in Fig. 5(a); of course, the grating should bepositioned nearly telecentric on the object side, toreduce vignetting.

The center of the slit curve always lies on the OXaxis, under the condition that the mirror surface istangential to the OY axis, Figs. 2 and 5(b), at thepoint 0. This is the Ebert condition; to build a Czerny-Turner spectrometer with circular slits, the center ofcurvature of the collimating mirror must lie on the OXaxis at Ct. The position of the second mirror is arbitraryand is a function of optical or mechanical requirements.Figure 5(b), shows the distance D from the axis to thecenter of the slit, and is in all cases the distance fromthe center of the slit curve to the center of the slit.In Fig. 5(a) this is the distance from P to the slitcenter, where the three curves have a common tangentpoint.

Preliminary work on an exact, analytic expression forslit curvature indicates that the slit curvature, as afunction of the blaze normal, goes through all the conicsections from a point through an ellipse, circle, ellipse,parabola, hyperbola, and finally a straight line, as theangle X varies from zero to 7r/2. The effect of slit lengthupon aberration is illustrated in Figs. 6(a), (b), and (c)and Figs. 7(a), (b), and (c).

In comparing the three low-blaze instruments onecan see that as a point source moves up the entranceslit, the direction of the coma is to the right. If there iscorrection for coma in a small region near the center ofthe slit, the coma is not corrected at a point above orbelow the slit center. The coma as a function of slitlength is schematically illustrated in Fig. 7.

If R1$dR2, the radius of the exit slit is simply (R2/R1 )X (radius of the entrance slit).

VI. COMPARISON WITH OTHER INSTRUMENTS

If one compares the luminosity 8 -11 '1 3 (defined asIBA, see Sec. II) of the various types of grating

spectrometer-monochromators now in use to the Czerny-Turner monochromator, symmetric or asymmetric, theCzerny-Turner monochromator will be superior. TheEbert spectrometer (concentric, symmetrical Czerny-Turner spectrometer) has been used for some time withgratings whose blaze is 60O. To achieve good resolu-tion, the width of the exit pupil (aperture) must bereduced from any theoretically calculated value. If agrating which has a blaze angle of 760 is used in an

(a) (b) (c)

FIG. 7. This schematically illustrates a general curve for theexit slit. The arrows show the direction of coma, length is indicativeof the amount and zeros show the points of coma correction.Curve (a) is for a high-blaze symmetrical arrangement or a low-blaze symmetrical arrangement with large angle (X). It shows thelarge amount of coma near the central region and decrease towardsthe slit ends. Curve (b) is for a high- or low-blaze unsymmetricalarrangement in which there is absolute coma correction near theslit center, e.g., aligning the instrument by imaging a point source.The coma increases towards the slit ends. Curve (c) represents asymmetric arrangement of low-blaze angle and small angle (X),an unsymmetric, low blaze arrangement with large angle (X), ora high-blaze, unsymmetric arrangement. Coma is introduced nearthe slit center and is balanced towards the ends.

Ebert system a further reduction in aperture would berequired to maintain good resolution.1 6

The Fabry-Perot spectrometer'"s2,1 3, 24' 26 has adefinite luminosity advantage whenever the resolvingpower is > 106 over the large, high-blaze gratingCzerny-Turner spectrometer. The resolving power of agrating may be written as R= X/6X- 2 (W/X) sin cosx.The flux passing through a spectrometer using a gratingof width m25 cm would be severely limited if R-10 6,.The factor k (see Sec. II) would be -0. In the ultra-violet, 1000 A7X 4000 A, and infrared, 1 kZX<25 ,q,the Fabry-Perot spectrometer presents technical prob-lems of coatings, plate fabrication and scanning methodswhich must be brought into any comparison with theCzerny-Turner spectrometer. The visible is the bestregion of the spectrum for comparison, since this isadvantageous to the Fabry-Perot spectrometer. A validcomparison may then be made when R < 101.

Without going into detail on the theory of theFabry-Perot spectrometer, the equation which describesthe luminosity of a Fabry-Perot etalon is LF==AFQr,where A P is the utilized area of the plates, £2 is thesolid angle, andr is a transmittance. The transmittance,r, results from the product of three transmittances,7A, TF, and rE. Under near-optimum conditionsTEsO.S, rF0.8, and TAZ0.75 (a good seven-layercoating of ZnS-Na 3AlF 6 will have rA=0. 7 ). 2 4

From the above and from the geometry of theFabry-Perot etalon, we may write for the luminosityof a single etalon LF,=1.5AF/R, where R is the re-solving power. From Sec. II, the luminosity of theCzerny-Turner spectrometer may be written as

24 J. E. Mack, D. P. McNutt, F. L. Roesler, and R. Chabbal,Appl. Opt. 2, 855 (1963).

25 R. G. Greenler, J. Opt. Soc. Am. 47, 642 (1957).

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SHAFER, MEGILL, AND DROPPLEMAN

LG= 2krTfiG sin (p cosx/R. We shall assume that k= 0.75,r=0.6, f=0.04, sinSO=0.97, and cosx-1. The lumin-osity of the Czerny-Turner spectrometer will then beLa 00.O4(AG/R) where AG is the area of the grating.On the basis of equal resolving power, the luminosityratio of the Fabry-Perot etalon to the Czerny-Turnerspectrometer is then 50(ApF/A1 ).

When two or three etalons (the third and fourthetalons may be interference-absorption filters24 25) areadded to the optical train to make a Fabry-Perotspectrometer, reduction in luminosity due to furtherabsorptance by the coatings, ghost rejection, reflectionlosses, and automasking must be considered. The lumin-osity ratio will then be < (A F/IA G) to Z 10(l F/Il a).

From the preceding it may be inferred that in mostcases a high-blaze grating Czerny-Turner spectrometerhas superior luminosity; Stroke" has made a similarobservation.

In the infrared, 1 puXZ25 ;, the detector noise, atpresent, is greater than the statistical photon noisei.e., signal noise. In this region of the spectrum aMichelson interferometer, making use of the Fouriertransform techniques, has a signal/noise advantageover the large, high-blaze grating Czerny-Turnerspectrometer. 26 If infrared detectors could be made suchthat the noise equivalent power is < 10-2 W, this ad-vantage would be partly eliminated. In the wavelengthregion 4 ,u ZX< 13 p and for weak sources of radiation,we must also consider blackbody photons from thespectrometer and chopper. This is an added photonnoise and can be serious. In this spectral region centerednear 9 u, no matter how good the detector, the superi-ority of the Michelson interferometer may well remain.

A similar argument may be used to compare nmodu-lating spectrometers such as Connes' SISAMla 27 orGirard's grille spectrometers2 6 2 to the Czerny-Turnermonochromator. We shall refer only to Girard's grillespectrometer, since both are modulating spectrometers.

It may be of interest to compare the instrumentfrom Ref. 29 which has a focal length of 200 cm toa Czerny-Turner spectrometer-monochromator. TheCzerny-Turner may have a luminosity increase of 2to 45 by using a high-blaze echelle. The Girard grillehas an aperture width of 16.9 cm; the Czerny-Turnerwill work at the wavelength involved with an apertureof 25 cm. This is an increase in luminosity of 2.The length of the central grille in the grille spectrometer

26 We shall be interested only in resolution < 1 cm-' in the abovestated wavelength region. This value of resolution, to the author'sknowledge, has not been achieved by interferometers employingthe Fourier transformation techniques. In the spectral regionwhere photomultiplying tubes can be used as detectors (< 1 A) thesignal/noise advantage of these types of interferometers isnullified. The argument shall proceed under the assumption thathigher resolution will be obtained in the future.

27 J. Connes, J. Phys. Radium 19, 197 (1958); Symposium onInterferometry, National Physics Laboratory, Great Britian No.11, 409 (1960).

2 8A. Girard, Opt. Acta, 7, 81 (1960).29 A. Girard, Appl. Opt. 2, 79 (1963).

is 3 cm, whereas in the Czerny-Turner the slit lengthcan be 8 to 9 cm; this provides an increase of luminosityof 3. The total factor of luminosity increase is then8 to 180. There is a factor of 130 increase in the ar-rangement of Girard's instrument as compared to hisinstrument used as a monochromator. The luminosityadvantage of the grille spectrometer over the Czerny-Turner monochromator is then from -16 to 0.7. Ifthe f number is to be maintained, the above comparisonis not realistic, since a high blaze grating in the Czerny-Turner spectrometer would have a width of -50 cm.The maximum attainable width is 25 cm. If anechelle of 25 cm were used, the luminosity advantageof the grille spectrometer would be -6. The previouscomparison of luminosity advantage of -1 would bequite valid if a focal length < 100 cm were used.

In the Girard grille spectrometer only the modulatedwavelength is detected as signal, but a large part of thespectrum on either side of the modulated wavelengthis detected as statistical photon noise. The photon noiseis more serious to the Girard grille than to a Michelsoninterferometer since the Girard grille detects onespectral element at a time instead of all the spectralelements simultaneously; hence, in the infrared near 9 pand using weak sources of radiation, the Czerny-Turnerspectrometer should be far superior to the Girard grille.

Only if the Littrow system of Girard will allow theuse of a high-blaze grating, longer grille slits, and largeraperture, will the luminosity gain of the grille spec-trometer be a factor of 130 greater than a Czerny-Turner monochromator.3 0

VII. DISCUSSION

The use of high-blaze gratings in conjunction withthe asymmetric Czerny-Turner arrangement shouldproduce a luminosity increase over a symmetricalsystem such that it is quite comparable to interfero-metric and modulating devices. The use of sphericalmirrors for collimating and condensing makes itrelatively simple and economical to build.

For large apertures the resolution limit will not bedefined by diffraction, but from the residual coma(Sec. V and Fig. 4).

The above suggests that an investigation as to thefeasibility of using aspherical mirrors for collimatingand/or condensing may prove worthwhile; but careshould be taken, for an analysis which only considersthe image quality of one point source is rather useless.The luminosity is proportional to the slit length; hence,several points along a slit should be considered.

As was explained in Sec. IV, the relative optimumrelationship of the grating angle, radius of the second

`0 G. Stroke (private communication) informed us that he gaveGirard a grating blazed at 45'. Evidently Girard did not use it inthe previous referenced papers since the scanning range was37° <o<22' in a Littrow system. If the grille spectrometer is usedwith a grating of 45' blaze angle, then the luminosity advantageof the grille spectrometer over the Czerny-Turner monochromatorwould be ;30 to 1.3.

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July 1964 OPTIMIZATION OF CZERNY-TURNER SPECTROMETER

mirror (R2) and the off-axis angle of the second mirror(am') has not been solved; and until this is done it willnot be a simple matter to optimize completely thedesign of the asymmetric Czerny-Turner spectrometer.

A grating blaze angle of 760 has been used throughout,when flux or luminosity were calculated. A blaze angleof 64° would make little difference in the result ofcomparison; in fact, this angle was used for the initialcalculations, but was changed after reading Stroke"0

under the assumption that this type of grating maysoon be available.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

ACKNOWLEDGMENTS

The authors are greatly indebted to Professor H. N.Rundle, University of Saskatchewan, who, while at theNational Bureau of Standards, Boulder, Colorado,acted as a responsive sounding board and excellentcritic. The authors also wish to thank Jack W. Stunkel,Walter Harrop, David Hansen, A. V. Brackett, andAlice Keysor for their assistance.

We would like to thank Dr. D. MO. Gates withoutwhose support this paper could not have been completed.

VOLUME 54, NUMBER 7 JULY 1964

Resonant Modes of Optic Interferometer Cavities. I. Plane-Parallel End Reflectors*

LEONARD BERGSTEIN AND HARRY SCHACHTER

Department of Electrical Engineering, Polytechnic Institnte of Brooklyn, Brooklyn, New York 11201

(Received 7 November 1963)

A study is made of the resonant or normal modes of optic and quasioptic interferometer cavities withplane-parallel end reflectors. The solution of the integral equation governing the relation between thenormal modes and the geometry of the cavity is found by means of a series expansion of orthogonal functions.The terms of the series for the normal modes can be interpreted as Fraunhofer diffraction patterns char-acteristic of the geometry of the end reflectors. Various geometries, such as the infinite-strip, rectangular,and circular end reflector cavities, are considered and the results plotted and interpreted.

I. INTRODUCTION

FABRY-PEROT interferometers are used exten-sively as infrared and optical maser (or laser)

resonators. The interferometer consists essentially oftwo highly reflecting plane surfaces facing each otherand separated by a fixed distance. An active medium isimmersed in the region between the reflectors. Anelectromagnetic wave leaving one of the reflectingsurfaces will be amplified as it propagates through theactive medium toward the other reflecting surface. Atthe same time, there will be a decrease in the intensityof the wave due to scattering by inhomogeneities in themedium. A further decrease in the intensity will becaused by the diffraction spread of the wave. When thewave arrives at the other end of the resonator additionalpower will be lost in reflection due to the finite conduc-tivity and/or partial transmission of the reflector. Forpower-buildup (oscillations) to occur in the resonator,the total power loss must be less than the power gainedby travel through the active medium. The relativescattering and reflection losses (i.e., the ratio of thepower losses per unit cross-section area over the fieldintensity) are constant and depend only on the

* The work reported here is part of a dissertation submitted byH. Schachter in partial fulfillment of requirements for the Ph.D.degree at Polytechnic Institute of Brooklyn. The work wassupported by the Joint Services Technical Advisory Committeeunder Office of Aerospace Research Grant AF-AFOSR-453-63.

homogeneity of the interferometer medium and thereflector properties, respectively. The diffraction losses,however, depend on the field distribution within theinterferometer. They thus determine not only thestart-oscillation condition but also the field distributionin the interferometer during oscillations. For anarbitrary initial field distribution, the diffraction losseswill cause shifts in the distribution from reflection toreflection until a self-reproducing field distribution, ifone exists, will be set up. These steady-state distribu-tions or normal modes must be independent of the initialdisturbance and are properties of the geometry of theend reflectors only. The relation between the geometryof the end reflectors and the field distribution of thesenormal modes, the diffraction losses associated withthem, and the resonant frequencies of these modes are ofinterest in understanding the operation of masers usingFabry-Perot interferometers as resonant cavities.Furthermore, from the distribution of the electro-magnetic field within the interferometer, we candetermine the field distribution of the emerging beam,its angular spread, and the maximum intensity that canbe achieved by means of a focusing lens.

Schawlow and Townes' concluded that the Fabry-Perot cavity is resonant to uniform plane waves travel-

'A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940(1958).

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