//v - 35- 

FINAL REPORT 

To 

THE NATIONAL AERONATICS AM) SPACE AGENCY 

FOR 

HOLOGRAPHIC GRATINGS FOR SPECrOROGRAPHIC APPLICATIONS: 

STUDY OF ABBERATIONS 

( CONTRACT NO NCR-2 1 -027-0 1 0 ) 

MANJIT S. BHATIA 

PRINCIPLE INVESTIGATOR 

BOWIE STATE COLLKE 

BOWIE, MARYLAND 

JANUARY 1987 

(BIISA-CR- 180267) BCLGGgAPHIC GEATIMGS PCR N87-156E5 

IFECTFCGBBPHIC LFELlCBTZCbS: 5IULP GP 

AEBERATICBS Final ReFOXt (EGWie Ztate CCll-, #al) F CSCL 202 Unclas 

G3/35 43520 
a 1 

This is the final report of the project entitled "Holographic 

Gratings for Spectrographic Applications: 

conducted at Bowie State College with the help of faculty and 

students . Study of Abberations" 

This report is divided into the following independent parts, 

each of which can be studied separately: 

1. 

2. 

3. 

4. 

5. 

Computer Program for Designing Holographic Gratings for 

Seya-Namioka Type Monochranators: Part I: Minimization 

of Abberations Over a Desired Wavelength Range. 

Canputer Program for Designing Holographic Gratings for 

Seya-Namioka Type Monochranators: Part I: Correcting 

Abberations at Specified Wavelengths. 

Design of Astigmatism or Coma Corrected Holographic 

Gratings for Rowland Circle Spectrographic Mounts. 

Simultaneous Correction of Astigmatism. Coma. and First 

Spherical Abberation Term of Holographic Concave Gratings 

for Rawland Circle Spectrographs. 

Design of Abberation Corrected Holographic Toroid Gratings 

for Seya-Namioka Type Monochromators. 

This report concludes this project. 
COMPUTER PROGRAM FOR DESIGNING HOLOGRAPHIC GRATINGS 

FOR SEYA-NAMIOKA TYPE MONOCHROMATORS 

PART I: MINIMIZATION OF ABERRATIONS 

OVER A DESIRED WAVELENGTH RANGE 
CONTENTS 

INTRODUCTION ......................... 

SECTION 1 . HOLOGRAPHIC CONCAVE GRATINGS ............... THEORY AND BASIC EQUATIONS FOR DESIGNING 

Minimization of Aberration Terms ............... Determination of Instrumental Parameters 5. p'. K and A. ..... Determination of the Recording Parameters pc . pD. 7 and 6 

and Minimization of Astigmatism or Coma 

Modified Method ....................... ........... 

SECTION2 ........................... 

Program Description ..................... Nlain Program ........................ Subroutine INSTRM ...................... SubroutineASTGMS ..................... Subroutine COMA ...................... Subroutine PRFROM ..................... 

SECTION 3 . DATA SET AND DESCRIPTION OF DATA CARDS .... 

Data Set .......................... Description of Data Cards and Sample Data Card ........ 

SECTION4 ........................... 

REFERENCES ......................... 

Page 

1 

. 

2 

4 

6 

.8 

9 

10 

10 

10 

12 

12 

13 

13 

13 

14 

14 

15 

19 

iii 
1 

ILLUSTRATIONS 

Figure 

1 Schematic Diagram Showing the Geometry for 

Recording the Holographic Grating . . . . . . . . . . . . 

2 Schematic Diagram of the Optical System for 

Seya-Namioka Monochromator . . . . . . . . . . . . 

3 Main Program Flow Chart . . . . . . . . . . . . . . 

4 TypicalDataCard . . . . . . . . . . . . . . . . 

5 ............e............... 

Page 

6 

11 

15 

16 

iv 
I 1 

COMPUTER PROGRAM FOR DESIGNING HOLOGRAPHIC GRATINGS 

FOR SEYA-NAMIOKA TYPE MONOCHROMATORS 

INTRODUCTION 

A computer program has been developed based on the theory presented in papers 

by Creiner and Scheffer (Reference 1) and Noda-Namioka and Seya (Reference 2), 

lhat determines the optimuni holographic grating recording parameters as a 

function of optical instrumental parameters and desired waveleng-lh range for 

Seya-Namioka nionochromator mountings. The report is divided into two parts. 

In Part I, the holographic grating recording parameters are determined such 

that astigmatism or coma may be minimized over a selected wavelength range, 

aiid in Part 11, we determine optimum recording parameters such that astibmatisin or coma niay be corrected to zero at specified wavelengths within the 

desired havelength range. The program also simultaneously displays the perior~i~ailce of the holographic grating and equivalent conventional grating as a 

functioii of wavelcngth relative to optical aberrations of astigmatism, coma, and 

spllerical aberration. 

In tlh report two methods of designing holographic gratings for operation in 

negitive'orders only have been developed. The first method is the GENERAL 

MllTIIOD, arid with this method the instrumental constants are related only to 

the holographic grating. The second method is called the MODIFIED METHOD, 

iIJld with this method it is possible to design holographic gratings that minimize 

asiigmatisin or coma over 3 desired wavelength range for Seya-Namioka monochromators that are interchangeable with conventional gratings. This means 

that thc instrumental constants are the same for both holographic gratings and 

cor? ven t i ona 1 gratings. 

'The contents of this report are divided into four sections. Section I presents 

 he the01.y arid basic equations for designing holographic concave gratings such 

that mhhization of aberrations of astigmatism or coma over a desired wavelength range may be achieved. The basic equations are extracted from Refereilce 2 and are presented so that a better understanding of the computer program 

ma) be appreciated. For more detailed theoretical analysis see Reference 2. 

A dedc*ription of the cor,iputer program including all of the subroutines is presentcd in Section 2. Section 3 presents the input data, definition of terms and 

data card description. dectior? 4 presents an example for ushg the program 

and describes the output data in detail. 

1 
SECTION 1. THEORY AND BASIC EQUATIONS FOR DESIGNING HOLOGRAPHIC 

CONCAVE GRATINGS 

Since a detailed theoretical explanation relative to the design of holographic concave gratings is given in Reference 2, we only present a brief description for 

ease of understanding the development of the computer program presented in 

this report. 

Figure 1 shows a schematic diagram of the optical system which defines a rectangular coordinate system. We have used the same terminology as Noda, 

Namioka, and Seya (Reference 2) in order to avoid any ambiguity. Let the origin be at vertex 0 of the concave grating having a radius of curvature R. Let 

the x-axis be the normal to the grating at 0 and let the xy plane be defined by 0 

and two coherent point sources C and D used to record the interference fringes 

on a concave substrate. Points A, P, and €3 are self-luminous points on the 

entrance slit; a point on the grating and a point at the focus of the diffracted 

image from P of wavelength h in the m-th order, respectively. 

For the ray APB, the light path function F is given by 

F = F,,, + wFIoo + LFoll +;i I w2F2,,, 

+fL2F020 +~w~F,~~ 1 ++ wL~F,,, + wLF,,, 

+ f w4 F,,, + ;r 1 w2 L’ F,,, + ; 1 L4 F,,, + wL2 F,,, 

+;i 1 L2FO2, +i L’F,,, +T 1 w’LF,,, + * - 

and 

Fj k = M.. 1J k + ( $)lojk 

The M terms refer to the contribution of the conventionally ruled grating and the 

H terms refer to the contribution of the holographic grating. In equation (2), m 

is the order of the grating, X, is the laser radiation Wavelength, and X is the 

wavelength at which the grating is being employed. The terms Mijk and Hijk 

are defined in terms of the functions fijk as follows: 

for (ijk) = (200), (020), (300), (120), 

2 
I 

Figure 1. Schematic diagram showing the geometry for recording the holographic grating. The instrumental and recording parameters are 

deiined as follows: 

Instrument Parameters Recording Parameters 

ro is the object distance from the entrance ‘C 

I; ib tile iniage distance from [tie exit slit to 

slit IO the centcr of ~lle grdting. 

the cen12r of tlie grating. 

to the normal of the grating. 

the noriiral of tlic gu~ing. 

is the elitlance slit position and B is the 

exit At position. 

‘D 

a ib the angle of incident radiation relative 6 

0 IS thc diffracted image ai& reldtivr: to 7 

A 

is the distailcc of recording point source C to 

the center of the grating. 

is the distance of the recording point source D 

to the center UT the Bratine. 

is the recording angle for positioning point 

source L) relative to ihc normal of the grating. 

is the recording angle for positioning point 

source C relative to the normal of the grating. 

3 
for (ijk) = (+loo), (220), (040), where 

For equations (2)-(7), R is the radius of curvature of the grating and (rC , y ,0) 

and (r ,,, 6,O) are respectively the cylindrical coordinates of the point sources 

C and D. 

The grating equation is given by 

for the principal ray, the ray originating from the center of the entrance slit, and 

diffracted from 0 at an angle (3. u is the effective grating constant defined by 

where 6 > y . Note that the number of grooves per unit length is N = l/o. 

Minimization of Aberration Terms 

The theory described up to this point has been presented for the general case, 

We now apply the general theory to a specific application: the Seya-Namioka 

4 
mongchroniator. 

parameters are defined as follows (Reference 2); see Figure 2. 

For the Seya-Namioka monochronialor the basic instrumental 

ro,F,', = constant, 2~ = a. -bo = constant 

(10) 

a0 = K+O, Po = 0-K 

where ro is the distance from the center of the entrance slit to 0 and F0 is the 

distance from 0 to the center of the exit slit. 2~ is the angle AOB and 8 is the 

angle of grating rotation measured from the bisector of the angle 2~ and has the 

same sign as the spectral order m. Under conditions (lo), the relation between 

0 and h is given by 

- 

We denote FiJk for the ray AOB by Fijk, Then, the aberrations in the SeyaNaiiiioka monochromator can be reduced by minimizing the functions 

where i + j + k 2 2 over a predetermined scanning range, 0, GO G 8, 

h, < h < A, 

or 

where 

U' 

:I.. = HiJk = (--I Hijk 'Jk sin 6 -sin y 

and 0, , rnd 0, are - related to h, and X, through equation (11). This is equivalent 

to imposing on FijL 's the condition 

5 
z 

CONCAVE 

GRATING 

BISECTOR 

I 

rb' (PLANE p OF LAN EXIT E SLIT) 

Figure 2. Schematic Diagram of the Optical System 

for Seya-Namioka Monochromator 

Determination of Instrumental Parameters p, p', K and A,,, 

The optimum instrumental parameters are determined in such a way that equation (15) is satisfied for the horizontal focusing condition $'zoo. This is accomplished by solving the following equations simultaneously, 

a1200 a1200 

aK a*,,, 

-- -0, -- - 0. 

6 
Once the optimum values of p,p‘ and K are determined, then each one of the 

integrals I.. with i + j + k 3 2 becomes a function only of Aijk. Then, those 

values of Aijk Is which satisfy equation (15) are calculated from the equation Ilk 

a’ijk -- -0 

where 

Aijk = [Bijk(e2)-Bijk(0,)I / [2(8, -Ol)-(sin 202 -sin201)1 COSK (17) 

where 

and so on. 

7 
Determination of the Recording Parameters pc , pD, y and 6 and Minimization 

oi‘ Astibmatism or Coma 

The recording parameters are dependent upon the aberrations to be minimized. 

With a predetermined effective grating constant, the following equations are 

solved simultaneously for minimizing astigmatism and one coma-type aberration, 

and for minimizing coma-type aberrations 

Before making any attempt to solve equation (22) or (23) it is necessary to in- , 

vestigate the conditions under which equation (22) or (23) can have real solutions 

for pc qd pD because H,oo and Hlzo are quadratic in pc and pD and their values 

depend on A,,, and A,20 and therefore on values of 8, in I,,, and I,*,. The condition may be stated in such a way that two quadratic equations of pc or pD resulting from equations (22) and (23) should not have imaginary roots. To fulfill 

this condition, A,,, and A, ,, must satisfy the equation 

where for A300 

a= (siu 6 cos’ y - sin y cos2 6) cos * y, 

b = [ 2p cos y sin 6 + sin (6 - y) cos 61 cos y, 

c = p (p + cos 6) sin 6 t A,,, cos2 ti(sin 6 - sin y), 

8 
and for A,,o 

;L = cos4 y sin 6 - sin y cos' 6, 

b = [2p cos y sin 6 + sin (6 - y) cos3 61 cos y, 

c = p (p sin 6 + cos3 6) sin 6 + A,~~ cos4 6 (sin 6 - sin 7); (26) 

p = Azoo(sin 6 - sin y) + cos .y - cos 6. (27) 

When condition (24) is satisfied, then pc and pD are solutions to equations (22) 

and (23) 

Up to this point, we have been describing the general method of designing aberration corrocted holographic gratins for Seya-Namioka monochromators which 

a re imt interchangeable with conventional gratings. 

Modi [ied Nethod 

Tlie inodified inetliod of designing holographic gratings for the Seya-Namioka 

monociiromator has a practical advantage in that the holographic grating is 

interchangeable with a conventionally ruled grating having the same radius of 

curvature. This is possible provided the same instrumental constants 6, p' 

and K used for the conventionally ruled grating are used to design the holographic 

grating. The instrumental constants for the conventional grating are determined 

ky solving the equation 

or 

and 

Thc: design procedure given by equations (15)-(23) must be modified in part in 

order to accommodate condition (28). The modification required is to replace 

cqiirttion (16) with equation (28) aid equatior. (17) with (ijk) = (200) and the rest 

ot thc procedure remains the same. 

9 
SECTION 2 

Section 1 described the theory of designing holographic gratings that minimize 

aberrations over a selected wavelength range. In this section, we use the 

theory and present the computer program which consists of a main program and 

four subroutines. Figure 3 is a diagram of the flow chart for the program. 

PROGRAM DESCRIPTION 

The following is a presentation of the main program and algorithms for each 

subroutine in the computer program to show how the algorithms perform their 

specific tasks. 

MAIN PROGRAM 

The purpose of this program is to direct the flow of calculations depending upon 

which option is selected. If option LMN = 2 is selected, then the optimum value 

of the instrumental parameter angle 2~ will be calculated. However, if the 

program user chooses to specify an angle 2~ as an input value then option 

LRIN = 1 is used. 

The computer program has been developed to accommodate two holographic grating design methods; the general method and the modified method. If the general 

method is selected, then the option called IETA = 1 is used in the program. 

When using option IETA = 1, all the instrumental parameters are related only to 

the holographic grating. 

If the modified method is selected, then option IETA = 0 is used in the program. 

When using this option, all instrumental parameters calculated are the same for 

both holographic and conventional gratings. This means that the holographic and 

conventional gratings are interchangeable if they have the same groove frequency 

and radius of curvature. The only difference is that the holographic grating 

minimizes aberrations and the conventional grating does not. 

Once the design method has been selected and the LMN option has been chosen, 

the following steps of calculation are performed. 

STEP A: Read input cards. 

STEP B: Call INSTRM to calculate the optimum angle 2~, and then calculate Bijk and Ai,k (equation (17) (LMN = 2) ) or calculate the 

functions Bijk and Aijk using the input value of 2~ (LMN = 1). 

10 
iwui: x0, A,, A*, c, N, R 

in, IETA, LMN 

I 

CALL INSTRM 

CALCULATE INSTRUMENTAL 

PARAMETERS F,p'. C 

CALL PRFORM 

CALCULATE AND PRINT 

ABERRATIONS (Fijk'~) 

CALL ASTGMS 

CALCULATE RECORDING 

PARAMETERS y.6, pc. p,, 

CALL COMA 

CALCULATE RECORDING 

PARAMETERS y 6, Pc, Pg 

CALL PRFORM 

CALCULATE AND PRINT 

ALL ABERRATIONS (Fijk't) I 

Figure 3. Main Program Flow Chart 

11 
I 

STEP C: Call ASTGMS to calculate the recording parameters pc, pD , 7, 

and 6 to minimize astigmatism ijk = (020) and one coma term 

ijk = (300). 

STEP D: Call COMA to calculate the recording parameters Pc, PD , Y, 

and 6 to minimize coma aberrations (ijk) = (300) and (ijk) = (120). 

STEP E: Call PRFORM to display the numerical values of each aberration 

of the holographic grating and compare with that of the equivalent 

conventional grating as a function of wavelength. Note that the 

program automatically calculates the recording parameters for 

both the astigmatism minimization and coma minimization. This 

feature was included in the program so that the program user 

may select which aberration minimization will best produce the 

optimum minimization for all aberrations (astigmatism, coma 

and spherical) throughout the desired wavelength range. 

Subroutine INSTRM 

This subroutine obtains the optimum values of the instrumental parameters, 

namely p, p', 2~ and by simultaneous solution of equation (16). A modified 

form of Newton's method is used to obtain the solution. The quantities Bijk are 

then calculated using the optimum values of p, p', and 2~ which in turn give Aijk 

by equation (17). If a prescribed input value of 2~ were used, the selection of 

option LMN = 1 would by-pass the optimization procedures when calculating the 

other instrumental parameters. 

The values of the instrumental parameters as well as A300, A020, and A 120 are 

printed at the end of this subroutine. 

Subroutine ASTGMS 

The purpose of this subroutine is to obtain the recordingparameters such that 

astigmatism (ijk) = (020) and a coma type term (ijk) = (300) are minimized over 

the given wavelength range A, < X < A,. This is achieved by finding a simultaneous solution of equation (22). 

It is first verified that the quantity NX, is less than the maximum value of 

I(sin 6 - sin ?)I, namely 2. If not, an error message is printed stating "error, 

GAMA out of bounds. 

B 

The values of a, b, c of equation (25) or (26) are then computed and the discriminant is checked to ensure b' - 4ac 2 0. The solutions of the quadratic equation 

12 
are computed and are checked for positiveness; the calculations show that one of 

the two roots [ (-h & d b2 - 4ac )/2a I is always positive and the other always 

negative. 

The value of the angle 7 to the desired accuracy is determined by the iterated 

solution,of the equation (22) by Newton's meqod. The resulting values of the 

recording parameters 7 and 6 are printed at the end of this subroutine. 

t I. 

Subroutine COMA 

The purpose of this subroutine is to obtain the recording parameters such that 

the coma type terms (ijk) = (300) and (120) are minimized over a given wavelength rairgc A, < A < A, . This is achieved by finding a simultaneous solution 

of equation (23). 

It is first verified that the quantity NA, is less than the maximuni value of 

I(sin E - sin y) 1, namely 2. If not, an error message is printed stating "error, 

GAM.4 out of bounds. 

'l'hc values 01 a, b, c of equation (25) or (26) are then computed and the discriminant is checked to ensure that b2 - 4ac 2 0. The solutions of the quadratic 

equation are computed and are verified - for positiveness ; the calculations show 

that one of the hvo roots [(-b +A2 - 4ac )/2a 1 is always positive and the other 

always negative. 

The due of the angle y to the desired accuracy is determined by the iteratcd 

solution of the equation (22) by Newton's method. The resulting values of the 

recarding parameters y and 6 are printed at the end of this subroutine. 

Subrouthic PRFROM 

This siibrk)utine calculates the performance of the holographic grating with the 

rc.coL*d.ing parameters BS coniputed above and displays the various aberrations 

AS il fuxlior. of the wavelength in the form of a table. The performance of a 

coi,vmtio.ial (met. hanically ruled) grating with the same groove frequency is also 

shown in the same table for comparison. 

SL'CTION 3. DATA SET AND DESCRIMION OE' DATA CARDS 

This scctioli aiscases the ingat data required for a successful operation of the 

p rogr.am. 

13 
1 

(a) DATA SET 

The following quantities are required as an input for this program: 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

a. 

9. 

1 0. 

A, = the wavelength of recording laser light. 

A, = the lower limit of wavelength. 

A, = the upper limit of wavelength. 

Ah = the wavelength interval selected for displaying the performance 

of the grating as a function of wavelength. 

C = the angle 2~ (value 0.0 is used with option LMN = 2). 

N = the number of lines per mm (a = 1/N, the grating spacing 

in mm). 

R = radius of curvature of the grating. 

m = order of the diffraction where m takes on negative values for 

the program. m = -1, -2, etc. 

IETA = parameter to specify the grating design method. (IETA = I) 

specifies the General Method. 

fied Method. 

(IETA = 0) specifies the ModiLMN = parameter to specify whether the angle 2~ is to be fed in as 

input (LMN = 1) or is to be computed by the optimization procedure (LMN = 2). 

It should be noted I1Lh!INf1 and ltIETA1l must be specified in the input for each 

data set for a successful run. 

@) DESCRIPTION OF DATA CARDS AND SAMPLE DATA CARD 

The data set consists of ten paramqters described in (a), The first seven parameters are to be pimched in the order shown in (a), starting in column 1 in 3'10.0 

format (i. e. , ten columns are reserved for each of the six parameters and each 

of them must include a decimal). 

The parameters m (-1, -2, -3, . . .), IETA (0,l) and LMN (1,2) take only 

single digit integer values. Two columns are therefore reserved for each of 

14 

-.. -. 
(4) Operating Range = 0 A to 4000 A. 

(5) Order m = -1. 

(6) Determine the optimum angle 2~. 

Solution (1) indicates that the modified method is used IETA = 0. (6) indicates 

that the option LMN = 2 is to be used. 

A data card is punched using the data supplied above and punched in the order 

listed below : 

A, = 4579.3 

x, = 0.0 

A, = 4000.0 

Ax = 500.0 

C = 0.0 (0.0 is punched on the input card when the optimum angle 2~ 

is to be determined) 

1 

N = 600.0 

R = 1000.0 

m = -1 

IETA = 0 

, LMN= 2 

l.&e that the waveA2ngL. input values are in angsdoms, N in l/mm, and R in 

millimeters. The angle 2~, where 2~ is C on the data card, is in degrees in 

decimal form. 

The output data for the example cited is shown in Figure 5 and described as 

follows : 

Line 1 shows the input parameters; order = -1 is the order of the grating, 

N = groove frequency in grooves per mm, R = radius of curvature 

of the grating in mm, IETA = 0 identifies the design method and 

LMN = 2 shows that the optimum angle 2~ has been calculated. 

16 

PRECEDING PAGE BLANK NOT FILMED 

- . - . - - - . - .. - - . - - . __ - 
I 

ORIGINAL PAGE IS 

Of POOR QUALITY 

I. 

4 

OJ 8 

xa 

I1 

0 0- a c1 

0 

, Y 

c: 

J 

* 

n 

h 

- 

. 

U 

t 

8 

h c 

P 

5 

w 

U 

L.7 c 

n .e .a 

an 

.N 

.. .* vo 

D. 

-1 

'I 

.C .* 

ac 

a. 

ad 

00 

01 

17 
Line 2 shows the input values for A,= LAMDAO (laser wavelength), 

X, = LAMDAl (the lower wavelength limit), and X, = LAMDA2 

(upper wavelength limit). 

Line 3 is self-explanatory. 

Line 4 shows the calculated instrumental parameters; C = 2~, = RHOA, 

P' = RHOB, rb = SMLR in mm, and 7, = SMPRP in mm. These 

parameters are defined in the text. 

Line 5 shows the type of aberration that has been minimized: Astigmatism = ASTG. 

Line 6 is self-explanatory and refers to line 7. 

Line 7 shows the optimum recording parameters; 7 = GAMA, 6 = DLTA, 

pc= RHOC and pD = RHOD. These parameters have also been 

defined in the text. 

Line 8 shows the titles for the table showing the performance of the grating as a function of wavelength. The table also shows the comparison of the holographic grating with the conventional grating having 

the same instrumental parameters. 

X = LAMDA is the wavelength at which the grating is evaluated. 

Fzoo x lo' is the horizontal focus and astigmatism of the first term. 

F,,, is the vertical focus and astigmatism of the second term. 

F,,, is the first coma term. 

F1,, is the second coma term. 

F400 is the first spherical aberration term. 

F2,, is the second spherical aberration term. 

FO4, is the third spherical aberration term. 

Line 9 shows the holographic grating performance (Modified Method), the 

value of the wavelength and the values of the aberrations. 

18 
Line 10 

Line 11 

Line 12 

Line 13 

shows the performance of a conventionally ruled grating (MECH. 

RULED). 

shows that the following information refers to minimizing coma 

type aberrations for the same instrumental parameters shown in 

Line 4. 

is self-explanatory and refers to Line 13. 

shows the recording parameters 7 = GAMA, 6 = DLTA, pc = RHOC 

and pD = RHOD required to fabricate a grating to minimize coma. 

Lines 14, 15, and 15 are defined in the same way as Lines 8, 9, and 10 

above. The values in this table refer to a grating designed to 

minimize coma over the desired wavelength. 

RE FEREFJC ES 

1. H. Greiner and E. Schaeffer, Optik 16, - 350 (1959) 

2. H. Noda, T. Namioka and M. Seya, J. Optical Society of America 64, 

1043 (1974) 

U. '-I T. Namioka, H. Noda and M. Seya, Science of Light - 22, 77 (1973) 

19 
CONTENTS 

Page 

INTRODUCTION.. .................................... 1 

SECTION 1. Theory and Basic Equations for Designing 

Holographic Concave Gratings ................... 2 

Minimization of Aberration Terms ......................... 5 

Determination of Instrumental Parameters p, 2; K and A ........ 7' 

Determination of the Recording Parameters p,, p,, y and 6 to 

Correct for Astigmatism or Coma at a Specified Wavelength. ....... 9 

Modified hlethod. ..................................... 10 

SECTION2.. ....................................... 11 

PROGRAM DESCRIPTION. .............................. 11 

MAIN PROCRAhl.. ................................... 11 

Subroutine INSTRM. ................................... 13 

Subroutine ASTGhlS ................................... 13 

Subroutine COhlA. .................................... 14 

Subroutine PRFORM. .................................. 14 

SECTfON 3. DATA SET AND DESCRIPTION OF DATA CARDS. .... 15 

SECTIONi. EXAMPLE AND DESCRIPTION OF DATA.. ......... 17 

REFERENCES....................................... 20 
COMPUTER PROGRAM FOR DESIGNING 

HOLOGRAPHIC GRATINGS FOR 

S E Y A - N AM I OKA T Y P E MOM 0 C H RO MAT 0 RS 

Part 11: Correcting Aberrations at 

Specified Wavelengths 
COMPUTER PROGRAM FOR DESIGNING 

HOLOGRAPHIC GRATINGS FOR 

SEYA-NAMIOKA TYPE MONOCHROMATORS 

PART II: CORRECTING ABERRATIONS AT 

SPECIFIED WAVELENGTHS 

lNTHODUCTION 

A computer program has been developed based on the theory presented in papers 

by Greiner and Schaeffer (1) and Noda, Namioka, and Seya (2), that determines 

thp_ optimum holographic grating recording parameters as a function of optical 

instrumental parameters and desired wavelength range for Seya-Namioka type 

mouocbromator mountings. In the first report, Part I, the holographic grating 

recording parameters are determined such that astigmatism or coma may be 

minimized over a selected wavelength range. In this report, Part 11, we determine optimum recording parameters such that astigmatism or coma may be 

corrected at specified wavelengths. By corrected, we mean that the aberrations, 

astigmatism or coma, are zero at specified wavelengths. The program also 

simultaneously displays the performance of the holographic grating and equivalent conventional grating as a function of wavelength relative to optical aberrations 

of astigmatism, coma, and spherical aberration. In this report, two methods of 

desib’nng holographic gratings for operation in negative orders only have been 

developed. The first method is the general method, and with this method the instrumental constants are related only to the holographic grating. The second 

method is the modified method and with this method it is possible to design holographic gratings that correct astigmatism or coma at specified wavelengths within 

the desirea wavelength range for Seya-Namioka monochromators that are interchangeable with conventional gratings. This means that the instrumental constants are the same for both holographic gratings and conventional gratings. 

. 

The contents of this report is divided into four sections. Section I presents the 

theory and basic equations for designing holographic concave gratings such that 

correction for aberrations of astigmatism or coma at a specified wavelength may 

be achieved. The basic equations are extracted from Ref. 2 and are presented so 

that a better understding of the computer program may be appreciated. However, we have modified ths equations for minimizing astigmatism or coma such ” 

that they will now correct astigmatism or coma at specified wavelengths A*. A 

descriptioniof the computer program including all of the subroutines is presented 

in Section 2. Section 3 preeents the input data, definition of terms, and data card 

description. Section 4 presents an example for using the program and also describes the output data in detail. 

1 
SECTION I. Theory ami Basic Equations for Designing 1 Holographic Concave Gratings. 

Since a detailed theoretical explanation relative to the design of holographic concave gratings is given in Ref. 2 we only present a brief description for ease of 

understanding the development of the computer program presented in this report, 

Figure 1 shows a schematic diagram of the optical system which defines a rectangular coordinate system. We have used the same terminology as Noda, Namioka, 

and Seya (Ref. 2) in order to avoid any ambiguity. Let the origin be at vertex 0 of 

the concave grating having a radius of curvature R. Let the x-axis be the normal 

to the grating at 0 and let the xy plane be defined by 0 and two coherent point 

sources C and D used to record the interference fringes on a concave substrate. 

Points A, P, and B are self luminous points on the entrance slit, a point on the 

grating and a point at the focus on the diffracted image from P of wavelength 

in the mth order, respectively. 

For the ray APB, the light path function F is given by 

and 

(1) 

1 1 1 1 

4 4 2 t - w2FZO2 t- l2FO2, t - 13F0,, t 5 W' lF211 t . . . 

The M terms refer to the contribution of the conventionally rulled grating and 

the H terms refer to the contribution of the holographic grating. In Eq. 2, m is 

the order of the grating, A, is the laser radiation wavelength and A is the wavelength at which the grating is being employed. The terms Mijk and Hijk are defined in terms of the functions fijk a6 follows: 

2 
for 

where 

f2,*(Fd = (p COSQ - 1) cosa, 

fOzo(p, a) = P - cosa, 

x, = rc cos yI y, = rc sin y, xD = rD cos 6 , yD = rD sin 6. (7) 

For equations (2)-(7), R is the radius of curvature of the grating and (rc, y, 0) 

and (r,,, 6, 0) are respectively the cylindrical coordinates of the point sources 

C and D. 

The grating equation is given by 

o(sinao+sin~o)=mh (8) 

for the principal ray, the ray originating from the center of the entrance slit, 

and diffrzcted from 0 at an angle p. o is the effective grating constant defined by 

3 
Figure 1. Schematic diagram showing the geometry for recording holographic 

gratings. The instrumental and recording parameters are defined aa follows: 

(Instrumental Parameters): 

r3 

ri 

J. 

p 

A 

is the objective distance from the entrance elit to the center of the grating. 

is the image distance from the exit slit to the center of the grating. 

ie the angle of incident radiation relative to the normal of the grating. 

is the diffracted image angle relative to the normal of the grathg. 

is the entrance slit position and B is the exit elit position. 

(Recording Parameters): 

rc 

rd 

S 

is the distance of recording point source C to the center of the grating. 

is the distance of the recording point source D to the center of the grating. 

is the recording angle for positioning point source D relative 

to the normal of the grating. 

is the recording angle for poeitioning point source C relative 

to the normal of the grating. 

j 

4 
where 

A0 

u= 

sin 8 - sin? 

Note that the number of grooves per unit length is N = l/u. 

Minimization of Aberration Terms 

The theory described up to this point has been presented for the general case. 

We now apply the general theory to a specific application; the Seya-Namioka 

monochromator. For the Seya-Namioka monochromat the basic instrumental 

parameters are defined as follows; (reference 2) see Figure 2. 

- 

ro, r; = constant, 2K = a. - ,ljo = constant 

where ro is the distance from the center of the entrance slit to 0 and i$ is the 

distance from 0 to the center of the exit slit. 2K is the angle AOB and 0 is the 

angle of grating rotation measured from the bisector of the angle 2K and has the 

same sign as the spectral order m. Under conditions (lo), the relation between 

d and A is given by 

(11) 

20 

m 

A= - cosKsin8 

We denote F, jk for the ray AOB by i?’i jk , Then, the aberrations in the SeyaNaniioka monochromator can be reduced by minimizing the functions 

where 

itJtKz2 

5 
z 

4 

CONCAVE 

GRATING 

X (GRATING NORMAL) 

Figure 2. Schematic diagram of the optical system for 

Seya-Namioka monochromator 

6 
< < 9 over a predetermined scanning range, 8, = 8 2 d2 or A 5 A = A, where 

RR p =- I Fl =y=- -R 

=0 r ri * 

Hi jk 

Aijk - sin 8 - sin y =(e) Hijk 

and 8, , and d2 are related to A, and A2 through equation (11). This is equivalent to iniposing on F, 's the condition 

. 

I., P le,- FTjk d8 = minimum 

8Jk 

*1 

Determination of Instrumental Parameters T;, T;', K and A,,, 

The optimum instrumental parameters are determined in such a way that Eq. (15) 

is satisfied for the horizontal focusing condition Fzoo . This is accomplished by 

solving the following equations simultaneously, 

a1 200 - 

-- 0, 200- -- 0, 3F ap 

a1 200 

arc aA2oo 

-= 0. aI 200- 

-- 0, 

Once the optimum values of p, F' and K are determined, then each one of the 

integrals Iilk with i + J + K 2 2 becomes a function only of Ai jk . Then, those 

values of Ai jk Is which satisfy Eq. (15) are calculated from the equation 

where 

7 
where 

B2,,(8) = (p/3) [cos8 t cos(8 t K) cos(28 t K 

t 3 sin(8 t K) sinK] 

t (p'/3) (cos0 t COS(^ - K) COS(^^ - K) 

- 3 sin(8 - K) sinK] - cos28 cosK, 

B,,,(3) = (;?/4), [sin(48 t 3K) 

- 4cos2(8tK)sinK- 48 cosK] 

+ (p'/4), [sin(48 - 3K) 

t 4 cos(8 - K) sinK - 40 cosK] 

t (p/6) [3sin(B t 2K) - sin(38 t 2K)l 

t (p'/6) [3sin(0-2K)-sin(38- 2K)], 

1 

2 B,,,(d) =-p2[sin(28 t K) - 28 cosKl 

+fpa2 [sin(28 - K) - 28 cosK] 2 

t (75/6) [3 sin(8 + 2K) - sin(38 t 2K)l 

t (3'/6) [3 sin(8 - 2K) -sin(38 - 2K)l , (21) 

and so on. 

8 
Determination of the Recording Parameters p, , pD, -,, aud 6 to Correct 

for Astigmatism or Coma at a Specified Wavelength 

The recording parameters are dependent upon the aberrations to be corrected 

and the selected correcting wavelength A*. With a predetermined effective grating constant, the following equations are solved simultaneously for correcting 

astigmatism and one coma type aberration, 

and for correcting coma-type aberrations 

Before making any attempt to solve Equations (22) or (23) it is necessary to investigate the conditionii under which Equations (22) or (23) cax~ have real solutions 

for pC and pD because H,,, and H,,, are quadratic in pc and P, and their Values depeixl on A,,, and AI,, and therefore on values of ~2 in I,,, and II2,. 

The condition may be stated in such a way that two quadratic equations of p, or 

9 
pD resulting from equations (22) and (23) should not have imaginary roots. To 

fulfill this condition, A,,, must satisfy the equation , 

b2 - 4ac > =O 

where for A 

300 

a = (sin& cos2y- sinycos26) cos2y, 

b= [2pcosysin6 +sin(&-y)cosb;] cosy, 

When condition (24) is satisfied, then p, and pD are solutions to Equations 22 and 

23. Up to this point, we have been describing the general method of designing 

aberration corrected holographic gratings for Seya-Namioka monochromators 

which are not interchangeable with conventional gratings. The next paragraph describes the method where the conventional and holographic gratings are interchangeable . 

Modified Method 

The modified method of designing holographic gratings for the Seya-Namioka 

monochromator has a practical advantage in that the holographic grating is interchangeable with a conventionally ruled grating having the same radius of curvature. 

This is possible provided the same instrumental constants ii, and K used for 

the conventionally ruled grating are used to design the holographic grating. The 

instrumental constants for the conventional grating are determined by solving the 

equation 

I200 de = minimum 

or 

-- -0 aI 200 200 

2-p a-p ' 

-= 0, 

and 

a1 200 

-= 0. 

3K 

10 
' The design procedure given by Equations (15)-(23) must be modified in part in 

order to accommodate condition (27). The modification required is to replace 

Eq. (16) with Equation (27) and Equation (17) with (i j k) = (200) and the rest of 

the procedure remains the same. 

SECTION 2 

Section 1 described the thcory for designing holographic gratings that correct 

astigmatism or coma at specified wavelengths within a desired wavelength range. 

Lu this section, we use the theory and present the computer program which consists of a maid program and fo-a subroutines. Figure (3) is a diagram of the 

flow chart for the program. 

PROGRAM DESCRIPTION 

The following is a presentation of the main program and algorithms for each subroutine in the computer program to show how the algorithms perform their specific ts'ks. 

MAIN PROGHAM 

The purpose of this program is to direct the flow of calculations depending upon 

which option is selected. If option LMN = 2 is selected, then the optimum value 

of the instrumental parameter angle 2K will be calculated. However, if the programmer chooses to specify an angle 2K as an input value, then option LMN = 1 

is used. 

The computer program has been developed to accommodate two holographic 

grating design methods; the general method and the modified method. If the 

general method is selected, then the option called IETA = 1 is used in the program. Wheli using option IETA = 1, all the instrumental parameters are related 

only to the holographic grating. 

If the modified rnethcd is selected, then option IETA = 0 is used in the program. 

When using this option, all calculated instrumental parameters are the same for 

both holographic and conventional gratings. This means that the holographic and 

conventional gratings are interchangeable if they have the same groove frequency 

and radius of curvature. Tha only difference is that the holographic grating corrects aberrations to zero and the conventional grating does not. 

11 
I 

I i 

CALL INSTRM 

CALCULATE I NSTR UMENTAL 

PARAMETERS p, p', C 

CALL ASTGMS 

CALCULATE RECORDING 

PARAMETERS y, 6, p,, p,, 

~ ~~ ~~ ~~ 

CALL PRFORM 

CALCULATE AND PRINT 

ABERRATIONS (Fli k's) 

CALL COMA 

CALCULATE RECORD I NG 

PARAMETERS y, 6, p,, pD 

r CALL PRFORM 

CALCULATE AND PRINT 

ALL ABERRATIONS (Fiik's) 

Figure 3. Main Program Flow Chart 

12 
Once the design method has been selected and the LMN option has been chosen, 

the following steps of calculation are performed. 

STEP A: Read input cards. 

STEP B: Call INSTRM to calculate the optimum angle 2K, and then calculate Bijk 

and Aijk Eqs. (17) (LMN 2) or calculate the functions Bijk and Aijk using 

the input value of 2K (LEIIIN = 1). 

STEP C: Call ASTGMS to calculate the recording parameters p,, pD, y, and S 

to correct astigmatism ijk= (020) and one coma term ijk = (300) at A*. 

STEP D: Call COhlA to calculate the recording parameters p,, pDly, 6 to correct coma aberrations (ijk) = (300) and (ijk) = (120) at A*. 

STEP E: Call PRFORM to display the numerical values of each aberration for 

the holographic grating and compare with that of the equivalant convdntiond grating as a function of wavelength. Note that the program 

automatically calculates the recording parameters for both the astigmatism correction and coma correction. This feature was included in 

the program so that the program user may select which aberration 

correction will best produce the optimum performance for all aberrations (astigmatism, coma aid spherical) throughout the desired wavelength range. 

Subroutine IXSTRM 

This subroutine obtains the optimum values of the instrumental parameters 

namely,-p, P', 2K and A200 by simultaneous solution of Eqs. (16). A modified 

form of Newton's method is used to obtain the solution. The quantities Bijkls are 

then calculated using the optimum values of j5, p' and 2K which in turn give Ai k 

by Eq. (17). If a prescribed input value of 2K were used, the selection of option 

LMN = 1 would by-pass the optimization procedure when calculating the other 

instrumental parameters. 

The values of the instrumental parameters as well as A, oo, A,, , , and A, , 

printed at the end of this subroutine. 

are 

Subroutine ASTGMS 

The purpose of this subroutine is to obtain the recording parameters such that 

astigmatism (ijk) = (020) and a coma type term (ijk) = (300) are corrected for a 

13 
specified wavelength A*. This is achieved by finding a simultaneous solution of 

Eq. (22). 

It is first verified that the quantity NA, is less than the maximum value of 

I (sin 6 - siny) I ; namely 2. If not, an error message is printed stating I1error, 

gamma out of bounds." 

The values of a, b, c of Eqs. (25) are then computed and the discriminant is checked 

to ensure b2 - 4ac 2 0. The solutions of the quadratic equation are computed and 

checked for positiveness; the calculation shows that one of the two roots I-b + 

-/2a] is always positive and the other always negative. 

The value of the angle y to the desired accuracy is determined by the iterated 

solution of Eq. (22) by Newton's method. The resulting values of the recording 

parameters y and 6 are printed at the end of this subroutine. 

Subroutine COMA 

The purpose of this subroutine is to obtain the recording parameters such that 

the coma type terms (ijk) = (300) and (120) are corrected at a specified wavelength A*. 

It is first verified that the quantity N A , is less than the maximum value of 

I (sin S - siny) 1; namely 2. If not, an error message is printed stating "error, 

gamma out of bounds." 

The value of a, b, c of Eq. (25) are then computed and the discriminant is checked 

to ensure that b2 - 4ac 2 0. The solutions of the quadratic equation are computed 

and verified for positiveness; the calculation shows that one of the two roots 

[ -b + im/2a] is always positive and the other always negative. 

The value of the angle y to the desired accuracy is determined by the iterated 

solution of Eq. (22) by Newton's method. The resulting values of the recording 

parameters y and 6 are printed at the end of this subroutine. 

Subroutine PRFORM 

This subroutine calculates the performance of the holographic grating with the 

recording parameters as computed above and displays the various aberrations 

as a function of the wavelength in the form of a table. The performance of an 

oquivkent conventional (mechanically ruled) grating with the same groove frequency is also shown in the same table for comparison. 

14 
SECTION 3. DATA SET AND DESCRIPTION OF DATA CARDS 

This section discusses the input data required for successful operation of the 

program. 

(a) DATA SET 

The following quantities are required as input data for this program: 

(1) A, = wavelength of laser recording radiation, 

(2) A = lower wavelength limit. 

(3) A = upper wavelensh limit. 

(4) 

(5) A* = wavelength at which aberrations will be corrected. 

(6) 

(7; 

(8) 

(9) LMN = Parameter to specify whether the angle 2K is to be fed in as an 

Ah = wavelength interval at which aberrations will be displayed. 

C = angle 2K (value 0.0 is usdd with option LMN = 2). 

N = numbar of lines per mm. (Q = 1/N; grating spacing). 

R = radius of curvature of the grating in mm. 

input (LMN = 1) or is to be computed by the optimization procedure 

(LMN = 2j. 

(10) 

(11) IETA = Parameter to specify the grating design method: 

m = order of diffraction where m takes on negative values. 

(IETA = 1) specifics the general method 

(IETA = 0) specifies the modified method. 

Note that ltLMNtl and “IETA“ must be specified in the input for each data set for 

a successful run. 

(b) DESCRIPTION OF DATA CARDS AND SAMPLE DATA CARDS 

The dzta set consists of elevan parameters described in (a). The first eight 

parameters are punched on the first card in the order stated in (a) starting in 

column 1 in F1O.O format. For example, ten columns are reserved for each of 

the eight paraimeters and each must include a decimal. The parameters LMN 

(1,2), m(-1, -2, . . . . .) and IETA (0, 1) are punched on the second card using 

only single digit integer values and two columns are reserved for each of these 

parameters starting in column 1. Figure 4 illustrates typical data card6 with 

the input values shown at the top. 

15 

Note that as many data sets as desired may be put into a single program run 

pkovided that, the values of all eleven parameters are punched for each data set 

which consists of two cards. 

SECTION 4 

This section presents a ty-pical example for using the program and also describes 

the output data in detail. 

Example: Determine the optiinum instrumental and recording parameters for 

the design of 3 holographic grating that corrects astigmatism or coma to zero 

at 2500 A having the fallowing specifications: 

(1) Groove frequency = 600 L/mm 

(2) Radius of curvature = 1000 mm 

(3) Operating range = 0 A to 4000 A 

(4) Order m = -1 

(5) Determine the optimum angle 2K 

(6) This holographic grating is not interchangeable with a conventional grating. 

Solution: (5) indicates that the option LMN = 2 will be used. (6) specifies that 

the General Method of design will be used (IETA = 1). A data card is punched 

using the data supplied above and punched in the order listed below: 

A, =4579,3 

A, = 0.0 

AX = 500.0 

X* = 2500.0 

i2 = 4000.0 

C = 0.0 (0.0 is punched on the input card when the optimum angle 2K is 

N = 600.0 

R = 1000.0 

m = -1 

t3 be determined). 

LMN = 2 

IETA = 1 

Note that the wavelength input values are in angstroms, N in lines per mm and 

R in mm. The angle 2K, where 2K = C on the data card, is in degrees in decimal 

form. 

The output data for the example cited is dhow in Fig. 5 and described as follows: 

17 
Line 1; shows the input parameters; order = -1 is the order of the grating, 

N = groove frequency, R = radius of curvature of the grating, IETA = 1 identifies the design method, and LMN = 2 shows that the optimum angle 2K has been 

calculated. 

' . 

Line 2; the input values for A , = LAMDAO (laser wavelength) , A = LAMDAl 

(lower wavelength limit) , A = LAMDA2 (upper wavelength limit) and A* = LAMDAC (correcting wavelength). 

Line 3; is self explanatory. 

Line 4; shows the calculated instrumental parameters C = 2K,p = RHOA, p' = 

RHOB, r; = SMLR in mm and Ti = SMPRP in mm. These parameters are defined in the text. 

Line 5; shows the type of aberration that has been corrected; Astigmatism = ASTG. 

Line 6; is self explanatory and refers to line 7. 

Line 7; shows the optimum recording parameters y = GAMA, S = DLTA, p,= 

RHOC and p,= RHOD. These parameters have also been defined in the text. 

Line 8; shows the wavelength (A = LAMDA) at which the grating is evaluated and 

the aberration terms described as follows: 

F,,, x lo-, is the horizontal focus and astigmatism of the first term. 

F02, is the vertical focus and astigmatism of the second term. 

F,,, is the first coma term. 

FIZO is the second coma term. 

F,,, is the first spherical aberration term. 

F;,, is the second spherical aberration term. 

F,,, is the third spherical aberration term. 

Line 9; shows the holographic grating performance (General Method), the evaluating wavelength and the values of the aberrations. 

Line 10; shows the performance of the equivalent conventional grating (MECH. 

RULED). 

18 
Y 

I 

ORIGINAL PAGE IS 

OF POOR Qum 

0 . 

c 

n 0 . 

0 U 

0 

0 

0 

2 

L 

C 

- 

b 

Y 

c 

Y 

'. 

0 

L 

L 

L 

...... 

""C" roc.- Ylr%C--. . OU.-r..I-..-. -.E.aur .- 0 "C Y Y Y -c -V6.01.&*'.-. - .......... .................. 

QOOCCOOOOCOOUCCCCC 

L 

- 

. 

-- . Y Y 

n " 

c I. 

r. 

V. *. 

Y. 

U. 8 

.a u LU 

YC 

-d 

rs .. 

e. 

L 4u 4 

nil 

ra 

tz *n 

ru U .L 

NU 

Y. 

Y. 

.. .. 

N 0 

- Y L ...... 

r" I .r ..... "'I -..,*I-.* -co.YYLL.. 

............ -. .. -Y ...... PC 3 .C -e num..--m %-* 

*h ...................... 

.Y - 

: .5 n " : . ; - ": n, ;>::*:q':::';: 00oOOUOOOouYcCOOUC 

I .., ,* n C. oo~~oo~~~nc0~00~0fC I,,* l1llIl 

0 - 

A 

0 ~DOUOUOUO 

"- c O"OO0 -0 0 

n an oooooooo n rli "'L..IO.IO.'O - - h , n c. 

.- ......... 

AL 0. 

C" 

02 

. 

a 

4c 

EN 

ON 

en 

on 

- 

E2 

om 

.L ." 

.e 0. .I 

. . 

i 

.I* 

..O 

.a. 

-1- 

C. R4 YA 

19 
I 

, Line 11; shows that the following information refers to correcting coma type. 

. aberrations for the same instrumental parameters shown in line 4. 

Line 12; is self explanatory and refers to line 13. 

Line 13; shows the recording parameters 7 = GAMA, 6 = DLTA, p, = RHOC, 

p,= RHOD required to fabricate a grating to correct for coma. 

Lines 14, 15 and 16 are defined in the same way as lines 8, 9, and 10 above. 

The values in this table refer to a grating designed to correct for coma at a 

specified wavelength. In this example the correcting wavelength is 2500A. 

REFERENCES 

(1) H. Greiner and E. Schaeffer, Optikl6, 350(1959) 

(2) H. Noda, T. Namioka and M. Seya, J. Optical Society of America 64, 1043 

(1974) 

(3) T. Namioka, H. Noda and M, Seya, Science of Light. 22, 77(1973). 

20 
DESIGN OF ASTIGMATISM OR COMA CORRECTED HOLOGRAPHIC 

GRATINGS FOR ROWLAND CIRCLE SPECTROGRAPHIC MOUNTS 
CONTENTS 

Page 

INTRODUCTION .......................... 1 

THEORY AND BASIC EQUATIONS FOR DESIGNING HOLOGRAPHIC 

CONCAVE GRATINGS ........................ 1 

Holographic Grating Design for the Rowland Circle Spectrographs ..... 5 

OVERALL LOGIC .......................... 6 

DESCRIPTION OF COMPUTER PROGRAM ............... 7 

INPUT DATA. DEFINITION OF TERMS. AND DATA CARDS ........ 9 

Dataset ............................ 9 

Data Cards ........................... 10 

PROGRAM EXAMPLE AND DATA DISPLAY. .............. 10 

REFERENCES ........................... 15 
DESIGN OF ASTIGMATISM OR COMA CORRECTED HOLOGRAPHIC 

GRATINGS FOR ROWLAND ClRCLE SPECTROGRAPHIC MOUNTS 

1 NT RODUCTION 

For niany spectroscopic instruments such as spectrographs, nionochromators, spectrophotometers, and similar instruments employing concave optical rellecting diffraction gratings, 

a major inherent optical disadvantage is that the diffracted image size (exit slit) is distorted 

and niany times larger than the objective six (entrance slit). This phenomenon is basically due to astigmatism caused by, i11 largz part, ilie conventional concave diffraction 

grating used in the instrument. In order to detect all the cncrgy in such large diffracted images, very large area detectors must be used. These detectors, if available, would 

be very large in size and would be unacceptable for use especi;Llly in flight spectroscopic 

instruments. At the present time, sinal1 area detectors are used with milch of the energy 

beiiig lost. A solution to this problem is to use the aberration corrcctcd holographic gratiiigs that correct for astigniatism. 

Thc purpose of this report is to present the design 01' liolograpllic gratings that will in 

many cases solve the problem of astigmatism and in otlier cam the problem of coma 

which causes a loss i!i resolving power in spectroscopic instrunicnts. 

The first section of tliis report presents the theory and basic equations for dcsigning holographic diffraction gratings. The remaining sections arc' devoted to the developinen1 of 

a computer program that calculates tlie necessary geonietrical data which is one of many 

requirements that makes the fabrication of holographic gratings a reality. The computer program hds been developed such that optimum recording parameters BS a function 

of instrument parameters are determined that make astigmatism or coma zero at specified ~.;ivelcngths for spectrographs based on the Rowlatid circle mount. Wlien optimum 

recording paraniet ers have been determined, the program will display, for the purpose 

of comparison, numerical tables of the performance for the holographic grating and 

equivalent comentional grating as a function of wavelength. This namerical comparison 

makes it possible to analyze the data and decide whethcr or not the holographic grating 

performalice is better than thc equivalent conventional grating rc1:itive. to optical aberrations. 

TlIEORY AND BASIC EQUATIONS FOR DESIGNING HOLOGRAPI-lIC SPHERICAI. 

CONCAVE GRATINGS 

Figure 1 shows the optical system for using and recording holographic gratings. The rect2ngular coordinate system is defined as follows: I is the origiii at the vertex of the grating blank and tlie x-axis is normal to 1. The x-y plane is defined by I and the two recording 

point soiuces C and D. Points A, U, and B are respectively, a self luminous point on 

the entrance slit, a point on the grating surfice, and a point 011 the focused difl'racted 

image from M at a wavelength A in the nth order. As a rcsult of this geonietry the 

aberrant optical path functioii for a conventional grating may be written as 

1 
. 

(GRATING BLANK) 

Figure 1. Schematic diawam of the optical system. A is a self ItilrliilOLlS point 011 the 

entrance slit, M is a point on the nth groove and B is the SPL'C~T~II image point. C and 

D are the rccording point light source positions. The pcrformancc am1 rccording piiraincters are defined as follows: 

Performance Parameters: 

LA is the distance from A (entrance slit) to the center of I of tlic grating. 

Lg is the distance from the diffracted iniagc B to the cziltcr 1 01' the grating. 

(Y is the angle of incidence. 

p is the diffraction angle. 

z is'the object height. 

z' is the image height. 

Recording Pard meters: 

Lc is the distance from the point source C to thc center I of the gratiiig. 

LD is the distance from the point source D to the center I of the grating. 

y is the position angle for point source C rdatiw to the llorl1li~l to thc grating. 

S is the position angle for point source D rclutive to the norind to rlic grntiiig. 

2 
AM = AM+MB-(lA+lB)-KNX (1) 

where I( is the order of diffraction, X is the diffracted image wavelength and N is the grating 

groove frequency. 

When a grating blank coated with the photoresist is illuminated with coherent spl~erical waves 

originating from points C and D, grooves are recorded, after developing, in the photoresist in response to the interference fringes produced at the site of the grating blank. 

The interference fringes are formed according to the equation 

MC - MD -(IC -ID) N= 

10 

where X, is the recording laser wavelength. 

Substituting eq. (2) into eq. (1) gives the aberrant optical path function 

AM = MA+MB-(IA+LB)-= ((Mc-MD)-(Ic-ID)\ x, (3) 

for the holographic grating. 

The quaptities on the right side of AM in eq. (3) are derived based on the coordinate system 

described in Figure 1 and the equations of a sphere. Since the rcsults of thesc derivations are lengthy, we suggest that the reader see reference 1 for more detailed analysis. 

Therefore, it sufficies to say that after a series cxpansioli is performed, the following 

aberrant optical path function AM is obtained. 

AM = -Y[sinor+sinp-KX (siny-sin6) I +-[A11 Y2 

Xo 2 (4) 

+y2z* IS21 +z4 [S,] +... 

where Y is the width and Z is thc height of the grating and 

3 
-( 

s2 =(& 

+- 41 

4 
Note that in the expressions of equation (S), LA, Lg, cn and @ are the terms for the conventionally ruled grating, and Lc, LD, y, and 6 are the terms related to the holographic contribution. R is the radius of curvature of the grating, K is the order of the diffracted 

wavelength, and A, is the recording wavelength. The aberration terms are defined as 

follows; AI is the horizontal diffracted image focus, A2 is the astigmatism or vertical 

focusing term, C1 and C2 are coma terms and SI , S2, and S3 are splierical aberration terms. 

The conventional gating equation is given by 

and for the holographic grating u is the effective grating constant and is defined by 

equation 

x, 

sin 7 - sin 6 u= wherc y > 6 

The preceding information describes the basic general theory for holographic gratings. What 

follows is an application of the general theory for I specific spectroscopic instrument; correcting astigmatism or coma at a specified wavelength for a Rowland circle spectrographic 

mount. 

HOLOGRAPHlC GRATING DESIGN FOR THE ROWLAND CIRCLE SPECTROGRAPH 

In order to design a holographic concave grating that has the Kowland circle as its focal curve; it is necessary to make the horizontal focusing tcrin A, and the coma term 

C1 zero at all wavelengtlis in equation (5). This condition is satisfied by 

LA = R cosa,Lg =R cosP,Lc =R cosyand LD = R cos 6 (8) 

The aberrant optical path function for condition (8) will now have the form 

AM =-Y[sin cn + sin@-- sin y-sin61 J +3_~[A2'1 22 

and the expressions for astigmatism A2' arid coma C2' in equatiorl (5) reducc to 

(9) 
and 

where X* is the wavelength at which the aberrations are to be corrected to zero and /3* is 

calculated from 

o (sin a* + sinP*) = )(A* (12) 

The recording parameters 7,S, Lc, and LD are determined for a chosen set of values u, A*, 

and a* to eliminate astigniatism A; or conia CZ at a specit'ied wavelengtli A*. ~f astigniatisin A*' is desired to be eliminated at X*, then the following set ot' equations must be 

solved; 

@ 

x, 

sin 7 - sin 6 

and equation (I 0), A2' = 0. For this case, the recording angles y and 6 are chosen such 

that astigmatism A2' is made zero at A*. 

o= (7) 

It' one chooses to eliminate coma at a wavelength X*, then the following set of equations must be solved; (13, (7) and (1 l), C,: = 0. For this case, the recording angles 

-y aiid S are chosen such that C2' is made zero at A*. 

When tlie recording parameters 7 and S have been determined for a given set of u, a*, 

and X*, then the values of LA, Lg, LC and LD are calculated from equation (8). The 

remaining aberrations SI', S2', and S3' in equation (5) are calculated after the aberrations A2' and C2' have been made zero at X* for the values deterniined for 7 and 6. 

The aberration calculations deterniined by the coniputer for astigmatism, coma, and 

spherical are independent of Y, Z, R, and the constants in the denominators ol' the expressions for the coefficients. In other words the values of tlie aberrations are pure 

nu in bers. 

OVERALL LOGIC 

In the first section, we presented the general theory of holographic concave gratings and applied the theory to designing concave holographic gratings, corrected for astigrnatism or 

coma, for a spectrograph based on the Kowland circle mount. 

In this section, we present the logical steps of the computzr program that lcads to the 

design of holographic gratings. 

6 
Step 1 requires that one selects the groove frequency, angle of incidence, correcting 

wavelength X*, and the radius of curvature R for the holographic grating. 

Step 2; Based on the data of step 1, determine the angle of diffraction @* at A*. 

Step 3; The expression for astigmatism A2' equation (10) is then scanned as a function 

of the recording angle 7 to determine the optimum angles 7 and 6 that will make A2' = 0 

at X*. Successive values of 7 are obtained by an equation of the form 7 = 7, + 47 where 

A7 is an incremental angle and chosen to equal 0.25". Therefore the accuracy of 7 is dependent on the value of A7; the smaller the incremental angle A7 the better the accuracy 

of 7. 

Step 4; When the optimum recording angles 7 and 6 have been determined for correcting 

astigmatism A7' at X*, then the performance of the gating relative to all aberrations as a 

function of wivelength over a chosen wavelength range is calculated for the holographic 

grating and displayed in the form of a table. For comparison, t!ie performance of an 

equivalent conventional grating is also calculated and displayed for the same wavelength 

range. 

The same logical process is employed for correcting coma C2' (equation ( 1 l)), however, 

the optimum recording parameters are related to correcting conia. 

A unique feature of the program is that it automatically calculates the optimum recording parameters for both astigmatism and coma corrections and then displays complete 

data for each aberration correction. This featllre was incliided in the program so that the 

program user inay select which aberration correction will best provide the optimum pert'ornrance for all aberrations (astigmatism, coma, and spherical) throughout the desired 

wavelength range. 

DESCRIPTION OF COMPUTER PROGRAM 

The previous section gave the logical sequelice for setting up the computer program. This 

section describes the computer program which consists of a main program and two subroutines. 

The main program directs thz tlow of calculations and calls upon the subroutines to perform thcir various functions. The flow diagram (Figure 2) shows that the following events 

will be ~ performed. 

1. Input data cards are read. 

2. Call CORECT. CORECT is a subroutine that determines the optimum recording 

parJmetzr values for 7 and 6 such that astigmatism is corrected to zero at A*. This 

is achieved by simultaneous solution of equatioiis (1 2), (7), and (10). Then this subroutine also automatically determines the optimum recording parsmeter values for 7 

7 
3. 

4. 

and 6 to correct coma to zero at A*. This task is accomplished by simultaneous solution of equations (1 2), (7), and (1 l). 

Call PRFORM. PRFORM is a subroutine that calculates the numerical values for all 

aberrations as a function of wavelzngth for both holographic and conventionid gratings. 

Output. This function displays the numerical values of the input data, recording 

parameters, and all aberrations for the holographic and conventional gratings as a function of wavelength in the form of a table. 

MAIN PROGRAM FLOW DIAGRAM 

START i-1 

~~ ~ 

CALL CORECT 

Determines values for correcting astigmatism ad coma. 

CALL PRFORM 

Calculates values for all aberrations as a function 

of wavelength. 

1 OUTPUT Displays complete data in the form of a table 

Figure 2. Main Program Flow Diagram. 

8 
' INPUT DATA, DEFlNlTlON OF TERMS AND DATA CARDS 

This section describes the input data and defines the pardmeters that will be punched 

on the data cards so that successful operation of the program will be achieved. 

DATA SET 

The data set consist of 11 input parameters that must be properly presented on the 

data cards in order to make proper use of the program. The parameters are defined 

as follows: 

1. R = radius of curvature in mm. 

2. K = order of diffraction, where K = i1 , *2, etc. The choice of pegative or positive orders is at the discretion of the program user. 

3. N = groove frequency in grooves/mm. 

4. X, = laser recording wavelength in angstroms. 

5. X* = correcting wavelength in angstroms. 

6. XL = lower wavelength limit in angstroms for displaying the performance of the 

grating. 

7. AX = wavelength interval in angstroms for displaying the perforniance of the 

grd ting. 

8. Xu = upper wavelength limit in angstroms for displaying the performance of the 

grating. 

9. CYL = lower angle of incidence in degrees decimal form. 

10. cuu = upper angle of incidence in degrees decimal form. 

11. Acu = hicremcntzd angle of incidence in degrees decimal form. 

cu~, cuu, and Acu require some explanation. The incremental angle of incidence Acu 

provides the program user the opportunity to obtain optimum recording parameters for 

successive angles of incidence at specified increments without punching data card sets 

for each desired angle of incidence, For example; if cuy~ and cuu are given respective 

input values 0.0" and 10.0" and Au = 5.0', then optinrum recording parameters will be 

obtained Cor OL = O.Oo, 5.0°, and 10.0'. Complete data will be displayed for each of 

the three angles of incidence. If the proyam user selects one input angle of incidence, 

then cu~ and cuu have the same input value and ACY = 0.0". 

9 
DATACARDS 

The order of the first eight input parameters must be punched in the order shown in the 

DATA SET section starting in column 1 in format F7.1 with seven columns reserved for 

each of the eleven parameters on the first data card and each parameter must include a 

dccimal. The parameters qy cru, and ALY are punched on the second data card in that 

order with seven columns reserved for each of these three parameters in F7.1 format and 

each parimeter must include a decimal. Figure 3 sliows a typical two card data set to 

illustrate the description stated above. 

Figure 3. 

I 

NOTE: 1. The program user niay include as many data sets as required in a single run 

of the program. 

2. The values of all 11 input parameters must be punchatl on all data card sets. 

10 
. 

PROGRAM EXAMPLE AND DATA DISPLAY 

This section presents a typical example of the use of the program and also describes the output data in detail. 

Example: Determine the optimum recording parameters 7 and 6 such that astigmatism or 

coma is made zero at 2500A for angles of incidence CY = 0.0" to 25.0" in the incremental 

angles of 2.0". The grating has 1200 L/mm with a radius of curvature of 1000 mm and will 

operate in the Is' order. The wavelength performance range is from 1 SOOA to 4000A in 

5OOA intervals. 

Solution: A data card set is punched using the data supplied in the example and punched 

in the order listed as follows: 

First Card 

1. K = 1000.0 mm 

2. E; = +1.0 

3. N = 1200 grooves/mm 

4. x, = 4579.3A 

5. A* = 2500.OA 

6. XL = I500.0A 

7. Ah = 500.0A 

8. xu = 4000.0A 

Second Card 

9. CYL = 0.0" 

IO. CYU = 25.0" 

11. Aa = 2.0" 

The output data for the example cited is shown in Figure 4. Output data for only one 

of the angles of incidence will be presented for the purpose of illustration. 

A detailed description of the output data for correcting astigmatisni or coma at 2500A 

is as follows: 

11 
I 

i 

i 

I 

I 

! 

I 

i 

1: 

1 

I 

I 

1 I

! 

i 

I 

i 

! 

I 

! 

I 

i 

I 

i 

I 

I I' I 

12 
INI 

OlOO 

I 

I 

I 

I 

I 

I 

I 

! 

I 

I 

i 

la 

/I 

jj 

ii 

i 

I 

I 

I 

i 

I 

I i

i I: 

! 

I 

(Y 

ul 

LY 

u 

" 

i" 

h 

5 

E 

E 

0 

..-( CI 

2 

13 
Line 1 is self explanatory and refers to the next two lines. 

Line 2 shows the equations for calculating LA, LB, LC and LD. 

Line 3 shows the radius of curvature R, recording wavelength X, = RLAM, grating order 

K, and the groove frequency N. 

Line 4 shows that astigmatism is being corrected and is indicated by option = 1. Option 

= 1 is an internal part of the program and is not an input parameter. 

Line 5 sliows the number of solutions available at the correcting wavelength X*. 

Line 6 indicates that tlie following data refers to the first solution. 

Lint: 7 sliows the values of the correcting wavelength X* = LAMDA, angle of incidence 

(Y = ALFA, diffracted inrage angle p = BETA, recording angle 7 = GAMA and recorditig angle 6 = DELTA. 

Line 8 shows the v;rlues of the aberrations corrected at X*. Note the value for astigmatism A, - is essentially zero. 

Line 9 shows the values of the aberrations for the conventional equivalent grating at X*. 

Notice how much greater the value of astigmatism is at X* compared to the value 

for the holographic grating in Line 8. 

Line 10 indicates that the data in Lines 11 and 12 refer to tlie holographic grating. 

Line 11 shows the titles for the performance of the holographic grating. LAMDA is the 

wavelength at which the grating is evaluated relative to the aberrations. BETA is 

the diffraction angle. The aberration titles AI, A2, C1, C2, S1 , S2 and S3 have 

been defined in the text. 

Line 12 sliows the values of the wavelength, diffraction angle, and aberrations for the 

performance of the grating. Note the values of tlie aberrations at 2500A and compare with Line 8. 

Line 13 indicates that the table that follows refers to the performance of an equivalent 

cotiveii t ional grating. 

Lines 

Lines 

4 and I5 refer to the conventional grating. Compare the values of this table with 

the values in the holographic grating performance table. Note the reduction in 

thc aberration values for the holographic grating. 

6 indicates that the following data is for correcting coma and is indicated by option = 2. Option = 2 is inherent to the program and is not an input parameter. 

14 
Line 17 indicates that for the angle of incidence shown in Line 19, there are two solutions for correcting coma. 

Line 18 shows that the first solution will be evaluated for coma. The output data for 

the second solution is not shown. 

Lines 19 through 27 are defined in the same way as Lines 6 through 15 except that they 

refer to coma. 

REFERENCES 

Pieuchard, G. and J. Flamand, Goddard Space Flight Center Report, Contract No. NASW 

2146, March, 1972. 
SIMULTANEOUS CORRECTION OF ASTIGMATISM, COMA, AND 

FIRST SPHERICAL ABERRATION TERM OF HOLOGRAPHIC 

CONCAVE GRATINGS FOR ROWLAND CIRCLE 

SPECTROGRAPHS 
SIMULTANEOUS CORRECTION OF ASTIGMATISM, COMA, AND 

FIRST SPHERICAL ABERRATION TERM OF HOLOGRAPHIC CONCAVE 

GRATINGS FOR A ROWLAND CIRCLE SPECTROGRAPH 

INTRODUCTION 

When designing spectroscopic instruments such as spectrographs, monochromators, spectrophotometers and other instruments employing concave optical reflecting diffraction 

gratings, one of the most important optical design criteria is to design the optical system 

such tliat optical aberrations are reduced or eliminated over the desired wavelength range 

of operation. The most notorious aberration is normally astigmatism because the diffracted images are magnified in the vertical direction. As a result, large area detectors are 

required to sense all thc energy in the diffracted iniage or, as is the usual case, available 

small area detectors are used at the expense of losing a great deal of energy in the diffracted image. 

However , with the advent of aberration corrected concave holographic optical reflecting 

gratings, astigmat'ism and other aberrations have been reduced or eliniinated at specified 

wavelengths. Recent publications have shown that for certain spectroscopic instrument 

designs irrcorporating corrective type hologrqhic gratings either astigmatism or coma may 

be minimized over a desired wavelength range or astigmatism or coma may be eliminated 

at specified wavelengths.( s293~4) However, the aberrations can not be corrected siniultancously at specified waveicngths. The purpose of this paper is to present the theory and 

basic equations that lead to the design of concave holographic diffraction gratings for 

Rowland circle spcctrogaphs tliat do eliminate astigmatism, coma, and the first spherical 

aberration term simultaneously at specified wavelengths. However, the simultaneous elimination or correction to zero of these aberrations occurs at wavelengths that are greater 

than or equal to the laser recording wavclength. 

A coqiputer progani was also developed to numerically analyze the theory, determine 

the optimum instrunient performance and recording parameters necessary for grating 

fabrication. The program also displays the performance, for comparison, of both the 

holographic grating and equivalent conventional gating as a function of wavelength. 

THEORY AND BASIC EQUATIONS FOR DESIGNING HOLOGRAPHIC SPHERICAL 

CONCAVE GRATlNGS 

Figure 1 shows the optical system for using and recording a concave spherical holographic 

gating. The rectangular coordinate system is defined as follows: I is the origin at the 

vertex of the grating blank and the x-axis is normal to I. The x-y plane is defined by I 

and the two recording point sources C and D. Points A, My and B are, respectively, a 

self luminous point on the entrance slit, a point on the grating surface, and a point on 

the diffracted image from M at a wavelength X in the nth order. A knowledge of the 

1 
.- 

I 

(GRATING BLANK) 

X 

Figure 1. Schematic diagram of the optical system. A is a self luminous point on the 

cntrancc: slit, M is a point on the nth groove and B is the spectral image point. C and 

D are the recording point light source positions. The performance and recording paranictcrs are defined as follows: 

Performance Parameters: 

LA is the distance from A (entrance slit) to the center of I of the grating. 

LB is the distance from the diffracted image B to the center I of the grating. 

CY is the angle of incidence. 

/3 is the diffraction angle. 

z is the object height. 

z’ is the image height. 

Rccording Panme t ers: 

LC is the distance from the point source C to the center I of the grating. 

LD is the distance from the point source D to the center I of the grating. 

7 is the position angle for point source C relative to the normal to the grating. 

6 is the position angle for point source D relative to the normal to the grating. 

2 
geometry of the optical system, Figure I, leads to writing the aberrant optical path function AM for the conventional grating as 

(1) AM = AM+ MB -(IA+ IB) - KNh 

where K is the order of diffraction, X is the diffracted wavelength and N is the gating 

groove frequency. 

When a grating blank coated with a photoresist is illuminated by coherent spherical wave 

fronts originating from points C and D, grooves are recorded, after development, in the 

photoresist in response to the interference fringes produced at the site of the gating 

blank: The interference fringes are formed according to the equation 

MC - MD -(IC - ID) N= 

LO 

where Xo is the recording laser wavelength. 

Substituting eq. (2) into eq. (1) gives the aberrant optical path function 

AM = MA + MB -(IA+ 1B) -g [(MC -MD) - (IC -ID)} x, (3) 

for the holographic gating. 

The quantities on the right side of AM in equation (3) are derived based on the coordinate 

system described in Figure 1 and the equation of a sphere. Since these derivations are 

lengthy, we suggest the reader see reference 4 for a more detailed analysis. Therefore it 

suffices to say that after a series expansion is performed, the following result for AM is 

obtained. 

+YW [s2i +24 [s3] +... 

where Y is the width and Z is the height of the grating and 

3 
, 

4 
. 

Note, that in the expressions for equation (5), LA, LB, cy, and 0 are the terms for the 

conventioiially ruled griating and Lc, LD, 7, and 6 are the terms related to the holographic contribution. R is the radius of curvature of the grating and K is the order. A, is 

tlit' recording laser waveleilgth. The expressions in equation (5) are defined as follows: 

A1 is the horizontal focusing term, A2 is the astigmatism or vertical focusing term, C1 

and C? are coma terms and SI, S2, 53 are spherical aberration terms. 

Tlie gating equation is given by 

u (sin Q + sin p) = KX ( 6) 

and for the holographic gating, 0 is the effective grating constant defined by 

where y > 6 x, 

sin y - sin S u= (7) 

The above description presents the general theory and basic equations for designing aberration corrected holographic gratings. We will now apply the theory to design a hologaphic grating that simultaneously corrects astigmatism, coma, and the first spherical 

aberration term at a specified wavelength. 

HOLOGRAPHIC GRATING DESIGN FOR ROWLAND CIRCLE SPECTROGRAPH 

A1 and C1 in equation (5) must be made zero in order to design a holographic grating 

tliat has the Rowland circle as its focal curve. The solution for this condition is: 

LA = R cosa, LB = R cos& Lc = R cosyand LD = R cos 6 (8) 

Substitution of equation (8) into the aberrant optical path function reduces equation (4) 

to 

where the expressions in equation (5) take the form 

5 
sin?. 'Y sin2 6 

cosa* cos@* cos 6 

SI' = - 

sin2 ot sin2 @* 

s3' = - (2 cos*a* - 1) + - (2 cos2 p* - 1) COS^ 01 COS^ p* 

sin2 6 (2 cos2 7- 1) - - (2 cos2 6 - 1) COS^ 6 

A* is the wavelength at which the aberrations are corrected to zero and p* is calculated from 

u(sina*+sinp*) = KX* (1 1) 

1 

Note that the groove frequency of the grating is given hy N = - . 0 

SIMULTANEOUS CORRECTION OF ASTIGMATISM, COMA, AND FIRST SPHERICAL 

ABERRATION TERM AT A* 

The simultaneous elimination of astigmatism, coma, and first spherical aberration term at 

A* depends on the instrument and recording parameters. For given instrumental parameters o and A* there are specific angles of incidence a* for which the recording angles 7 

arid 6 are chosen such that astigmatism, coma, and first spherical aberration term are 

sitnultaneously made zero. These aberration corrections are accomplislied by simultaneously solving the following set of equations: 

u (sin a* + sin @*) = KX*, (1 1) 

A0 

siny-sin 6 o= 

6 

(7) 
and A2' = C2' = 0 in equation ( 10). Since the expression fsr SI' is identical to Az', the first 

splierical aberration term SI is autoinatically zero. A numerical analysis of the above tJieory 

shows that solutions exist only for X*> 1,. The numerical analysis also shows that there 

;ire one or more angles of incidence a* for each correcting wavelength A*, depending upon 

groove frequency for which A2' = C2' = SI' = 0. Figure 2 shows some of a family of 

curves which for a given groove frequency and correcting X*, the angles of incidence a* 

and the recording angle 6 may be obtained such that A2' = C2' = SI' = 0. The groove 

frequency is shown for each curve and the correcting wavelengths are indicated at different points along that curve. For example, at a. groove frequency of 1200 L/mm and a 

correcting wavelength at 5000 A, there are three angles of incidence (5.07', 30.77' and 

43.16') and two corresponding recording angles S(-2.83' and 7.01') where the aberrations 

may be simultaneously corrected. It is interesting to note that as the grating groove frequcncy increases, the number of incident angles a* and range of correcting wavelength 

decreases. 

The diffracted angle p* and other recording angle 7 are calculated by equations (1 1) and 

(7) respectively. After the instrument and recording angles have beeq determined, the 

reniiining instrument parameter values LA, LB, and recording parameter values Lc, LD 

are calculated using expressions in equation (8) along with the value of the radius curvature R. 

COMPUTER PROGRAM 

In the first part of this paper, we have presented the theory and basic equations oecessary 

to design holographic gratings that are aberration corrected at specified wavelengths at or 

above the laser recording wavelength for Rowland circle spectrographiq mounts. We have 

also presented data (Figure 2) that shows the different angles of incidence that could be 

used to design an optical system based upon a specitied groove frequency anci correc;iiug 

wavelength that eliminates astigmatism, coma, and first spherical aberration term sirnult~neously. Now we present the computer progrqm that made it possible to numerically 

analyze the theory and calculate the optimum performance and grating fabrication parameters. T,he program also displays the performance of the holographic grating and equivalent conventional grating relative to all the aberrations as a function of wavelength. 

DESCRIPTION OF COMPUTER PROGRAM 

The computer program consists of a main program and four subroutines. The qain program djrects the flow of calculations and calls upon the subroutines to perform their 

various functions. The flow diagram (Figure 3) shows the events that will be performed. 

The algorithm may be summarized as follows: Assume that the radius of curvature, growe 

frequency, and correcting wavelength has been selected, then an angle of incidence "1 

0,Q' is chosen and the corresponding diffracted angle fl is calculated. Based on the above 

7 
8 
CALFA2 

CALL COMA 

CALL PRFQRM 

Figure 3. Program Flow Diagram. 

9 
parameters, the recording parameters 71 and 61 are determined such that A2 = 0. The 

above procedure is repeated at another angle of incidence c9, which is determined by an 

cquation of the form "2 = cy1 + h; where cy1 is the previous angle of incidence and Aa 

is a very sniall incremental angle. At this point, there are two sets of parameters, CUI, 01, 

71, 61, and "2, &, 72, 62 for which A2 = 0. Note that A2 can be made zero at any 

angle of incidence. For these two sets of parameters, two coma values for C2 corresponding to C2(crl) and C2(q) are calculated. If the values of the C2 terms have opposite signs, 

then there exists a value a between a1 and cy2 for which A;! = C2 = 0 for the same values 

of u, 0, y, and 6. The values of 01 and associated parameters are determined by a modification of NewioIl's method. 

However, if C2(crl) and C2(a2) have the same sign then there isn't any value of a between 

cy1 and cy2 for which A2 = Cz = 0. When this happens, we set a3 = 42 + Ao~ and calculate 

the corresponding angles p3, 73, and 63 such that A2(a3) = 0. Then Cz(a3) is calculated 

and the sign is compared with C2(a2). Again, if the signs are opposite there is another 

value of a between (112 and a3 such that A2 = C2 = 0. This process is repeated until all 

angles of incidence between 00 and 90" have been evaluated and only those angles of incidence are selected which satisfy the condition that for a certain cy there is a p, 7, and 6 

that produces A2 = C;! = 0 for a specified groove frequency, radius of curvature and correcting wavelength. Since the equations for A2 and SI are identical, then S1 is also zero. 

The algoritlun is supported by four subroutines. Subroutine ASTGSM calculates the 

pmrncters required to make astigmatism zero at angles of incidence a. 

Subroutine FUNCAC calculates the values of the coma terms based on the parameters 

used to inake A2 = 0. Then it compares the signs of the coma terms; if opposite this 

subroutine will determine the angle of incidence and related parameters that make coma 

zero. If tltc signs are the same, it returns the problem to subroutine ASTGSM and the 

process is repeated. 

Subroutine Coma calculates the values of ai, p, 7 and 6 such that C2 = 0, when A;! = 0 

for the sinie set of parameters. 

Subroutine PRFORM calculates and displays in the form of a table the numerical values 

of each abcrration for the hologaphic grating and equivalent conventional grating as a 

function of wavelength. 

INPUT DATA, DEFINITION OF TERMS AND DATA CARDS 

This section describes the input data and defines the parmeters that will be punched on 

the data cards so that successful operation of the program will be achieved. 

(A) DATA SET 

The data set consists of 12 parameters that must be properly presented on the data cards in 

order to successfully use the computer program. These parameters are defined as follows: 

10 
1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

(B) 

R = radius of curvature in mm. 

N = groove frequency in grooves per mm. 

A, = laser recording wavelength in angstroms. 

X* = correcting wavelength in angstroms. 

XL = lower wavelength limit in angstroms for displqying the performance of the 

gating. 

Xu = upper wavelength limit in angstroms for displaying the perfomance of the 

grating. 

AX = wavelength interval in angstroms for displaying the performance of the grating. 

K = order of diffraction, where K = *l, k2, etc. The choice of negative or positive orders is at the discretion of the program user. 

Y = grating width in mm. 

2 =, gating height in mm. 

Aa = incremental angle in degrees decimal form. The smaller the incremental angle, 

the more accurate the successive values of a. 

IOPTION = Selects whether or not the coefficients Y and Z should be included in 

the aberration values. If IOPTION = 1 is selected, then the aberrations are calculated withniit the rncfficlentscalculated include the coefficients. 

If !W!’!CN = 2 is BP!Pc!PQ, the2 the zk.?sfic:.?: 

DESCRIPTION OF DATA CARDS AND SAMPLE DATA CARDS 

Tlie order of the first eight input parameters must be punched in the order shown in 

(A) starting in column 1 in F1O.l format with 10 columns reserved for each of the 

scven parameters and each parameter must include a decimal. The eighth parameter K 

is in I2 format and punched in columns 71, 72. The next four parameters are punched 

on the second card where parameters Y, Z and Aa are format FIO.l and IOPTION is 

in I2 format. Figure 4 shows a typical two card data set to illustrate the description 

stated above. 

11 
1000.0 1200.0 4579.3 5000.0 9Ooo.o 4500.0 600.0 1.0 \ 

I 

Y z IOPTION 

/ 2 

Figure 4. 

NOTE: 1. The progam user may include as many data sets as required in a single 

run of the program. 

2. The values of all 12 input parameters must be punched on all data card 

SI: ts. 

12 
EXAMPLE AND DESCRIPTION OF DATA 

A typical example of the program and,detailed description of the output data is as follows: 

Example: Design a holographic diffraction grating that simultaneously corrects for astigmatism, coma, and the first spherical aberration term at 5000 A for a Rowland circle 

spectrograph. The grating has 1200 L/mm and a 1.0 meter radius of curvature. Determine the performance of the holographic grating and equivalent conventional gating 

throughout the wavelength range 4500 A to 9000 A at 500 A intervals. 

Solution: Data cards are punched using the data supplied in the example and punched 

in the order listed below: 

First Card 

1. R= 1OOOmm 

2. N = 1200 L/mm 

3. A, = 4579.3 R 

4. A, = 5000 A 

5. xu = 9000 A 

6. AL =4500 A 

7. AA= 500A 

8. K = +1.0 

Second Card 

9. Y = 1.0 

10. z= 1.0 

11. Aa = 0.5" 

12. IOPTION = 1 

The output data for the example cited is shown in Figure 5. Output data for only one 

of the angles of incidence will be presented for the purpose of illustration. 

A detailed description of the output data for simultaneously correcting astigmatism, coma, 

and the first spherical aberration term at 5000 A is as follows: 

I 13 
* 

0 * 

t * 

t * 

0 

0 

4 

Y 

4 

* 

t 

t 

Y * 

t 

U 

t 

Y 

t 

t 

U 

it 

t 

c 

0 

0 

0 

Y * 

Y 

4 

a *. *. *. *. *. 

0. 

e. *. 

., 

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14 
Line I: 

Line 2: 

Line 3: 

Line 4: 

I 

Line 5: 

Line 6: 

Line 7: 

Line 8: 

Shows the correcting wavelength LAMDA = A,, angle of incidence ALFA = a, 

diffracted wavelength BETA = p, recording angle GAMA = y and recording 

angle DELTA = 6. 

Self-explanatory. 

Sliows the symbols for the wavelength and each aberration. The symbols for 

the aberrations have been defined in the text. 

Shows the correcting wavelength and values for each aberration at the correcting 

wavelength. 

Self-explanatory. 

Indicates that the holographic grating has been evaluated. 

Shows the titles of wavelength = LAMDA and the aberrations AI, A2, C1, C2, 

SI, S2 and S3. The table shows the values OF each aberration as a function of 

wavelengt h. 

Indicates that Line 9 and the corresponding table are for the evalpation of the 

equivalent conventional grating. The equivalent conventional grating is defined 

as laving the same groove frequency and radius of curvature as the holographic 

gating. However, the conventional grating cannot correct aberrations. Note 

the reduction in aberration values as a function of wavelength as compared to 

the conventional grating. For example, at 5000 A the value of astigmatism A2 

for the holographic grating is 0.3 16973 x and that for the conventional 

grating is 0.31292. This shows that astigmatism is significantly reduced by a 

f;iitGi of !04 fer !he hc!cgrnphic grating at 5000 A. Note that C2 and SI are 

also reduced by the same order of magnitude. The remaining spherical aberration terms are also reduced by a factor of 102 for the same wavelength. Also 

note that the aberrations increase for dzcicasifig and Increasing wavelengths 

about the correcting wavelength at 5000 A. However, the increase in aberrations remains less than that of the equivalent conventional grating. 

In summary, the theory and basic equations have been presented to design concave spherical holographic reflecting diffraction gratings that are corrected to zero at specified wavelengths at or above the laser recording wavelength. A numerical analysis shows that there 

are specific angles of incidence based on the selected groove frequency and correcting 

wavelength that simultaneously corrects the aberrations A2, C;! and SI. A computer program has been developed so that application of the theory may be applied to practical 

spectroscopic instrument design requirements for space and ground based laboratory 

applications. 

15 
' 

REFERENCES 

1. 

2. 

3. 

4. 

Computer Program For Designing Holograpluc Gratings For Seya-Namioka Type 

Monochromators. Part 1 : Minimization of Aberrations. Over a Desired Wavelength 

Kange., A. J. Caruso and M. S. Bhatia, GSFC X-720-77-79, Nov. 1976. 

Computer Program For Designing Holographic Gratings For Seya-Namioka Type 

Monochromators.. Part 11: Correcting Aberrations at Specified Wavelengths., A. J. 

Caruso and M. S. Bhatia, GSFC X-720-77-145, May 1977. 

Dcsign of Astigmatism or Coma Corrected Holographic Gratings for Rowland Circle 

Spectrographic Mounts, A. J. Caruso and M. S. Bliatia, GSFC X-722-78-3, February 

1978. 

Goddard Space Flight Center Report, Contract No. NASW 2146, March 1972. 

16 
DESIGN OF ABERRATION CORRECTED FIOLOGRAPHIC TOROID 

GRATINGS FOR SGYA-NAMIOKA TYPE MONOCHROMATORS 
CONTENTS 

Page 

Introduction ................................................................ 1 

Section 1 ................................................................. 2 

Theory and Basic Equations for Designing Holographic 

Torodial Concave Gratings ............................................ 2 

Minimization of Aberrations .......................................... 7 

Determination of Instrumental Parametersp. p'. K. q and AZoo ............... 9 

Determination of the Recording Parameters pc. p~ . y. and 6 and 

10 

ModifiedMethod ................................................... 12 

Minimization of Astigmatism and Coma .................................. 

Section 2 ................................................................. 13 

Program Description ................................................. 13 

Subroutine INSTRM ................................................ 14 

Subroutine ASTGMS ................................................ 14 

SubroutineCOMA .................................................. 14 

SubroutinePRFkOM ................................................ 16 

Section 3 Data Set and Description of Data Cards .................................. 16 

(A)DataSet ....................................................... 16 

(B) Description of Data Cards and Sample Data Cards ....................... 17 

Section 4 Program Example ................................................... 17 

References ................................................................. 22 

iii 
, 

ILLUSTRATIONS 

Page 

Figure 

1 Schematic diagram of a toroid ............................................. 2 

2 Schematic diagram showing the geometry for recording the holographic grating ....... 4 

3 Schematic diagram of the optical system for Seya-Namioka Monochromator ......... 8 

4 Main Program Flow Diagram .............................................. IS 

5 Typical Data Card Set ................................................... 18 

6 DataDisplay .......................................................... 21 

iv 
DESIGN OF ABERRATION CORRECTED HOLOGRAPHIC TOROID 

GRATINGS FOR SEYA-NAMIOKA TYPE MONOCHROMATORS 

Part I: Minimization of Aberrations For 

Desired Wavelength Range 

A. J. Caruso 

h1.S. Bhatia 

Seya-Naniioka type monochromators are designed such that wavelength selection is obtained by 

rotating a concave reflecting grating about its vertical axis. Diffracted image focussing is achieved 

by one optical element; nornially a concave spherical reflecting diffraction grating. The angle of 

deviation, defined as the angle between the entrance and exit slits relative to the vertex of the 

grating, is a fixed value, normally 70'. At this fixed value most of the optical aberrations are 

reasonably acceptable except for severe astigniatisni which is caused by the mechanically ruled 

concave spherical grating. With the advent of ruling holographic aberration correctcd concave 

spherical reflecting diffraction gratings, it has become possible to design Seya-Namioka type monochroinators such that astigniatism or coma may be corrected to zero at specified wavelengths or 

minimized over a desired wavelength range (ref, 1, 2, 3, 6). These holographic grating designs 

have resulted in a significant improvement for the output efficiency of Seya-Naniioka monochromators for wavelengths longer than 100 nm. However, for spherical concave gratings with angles 

of deviation greater than 70' holographic grating theory shows that aberration correction cannot 

be achieved for wavelengths shorter than 80 nm. Further, at 70' angle of deviation the output 

efficiency is also decreased because grating coating materials reflectivity is significantly reduced 

at wavelengths below 10.5 nm. 

In order to over come this problem, we have substituted the toroid geometry for the grating 

h!:lnk iI! place nf the concave spherical geometry. (reference 5) The major advantage of the toroid 

geometry is that astigniatism can be eliminated at specified wavelengths and also dr;imatically minimized over a wide wavelength range by properly determining the correct toroid vertical radius of 

curvature and horizontal radius of curvature as a function of the angle of incidence and the required 

holographic ruling geometry. Another advantage is that at larger incident angles the reflectivity 

of the same grating coating material increases at wavelengths less than 10.5 nm. 

The purpose of this report is to present the theory and design equations for holographically ruled 

toroidal diffraction gratings that minimize aberrations for Seya-Namioka type monochromators 

with angles of deviation larger than or equal to the normal 70" for wavelengths shorter than 100 

nm. 

This report is divided into four sections. Section 1 presents the theory and design equations for 

designing aberration corrected holographic toroidal gratings. Two methods have been developed: 

the general method requires that the instrumental parameters be related only to the holographic 

toroid grating design. 

that the instrumental 

The second method is called the modified method. This method requires . 

parameters be the same for both the conventional and holographic toroidal 

1 
I 

gratings. The modified method therefore designs such that the holographically ruled toroidal 

grating is interchangeable with the conventionally ruled grating providing they have the same 

groove frequency, and radius of curvatures R and pt. 

Section 2 presents a description of the computer program. The program has been developed such 

that for a given groove frequency and wavelength range, the optimum instrumental and holographic 

ruling parameters are determined for the toroidal concave grating. The program also displays the 

performance of the holographic toroidal grating and equivalent conventional toroidal grating, as 

a function of wavelength relative to optical aberrations such as astigmatism, coma, and spherical 

aberrations. Section 3 presents the input data, definition of terms and data card description. 

Section 4 presents an example for using the program and describes in detail the output data. 

Scction 1 

Theory and Basic Equations for Designing Holographic Toroidal Concave Gratings 

A toroid is produced when a circle is rotated about an axis which lies in the plane of the circle but 

does not pass through the midpoint of the circle (Fig. 1). Figure (1) also shows a scclion (toroidal 

grating blank) cut out of the toroid for which there are two radii of curvatures: a long horizontal 

I 

Figure 1. Schematic diagram of a toroid. R and pt are the radii of the toroidal surface. Cut 

out section is typical of a toroid grating blank 

2 
. radius K and a short vertical radius pt. If pt = R, then a spherical concave surface will be generated; ie, when the axis of rotation passes through the midpoint of the circle. As a result, a11 inferences which are valid for the toroid are also valid for the sphere when pt = R. 

We now proceed with the theory and basic equations for designing holographic toroidal gratings 

with the above concept in mind. 

Figure (2) shows a toroidal grating blank in an optical schematic diagram which defines a rectangular coordinate system. Let the origin be at vertex 0 of the concave toroid grating blank which 

has a horizontal radius of ciirvature R 3nd a vertical radius of curvature pt. Let the x-axis be defined by 0 and two coherent point light sources C and D which are used to produce the interference fringes on the concave toroidal blank surface. Points A, P, and B are self luminus points 

on the entrance slit, a point on the grating and a point at the focused diffracted image from P at 

wavelength h in the m* order respectively. A knowledge of the geonietrical optical system of 

Figure 2 leads to writing the abcrrant optical light path function F for the conventional grating 

;IS 

F = AP + PB + nin h (1) 

where n is the groove frequency, m is the diffracted order and h is the diffracted wavelength. 

Whcn a grating substrate coated with a photoresist layer is illuminated by coherent spherical wavefronts originating from points C and D, grooves are recorded and formed after development in 

the photoresist in response to the interference fringes produced at the site of the grating surface. 

The interference fringes are formed according to equation 

where n is the groove frequency and A, is the laser wavelength radiation. 

Substituting equation (2) into equation (1) gives the aberrant optical light path function 

mh 

F = AP t PB +- ((CP - DP) - (CO - DO)] A0 

(3) 

for the holographic grating. The terms on the right side of F in equation (3) are derived bqsed on 

the coordinate system described in Figure (2). By substituting the trignometric terms for the 

toroid and other required terms, in equation (31, the following aberrant optical light path function 

F is deriJed after a power series expansion is performed. 

F=Fooo +WFloo +LFoll +!hW2F,oo +!hL2F02, 

-. . .. : 

3 
. I 

EXIT SLIT 

IMAGE PLANE 

B 

I’ligiire 2. Schematic diagram showing the geometry for recording the holographic grating. The 

instrumental and recording parameters are defined as follows: 

Instrument Parameters 

ro 

rb 

a 

0 

A 

is the object distance from the enirance 

slit to the center of the grating. 

is the imagc distance from the exit sli! to 

tlir ccnier of the grating. 

is the angle of iiicident radiation relative 

to thc normal of the grating. 

is the diffracted image angle relative to 

the normal of the grating. 

is the entrance slit position and E is the 

exit slit position. 

Recording Parameters 

rc 

rD 

6 

7 

is the distance of recording point source C to 

the center of the grating. 

is the distance of the recording point source D 

to the center of the grating. 

is the recording angle for positioning point 

source D relative to the norinal of the grating. 

is the recording angle for positioning point 

source C relative to the normal of the grating. 

4 
- and 

where the subscripts ijk Of Fijk are exponents of WiIJzk except for FOG,. 

The terms Mijk and H,,I; are defined in terms of the function fijk as follows 

for ijk = (200) and (300) 

and 

for ijk = (020) and ( 120) 

and 

where the expressions for the conventional toroid grating are 

f200 (p, a) + f(p', 0) = cow (pcos Cw-1) + cosp (p'cosp-1) 

fao0 (p, a) + fsoo (p', P) = p - corn-pcow (pcosarl) [cow(pcoscrl) - 4 psin2al (8e) 
where 

6 
R 

Pt 

and q=- where R is the horizontal radius and pt is the vertical radius of the toroid. 

The term F, oo in equation 4 is the well known grating equation and is given as 

mh = (I (sin a + sin p) ( 10) 

for the principal ray, the ray originating from the center of the entrance slit and diffracted from 

0 at an angle 0. In order to holographically rule the grating with the required groove frequency, 

the effective grating constant is defined as 

ho 

sin 6 - sin 7 

a= 

where 6 > 7. 7 and 6 are the recording angles. 

Note that the number of grooves per unit length is n =-. 

The individual expressions for equations (8a) thru (8n) have physical significance as to the formation and imperfections of the diffracted image. The expressions control the various conditions 

for the foci and image deficiencies in the following order; 8a and 8h govern the primary focusing 

condition on the Rowland Circle, 8b and 8i determine the amount of astigmatism relative to the 

diffracted image, 8c, 8j, 8d, and 8k Coma, 8e and 81 higher order focusing condition on the Rowland Circle and 8f, 8g, 8m and 8n the toric aberrations. It is interesting to note that when one 

compares these expressions of the toroid with those for the equation of a sphere Ref (I), there 

are some expressions that are identical for both. (toroid & sphere). For example, all the expressions 

which do not include q are identical with those expressions for the sphere. It is only those expressions that involve q that account for the difference between the two gratings. Note that the 

difference between the two gratings vanishes when pt = R. 

1 

a 

Minimization of Aberrations 

The theory described up to this point has been presented for the general case. We now apply the 

general theory to a specific application: The Seya-Namioka monochromator. This type of monochromator has a single reflecting and wavelength dispersing element; the grating. Wavelength sclection is achieved by rotating the grating about the vertical axis at the front surface of the center of 

the grating. The basic instrumental parameters for this type monochromator are defined as follows; 

see fig 3. 

To, I$, = constant, 2 K = (Y, - 0, = constant 

a0 =K + 6,& = 6 -K 

wherc ro is the distance from the center of the extrance slit to 0 and 5 is the distance from 0 to 

the center of the exit slit. 2K is the angle of deviation AOB and 6 is the angle of grating rotation 

measured from the bisector of the angle 2K and has the same sign as the spectral order m. 

7 
\ 

TOROID 

CONCAVE 

GRATING 

a f 

/ SLIT 7 

Y X (GRATING NORMAL) 

Figure 3. Schematic diagram of the optical system for Seya-Nanlioka monochromator 

Undcr conditions (1 2), the relation between 8 and h is given by 

h =(:)cos k sin 0 

In order to reduce the aberrations in the Seya-Namioka monochromator, we denote the fiinction 

Fijk for the ray AOB by FiJK. We then minimize the fuiiction 

where i + j + k > 2 over a predetermined scanning range, 8 G 0 Q 82 or A, < X < X2 \vhere 

8 
and 8, and O2 are related to XI and X2 through equation ( 13). This is equivalent to imposing on 

Fil K the condition 

- 

e81 

Iijk =I ' F2 1J k d8 =minimum (17) 

Determination of Instrumental Parameters 9, F, K, q and Azoo 

The optimum instrumental parameters are determined in such a way that equation (1 7) is satisfied 

for the horizontal focusing conditi~n~~~~. This is accomplished by solving the following equations simultaneously, 

1 

With the values of p7 ir' and K determined, we proceed to determine the value of q, where q =- P' 

P has been derived by Greiner and Schaffer (ref 4) as 

2 cos K (sin (6, -8 )) p t. 

(P-+b)(62 -61) 

where K is the half angle between the entrance and exit slits relative to the vertex cf the grating, 

O2 and 8, are the limits of the angular rotation of the grating and 

q is a very important parameter in that it determines the optimum geometry of the toroid grating 

for the desired wavelength range. Since R is a selected value and the optimum q has been determined by ( 18a), then the vertical radius of curvature of the toroid grating may be calculated using 

equation 

1R 

(I=-=- 

p pt 

- -* When the optimum values of p, p , K and q are determined, then each one of the integrals I ijk 

with i + j + k > 2 becomes a function only Of Aijk. Those values of Aijk *s which satisfy equation 

( 17) are calculated from equation 

9 
4 

where 

and 

atid so on. 

B~~~ (e) = (&%OS e + cos (e + K) cos (28 + K) 

+ 3 sin (8 + K) sin K] 

+ (75'/3) [cos e + cos (e - K) cos (20 - K) 

- 3 sin (8 - K) sin K] - cos 28 cos K 

Bo20 (0) = 2 (P+ij') COS 8 - q cos 2 0 cos K 

B3 oo = (p?4)2 [sin [48 + 3K] - 4 cos 2 (8 + K) sin K - 40 cos K] 

+ (P'/4l2 [sin I48 - 3Kl + 4 cos 2 (0 - K) sin K - 40 cos K] 

+ (iT/6) [3 sin (8 + 2Kl - sin (38 + 2K)I 

+ (p"16) [3 sin (8 - 2K) -sin (38 - 2K)l 

B,20 (8)=(1/p)2 [sin(28+K)-28 cosK1 

+ (!/7,7')2 !sin (20 - Q - 05 K! 

qa 

6 

+ -13 sin (0 + 2K) - sin (38 + 2K)l 

ClP' +- [3 sin (0 - 2K) -sin (38 - 2K)] 6 

1)ctcniiination of the Recording Parameters pc , Pd , 

and Coilla 

The recording parents depend upon the aberrations to be minimized. With a predetermined effective grating constant and wavelength range, the following equations are solved simultaneously for 

minimizing astigmatism and one coma-type aberration. 

and 6 and Minimization of Astigmatism 

10 
A0 sin 6 -sin y=- 

and for minimizing coma-type aberrations 

x 

sin 6 - sin y =- 

0 

Before making any attempt to solve equations (25) or (26) it is necessary to investigate the conditions under which equations (25) or (26) can have real solutions for pc and p~ because H3oo 

and HI 

on the values of 

ratic equations of pc or p~ resulting from equations (25) and (26) should not have ipaginary 

roots. To fulfill this condition, A3 00 and AI 

are quadratic in pc and p~ and their values depend on A3 oo and Ai and therefore 

in I,oo and It 20. The condition may be stated in such a way that two quadmust satisfy the equation 

where for A3 oo 

a = (sin 6 cos2 y - sin y cos2 6) cos2 y 

b= (2p COST sin 6 + sin(6 -7) cos 61 cosy 

c = p(p + cos y) sin 6 + A3 oo cos2 6 (sin 6 - sin y) 

(28) 

11 
and for A, 

a = cos4 y sin 6 - sin 7 cos4 6 

b= [2p cosy sin 6 + sin (6 -7) cos3 61 COST 

c = p (p sin 6 + cos3 6) sin 6 + A I 

(29) 

cos4 6 (sin 6 - sin 7) 

p = A2 00 (sin 6 - sin y) + cos 7 - cos 6. 

when condition (27) is satisified, then pc and p~ are solutions to equations (25) and (26). 

Up to this point, we have been describing the general method of designing aberration corrected 

holographic toroid gratings for Seya-Namioka type monochromators which are not iiiterchangeiible with conventional toroidal gratings. We will now deal with the design method where bdth 

types of gratings are interchangeable. 

Modified Method 

The modified method of designing holographic toroidal gratings for the Scya-Narnioka moiie 

chroinator has a practical advantage in that it may now be designed so that it is interchangeable 

with the conventionally ruled toroidal grating having the same groove frequency and radius of 

curvatures R and p,. This is possible provided the same instrumental constants P, p', q and K 

used for the conventionally ruled gratings are used to design the holographic toroidal grating. The 

instrumental constants for the conventional toroidal grating are determined by solving the 

equation 92 - 

IZo0 =/ Mioo de = minimum 

or 

and 

- =O a12 0 0 

aK 

The design equations given by equations ( 17)-(26) must be modified in part in order lo nccominodate condition(30). The modification required is to replace equation (1 8) with cqantion (30) 

and equation (20) with (ijk) = (200) and the rest of the procedurc rcniains llic same. 

The design of toroidal holographic gratings and optical systems for Seya-Naniiokrt tnoriochromators have been presented. Design equations for minimizing aberrations such as astigniatisni and 

coma have been derived. By now, you must have observed the the tlesign equations for concave 

toroid gratings arc very similar to the design equations for concavc spherical pr:itings. In fiict, if 

12 
1 U 

! 

the parameter q has a value of one; ie pt = R, then the toroidal equations are identical to those 

equations for the sphere. The advantage of the concave toroid grating relative to the concave 

spherical grating is that astigmatism can be minimized for larger angles of incidence and at shorter 

wavelengths where under the same geometrical conditions the concave spherical grating cannot. 

Section 2 

Section 1 described the theory of designing holographic concave toroidal gratings that minimize 

aberrations for a selected wavelength range. In this section, we will apply the theory and present 

a computer program that will calculate all the necessary parameters for using and ruling the holographic toroid grating for the Seya-Namioka monochromator. 

Program Description: 

The computer program consists of a main program and four subroutines. Fig (4) is a flow chart 

diagram of the program. 

The purpose of the main program is to direct the flow of calculations depending upon which option 

is selected. If option LMN=2 is selected, then the optimum value of the instrumental parameter 

angle 2K will be calculated. However, if the program user chooses to specify an angle 2K as an 

input value, then option LMN=l is used. 

Two toroidal holographic grating design methods have been developed: the general method and the 

modified method. If the general method is chosen then the option called I ETA=I is used. When 

using this option, all instrumental parameters are related only to the toroidal holographic grating. 

IIowever, if the modified method is selected, then the option IETA=O is used in the program. 

When using this option, all instrumental parameters calculated are the same for both the 11010- 

o- nrmhir -,~ ___- 2nd -__ - rnnvrntinns! - - __ - .. .. . .. . gratings. This means that the holographic and conventional toroid 

gratings are interchangeable if they have the same groove frequency and radii of curvatures. (R 

and pt). The difference is that the holographic toroid grating minimizes aberrations and the conventional toroid grating only reduces astigmatism. When the design method has been seiected and 

the option has been decided upon, the following steps of calculation are performed: 

Step A: Read input cards 

Step B: Call INSTRM to calculate the optimum angle 2K, ii, jT’ and pt. Then calculate Bijk and 

Aijk from equation (20) where LMN=2 or calculate the functions Bi,k and Aijk using the selected 

input value for 2K when LMN=l. 

Step C: Call ASTGMS to calculate the recording parameters pc, p~ , y and 6 to minimize astigmatism ijk = (020) and one coma term Uk = (300). 

Step D: Call COMA to calculate the recording parameters pc, p~ , y and 6 to minimize coma 

aberrations ijk = (300) and ijk = (1 20). 

13 
Step E: Call PRFORM to display the numerical values of each aberration of the holographic toroid 

grhig aid compare with that of the equivalent conventional grating as a function of wavelength. 

Note that the program automatically calculates the recording parameters for both the niininiization of astigmatism and coma. This feature was included in the program so that the program user 

may select which aberration minimization will best producc the optimum minimization for all 

abcrra tions through out the dcsircd wavelength range. 

The procedure stated above is supported by four subroutines. 

Subroutine INSTRM 

Subroutine INSTRM determines the optimum values of the instrumental parameters p, P‘, 2K, and 

Azo0 by siniultaneous solution of equation (18). A modified form of Newton’s method is used to 

obtain tlic solution. With a knowledge of p, p” and 2K the optimum vertical radius of curvature 

pt for thc toroid’ is calculated. After all the optimum instrumental parameters are determined, 

then the values of the Bijk are calculated which in turn are used in equation (20) to determine 

the values of Aijk ’s. Note that if a prescribed input value of 2K were used, the selection of option 

LMN=I would by-pass the optimization procedures when calculating the other instrumental paramctcrs. 

Subroutine ASTGMS 

The purpose of this subroutine is to obtain the recording parameters such that astigmatism (ijk) 

= (020) and one coma term (ijk) = (300) are minimized over the given wavelength range X1 G 

X < h2. This is achieved by simultaneous solution of equation (25). Before proceeding with the 

solution, it is first verified that the value of NXo is less than the maximum value of Ksin 6 - sin 

r)l, nrlmcly 2. If not, an error message is printed stating “error, GAMA out of bouilds.” Now, 

if NXo is less than 2 then the values of a, b, and c of equation (28) or (29) are computed and the 

L~ixrii~ii;;:c isc!xcke:! to e::sxp !ht h2 - 4.c IO. The wlutions of the quadratic equations are 

ccmputcd and checked to make sure that the positive value has been used, because the calculations 

show that one of the two roots [(- b +,/(b2 - 4ac)/za] is always positive and the other is always 

negative. When the discriminate has been verified to be positive, then the optimum vililtes of 

pc, p~ , and 6 are determined by the iterated solution of equation (25) using Newton’s method. 

Subroutine COMA 

Tlie purpose of this subroutine is to obtain the recording parameters such that the coma terms 

ijk = (300) and ijk = (1 20) are minimized for a given wavelength range A, G X Q X2. This is 

achieved by a simultaneous solution of equation (26). We use the same procedure used in subroutine ASTGMS to verify that the discriminant is positive and then proceed to deterniinc the 

optinium values pcr p~, 7, and S by the iterated solution of equation (26) using Newton’s 

method. 

14 
. 

START 

MAIN PROGRAM FLOW DIAGRAM 

I 

CALL PRFORM 

CALCULATES VALUES FOR ALL 

ABERRATIONS AS A FUNCTION OF 

WAVELENGTH, DISPLAYS OUTPUT. 

2 

END 

XL , A,, Ax, C, N, R , K, 

IETA, LMN, QTI, QTF, 

DQT, IQ 

CALL INSTRM 

DETERMINES OPTIMUM C AND p,p', pt 

I 

CALL ASTGMS 

COR R ECTl N G ASTIGMATISM 

DETERMINES pc, PO, ~,6 FOR 

CALL COMA 

CORRECTING COMA. 

DETERMINES pc, PO, 7,6 FOR 

Figure 4. Main Program Flow Diagram 
Siibroutinc PRFROM 

This subroutine calculates the performance of the holographic toroid grating using the calculated 

instrumental and recording parameters and then displays the values of each aberration as a function 

of wavelength in the forin of a table. The performance of a conventional (mechanically ruled) 

grating with the same groove frequency is also shown in the table for comparison. 

Section 3. DATA SET AND DESCRIPTION OF DATA CARDS 

This section discusses the input data required for a successful operation of the program. 

(A) DataSet 

The following parameters are required as input for the program. 

1. A, = wavelength of recording laser light in angstroms. 

2. h~ = lower wavelength limit in angstroms. 

3. A, = upper wavelength limit in angstroms. 

4. Ah = wavclength interval in angstroms which aberrations will be displayed. 

5. C = anglc 2K (0.0 is used with option LMNz2). 

6. N = groove frequency (number of lines per mm). 

7. K = Radius of curvature in mm. 

8. m = order of diffraction where m = f 1, ?: 2, etc. 

IO. QTI = Initial value of the torus parameter q,. 

12. QFT = Final value of torus parameter qf. 

13. DQT = lntcrval at which the torus parameter Aq will change. 

14. IQ = I’aramctcr to specify the torus radius options. 

IQ=2;Torus parameter q is selected by user. 

IQ= 1 ;Toroid radius p, in mni is automatically computed 
. b 

- Note: If IQ=I is selccted, then QTI, QTF and DQT are NOT punched on the data card. The 

computer automatically bypasses these parameters and calculates the optimum q values and 

pt. However the value 1 for IQ must be punched in the appropriate column on the card. 

If IQ=2 is selected, then QTI, QTF, and DQT must be punched on the data card in the 

correct order. For this case, the computer will calculate the pt based on the values selected 

for the range of q's desired. 

(B) DescriPtion of Data Cards and SamDle Data Cards 

The first seven input parameters must be punched on the data cards in the order shown in (A) 

starting in column I in F 10.1 format with ten columns reserved for each of the seven parameters 

and each parameter must include a decimal. The next three parameters are in I2 format starting in 

column 71. The parameters QTI, QTF, DQT, and IQ are punched on the second card in format 

F 10.0 and I2 respectively. Fig (5) shows a typical two card data set to illustrate the above 

description. 

Note: (I) The program user may include as many data cards for a single run as desired. 

(2) The values of all 14 parameters must be punched for each additional set. 

Section 4 

This section presents a typical example for using the program and also describes the output data in 

de tail. 

Example: Determine the optimum instrumental and recording parameters for designing a concave 

toroid holographic grating for a Seya-Namioka monochromator that minimizes astigmgtism or coma and has the following specifications: 

1. Interchangeable with conventional concave toroid grating. 

2. Groove frequency = 550 L/mm 

3. R = 1,000 mm 

4. Operating wavelength range = 0.OA to 1 GOOA. 

5. Order=-1 

6. C = 142' 

Solution: 

(I) indicates that the modified method is used; IETA=O. (6) says that option 

LMN= 1 is used. 

17 
. b 

4579.3 0.0 1600.0 200.0 142.0 550.0 1000.0-1 0 1 

0lll0ll11111l1l I 111l11 I 1111l1 I I11111111 111111l I IIIII I IO ll101 

111111111111111111111111 III11I11I1I111111111 111111111111111111 111111 111 1111 

11111111111111111111111111111~11111 11121111111 11111111111111211111111111111111 

I I I I I 1 1 I I m 11 11 11 11 IS 11 11 11 II n 11 11 11 11 n n 11 n n Y II u IS n :: I 11 Y n a 11 a u Y a a 11 a a Y 11 Y Y n n Y SI n Y II 11 P u u n I II I II n 11 11 n u n n 11 I n I 

11331133 33313113 33133!311 311311113 311113111 133311333 131111333 11135311111 

4444 (1(111(((1(4(4(44~(((11(11((((4((((((11(4 ((((1(444((1(41414(1((11((1110(( 

52355 55555555555555555555555555s5555555555553555555s55 55555555555555555555555 

C6C66~~lC66l6ClC6666~6C6l 6l66llCC6ClClllllll66C66~lCCl~6ClC~ClCl666lCCClll~llCl 

I 111111 1~11111111111111111~111111~11~111~111111111111111111111111111111111111111 I 

I 1111111l 1l1111l11 ll1ll1l11 11l111111 111111111 111111111 11l1l1111 1l1l1ll111l I 

99¶¶9¶1 99¶¶¶9¶¶¶¶1¶¶¶¶1¶¶¶¶99¶¶99¶¶1¶¶¶~11¶¶¶¶¶¶¶¶1¶1¶¶191¶1~¶¶~¶¶9~1¶¶¶¶¶1¶¶¶9 I I i I I I I I I II II 11 11 II 11 IS II II II 11 11 11 11 II IS n II n 11 Y II 12 IS n IS I SI m n L 11 11 JI u IS Y II II 11 n $1 11 11 n $5 Y SI n :I u 11 11 11 SI IS II 11 u 11 II 11 11 11 II I) II II II IS II 

",CL-50., 

1 .o 1 .o 2.0 1 

OT 

I0llIl0ll 1Il10111 l111IOl1 1l1l1ol11l1l1111lllelllll)l)ollllleeellnlllnlllllll 

IIIIIII IIIIIIIII 1111111111111 111111111111111111111111111111111111111111111111 

111:11111111111111111111111 1111111111111111111111111111111111111111111111~11111 

I I I I s 1 I I 1 II II II IS u IS 11 II 11 II 11 11 11 11 11 n II 11 II n Y 11 u u n n I 11 n I a 11 u 1111 a a II II II B SI Y u n n n SI n Y a 11 II II II u Y II u II m II n 11 Y n n II II n II 

11133131 1113133!1 111331113 11~11111~31111113111111311131331333111131~11~11113~ 

((4((1((1(1(11(((((((((1(((1((1((11(((((~(((((~(~(4~4~44((((((4((((((((4(4444((( 

53555555555535555555555555555555555555555555555555555555555555555555555555555555 

6666666Cl6C666CC6ClCC~6l6666CClC6CC~CCCllClllClCCl66666l~~66lC66C6l666666l666C6l6 

I 11111111111111111111~11111111111~1111111111111111~~11~11111117~7~111111111111111 j 

I iiieiiti IIIIIIIII IIIIIIIII IIIIIIIIIIIIIIIIIIIIIII~IIIIIIIIIII~IIIIIIIIIIIIIII j 

J¶9¶99¶¶99¶¶¶111¶9¶¶~¶9¶¶¶¶¶¶¶¶¶¶1¶¶19¶~1¶¶¶11¶¶1¶1¶¶¶¶1191¶19¶9199¶91¶¶1¶¶¶¶¶9¶ 

I 1 I 1 I I 1 I I 11 11 11 11 II 15 11 11 11 11 n 11 11 11 11 n n 11 11 n n 11 11 11 n n I II Y n a II 11 11 u 11 11 11 u II Y $1 $I SI Y ss u $1 n SI u 11 81 11 Y 13 u 11 u H 11 I1 I1 I1 I1 IS n I1 I1 n II 

ma-sow 

Figure 5. Typical Data Card Set 

18 
The first data card is punched using data supplied above and punched in the order listed below: h 

A, = 4579.3 

AL = 0.0 

A,, = 1600.0 

Ah = 100.0 

C = 142.0 

N = 550.0 

R = 1000.0 

K = -1 

IETA = 0 

LMN= 1 

The following is punched on the second data card: 

QIT= 1.0 

DQT = 1.0 

QTF = 2.0 

IQ= 1 

The output data for the example cited is shown in fig 6 and described as follows: 

Line 1 Shows the input parameters; order = -1 is the order of the grating, N = groove frequency 

in grooves per mm, R = horizontal radius of curvature of the grating in mm. IETA = 

identifies the Jcsign method and LMN = 1 shows that the angle C has been specified. 

Line 2 shows the input values for X, = LAMDA 0 (Laser Wavelength), AL = LAMDA 1 

(lower wavelength limit), Xu = LAMDA 2 (upper wavelength limit), 

Line 3 is self-explanatory. 

Line 4 shows the calculated instrumental parameters; C = 2K, ji= RHOA, p" = RHOB, To = SMLR 

in mm, and = SMLR in mm. These parameters are defined in the text. 

19 
. b 

Line 5 shows the values of the torus parameters: Q = q value defined in the text, pt = value of 

.- tkc vertical toroid radius. 

Line 7 is self-explanatory and refers to line 8. 

Line 8 shows the optimum recording parameters; 7 = GAMA, 6 = DLTA, pc = RHOC, and p~ = 

RHOD. These parameters are defined in the text. 

Line 9 shows the titles for the table showing the performance of the grating as a function of wavelength. The table also shows the comparison of the holographic toroid grating with the 

toroid conventional grating. 

A = LAMDA is the wavelength at which the grating is evaluated. 

F, , X I O4 is the horizon tal focus. 

F, is the vertical focus and astigmatism. 

F, is the first coma term. 

F, is the second coma term. 

F4 ,, is the higher order focusing term. 

F, , is the toric aberration. 

F040 is the toric aberration of a higher order 

Line IO shows the holographic grating performance (Modified Method), the value of the wave- 

:Ciigi;i and the vaiiies of the aberrations. 

Line 11 shows the performance of the conventionally ruled grating (MECH. Ruled). 

Line 12 shows that the following information refers to minimizing coma type aberrations for the 

same instrumental parameters shown in line 4. 

Line I3 is self explanatory and refers to line 14. 

Line 14 shows the recording parameters 7 = GAMA, 6 = DLTA, pc = RI-IOC, and PD = RHOD 

required'to fabricate a toroid grating to minimize coma. 

Lines IS, 16, and 17 are defined in the same way as lines 9, 10, and 1 1 above. The values in this 

table refer to a grating designed to minimize coma. 

20 
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Figure 6. 

21 I 
Refcrences: 

1. A. J. Caruso and M. S. Bhatia, GSFC X-Document X-720.77-79, Nov. 1976 

- v 

2. H. Noda, T. Namioka and M. Seya, J. Optical Society of America, 64, 1043 ( 1974) 

3. A. J. Caruso and M. S. Bhatia, GSFC X-Document X-720.77-145, May 1977 

4. H. Greiner and Schaefer, Optik 16,351, (1959) 

5. H. Haber, J. Optical Society of America. 40 153, March 1959 

6. A. J. Caruso and M. S. Bhatia, GSFC X-Document X-722-78-3 Feb. 1978 

, 

22