Spectral spread function of a double-pass parabolized Ebert monochromator

MIU63:1

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Spectral spread function of a double-pass parabolized Ebert monochromator Mirta A. Gil and Guillermo 0. Mattei We have studied the influence of the residual coma and the width of the intermediate slit on the spectral spread function of a double-pass parabolized Ebert monochromator.

It has been demonstrated by Gil and Simon' that, although from the geometrical point of view the coma present in a parabolized in-plane double-pass Ebert monochromator doubles that existing in the equivalent single-pass system, this doubling cannot be inferred from their corresponding spectral spread functions (SSF's). In that study the authors suggested that the intermediate slit (which acts as a low-pass spatial filter) should have a decisive influence not only on the intensity variations but also on the width, asymmetry, and displacement of the SSF's maximum intensity. The purpose of this Note is to evaluate the SSF's of that double-pass system for different intermediate slit widths and then to study its influence on those functions. For the evaluation of the SSF we have applied the same methodology used in Ref. 1, i.e., our starting point was the well-known Fourier relationships between the light distribution in different parts of the optical system [see Eqs. (1) and (2) of Ref. 1]. Only the conventional symmetric design with the grating placed at the focal point of the mirror is considered here. The construction parameters that we have chosen are the same as the ones used in Ref. 1, i.e., a grating of 1200 lines/mm and 10 cm in diameter, a mirror with a focal length f of 1 m, and 3-cm-long curved slits placed 9 cm from the center of the grating (which implies equal off-axis angles, = ' = 0.045). The curvature of the entrance slit p was selected so that the curvature of the spectral lines p' is independent of the wavelength.2 4 This design exhibits only residual coma,2 and thus the aberration function in terms of dimensionless coordinates (x, y) normalized to the radius of the pupil Departamento de Fsica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428Buenos Aires, Argentina. Received 21 November 1991.

0003-6935/93/010027-03$05.00/0. c 1993 Optical Society of America.

is [see Eq. (4) of Ref. 1] D(x,

R 3 cos 3 ' = 2f 2 (t-1)

x (30x3 + Oy

2

-1/ 2popo2X3-/2poI0

2

xy 2

-

x2 y) (1)

where R is the radius of the grating, t = cos 4)/cos 4)' is the anamorphosis factor, 4)and 4)'are the incident and diffraction angles on the grating, Po = f/2p', and 80o is the field angle.

Evidently ¢F(x,y) depends on Xthrough the factors t and cos 4)',and its explicit dependence can be easily evaluated from the grating equation and the relationship

4)'- 4 = 40 [see Eqs. (5) and (6) of Ref. 1].

Assuming that the pupil is uniformly illuminated and determining the transmission function for the slit acting as a filter in terms of the intermediate slit characteristics (width w and curvature p"), we have numerically calculated, for the first stage of the system, the amplitude distribution on the image of the pupil resulting from each object point as follows [see Eq. (9) of Ref. 1]:

sin[c(x -x')]2

D(x',y') =4rC() X (x - x')

E[x,y' - 2avo(x- x')]

exp[-i(x - x') (uo - avo2)]dx, (2)

where x' and y' are the normalized coordinates over the image of the pupil; C(X) is the complex function depending on and on the parameters of the system; E(x,y) is the amplitude distribution over the entrance pupil,

E(x,y) =

fB(x, y)exp

tr 4(x, y)]

over the pupil elsewhere

1 January 1993 / Vol. 32, No. 1 / APPLIED OPTICS

27

where B(x, y) is a complex function (which is constant only for the first stage); u0 and v0 are angular frequencies corresponding to the Gaussian image point; c is the normalized half-width of the intermediate slit,

160.00

I/X 0[a.u] 3

120.00

c = (TrR cos +'/Af)w;

and a is related to the slit's curvature, a

=

(1/2p")(Xf/2wrR cos

80.00

4,').

In a similar way, namely, by using Eq. (2) again, we numerically calculated the amplitude distribution that originated at the second stage of the system. In this case c represents the normalized half-width of the exit slit, and a is related to its curvature p'. One must also take into account the fact that the pupil is no longer uniformly illuminated and that the values

40.00

T

of D(x', y') resulting from the first stage represent this nonuniformity. Hence we obtained the intensity distribution of the double system in the plane

28

APPLIED OPTICS / Vol. 32, No. 1 / 1 January 1993

i

.350 08

Ae[nm] 100.00

I/A,[a.u]

(x', y'). It is found that the intensity variations

caused by v0 are negligible, and therefore to calculate the SSF we need to consider for each grating position only the contributions originating at the object points distributed over the entrance slit width and then integrate over x ', y'. Figure 1 shows the SSF's of the double-pass system for three different wavelengths (X0 = 350, 500, and 750 nm). In each of the three cases it is assumed that the entrance and exit slits have the same width (50 .lm), whereas for the intermediate slit three different widths are considered (25, 50, and 75 [um). We selected these values taking into account that the widths of the slits and the coma image width are of the same order of magnitude, and the coma image for the single-pass system in Ref. 2 was 20-40 pumin the 350-750-nm spectral range (and consequently it should double for the double-pass system). Table 1 summarizes the main characteristics of the curves shown in Fig. 1: the deviation A between the value of the effective wavelength Xe corresponding to the maximum intensity (IM)and X0 varies with w and also with Xo. When w is equal to the entrance- and exit-slit widths (50 Am), A is of the same order (or smaller) than that obtained for w • 50 pum. On the other hand, A increases with increasing Xe, i.e., the greater the coma the larger A is. It is also observed that the value of IMfor w = 25 [Lmis, for all Xo, 57% of Tv for w = 50 pLm,while the values of IMfor w = 75 plm are 4, 7, and 14% greater than the IM values corresponding to w = 50 .lmand X0 = 350, 500, and 750 nm, respectively. If w remains constant, IM decreases with decreasing Xo. It is found that the half-widths (AX)increase with increasing w. The displacements observed for the SSF's from their classical triangular or trapezoidal forms show the diffraction effects caused by the intermediate slit. Figure 2 shows the SSF's of a parabolized singlepass Ebert monochromator for Xo= 500 nm and with the same parameters of the previous double-pass

I rrr

4

80.00

3 60.00

40.00

20.00

A[nm] 50.00

40.00

A*(nm]

Fig. 1. Spectral spread function of the double-pass system for three different wavelengths (Xo = 350, 500, 750 nm).

The abscis-

sas represent the effective wavelengths (Xe),and the ordinates represent the intensities relative to \o (i.e., I/Xo). In each case the curves labeled 1, 2, and 3 correspond to intermediate slits of width w = 25, 50, and 75 pum,respectively.

Table 1.

Main Features of the Curves Shown in Fig. 1,

w (plm)

Xo(nm)

A (nm)

IM (a.u.)

AX (nm)

25 50 75 25 50 75 25 50 75

350 350 350 500 500 500 750 750 750

0.0083 0.0062 0.0056 0.0082 0.0061 0.0109 0.0102 0.0077 0.0128

267 468 487 235 408 438 184 325 371

0.0414 0.0430 0.0472 0.0409 0.0428 0.0468 0.0388 0.0410 0.0443

aW;Intermediate-slit

width; Xo,wavelength; A = XeM - Xo,devi-

ation; IM, maximum intensity in arbitrary units; AX,half-width. XeMrepresents the effective wavelength corresponding to IM.

system, except for the exit-slit widths that have been

fixed at w = 25, 50, and 75 jim.

In this case the

SSF's show deviations from their classical triangular or trapezoidal forms, which is an indication that the coma and the exit slit influence the SSF's as well. Table 2 summarizes the main features of the curves in Fig. 2: A varies with w (it increases when the exit-slit width is larger or smaller than the entranceslit width); IM also varies with w, in particular the values of IM corresponding to w = 25 and 75 jIm are, respectively, 49% smaller and 20% greater than the value of IM for w = 50 jlm; AXincreases with w. These results and those from Ref. 1 (where A increases with X0) show that both the intermediate slit of the double-pass system and the exit slit of the single-pass system act in a similar way, according to their influence on the corresponding SSF's. When comparing the SSF's represented in Figs. 1 and 2 we must bear in mind that the curves in Fig. 2 correspond to the same spectral line but to different exit slits, whereas the curves in Fig. 1 are associated with spectral lines of different widths (according to the filtering produced by the intermediate slit) and fixed exit slit. On comparing the values in Table 1 (where X0 = 500 160.00

I/A0 [a-u.]

120.00

3 80.00-

/

Table 2.

w (m) 25 50 75

Main Features of the Curves Shown in Fig. 2a

X0 (nm)

A (nm)

IM (a.u.)

AX (nm)

500 500 500

0.0086 0.0043 0.0086

254 503 603

0.0409 0.0436 0.0565

aw, Exit-slit width; Xo, wavelength; A =

eM - Xo, deviation; IM,

maximum intensity in arbitrary units; AX,half-width. XeMrepresents the effective wavelength corresponding to IM.

nm) with those in Table 2 (where w = 50 p.m), i.e., when comparing equivalent single- and double-pass systems, we found that the SSF's of the double-pass system have a greater A and a smaller value of IM, whereas AXremains approximately constant. The results analyzed here can be summarized as follows: the SSF of a double-pass system shows deviations from the classical triangular or trapezoidal form that are due not only to the residual coma but also to the width of the intermediate slit. However, A increases with increasing Xo(as the coma increases), and it also varies with the width of the intermediate slit. Although, from purely geometrical considerations, the A value associated with the SSF of the doublepass system should approximately double the value of that corresponding to the equivalent single-pass system, in the case where the intermediate slit width coincides with the entrance- and exit-slit widths, the observed value is less than the expected one, while A increases as the width of the intermediate slit becomes bigger or smaller than that of the other two. This shows that the intermediate slit has a decisive influence on the asymmetry curves. On the other hand, it is observed that the intensity variations not only are a result of the intermediateslit width (a classically expected result since this slit modifies the spectral-line width with respect to the exit slit) but are also affected by both the coma and diffraction effects. The greater X0 is, i.e., the greater the coma and diffraction effects, the smaller IMis. Finally, since the intermediate slit modifies the width of the spectral line, the half-width also depends on the slit width. This work was supported by the Consejo Nacional de Investijaciones Cientificas y Tcnicas (CONICET) and the Universidad de Buenos Aires, Buenos Aires, Argentina (UBA). M. Gil is a member of CONICET, Buenos Aires. G. Mattei is supported by a fellowship from UBA. References

40.00

1

1. M. A. Gil and J. M. Simon. "Double-pass

monochromator:

parabolized

Ebert

spatial filtering analysis of the aberrations,"

Appl. Opt. 26, 2906-2911 (1987).

2. M. A. Gil and J. M. Simon, "Coma compensation in a parabo-

0.00 -

499.80 499.85 499.90 499.95 500.00 500.05 500.10

Ae[nm]

Fig. 2. Spectral spread functions for the single-pass system for Xo= 500 nm. Curves 1, 2, and 3 correspond to exit slits of width w = 25, 50, and 75 jim, respectively.

lized Ebert monochromator,"

Appl. Opt. 18, 2280-2285 (1979).

3. G. W. Stroke, "Diffraction gratings," in Optical Instruments, S. Fligge, ed., Vol. 29 of the Encyclopedia of Physics (SpringerVerlag, Berlin, 1967), pp. 426-754.

4. M. A. Gil and J. M. Simon, "Aberrations in plane grating spectrometers," Opt. Acta 30, 777-806 (1983). 1 January 1993 / Vol. 32, No. 1 / APPLIED OPTICS

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