Abstract

Astigmatism is left uncorrected in traditional Czerny–Turner spectrometers with spherical mirrors, which leads to low throughput of modern instruments applying line-array detectors. By gradually varying the sagittal curvature, freeform mirrors are introduced to suppress astigmatism for a wide wavelength range simultaneously. So as not to reduce spectrum resolution, further calculations of the light path are performed for extra coma compensation. A design example is presented with optimized parameters. The ray-tracing result has revealed a reduction in sagittal spot size from several hundred micrometers to around 10μm in the wavelength range from 200nm to 800nm.

© 2009 Optical Society of America

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References

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  1. W. G. Fastie, “High speed plane grating spectrograph and monochromator,” U.S. patent 3,011,391 (5 December 1961).
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    [CrossRef] [PubMed]
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  9. I. M. Gulis and A. G. Kupreev, “Astigmatism correction for a large-aperture dispersive spectrometer,” J. Appl. Spectrosc. 75, 150-155 (2008).
    [CrossRef]
  10. J. Reader, “Optimizing Czerny-Turner spectrographs: a comparison between analytic theory and ray tracing,” J. Opt. Soc. Am. 59, 1189-1196 (1969).
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  11. ZEMAX is a trademark of Zemax Development Corporation, Bellevue, Washington 98004, USA.
  12. ZEMAX User's Guide, 225-226, Version: 24 July 2002.
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    [CrossRef]

2008 (1)

I. M. Gulis and A. G. Kupreev, “Astigmatism correction for a large-aperture dispersive spectrometer,” J. Appl. Spectrosc. 75, 150-155 (2008).
[CrossRef]

2007 (1)

2002 (1)

2000 (1)

1983 (1)

1980 (1)

R. W. Esplin, “Use of curved slits to increase throughput of a Hadamard spectrometer,” Opt. Eng. 19, 623-627 (1980).

1969 (1)

1966 (1)

1964 (1)

1945 (1)

Beutler, H. G.

Dalton, M. L.

Droppleman, L.

Esplin, R. W.

R. W. Esplin, “Use of curved slits to increase throughput of a Hadamard spectrometer,” Opt. Eng. 19, 623-627 (1980).

Fastie, W. G.

W. G. Fastie, “High speed plane grating spectrograph and monochromator,” U.S. patent 3,011,391 (5 December 1961).

Futamata, M.

Gil, M. A.

Gulis, I. M.

I. M. Gulis and A. G. Kupreev, “Astigmatism correction for a large-aperture dispersive spectrometer,” J. Appl. Spectrosc. 75, 150-155 (2008).
[CrossRef]

Katakura, K.-I.

Kupreev, A. G.

I. M. Gulis and A. G. Kupreev, “Astigmatism correction for a large-aperture dispersive spectrometer,” J. Appl. Spectrosc. 75, 150-155 (2008).
[CrossRef]

Longworth, J. W.

Megill, L.

Nayyar, V. P.

Nicolosi, P.

Pelizzo, M-G.

Reader, J.

Rimington, N. W.

Schieffer, S. L.

Schroeder, W. A.

Shafer, A.

Simon, J. M.

Takenouchi, T.

Villoresi, P.

Appl. Opt. (5)

J. Appl. Spectrosc. (1)

I. M. Gulis and A. G. Kupreev, “Astigmatism correction for a large-aperture dispersive spectrometer,” J. Appl. Spectrosc. 75, 150-155 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Eng. (1)

R. W. Esplin, “Use of curved slits to increase throughput of a Hadamard spectrometer,” Opt. Eng. 19, 623-627 (1980).

Other (3)

W. G. Fastie, “High speed plane grating spectrograph and monochromator,” U.S. patent 3,011,391 (5 December 1961).

ZEMAX is a trademark of Zemax Development Corporation, Bellevue, Washington 98004, USA.

ZEMAX User's Guide, 225-226, Version: 24 July 2002.

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Figures (13)

Fig. 1
Fig. 1

Optical structure of a typical Czerny–Turner spectrometer with coordinates for each element.

Fig. 2
Fig. 2

Freeform surface shape of M 2 .

Fig. 3
Fig. 3

Collimating part and the camera part of the light path.

Fig. 4
Fig. 4

Light traveling from the freeform collimating mirror.

Fig. 5
Fig. 5

Relationship among w 1 , w 1 , and w g .

Fig. 6
Fig. 6

Polynomial evaluation of R 2 S ( ω 2 ) .

Fig. 7
Fig. 7

Discrete grid points on M 1 and M 2 for extra coma estimation.

Fig. 8
Fig. 8

q value estimation.

Fig. 9
Fig. 9

Ray-tracing result of the design with freeform M 1 and M 2 .

Fig. 10
Fig. 10

Variation of spot X sizes with q.

Fig. 11
Fig. 11

Ray-tracing results of (a) spherical M 1 and M 2 , (b) spherical M 1 and toroidal M 2 , and (c) spherical M 1 and freeform M 2 ( q = 0 ).

Fig. 12
Fig. 12

Geometrical relationship of Δ 2 .

Fig. 13
Fig. 13

Δ 2 values over M 2 .

Tables (3)

Tables Icon

Table 1 Parameters of the Original Structure with Spherical Mirrors a

Tables Icon

Table 2 RMS Spot Sizes for Different Wavelengths

Tables Icon

Table 3 Spot Size Data for the Results in Fig. 11

Equations (38)

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sin β sin α = R 2 T 2 cos 3 β cos 3 α g R 1 T 2 cos 3 α cos 3 β g ,
Δ f = ( f 1 S f 1 T ) + ( f 2 S f 2 T ) ,
f 1 T = ( R 1 T / 2 ) cos α , f 1 S = R 1 S / ( 2 cos α ) , f 2 T = ( R 2 T / 2 ) cos β , f 2 S = R 2 S / ( 2 cos β ) .
R 2 S ( ω 2 ) = b 0 + b 1 ω 2 + b 2 ω 2 2 + b 3 ω 2 3 + ,
R 2 S ( ω 2 ) = b 0 + b 1 ω 2 .
R 1 S ( ω 1 ) = a 0 + a 1 ω 1 + a 2 ω 1 2 + a 3 ω 1 3 + .
R 1 S ( ω 1 ) = a 0 + a 1 ω 1 ,
Δ 1 = Δ 1 ( ω 1 , l 1 ) R 1 T 2 ω 1 2 l 1 2 [ R 1 T R 1 S ( ω 1 ) ( 1 1 l 1 2 R 1 S ( ω 1 ) 2 ) ] 2 ω 1 2 ,
Δ 2 = Δ 2 ( ω 2 , l 2 ) R 2 T 2 ω 2 2 l 2 2 [ R 2 T R 2 S ( ω 2 ) ( 1 1 l 2 2 R 2 S ( ω 2 ) 2 ) ] 2 ω 2 2 .
K = cos α g cos β g K 1 + K 2 ,
K 1 = K 1 + δ K 1 , K 2 = K 2 + δ K 2 .
δ K = cos α g cos β g δ K 1 + δ K 2 = 0.
( R 1 T ξ 1 ) 2 + ω 1 2 + l 1 2 = R 1 T 2 .
ξ 1 = ξ 1 + Δ 1 ;
[ A P 1 ] = { ( z A ξ 1 ) 2 + ( x A + ω 1 ) 2 + ( y A l 1 ) 2 } 1 / 2 .
[ A P 1 ] = { ( z A ξ 1 ) 2 + ( x A ω 1 ) 2 + ( y A l 1 ) 2 } 1 / 2 .
[ A P 1 ] [ A P 1 ] [ A P 1 ] ξ 1 Δ 1 Δ 1 z A { ( z A ξ 1 ) 2 + ( x A ω 1 ) 2 + ( y A l 1 ) 2 } 1 / 2 ( as     z A ξ 1 z A ) .
z A = r A cos α A , x A = r A sin α A , y A = y A .
[ A P 1 ] [ A P 1 ] Δ 1 cos α A { 1 + y A 2 r A 2 2 r A ( ξ 1 cos α A + ω 1 sin α A ) 2 y A l 1 r A 2 + 2 R 1 T ξ 1 r A 2 } 1 / 2 .
[ A P 1 ] [ A P 1 ] Δ 1 cos α { 1 4 tan α ω 1 R 1 T } 1 / 2 .
[ P 1 B ] [ P 1 B ] Δ 1 cos α B { 1 + y B 2 r B 2 2 r B ( ξ 1 cos α B + ω 1 sin α B ) 2 y B l 1 r B 2 + 2 R 1 T ξ 1 r B 2 } 1 / 2 .
[ P 1 B ] [ P 1 B ] Δ 1 cos α .
δ F 1 = ( [ A P 1 ] [ A P 1 ] ) + ( [ P 1 B ] [ P 1 B ] ) Δ 1 cos α ( 1 + 1 { 1 4 tan α ω 1 R 1 T } 1 / 2 ) .
δ K 1 = δ F 1 ω 1 ,
ω 1 = ω 1 cos α = ω g cos α g , ω 1 [ w 1 , w 1 ] , ω 1 [ w 1 , w 1 ] , ω g [ w g , w g ] ,
Δ 2 = Δ 2 ( ω 2 , l 2 ) R 2 T 2 ω 2 2 l 2 2 [ R 2 T R 2 S ( ω 2 ) ( 1 1 l 2 2 R 2 S ( ω 2 ) 2 ) ] 2 ω 2 2 ,
δ F 2 Δ 2 cos β ( 1 + 1 { 1 4 tan β ω 2 R 2 T } 1 / 2 ) ,
δ K 2 = δ F 2 ω 2 ,
ω 2 = ω 2 cos β = ω g cos β g , ω 2 [ w 2 , w 2 ] , ω 2 [ w 2 _ λ , w 2 _ λ + ] , ω g [ w g , w g ] .
b 0 = 121.13 , b 1 = 1.1821
a 0 = R 1 T = R 1 , a 1 = q b 1 ,
S = i j k { V i j k × [ cos α g cos β g ( k ) δ K 1 ( i , j ) + δ K 2 ( k , j ) ] 2 } .
ω 1 = cos α g cos β cos β g cos α ω 2 1.0064 ω 2 ( at     λ = 500 nm ) .
Δ 2 = ξ 2 ξ 2 = ( ξ 2 R 2 T ) ( ξ 2 R 2 T ) .
ξ 2 R 2 T = R 2 T 2 ω P 2 l 2 2 .
ξ 2 R 2 T = [ R 2 T R 2 S ( ω 2 ) ( 1 cos τ ) ] cos θ , ω P = [ R 2 T R 2 S ( ω 2 ) ( 1 cos τ ) ] sin θ , l 2 = R 2 S ( ω 2 ) sin τ .
ξ 2 R 2 T = [ R 2 T R 2 S ( ω 2 ) ( 1 1 l 2 2 R 2 S ( ω 2 ) 2 ) ] 2 ω P 2 .
Δ 2 = ( ξ 2 R 2 T ) ( ξ 2 R 2 T ) R 2 T 2 ω 2 2 l 2 2 [ R 2 T R 2 S ( ω 2 ) ( 1 1 l 2 2 R 2 S ( ω 2 ) 2 ) ] 2 ω 2 2 .

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