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(1)Mem. School. E.O.S.T. Kinki University No.8: 1--48 (2001). 1. Geometrical-Optical Theory of Diffraction Grating on Ellipsoidal Surfaces for General Incident Angle Fundamental Theory and Optical Ray Tracing Calculations Minoru Hisa Abstract Theoretical determination of optical characteristics, in particular aberration of a spectral image, of a spectroscope using a diffraction grating is indispensable for designing of a spectroscope. In this paper the extent of the aberration is analytically treated using Fermat's principle for development of a diffraction grating capable of complete elimination of astigmatism, as well as design and preparation of a spectroscope using the grating, which has led us to a satisfactory result for trial of experimental verification. This study is summarized as follows. ( 1) Traditional diffraction grating theory has occasionally caused errors in the process of theoretical calculation because off-plane mounting has been treated as either a theoretical extension or an approximation of in-plane mounting. In this study, we have developed a theory independent of either inplane or off-plane because the calculation was based on general incident angle. As the result, an in-plane mounting process capable of complete elimination of astigmatism was found. In addition, we found a novel system of off-plane mounting that can completely eliminate astigmatism, which had not been found by traditional theory of diffraction grating on a spherical surface. ( 2) W. T. Welford has proposed the process of optical tracing, but he has not carried out a concrete orbit calculation. The example of embodiment of lighttracing calculations in an optical system using non-spherical diffraction grating has also been scanty. In this study, we have obtained satisfactory result by carrying out ray-tracing calculations based on general incident angle for diffraction grating on ellipsoidal and toroidal surfaces. ( 3) We designed and prepared a spectroscope eliminating astigmatism using diffraction grating on a toroidal surface as an approximation to an ellipsoidal surface to experimentally verify the theory. We have obtained experimental results perfectly in accordance with the theory as far as elimination of astigmatism and line dispersion is concerned. This verifies the validity of the theory of diffraction grating on ellipsoidal surfaces for general incident angle, and also practical applicability of the spectroscope for spectroscopic study using a non-spherical surface. In particular, the spectroscope having no astigmatism in in-plane mounting is useful for experimental works and.

(2) 2. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001). that free from astigmatism in off-plane mounting is useful for such spectroscopic outdoor experiments as cosmic observations. Also the usefulness of diffraction grating on a non-spherical surface has been verified. CHAPTER 1. GEOMETRICAL-OPTICAL THEORY OF DIFFRACTION GRATING ON ELLIPSOIDAL SURFACES FOR GENERAL INCIDENT ANGLE -FUNDAMENTAL THEORYSection 1. Introduction. The process for theoretical determination of optical characteristics of a spectroscope using diffraction grating, in particular the aberration of spectral image, is roughly divided into geometrical-optical method convenient for designing of spectroscope and physical-optical method suitable for precise discussion of e.g. a profile of the spectrum lines. The study of the latter method, however, has be en made only poorly because it includes such very difficult problems as diffraction phenomenon, interference capability and light polarization. The geometrical optical method includes light tracing method traditionally employed for lens designing etc., which has rapidly developed owing to the recent progress of electronic computers, and a method using Fermat's principle. Fermat's principle permits analytical treatment of the extent of aberration and also designing of optical system that possesses required optical characteristics. One can also understand the contribution of the aberration in each optical system. Fermat's principle was applied to optical systems by Hamilton, and was applied to diffraction grating by Zernike both for the first time. After that Beutler developed the theory of diffraction grating in a similar way as Zernike did, but he could not complete the study. The attempt was succeeded and completed by Beutler's, but Namioka pointed out that Beutler's theory involved an error and he revised and generalized the theory. A large number of studies have been made on spectroscopes using diffraction grating, but none of then1 has been able to eliminate astigmatism of spherical diffraction grating excepting for a few mountings. Therefore attempts have be'en made to solve the problem using diffraction grating on non-spherical surface, and many theories have been proposed, which include Namioka's theory of diffraction grating on ellipsoidal surface. This theory, however, treats off-plane mounting as an extension of the theory of in-plane mounting, and thus includes many factors that might cause gross errors. In this study, we developed the theory of diffraction grating on ellipsoidal surfaces for general incident angle based on the following principles. ( 1) Theoretical calculation was performed faithfully based on the Fermat's. principle, which was used only to define the path of diffracted light. ( 2) The function of optical path was serially expanded based on the radius of the projected point P' on the coordinate plane of the diffraction point P, and was sorted according to the degrees..

(3) 3. Symbols and letters necessary for the following discussions are lumped together in the table summarizing means of description.. y. P'(w,l,o). "\X,y /,t., * *,a) <.f. ....... ....... ....... ....... ..... ...... ...... ....... .... ....... z ..... Fig.- 1. Fig. - 2.

(4) 4. Memoirs of The School of E.O.S. T. of Kinki University No. 8 (2001). The following relationship. 1S. valid also with regard to image spatial quantity.. x = r cos f y = r cos T} z = r cos r cos f =sin r cos e cos T} =sin r cos e =sin s cos f = sin T} sin a = cos s sin a cos r = sin T} cos a = cos s cos a cos 2 f +cos 2 T} +cos 2 r = 1 cos 0 = cos f cos if> + cos T} sin if> = sin (cos (if> - e ). (a - 1). (a - 2) (a - 3) (a - 4) (a - 5) (a - 6) (a - 7). (a - 8) (a - 9) (a -10). Means of Description (cf. Fig. - 1 and Fig. - 2 ) Object spatial quantity. Image spatial quantity. Quantity on plane of grating. Origin: 0 On the pupil plane Z axis: Normal line of the grating Y axis: Parallel with the grooves at 0 X axis: Perpendicular to the grooves at 0 X, Y, and Z lie rectangular inright-hand relation. Object spatial point (Point light source): A (x, y, z) Projection of A on XY pIa ne: A' (x, y, 0) Projection of A on XZ pIa ne: A' (x, 0, z). Origin, Z, Y, and X axes: same as the left. An asterisk * is attached to each quantity to express an. image spati~l quantity, although the coordinate system is based on the ima ge spatial quantity. Object spatial point (image point) : A" (x', y", z") Projection of A" on x" y" plane: A"' (x', y', 0) Projection of A' on x" z" plane: A"' (x", 0, z·). Origin, Z, Y, and X axes: same as the left. Zp axis: Falls on Z axis. L axis: Falls on Y axis. W axis: Falls on X axis W, L, and Zp: Fall on X, Y, and Z. Point on plane of grating :P(xcosif>, X sin if> , Zp) Projection of P on WL plane: P' (X cos if> , xsinif> , 0). f ; LXOA. f' ; LX'OA'. if> ; LWOP'. LYOA r ; LZOA e ; LXOA' o ; LAOP' a ; LZOA' S ; LA'OA. T}' ;. T};. r ; OA. LY'OA". r' ; LZ'OA" e' ; LX'OA" o' ; LA"OP' a" ; LZ'OA" s" ; LA"OA". X ;. OP'. Zp=PP'= c'. f (2n~3:!!ln. n. 2. Kn(if». n"'i. Wherein K ( if> ) = k - k' cos2 if> = cos: if> a. + sin: if>. b (a-ll). r" ; OA" Here. 1), 1(1 1) k=21(1 if+;: ,k =2 if-;: In these equations, a, b, and care the diameters in the direction of X, Y and Z axes, of the ellipsoidal surface, respectively..

(5) 5 Section 2. Optical Path Function. The origin of rectangular coordinate is defined on the center of the diffraction grating, and Y and Z axes respectively assume the direction of the tangent line at the groove on the origin and the direction of the normal line at the origin of the plane of the diffraction grating. X axis is assumed to lie rectangular to both of Y and Z axes. A (x, y, z), A*(x*, y*, z*), and P (w, l, zp) are light source point, image point, and an arbitrary point on the plane of diffraction grating, respectively. Here, w assumes a discontinuous value and the ratio of wand grating constant d (w/ d) should assume only an integer, but is treated as a continuous function when it is used to introduce the expression of diffraction grating. For simplicity, a groove is regarded as a single line without width. The grooves of the diffraction grating on an ellipsoidal surface are assumedly grooved so that they give a straight line of a constant interval when projected ori the contacting plane (X-Y plane) at the origin of the grating. Assuming that the light from an object A is diffracted at a point P on the plane of diffraction grating and reaches the image point A *, the optical function F is given by adding a term expressing the effect of diffraction to the light path -length AP+ PA *. ( 1). F =AP+PA*+ wrnJ.../d. where. ( 2) l. w=xcos~.. = xsin~. (3). Thus using equations (2) and (3), we obtain equation (4). Ap 2= 7 2+ X2 -2 7 xcoso-2 7zpcost + z~. (4). Since the point of diffraction lies on an ellipsoidal surface, equation (5) IS derived. Thus the light path length is expressed by equation (6), where L is defined by equation (7). ( 5). 1. AP= 7 { 1 -. ~ ( 2:!: ~ !! L n } n=l. = 7 { 1 - ~ L ~ ~ L 2 - 116L 3 - 1~8 L 4 ••••••••• }. = 7- xcosO' -. ~2. - L2. { cK( <6) cost -. ~sin2 0'. {.£K( <6)cosO'cost- ~sin2 O'cosO'} 7 7 .. - L{ CK2 (~)·cost- £K (<6) sin 8 . 7 2. -. (6). }. 2. O'cosO'. ~CK( <6)( 1 - 3 C~S2 0') cost 7. +;. 1 - 5 cos 2 If )sin 2 If } + 0 ( X. 5. 3 (. ).

(6) 6. Memoirs of The School of B.O.S. T. of Kinki University No.8 (2001). (7). Here O( X 5) means terms above X 5. Similarly, equation (8) is obtained. P A *= y *- xcoso * 1 . 2.t' -X - { cK( cp)cost*--.sm u *} 2 y 2. - r{~K( ¢»coso*cost*- ~ sin 0*coso*} 2 y* y 2. - L8 { CK2 ( cp) cost *-. (8). C: K Z ( cp) sin 2o*coso*. y. 2. 2. y*zCK( ¢» ( 1 - 3 cos 0*) cost*. ,+ 1 - 5 cos 2 0*)sin2 o*} + 0 (X 5 ) . ~3( y. As a result, the light path function F is expressed by equation equation (9) according to the order of X. Each term is listed in equations (10) and (11). F. = 10 + 1. 5. 1. + 1 2 + 13 + 1 4 + 0 ( X. (9). ). 10=y+y*. (10}. 11 =- X(-coso+coso*) + Axcoscp 2. sino sino * } X cK( cp)(cost+cost*)--y--~ 1 =-2 {. 2. 13 =. X - 2. 3 {. C yK( cp) cosocost +. sin Z oCoso y2. C 1* K( cp) coso *cost *. sin Z o*coso*. }. y*2. 2 X4 { CK2 ( ¢>)( cost + cos t *) - C2 K2 ( cp) ( -sin t siny2*t * .) 14 = -"8 y- +. -~CK(cp)( 1-3 cos 20)cost y + ~( 1 - 5 cos 2 0)sin 2 0 y. y. ;2. (11). cK(cp)( 1-3cos20*)cost*. 2 2 + _1_( y*3 1 - 5 cos 0*) sin 0*. Here A = m A / d where d, A and m mean the interval of the grooves, the wavelength of the diffracted light, and the order of the diffracted light, respectively..

(7) 7. Section 3. Direction of Main Diffracted Light and Focal Distance. Generally, Fermat's principle suggests the diffracted light to have' a direction so that the optical path function F assumes a constant value with regard to the coordinate of the point P on the plane of grating. Thus by application of this principle to' the optical path function, equation (12) is obtained.. of=. of 'dX+.l of ·d¢= 0 aX x o¢. (12). This is transformed into a set of ,binary simultaneous equations having degree of freedom equal to the point P' on a pupil plane, in order that the equation is valid irrespective of the value of X and if>. It requires that equations (13) are simultaneously valid. Since these equations are still functions of X and if> (power series), it is transformed into equation (14) and is approximately solved in the order of nSIng power. F X= of = 0 aX. (13). F;=~=O Xo¢. (14) F;=. x~¢(fo+fl+f2+f3+f.+""")=O. 3. 1 Zero order approximation. Zero order approximation is given by equation (15). Here the light path function F does not include K (if» and accordingly is independent of the term related to the curvature of the grating surface, approximating a plane. To this is applied Fermat's principle.' F::::fo+fl. (15). Since fa, and accordingly f land f o<P are constant and do not affect the partial differentiation calculation, 0 F = 0 is transformed into two equations including f lX and ft<P equation (16). In order that these equations satisfy any value of if>, equation (17) should be valid..

(8) 8. Memoirs of The School of B.O.S. T. of Kinki University No. 8 (2001). !Il =. (-cos~-cos~:+A)cos~. + ( - COS7] - COS 7] :) sin~ = 0. (16). 11'$=-( -cos~-cos~:+A)sin~ +( -COST) -COST}:)cos~= 0. [. COS~ + COS'70*= 0. (a). cos~ + cos~o*= A. (b). (17). Equation (17) - (a) gives r) 0" = 7[ - r), and the direction of OA" that simultaneously satisfies both of equations (17) - (a) and (17) - (b) is indicated in Figure 3. Equations (17) - (a) and (17) - (b) are also expressed by equation (18) using equations (a- 5) and (a- 6), too.. Fig.- 3. 80*=- 8. .. . *. (18). A m).. Sma +smao = COS8 = dCOS8. Thus the direction of OA *, r) 0* and f 0" is determined. Since OA" can be considered the light diffracted at the center 0 of the diffraction grating, i.e. main diffracted light, zero order approximation can be understood to define the direction of the main diffracted light. Hereafter, quantities related to main diffracted light are to be expressed by adding suffix o..

(9) 9. 3.2. The First Order Approximation. Now that the direction of main diffracted light has been determined by 0 th order approximation, we proceed to higher approximation of the light path function equation (19) including X 2 term, and apply Fermat's principle to it. (19) We are paying attention to the image point Ao* on the main diffraction light path, but since we have defined fl + flx and fa'" +it'" to be zero, equation (20) should be satisfied in order that 0 F= 0 . (20) Transformation of f2 gIves equation (21), where A, B, and C will gIven as shown in equation (22) .. .h = - ~ (Asin2 ¢>+Bcos2 ¢>+C). (21). 2 2 A = -COS~COS1/ + -.-cos~ :COS1/:. r. r. .. _ sin 2 tsin2 () + sin z t:sin2 -. y. y* z. 0: z. z. B = cos ~ - cos 1/ + cos ~:- cos 1/: 2. y. y*. 2. 2. _ sin t cos2 () + sin t :cos2 (): -. y. y*. 2 c k' (cost + cost:). 2 c k' (cost +cost t). (22). C = cos 2 s-~ + cos 2 1/ - 2 + cos 2 s-~ 0*+ c~s 2 1/ 0*- 2 + 2 c k ( cos t + cos t t) y y 1 +co~ t t + 2 c k ( cos t + cos t t) y .. Equation (20) is transformed into equation (23) using equation (21). The focus is to be determined by solving this equation. {X( Asin2 ¢> + Bcos2 ¢> + C) = 0 x( - Bsin2 ¢> + Acos2 ¢». =0. (23). The condition for equation (23) to be effective irrespective. of X value is shown by equation (24), wherein suffix J expresses the quantity related to focus..

(10) 10. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). Af. Sin2. ~f+Bfcos2 ~,r=-Cf {. - B,rSln2 ~,r+ A,rcos2. ~,r=. (24). 0. By eliminating sin 2 ¢ J and cos 2 ¢ J from equation (24) to investigate the relationship between quantity of object space and that of image space, equation (25) is obtained. A~+B~=C~. ~. (25). Now, equation (25) has to be considered separately on two cases In which CJ 0 and CJ~ O. Thus,. ( i) When CJ= 0 , The focal distance i j satisfies the following equation (26) directly from equation (25), where equation (27) is defined.. a' )'j2 + b' ),j+ c' = 0. + .£k' sin2 ~cos2 f) (cos~ + cos~:) + )'. (26). cos: ~ )'. (27). + c k' sin ~:cos2 f):( cos~+cos~:) - c k( 1 +cos 2 ~:) (cos~+cos~:) 2. When a' ~ 0, the focal distance i j is given by equation (28) by equation (26), and thus it has two different values. If we express the farther focus as iff and the nearer focus as r in, equations (29) is derived (cL Fig.- 4 ). -b'±Jb'2-4a' c'. "V*----2 'I fa'- - - - -. (28). *_ -b'+Jb'2-4a'c' 2 a'. )'ff-. *_ - b -. (29). ";~b-:-/-:-2---4-a-'-c-'. )'fn-. 2 a'. When a' = 0, equation (26) leads to equation (30) to gIve a single focus by equation (31). b'. )'/+ c' = 0. (30).

(11) 11. c'. y/=-v. (31). The next step is to determine ¢ I corresponding to these r / 'so Equation (24) gives equation (32) and thus equation (33) will be obtained. .. Af. cos2 rpf. = - Ji.c.. C. tan2 rpf= Bf. (32). f. 1 T an -1 -Af 2 Bf. In the case of. A.. _ 'f-'f--. + 2 n + 1 1C 2. In the case of In the case of. (33). Bf = 0. : rpf. =. (2n+ 1) 4. 1C. Where Tan- 1 means the main value and n is an arbitrary integer. This means that the diffracted light from point P on the grating plane intersected by the line of inclination angle ¢ I on the X-Y plane focuses on the main diffracted light. Substitution of r 1* contained in AI and BI with r I: and r n~ gives two values, ¢ In and ¢ nn respectively (cf. Fig.- 4) The in":plane mounting gives either ¢ f = 0 or ¢ f = J[ / 2 (cf. Fig.- 5). When CI = 0 Equation (25) gives A J = 0 and Bf = 0 which means that the diffracted light of the first order approximation should have a single focus irrespective of the values of X and ¢. This is called a stigmatic condition. (ii). y. z.

(12) 12. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). A. Fig.- 5 Section 4. Theory of Aberration. In the previous section we expanded the light path function F (X) in the order of increasing power of X and determined the direction of main diffracted light f 0*, r; 0* and r; 0* based on the 0 th order of X, as well as focus Ao* J based on t he first order of X. In cases other than A = B = C = 0, however, the diffracted light gathering on A; J is the only light that satisfies tan 2 ¢ = AJ/BJ and cos 2 ¢ = - BJ/CJ on the pupil plane of the diffraction grating. The light diffracted at other part of the grating reaches a point A * (f *, r; *, r; *) with a little distance from Ao* J. This is the aberration, which this section handles with. Since we have determined Ao· J based on the first order of X, this section studies the higher order of X. Since we have defined the directional cosines of the main diffracted light as cos f 0*, cos r; 0* and cos r; 0 in the previous section, the directional cosines of arbitrary diffracted light reaching A· ( f *, TJ *, r; *) are to be expressed by equation (34). Theref ore, these are expanded to give equation (35). cos( ~:' + ,u), cos(. 1]:' + a ),. cos( ~:' + €. ). (34) (35).

(13) 13. O( I-l 3) stands for terms above I-l 3 and thus O( I-l 3), O( a 3), and O( c 3) will be omitted for further calculations. Similarly, we will get equation (36) and (37), and by putting pas shown in equation (38), we obtain equation (39). * ) =cOST)o-aSlflT)o'-2"a * ... 12 COST) 0* cos ( T)o+a. COS(. t: + E ) = cos t : -. p=. ~~* -. 1. E. ~. sin t : -. E2. (36). cos t :. (37) (38). 1. 1. (39). --.-=-*( 1 +p) I'. Yf. 4. 1 Re-expansion of Light Path Function. The light path function F( X) (Eq. - (11)) is re-expanded using equations (35), (36), (37) and (38). In the re-expanded light path function Iu, i means the degree of X and j means the degrees of I-l, a, and c. Thus we obtain equation (40). The terms having i + j equal to or exceeding 3 are ignored. 11 = - x( coso. + cosO' *- AcostP). ./to =. - x( cosO'+cosO't-AcostP). 111 =. x( ,usin~ tcostP -. q. sin1Jo*sintP). (40). 121 = - ~ X2 [ -. E. CK( tP) sint t. +p(cosO't.fa = - ~ X3 [. C K(. + ;/ {- ,usin~ tcos¢ -. 1 )} ]. ¢) { ~ cosO'cost + ?osO'*cos t *}. _ sin 2 O'cosO' _ sin 2 0'*cosO'* ] 1'2. 1'*2. 130=- ~ x CK(¢){ ~cosO'cost+ ;:cosO'tcostt} 3. [. sin 2 0' tcosO' t ] 1':2. q. sinT) tsin¢.

(14) 14. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001). Now, /20 can be written as equation (41) using equations of (a- 4), (a- 5) and (a-g) based on equations (42). (41). +. B.1= sin2 t cos2 8. sin. 2. "I. t tcos2 * 8* 0. "If. 2 k' ( C. ~+. cos~. ~ *). cos~o. (42). 4.2 The First Order Aberration The first order aberration is used to know how far is the point Ao* ( f *, TJ *, 1; *) from Ao* ( f 0*, TJ 0*, 1; 0*), depending on the first order change of f.L, a, and E. We have already defined equation (43) in the first order approximation, and thus the light path functions further needed for calculation of the first order aberration are ill and /20, and of= 0 requires equation (44). Then we obtain equation (46) by solving the equation (44) based on equation (45). As described in Section 3 - 2, the condition for the equation (47) to be valid irrespective of the values of X and ¢ is equation (48) based on equation (24).. f/+ fl~ = 0,. ffl+ fl~ = 0. (43). 1.M+J2~=O. flt+f2:=. (44). 0. .ft~ = ,usin~ tcosrP - u sin7J tsinrP fl~ = -,using: tsinrP- usin7J tcosrP. f2~ = -. ; X {AfSin2 rP + Bfcos2 rP+ Cf }. f2~ =. ; X { Afcos2. ,u. =. -. X·A f. 2 sing: :COSrPf. (45). rP - B f sin2 rP },. sin ( At _ 'fJ. At. ). 'fJf. (46) u = 2 sin1] :sin¢>f.

(15) 15. 1/+ ,h:+12~ = 0. (47). /1o"'+,ht+ht= 0. Cf=-. Af 'n2'/" 'Yf. SI. cos2 rpf .. (48) A tan2 rpf= B~. B. cos2 rpf= - C~. Now, the length lA of astigmatic line is determined by the following. From equation (49), equation (50) is obtained. r~( coss t, COS7] t, costt) r*(cos~ *,. COS7] *, cost *). (49). At(xt, yt, zt). A*( x*, y*, z*). l!=( x*-xt) 2 +( y*- yt) 2 +( z*- zt). 2. = ,),*2 + ')'~2 -2 ')'*')'~( cos~ *cos~ t+ COS7] *COS7] t+ cost *costt). (50). Defining OA * as r *, and the angle between r * and r J* as Q, where putting r j as shown in equation (51), then equation (52) is given. (cf. Fig.- 6. OA * is approximated to be perpendicular to A *Ao*) ')'~:;:. ')'*cosQ. l2A -_. ')'f. (51). *2 ( -c-os-: 12-Q-. (52). And also, from equations (53) and (46), Equation (54) is obtained. cos. 2 (. s t+ J..I.) +cos 2 ( 1J t+ q) + cos. 2 (. t t+ E ) =. 1. (53).

(16) 16. Memoirs of The School of B. o. S. T. of Kinki U ni versity No.8 (2001). x. Fig. - 6 In the similar manner, cos 2 Q is given by equation (55). together with (46), (54), (55), and (52), lA is given by equation (56). t=. .. I.. '.. 2 SIn t 0 COS ~ 0. (,u2sin2~t+CT2sin27]t-,usin2~t-CTsin27]t) .. (54). COS 2Q= (cos~ ·cos~ t+ COS 7] ·COS7] t+ cost ·cost t) 2 (55). (56). Thus the maximal length of astigmatic line LA can be shown by equation (57). (57). As is deduced from two equations in equation (46), on the pupil plane of the grating, (J and J1 assume constant values when X sin (¢ ~ ¢,) assumes a constant value. On the pupil plane, X sin (¢ - ¢ ,) is the distance between point P' and the inclination line composed of ¢ = ¢ f. This indicates that the diffracted light from points on a line parallel with this inclination line (¢ = ¢ f) converges on a same point on the astigmatic line (ef. Fig.- 7 )..

(17) 17. r---. [Xsifl{q,-cpf}lmax. xsin{q,- q,f). X. ---. -. z. Fig.- 7 4.3 Condition to Avoid Astigmatism. The condition to avoid astigmatism is the solution that simultaneously satisfies A J = 0, BJ = 0 and CJ = 0 in equation (58) as is derived from equations (21) and (23). 2 2 A f = -COS~COSTJ + --.cos~ tCOSTJt. Y. Yf. (58) -2 c k' (cost +costt). C. f. = cos 2 ~ +cos 2 -2 + cos 2 ~:+cosTJ:-2 Y. Y;. + 2 ck(cost+cost:).

(18) 18. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). Section 5. I n-plane Mounting. The methods of mounting of diffraction grating include in-plane mounting and off-plane mounting. In this section we study the former and the latter will be handled with in Section 7. In in-plane mounting, r] = 7[ / 2 and from equation (17), r] * = 7[ - r] = 7[ / 2 . Using equations (a. 4), (a. 5), and (a. 7 ), Equation (59) is shown to be the condition to avoid astigmatism. Thus, when equation (60) holds the equations are solved under BJ= 0 and CJ= o to give equation (61).. (59- 1) •. 2. •. *. 2. B = SIn a + SIn ao -2 c k' (cosa +cosat) = 0 f. r~. "I. 1 +cos 2 at. ~~-:-----.:.-. "It. + 2 c k( COSa + COSa t) =, 0. (59- 2) (59- 3). sin2 a. Det=. (60). I 1 +cos a 2. 2. 1' = "I. ( cos at ~2) C b2 cosat-cosa 2. (61). 1_). c ( cos a __ b2 a2 1 r~= COsa -cosat. These are the solutions to eliminate astigmatism. Therefore, to satisfy r = r * for designing spectroscopic mechanism, we obtain equation (62) from equation (61). (62) This is the astigmatic condition to satisfy r = r * in in-plate mounting (cL Fig.- 8) Now, putting Rl and R2 for curvatures of tangential and sagittal directions of an ellipsoidal surface given by equation (63), respectively, w'e obtain equation (64), with which the each part of equation (61) is respectively transformed into the each part of equation (65), where g=b 2 / a 2 = R 2 /Rl. =1. (63).

(19) 19 3. 11 +( azp/aw) w-O I"z R -. l=O (a2zp/aw2)w=o. 1-. c. l"O. (64). r=R 1 (cosa-cosa:)/11- (cos 2 a:/g)l; rJ=Rl (cosa:-cosa) /1 1 -( cos 2 a /g) I. (65). g= b 2 / a 2 =Rz/Rl In the next step, equations (66) or (67) are given wherein the ratio of equation (65- 2) to equation (65- 1) is defined as h. (66). a. cos 2 g( 1 + h) / h. +. a:. cos 2 g( 1 + h) = 1. (67). Equations (66) and (67) are more general forms of astigmatic conditions. l. b. a:. az =. (68). COSiO + COsla:. 2. -90· .004 .015. z. .033. 0. ...... .056. I-. U. a:. .069. l.t.. l.t... .125. ~. ...... .164. 0. .207. l.t... 0. w. -45·. .250. -'. .293. z a:. .336. t.:l. .375 .411 .442 .467 .485 .496. ANGLE OF INCIDENCE. Fig.- 8.

(20) 20. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). Section 6. Rotation of Diffraction Grating and Scanning along Wavelength in Stigmatic I n-plane Mounting. We assume that the diffraction grating of the spectrometer is rotated by an angle w (cf. Fig.- 9) where L 808', the angle between incident slits and radiating slits' around the center of the diffraction grating, is constant and is equal to Q, and the diffraction grating is arranged so that 08=08'.. Fig.- 9 8uppose the rotation angle is measured based on a bisected line of 808', the incident angle a and the diffraction angle a 0 are respectively expressed by equation (69). a=Q/2+w at=-Q/2+w. (69). 8ince w = 0 means the 0 th order light, Equation (70) is given. a=-at=Q/2. (70). Here, in order that each of equations (65 - 1) and (65 - 2) assumes a finite value, g must be expressed by g = cos 2 (Q/ 2). Now, in order that the value h = r J * / r is constantly valid irrespective of the rotation angle w of the diffraction grating, dh/ dw = 0 has to be satisfied. 8ince a=Q/2+w, ao*=-Q/2+w, and equation (70) from equations (66) and (67), which gives equation (71). From dh/ d w = 0, and the based on equation (71), equation (72) will be given..

(21) 21 h cos( Q+ 2 w) + cos ( Q - 2 w ). = (2 g - 1 ) ( 1 + h). (71). h- 1 cotQ+tan2 w+ h + 1 = 0. (. Thus for equation (72) to be constantly valid irresI!)ective of (73) or (74) must be satisfied. Q = 7r/2,. h= 1. W. 72 ). value, equations (73) (74). Here, d 2h/ dw 2 = 0 is also satisfied, which means that h assumes a stationary value against the rotation angle w of the diffraction grating. Consequently equation (75) is given from equation (73), (74) should be valid, and then from equation (76), equation (77) is given. b2. R2. 1. g=7=~=2. (75). (76) (77) The relation between the rotation angle w of the diffraction grating and the wavelength A of the diffracted light is given by equation (79) derived from equation (78). sina +sina:=mA/d. (78). sinw= mA /( -/2 d ). (79). Now, defining AI for the wavelength to focus on Ao* on the image plane, the diffracted light of another wavelength A * reaches a point A * away by r * from the center 0 of the diffraction grating. The distance r * from the center 0 of the diffraction grating to A * is obtained as follows, defining a * for the diffraction angle of light of the wavelength A *. By adding equations (59- 2) and (59- 3), we obtain equation (80) . 2. cos a ---+ /'. cos 2 a:. *. /'f. =. COSa + cosa: Rl. (80). Defining AI for the wavelength of the diffracted light that focuses at a distance of r 1* at a diffraction angle of a 0*, then we obtain equation (81). Substituting WI for the rotation angle of the diffraction grating, equation (82) is derived. sinaf+sina:=mAf/d. (81) (82).

(22) 22. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). Now, in this case the diffracted light of another wavelength A '" reaches the point A * satisfying equation (83) (cf. Fig.-l0). (83). sina f+ sina *= mA */ d. G. Z Fig. -10 The equation (80) is also valid for the diffracted light of this wavelength A * to satisfy equation (84). cos a * I' * 2. +. COSa +Cosa * Rl. (84). Thus by transforming equation (84), we obtain equation (85). cosa* Rl. COSaf Rl. (85). In this case, we obtain equation r = (R t /12) sec W J then substituting of r = (R 1 /12) secwJ into (85), equation (86) which leads to equation (87), will be given. cos 2 a *. COSa * COsa f 2 - /'2COSWf COS af r* Rl Rl Rl 2 *_ Rl COS a * r - COSa*- (J2"coswf cosa f - 1 )cosaf. ~~----=. (86) (87). This relationship is important in photographic light measurement. 6 . 1 Dispersion. The direction of diffraction of a main diffracted light where the wavelength. IS.

(23) 23. A and degree af arder is m is given by equatian (88) fro.m equatian (17).. Icos. ~ + cos H A= mA / d. COS7]. + COST]:' A =. (88). 0. And far the diffracted light af the wavelength A' = A + 6. A. equatian (89) will be given then using equatians (88) arid. f. 0* A. =f. 0*),. + fl = f. 0* A. + 6. f. 0* A. ros~+cosHA.=mA' /d COST]. (89). + COST] tA' = 0. Then we abtain the angular dispersian using with the equatian (36) as shawn in the equatian (90). d~: I m I~ = dsin~:. (90). In in-plan maunting, equatian (90) will be expressed as the equatian (91).. m. dat. ---;r;:- =. (91). dcos a :. In additian, in the case af phatagraphic light measurement, it is impart ant to. have the relatianship an the difference between the diffractian angle a / far the main diffracted light af the wavelength AJ and the diffractian angle a * far the diffracted light af ather arbitrary wavelengths A *. the relatian is given by equatian (92) using the fallawing twa equatians: A*. sina*=d -. A. 1 {. 2d -72. 1-. m. (A. 1. )2 }r. (92). sin a + sin a * = m A * / d and a =7[/4 +wJ Section 7. Off-plane Mounting Free from Astigmatism by Diffraction Grating on Toroidal Surface. In this sectian we are gaing to. describe the aff-plane maunting free fram astigmatism using diffractian grating af the taraidal surface, to. canclude that it is applicable far practical device as has been the case af in-plane maunting.. 7 . 1 Conditions to Eliminate Astigmatism Equatian (25) af Sectian 3 suggests that CJ = 0 results in AJ = 0 and BJ = 0 , which gives equatian (92) fram equatian (22). Transfarmatian af equatian (93) gives equatian (94)..

(24) 24. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). A. f. = sin 2 ~sin2 8 + -=s:.::in::.-.2..i!.~...::.t=si:.:.;n2~8..::...t_ y. B = . : s:.: .:;in;:. .2. . =.s. .:;c. .:. . os.: ,: 2:.,. ;8_ + sin f. o. yj. 2. y. Stcos2 8: yj. (93) -2 c Ie' (coss +COSs:) = 0. +2 cle(cos~+cos~:)= 0. (94- 1). Bf. =. cos 2 S- cos y. 2. T]. + cos 2 ~ t--- cos 2 T] t yj. (94- 2). -..: 2 c Ie' ( cos S+ cos St) = 0. (94- 3). + 2 c Ie ( cos s+ cos st) =. 0. Then the solution simultaneously satisfying AJ = 0, BJ = 0 and CJ = 0 gives t he condition to avoid astigmatism. In order that 1 / rand 1 / r j have solutions other than 0 in each of equations (94 - 1) and (94 - 2), equation (95) has to be valid.. I. COSSCOST]. I COs 2S- COs 2 T]. COSs tCOST] t cos St- cos 2. 2. T]. t. I -0. Thus using equation (17), cos T/ + cos T/ 0* = 0, equation (95) into cos T/ (cos, + cos, 0*) (cos, cos, ~ -cos 2 T/) = 0, which gives equation (96). COST]. = 0. (95). IS. transformed. (96- 1). COSs + coss t= 0. (96- 2). coss coss t- cos 2 T] = 0. (96- 3). Here the equation (96- 1) means in-plane mounting, which was studied earlier. Also, equation (96- 2) is related to 0 th order light and so its discussion is omitted here..

(25) 25. Equation (96- 3) is transformed into equation (97). cos~ cos~ ~- cos TJ 2. = sint sint ~cos( 8 -. 8~). =0. (97). Here we omit the case of sin f = 0 or sin f ~ = 0 since it is the same as cos TJ = 0 . Equation (97) gives equation (98), which leads to equation (99). cos( 8 - 8 ~) = 0. (98). 0:=0- (2n+l) 2. If we assume. e. (99). 7(. in the first quadrant, equation (l00) IS gIven. Then equation. OOD is obtained. ()t= ()- ~. (100). 2. cos8:=sin(), sinO:=-cosO, sint=Acos() sintt=Asin(),. cos~=ACOS2 (), cos~:=Asin2. (). (10D. Based on these relations we obtain equation (102). 002-1). 002- 2). C. f. =. 2. 2. A cos 0-2. r. +. A2 sin 2 0-2 r~. (102- 3). +2 c k(.j 1 - A2 cos 2 () +.j 1 - A2 sin 2 8) = 0. Thus equation (103) is given from equation (102 - 1 ), and substituting equation (103) into equation (102 - 2) ,we will obtain equation (104). (103) 2 ck. I (. .j 1 - A2 cos 2 8 +.j 1 - A2 sin 2, 8 ) = 0. (104). From equation (04) f( '= 0 will be obtained, and from k'=l /2 (1 /b 2 -1 /a 2 ) we will get a = b. But, a 'toroidal' surface satisfying this relation means a ,spherical' surface. Thus by substituting 1 / r = tan 2 e / r j into equation (102 - 3 ), we obtain equation (105)..

(26) 26. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001). 1. yj. (105). V cos 8 ( J 1 - ..1 COs 8 2. 2. 2. +J 1 -. ..1 2 sin 8 ) 2. 1 - ..1 2 sin2 8cos 2 8. Since a= b, we obtain equation (107) taking equation (106) into consideration. c. 1. c _ 1 _ 1. (106). CT=Rt=l)2-~-R. (107) Also equation (108) is given from equations (103) and (107). (108) These solutions give off-plane mounting free from astigmatism (cf. Fig.-11).. y. Fig. -11 Left: Projection of incident and radiating light on pupil plane in stigmatic offplane mounting Right: Relation between light source point and image point in stigmatic off-plane mounting.

(27) 27. (cm). 8·1O· _ _ _ _ _ _~_-_ _~8.45·. 100. //'i. -// 1,/1/. '.,----/-,..:;..,- - ..-./. 25. :. so. 100. Y(cm). Fig. -12. Relation between the distances r from light source point to center of diffraction grating and r t from image point to center of diffraction grating.

(28) 28. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001) R-IOOcrn d-I/6000 ~""!::~00=.O="':"""'r-. _ _r-_ _y -_ _~_ _ _ _..,-_ _ ,e_eo..,e4-M.'4C111 .. A~:. /,A:.. (em). ~.. i'···\;··. 100·. ,>.::i;;!. 0,1. : . ~ :!.. i----+---+----+----t----t-----1. R-IOOcm. ..... 100·. .....,;A. .'./. '-'1000. I~o-. :. v* : :. If'. .... ,,'. ISO. d -1112000. >/~ ;; / .. ,I. i::. '.. '. ./. I. :. / , ::L ". ,iL. 1. /. /. ./. !. I. J. /. J. I. J. I. /. 100. J. /. I. ://100.0_1. :!J--I~'/l i'. t. /. :. '. /. I. J /. /. /. ..-t'7.17cm. I. J. lilZL. :. ;. .-. Bcdtg). .. 0". I'. o. le41) J:. i,;:L·. SO. /. 1000. 2000. eo'. 3000. ~(1). Fig. -13 Upper: Relationship among focal distance r J *, wavelength A of diffracted light, and the angle () between projection of incident light on pupil plane and x axis, in stigmatic off-plane mounting using spherical diffraction grating with 600/mm pitches and 100cm radius of curvature. Lower: Relationship among focal distance r /, wavelength A of diffracted light, and the angle e between projection of incident light on pupil plane and x axis, in stigmatic off-plane mounting using spherical diffraction grating with 1200/mm pitches and 100cm radius of curvature..

(29) 29 CHAPTER 2. OPTICAL RAY TRACING Section 1. Introduction. In general, optical ray tracing method means a method to confirm direction of light path as well as progress of light emitted from an arbitrary point of the surface of a substance, experiencing reflection, refraction, and diffraction on an optical device, through elaborate calculations based on Snell's principle and others. The calculation of optical ray tracing is roughly summarized into the following steps. ( 1) Coordinates of an object point (xo, Yo, zp) and directional COSInes of light (L, M, N) is formulated on an XYZ coordinate system. ( 2) An intersecting point (w, l, zp) with the plane of optical device S (w, l, zp) is determined. ( 3) The change of directional cosine on the direction of light after refraction, reflection or diffraction is determined. (4) The determined point of intersection and cosines are indicated on an XYZ ordinate. ( 5) Steps (1) through (4) are repeated on planes of succeeding optical devices. Methods by optical path function have been dominant for analyses of concave diffraction grating, and examples of analyses by optical tracing are relatively scanty. Methods of optical path function, however, do not include threedimensional treatment of resolution etc. and in consideration of supplementing this as well as of combination of concave diffraction grating with optical devices, we believe the study by optical tracing is indispensable. It was Rosendahl that for the first time applied and formulated optical ray tracing method to an optical system using concave diffraction grating. Later, Spencer et al. introduced a numerical formula necessary for optical ray tracing calculations considering the effect of grooves of a concave grating on assumption of a minute planar grating. Welford obtained directional cosines (L', M', N') of the diffracted light by simultaneous application of light path function and showed the possibility of accurate optical tracing, and of application also to the diffraction grating using planar grating or non-spherical blank. Welford, however, did not give any practical result of numerical calculations. Here, we report the result of numerical calculations of optical ray tracing of diffraction grating on ellipsoidal and toroidal surfaces using a revised Welford's method. Section 2. Analytical Calculation for Optical Tracing. By application of Fermat's principle to light path function F (w, l, zp), we obtain equation (l09)..

(30) 30. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001). aF _ w-x. aw -. AP. aF _l=1L al -. AP. +. zp-z. az,; AP. +. Zp - z . oZp AP al. aw. + w-x* +. zp-z PA*. . oZp. l - y*. Zp - z P A*. . ozp. P A*. +. P A*. +. oW. ol. +. ok =0. ow. + E..!£ = al. 0. (109- 1 ). (109- 2). Here K, =mn(w) A, where m is the degree of diffracted light, new) is the number of grooves of the diffraction grating from the origin 0 to a coordinate (w, 0, 0), and A is the wavelength of the diffracted light.. (110). 2.2. 1 Diffraction Grating on Ellipsoidal Surface. In general, an arbitrary point P (w, l, zp), on an ellipsoidal concave grating surface, satisfies (111). . (111). Equation (112) is derived from the formula of equation (111). Here R t = a 2 / c and R2=b 2/c. WZ. ZP =. Rz -R2 { 1 - ( Rl R2. +. lZ. R~. )}. t. (112). Considering the case of in-plane mounting shown in Section 1 (cf. Fig.- 9), we obtain equation (112) from equations (74) and (76), where r f = (R t /12) sec W f and W f is the rotation angle of the diffraction grating satisfying the following relation with respect to the noted wavelength Af. sinwf =m Af/(/2d). xo =/,sin(7l' / 4 +CLu)=(Rl / 2)( 1 +tanw.r) Zo =ycos( 7l' / 4. +wf)=(Rl / 2)( 1 -tanwf). .rl'=y~sin(wf-7l' /. (113). 4 )=(Rl / 2 )(tanwf-l). zt=y~COS(Wf-7l' / 4 )=(Rl/ 2)( 1 +tanwf). Now, if we express the directional cosines of the incident and diffracted light by L, M, N, and L', M', N', respectively, then equation (113) is given. Equation (115) is also derived from equation (112). Thus equation (109- 1) is transformed into equation (117) using equations (114- 1), (114- 2), (115- 1), and (115- 2)..

(31) 31. L -x-w -- AP'. L'. x*-w AP*. ,. M~ =AP. N. M'=Y*- l. N'. PA*. azp __ ~[ 1 _(~ +~ aw 2 Rl R2 Ri. Z-Zp. (114- 1). AP. z*-zp. (114- 2). PA*. )]-} I. 2W) \ - Rl R2. (115- 1 ). (115- 2). When the grooves of the diffraction grating are linear and grooved at equal interval d, equation (115) comes out.. ak. 3w. =mA. an. aw. =mA ~(W) =mA aw d d. (116). As well, equation (109- 2) is transformed into equation (118) using equations (114- 1), (114- 2), and (115- 2), where putting Ee, Fe, and Ge into each formulas resepectively,as shown in equation (118) and wherein suffixe means an ellipsoidal surface. Then equations (117) and (118) are transformed into equation (119), and we will obtain equation (121) from these equations. (117). (WZ Rl R2. ,. , l [ M +M +( N +N) . If; 1 -. L+L'+( N +N')Ee-Fe= 0. M+M'+(N +N')Ge= 0. w [ (W ' P Ee=Jf;. 1 - RIRz +Ri 2. mA. Fe=y. )J-t. P. + R~. )]-t. =0 .. (118). (119- 1). (119- 2). (119- 3).

(32) 32. Memoirs of The School of B.O.S. T. of Kinki University No. 8 (2001). W Ge=R; N+N'=. 1 [ 1 - (W2 ZZ )JRIR2 + R~ 2. (120). L+~:Fe =_M~~'. (121). If we express the value of the formula (121) as J, then we obtain three equations as shown in equation (122). -L' =-L-Ee' J +Fe. (122). M'=-M-Ge' J. N'=-N+J. Then, we will obtain equation (123) with addition of the each square's of these three equations. L'2 +M'2 +N'2 =L21-E~J2 +F~+2EeL' J -2F eL-2EeF eJ +M2 +2MGeJ +G~J + N2-2NJ +J 2. (123). Since the sum of directional COSInes of incident and diffracted light IS 1, so we obtain equation (124). (E~+G~+ 1 )J2 +2(EeL+MGe-EeFe-N)J +F~-2FeL=. 0. (124). By solving the equation with respect to J, equatIon (125) IS gIven. J. -(EeL+MGe-EeF e- N) E~+G~+ 1. (125) .±. J(EeL+MGe-EeF e-N)2 -(E~+G~- N)(F~-2F eL) E~+G~+ 1. Thus this value of J can be used to determine each value of L', M', and N'. Now, a plane that passes through Ao* (xo*, yo*, zo*) and stands vertical against r j is given by formula of equation (126) (126) Also, the formula of a straight line that passes a point (w, l, zp) and has directional cosines L', M', and N' is given by equation (127). X-W_~_Z-ZP. L'. -. M'. -. N'. (127). Then, we will obtain the coordinates of a point where the straight line intersects with the plane as shown in equation (126), using with combination of equations (126) and (127), in equation (128)..

(33) 33. (128). Z. *. N'(,rl2 +Z:2 -rlw)+L',rlzp L',rl+N'z:. So each value of x*, y*, and z* is incorporated into the coordinate system having the origin at Ao*, possessing X' axis within the XY plane, where OA * is Z' axis, and Y' axis is parallel to the Y axis, (cf. Fig.-14). Each values of X', Y', and Z' will given, respectively, as shown in equation (129) . X'. /2(z*-z:) sinWf-COSCc)f. (129). Y'=y* Z'=o. Fig.-14 Figures 15---17 are spot diagrams of the result of numerical calculations of optical ray tracing obtained using above X' and Y' under the conditions.

(34) 34. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001). described in each of the figure. In the next step, in order to eliminate aberrations of higher order including coma aberration, on a particular wavelength il. c, we consider on the number n (w) of grooves ofa diffraction grating. The condition to eliminate aberrations . of higher order including coma aberration is obtained when the difference between the path length r + r j of the light reaching the center of the diffraction grating and diffracted there, and the path length of the light reaching point (w, 0, zp) and diffracted there satisfies the following relationship for a particular wavelength il. c. (130) , where r 0 stands for the distance between the point light source and the center of the diffraction grating; Spot diagram of the image on the image plane by a point light source at each point (y = 0.3, 0.2, 0.10, - 0.1, - 0.2, - 0.3) of incident slit using ellipsoidal diffraction grating with straight grooves of equal intervals (number of grooves: 1200/mm) y= 0 means in-plane mounting. y = 0.1 indicates that the position of point light source is O.1mm above x-z plane parallel with y axis. Captions have the same meaning in the following figures. [em]. i/;,i,;c<, 0.3 I-r--t--i---+",:?R!fi:~;:,-I--l---I----l. 0.2 r---t---j--+_~ii;::i!.::~·-4_--+ _ _+------I ' ::::::!:.'. ! ".:l",. o.i. o. "j I. I. ....:1:·. -0.1. ':r ,I. -0.2 I-t--i--+----+::~),."II-'-+-+--+_~. "':"l" -0.3. I-t---r---t-~:'rl?:{~':'.'···--+--+--I-~. ""l" -0.3. -0.2. o. -0.1. 0.1. 0.2. 0.3. Ellipsoidal. Coma. A=1200 A d= (1 /12000) em. y=o. (131). CemJ. Fig.-15.

(35) 35. . :.::,::',. IOO~______~____~~~~~~------~. -100. I--------+-------~~:__:_~~------~. ................ Fig.-16. o. -100. 100. A =1200A.. d = (1 /12000) em. y=O. (132). Spot diagram of the image plane of in-plane mounting (y = 0) under the' same conditions as mentioned above. [emJ. ..... ..... . .. . . . . . . . '. '. .. '. '. -0.09. -0.1. -0.11. . .. . .. . .. ...... . . ..... . '. '. Fig.-17 -0:01. o. (em). A=1200A.. ::~ .11/12000). em. (133)'. Spot diagram of the image plane of off-plane mounting (y = O.1cm) under the same conditions as mentioned above..

(36) 36. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). In the equation (130), r 1=0 stands for the distance between the point light source and the point on diffraction grating (w, 0, zp); and r ~=o stands for the distance between the point on diffraction grating (w, 0, zp) and the image point. Equation (130) is transformed into equation (134), and by substituting each numeral with x, z, x*, z*, and ZP, respectively, we obtain equation (135). 1. n(w)Ac=mic. [v'2 RlseCWf.;\c- {(X-W)2+(Z-Zp,. z=o)2r. (134). ~{(x*-w)Z +(Z*-Zp,,~o)2. n( W )Ac =. -Rl(. m. 1. L[v'2 Rl. +. secw f. AC. tanwf.Ac)W. x ( 1- RlW2Rz. +. )i]i. - [R¥ T. -. (1-::. [~t )W. Z. }. f]. Sp.c 2 W f. AC -Rl Rz ( 1 -tanWf. Ac) +2Ri + {RIRz(. 1-tanwf.Ac). -2R~}. sec z Wf.;\c -Rl Rz( 1 + tanwf.;\c) + 2R~ (135). Thus (8 n (w) /8 w) A c is obtained by differentiating equation ( 135) with regard to w. Equation (136) will be given..

(37) 37. (136). x[ Rdtanwf.AC -1)- 2 (1 -~: )w +. R~R2. {Rl R2 ( 1 +tanwf.Ac)- 2 Ri }( 1. -R~~2)-t J]. Equation (136) indicates the relation between the number of grooves and the coordinate w per unit length of diffraction grating to eliminate aberrations of h igher order including coma aberration for a particular wavelength .A. c. If we assume that a spectroscope provided with grooves according to equation (136) receives light of general wavelength, and that the degree of the diffracted light of a particular wavelength .A. c is the same, the value of k should be equal to m.A. new) .A. c in the light path function F= AP+ PA * + If, • Accordingly, the term f) If, / f) w on applying Fermat's principle to this light path function should be equation (137), and thus from equation (136), (137) becomes to equation (138) validly. (137).

(38) 38. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001). (138). 1 [Ri sec 2 Wf,>..c +22. -. R 1 R 2 ( 1 +tanw ) + 2 R22. Therefore, when a diffraction grating on an ellipsoidal surface provided with grooves such that aberrations of higher order including coma aberration is eliminated is used for a particular wavelength A c, optical tracing for general wavelengths is to be carried out in the following way. Optical ray tracing numerical calculations, for general incident wavelength, should be carried out according to defined steps are carried out based on equations (109- 1 )' and equation (109- 2), assuming that equation (109 - 1 )' means the equation in which the term of a K / a w in equation (109 - 1) is substituted with equation (138) by equation (109- 2), and numerical calculations should be carried out based on wf,}.corresponding to the wavelength A in each equation (113). The spot diagram of the result of numerical calculations by this method is shown in Fig.-18--Fig.-20. The blank for the diffraction grating on an ellipsoidal surface not only contains difficult problems in production engineering based on current technology, but also we have yet no effective method of accuracy inspection after preparation..

(39) 39. Therefore in this study we decided to prepare diffraction grating on a toroidal surface because it can be approximated to the ellipsoidal surface in the vicinity of the center of grating. We have carried out the optical ray tracing and numerical calculations with the diffraction grating on a toroidal surface in the same manner as that on an ellipsoidal surface, and obtained the results shown in Fig.-21--Fig.-23. [em]. r--"'"-~-..---..----.----.-----.---~-. O.:!S. 1----I----+---+---li\IIII~--II----I---I----l. O•2 1__-+--_4_-4----ll\\lii'~--~---I--_+_--I iii 0.1 1__-+--_4_--1--\\1'1-- - j - - + - - - j - - _ _ l. o. 1__-~---_4_-~----I--I~-~-~~. -0.1. - 0.2. 1__--I---_4_--I------i!III'!)i----4--~--_+_--I. - 0.3. I---I----+---+----ifll!l~--,---,---+----l -0.3. -0.2. -0.1. o. 0.1. 0.2. 0.3. [em). Fig. -18. Ellipsoidal, Noma-Coma. A =1200A Ac=1200A do = ( 1 /12000) em. (139). y'=o. Spot diagram on the image plane by a point light source at each point (y = 0.3, 0.2, OJ, 0, -0.1, -0.2, -0.3) of incident slit. The incident light of wavelength of 1200 A is assumed to reach the diffraction grating on an ellipsoidal surface provided with grooves so as to eliminate higher order of aberration including coma aberration with respect to incident light with wavelength of A c=1200A. y= 0 means in-plane mounting. y=O.1 indicates that the position of point light source is O.1mm above x-z plane parallel with y axis. The symbol do stands for the interval of grooves at the center of the diffraction grating where w= 0 ..

(40) 40. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001). IOO~-------I-~~~--~-------~-------~. o i---------j--t+-i+H-1it. -100. I---------+-~~:.-----+_-----___j-----__r. Fig.-19 -10'0. o. 100. A =1200A Ac=1200A do={ 1 /12000) em y =0. [/Lm]. (140). Spot diagram of the image plane in in-plane mounting under the same condition as mentioned above. [em]. -0.09. -0.1. -0.11. Fig.-20 -0.01. o A =1200A Ac=1200A do=(1/12000) em y =0.1. 0.01. (em]. (141). Spot diagram of the image plane in off-plane mounting under the same condition as mentioned above..

(41) 41. . I I-~_-t-'-_-_-_-_-I0~- ~=_. .,__:~~=-+H;:::_I ~I. [em]. :::. o. -1._- - \ - - - - i. r-----1f--I---+--o!'i',--+--I--~-_I. -0.1. -'---I--I---;.itiii~- - - - o - - i - - - - - ;. -0.2 f--f--11----11--(1!~;.·- - + - - - 1 - - - 1 - - " " , -0.3 1---1---!--:-~:;;\:il.Joi!i!;:~:,,-·- i - - - + - - - - I - - - - !. -0;3. -0.2. o. -0.1. 0.1. 0.2. 0.3. Fig. -21 [em]. (142). ). =1200A d = (1 /12000) em y =0. Spot diagram on the image plane by a point light source at each point (y = 0.3, 0.2, 0.1, 0, -0.1, -0.2, -0.3) of incident slit using diffraction grating provided with grooves of equal intervals on a toroidal surface (Number of grooves: 1200/cm).. ... 100. 1------I-----....,-=-1-....,....7~ ~"'":--_!----_i. -I 00. I-----I-----:....,.....~-I~~~-r'---I----__f .',. .... Fig. -22 -100. o. 100. ),,=1200A d=(1 /12000) em. (143). y=o. Spot diagram of the image plane In in-plane mounting (y = 0) under the same condition as mentioned above..

(42) 42. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). [em]. -0.08~. _______. .~. ______-+______. ~. ______. ~. 'Wi,?> . . -0.121---------l------...:..:..--+--------1·---------;. -0.02. o. 0.02. [em). Fig.-23. Toroidal. Coma. A =1200A d = (1 /12000) em y =0.1. (44). Spot diagram of the image plane in off-plane mounting (y = O.1cm) under the same condition as mentioned above. The analytical calculation formula of optical ray tracing, for a toroidal diffraction grating with equal width of grooves which is spaced with equal intervals on the pupil plane, is given by equations 045- 2) 045- 2), and (1453) corresponding to respective equations 019- 1), 019- 2), and 019- 3).. (145).

(43) 43. x{:,. _l+(l_;~Z)t}]-t. x ( 1 + ta n(JU, AC. ) {. 1-. ~. (1 -. r. ~; t }. (146). Here t stands for a toroidal surface. When optical ray-tracing is carried out with using a diffraction grating having grooves on a toroidal surface so as to eliminate aberrations of higher order including coma aberration with respect to a particular wavelength A c, the same process as ellipsoidal surface can be used by substituting the equation (145- 2) with ,(146) with respect to a general wavelength A. The results of optical ray tracing are shown in spot diagrams In Fig.-24 to Fig.-26. [em]. 0.3. 0.2 0.1 _II. o -0.1. I---+---I---I.-..-fn --/----/---t---I. -0.2. 1--~--I--+--f/I!llt--+---I---t---t. -0.3 1 - - - + - - - ( - - ~/{II~·--I--'I----t---l. -0.3. -0.2. -0.1. o. 0.1. 0.2. 0.3. Toroidal, Non-Coma A =1200A Ac=I200A do = C1 /12000) em y =0.1. (147). [em]. Fig. -24.

(44) 44. Memoirs of The School of B.O.S.T. of Kinki University No. 8 (2001). Spot diagram on the image plane by a point light source at each point (y = 0.3, 0.2, 0.1, 0, -0.1, -0.2, -0.3) of incident slit. The incident light of wavelength A c of 1200 A. is assumed to reach the diffraction grating on a toroidal surface provided with grooves as to eliminate higher order' aberrations including coma aberration with respect to incident light having wavelength A. of Ac=1200A.. y= 0 means in-plane mounting. y=O.1 indicates that the position of point light source is O.1mm above x ~ z plane parallel with y axis. The symbol do stands for the interval of grooves at the center of the diffraction grating where w= O.. 100. o. -100. -'00. o. [I'm]. 100. Fig.-25. Toroidal. Non-Come. ).. =1200A ).. c=1200A. do= (1 /12000) em y =0. (147). Spot diagram of the image plane in in-plane mounting under the same condition as mentioned above..

(45) 45. [em]. -0.081--______1_________ 1·_________1_______. ~. - 0.1 1------+--=....... -0.12. I----------j-.::....-------I----------I----------I. o. -0.02. 0.02. Fig. -26. [em]. Toroidal. Non-Coma. A =1200A Ac=1200A do=Cl /12000)cm '!J =0.1. (148). Spot diagram of the image plane in off-plane mounting (y = O.1cm) under the same condition as mentioned above. For information, Fig.-27 and Fig.-28 show how the grooves of the diffraction grating on an ellipsoidal surface have to be changed depending on the change of w in order to eliminate higher order aberrations including coma aberration.. A. 12012.----.----r----.-----,. AC= 500 0 - AC= 1000 A " AC= 1200 ,. Ellipsoidal d. = 1/12000. A. 120061---4-~---~---~-~~. 12004t-----'t!----t------il-----. /2000'-----'---1.25 -2.5. o Fig.-27. 1.25. 2.5. W [em].

(46) 46. Memoirs of The School of B.O.S.T. of Kinki University No.8 (2001). 12012.----r-----,.------,--· n [em-I]. EllIpsoidal d.= 1/12000 1----1--------+-. 120111-~--1-. 12010 1- - - - ' - - - - 1 - - - - ' - - -.. .. 2.4. 2.45. 2.5. w[em]. Fig. -28 The figures show how the grooves of the diffraction grating on an ellipsoidal surface have to be changed depending on the change of w in order to eliminate higher order aberrations including coma aberration. Grating constant do: 1 / 12000cm, and wavelengths of the diffracted light: 500A, 1000A, and 1200A.. References. W. T. Welford. W. R. Hamilton F. Zernike H. G. Beutler T. Namioka. R. A, Sawyer G. R. Rosendahl . W. Werner. 1962, 1963, 1963, 1965, 1931, 1935, 1945, 1954, 1959, 1959, 1959, 1961, 1961, 1961, 1951, 1961, 1962, 1967, 1970.. Optica Acta, 9, 389 J. Opt. Soc. Am. 53, 766 Optica Acta, 10, 21 Progress in Optics, Vol. 4 Marthematical Papers (The University Press, Cambridge, England) in Festchrift, Pieter Zeeman'p.323 J. Opt. Soc. Am. 35, 311 Science of Light 3. 15 J. Opt. Soc. Am. 49. 446 J. Opt. Soc. Am. 49. 460 J. Opt. Soc. Am. 49. 951 Space Astrophysics J. Opt. Soc. Am. 51. 4 J. Opt. Soc. Am:. 51. 13 Experimental Spectroscopy 2 nd edition J. Opt. Soc. Am. 51. 1 J. Opt. Soc. Am. 52. 408 Applied Optics. Vol. NolO 'Imaging Properties of Diffraction Gratings' UITGEVERIJ WALTMAN-DELFT.

(47) 47. • 517t~O)~*Et-JtJf.f41V<:tj:,. *T,. AM A. fJ J. 7 ~ (~7t V:'; A'),. C,. ':J. ",. lliM A I). 1!E~f.f4, 11fftt~~~t 25 ~ ~ 'I tj:~mtJ ~~ffl~'I~ 7° I) A'A. I). ':J. :J I). J. - 7 -~. (:J I). J. - 7 - V :,; A'),. ,,~~ 'I tj:~~¥Z:fN C i- O)!&~,. 0n. iBlftJE:. c'c: 25 ~ 0 it:., 511tX*T c L -C tj:11fft ~c. J:: ~ 1ffitJT~O). iBlft~C.J:: ~@]tJTO)*~ ~ O):il~'I~ffl~'I~@]tJTmT~C.*}3ljc:~ ~o. ~~~C., 11fftiJ~2000A.I2.rf 0)1i~ffij~7U~~j~c: tj:, 7t~*TO)~iJ~j)tJ T517t~iJ~ ffl ~ 'I. ~ 1~;dC.,. 511tX. < -CTV, @]tJTm. ~0. @]tJTmT517t~~f.f41V<:T ~@]tJTf~Tc, 1iBO)ffflM7t~*TO)~1OJ~Et-J~ii~c.~T ~~~~ ~~iJ~, @]tJTf~T~MBC:25"'?-C, <:O)~~~tj:, ~,~~,c.~t:t~517t~~~t'j:?JJ~~, *rn!\mI1!~ fflUrJtIj~~,. i t:. tj: 7° 7. 25 ~ ~ 'I tj:, i- n 0 O)~~~,~ffl~~O)~~t, y:'; ~. A'"7 UiJtU~~ 0) ~~t tJ. c' ~C. fr5 ffl ~ n -C ~ 'I ~. @]tJTmT'C.~T~1iJf~tj:, )jf~Et-J~c. ~1l<, ~~. 0. " 0. :,;,JL:i!!1iY:Mffl~~,. 0. < 0)1iJf~~J!~ <: ciJ~c:~ ~o. Beutler tj:, 3 (j{5C~rs'I*JO)}~7tiJj~!& 0:tJk "? t:.~~~~~~.Im~ L t:.o ~~'-C, Namioka. tj:" Beutler o)I1!g~O)~ 0 ~rriE, <: n~tJt~ L-c{**1t L, i- 0)*5:!l~ ffl ~'-C*Jf L ~'~A 0)517t~~~~. L t:.o. L iJ) L, <: n 0 0) 11!~~c: tj:, off-plane mounting ,c. x1T ~ ~~~.Im ~ iJ~, in-plane mounting iJ). 0 O)ili{~~tlf ,c. tJ "'? -C ~, ~ 0) C:, i- 0)*5:!l c L -c,. ~~iJ~1:. t. ~. <: c ,c. tJ "'? t:.o. Werner tj:, @]tJTmTO)1Jf~A~f&~.Im~J0c:* L, off-plane mounting 'c.~ffl c: ~ ~ ~~~~.Im~. L,. ±@]tJT7tf!7~J:O)~#-UC.1&5R:T ~7t~~O)@]tJTmT1IDJ:O), @]tJT}~\O),JL8lJ)\iJ~,. 5R:11Lii'c. J:: 0 ~1tT ~ L. c ~J!lli L t:iJ~, ~. 1&. ili:{~O)mstiJ~~iJ) "? t:t:.661&~~tlf~c.~iElit~ ~. t:. L t:.o it:.,. L.nt?O)c'O)~~~,c.~~'-c~, J'Cr~O)7tO)~iBlft~c.x1T~A~~. ~J~h~·'c.X. < L. c O)c: ~ tJ ~ 'I~FJ;!?)j)(~~jG:£:'C.~~T ~Ml¥fJT~tlf C,. ~O)~~t}jm~ J!lli. r. )vJ0~*~UrJtUT. i- 0)*5:!l~c.~-j' <517t. L-C ~,tJ ~ '0. ~~'~@]tJTmT~ ffl ~'t:.517t~'~~ ~ '-C tj:, ~Fg113(~iJ~mn~. c ~ 'I '5 ~ltiJ~25 0, <: niJ~,. <: O)517t~O)XgO)~-:JC:25 ~o i- L. C:~F1i<1ID@]tJTmT~ffl~'I-C~F}~1&~~~~T ~ t:.66O).

(48) 48. Memoirs of The School of B.O.S. T. of Kinki University No. 8 (2001). 9~~~~,,&~~.~~h~~~hT~~0. tJ. iJ~-c: ~" Namioka moun ting. ~~ ~ ~:l. ~~. d: Gmpl)(1iii@]tJTf~r9~\fU ti{--t*~ tJ.. ~. 0) -c: ~ G iJ)" off-plane. G~:t& ~ \ iJ)" in-plane moun ting iJ~ i; 0):ilI {£L~tlfiJ~ i; IV<: '"? T ~ \ G0 i t:.". Fermat 0)~9i-:7't~~~~~~~ L T1~ i; h G= -:J 0) JJf3!A: 0) oj t" '±SlliiiJJIoJO)EfE~*11 i-~~"JJf3!A: i- ~¥ ~ ". z: h c ~~ tJ. JJIoJ 0) EfE~*1Ii-~~"JJf~ ~~ ttA LT1~ i; h G~¥i- ~ '"?. T~FJ~A){~O) tJ. ~ \~{lj:: c. L T ~ \ G~ ti". 1At*iJ~ i; 0) [!:!Jiii@]tJT~r9~\fU ~~J! i; h. *~\fUJt t~ ~ ~ \ T ti" Fermat 0) ~9i-:7't~~~ t~~~. G~\fUm c ~ G. L t:.~ t~1~ i; h G= -:J O)JJf3!A:" HP. t" '±S¥iiiJJ[OJO)EfE~*1Ii-~uJJf3!A:c" z: ht~~~:tJ.JJIoJO)EfE~*1Ii-~uJJf3!A:c i)tli ~ ii-T~¥~ \ t:.o. z: 0) z: c tijG~1~tBi-13-;t G~¥i-~~ t:'-g <: c i-~,* LT~\ Go :ilI$" Jovinn-Yvon *±t~ d: '"? T~~~o) holographic @]tJTf~ri-@]$i~ ii- G z: c T~¥Bift~~t>t:.. '"?. T~{~'t~O) d: ~\$g~JtiJ)~~~f'F~ ht:.. t~ d: G~F~1j3(~O)tJ. ~\~{lj::ti" ~~¥BlftO). c. c. ~\ oj ¥R~ ~ ~. t~ d:. '"?. G iJ)" Namioka. ~ O)hlV<: ~ lL~:J~{lj::-c:~ ~" it:." Jovin-Ivon. *± t~ d: ~ ~~ ~ h t:.$ @.Jf3 ~ jG~ t~~FJB1j3(~iJ)~~~ ~ h t:. c ~ \ oj ~ 0) -c: ti tJ. ~ \0 *~tfaJt-c:ti" ~{OJ:7't$0)~*~9-c:~. G Fermat. 0)~9t~}it~~t~1At~\" mPi{*iiiO)~{lj::. 0) ~ c -c:" ""CO)~7t~Ij!i-~¥~" in-plane mounting" off-plane mounting 0) ~\ThO)J~ ~i-~~ul~~A~~mPi~iii@]tJT~r9~Ji-~~L~0. i t:." :7't~17~j]!Jfj))\m'~d:~" mPi{*iii@]ffi~rc i' l:t'xf~~i i-fr~ \". i'. 0 -{. 0 -{. 7")viii@]ffi~rO)*5{~~'t~t~-:J~\T. 7" )vrm@]ffi~ri- ffl ~ \ t:.7t:7't~O)~j:JH~'~ -:J ~ \ T~~" z: 0) Ij!g\fU 0). c ,~" ~F:bi<rm@)ffit~ri-~~\T~¥Blft~Jl~,~vt.: '"? T~FBJj)(~i-*~'~~~ L1~G~ <*JT L ~ \ID! O)7t:7'tJJA:i-~$ L t:.o ~.

(49)