Abstract

The wavefront aberration polynomial and transverse ray aberration expansions will be derived for grazing incidence two-mirror Wolter telescopes including all paraboloid–hyperboloid and paraboloid–ellipsoid combinations. The reference sphere is determined with the help of the principal surface of the telescope, and the aberration polynomials will be given as functions of the coordinates of the ray intersection with the reference sphere. Third, and some of the fifth- and seventh-order aberration terms will be analyzed. Also, the well-known relationship between wavefront aberration polynomial and transverse ray aberration polynomials will be verified.

© 1988 Optical Society of America

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References

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  1. H. Wolter, “Mirror Systems with Glancing Incidence as Image Producing Optics for X-Rays,” Ann. Phys. 10, 94 (1952).
    [CrossRef]
  2. T. T. Saha, “General Surface Equations for Glancing Incidence Telescopes,” Appl. Opt. 26, 658 (1987).
    [CrossRef] [PubMed]
  3. L. P. Van Speybroeck, R. C. Chase, “Design Parameters of Paraboloid–Hyperboloid Telescopes for X-Ray Astronomy,” Appl. Opt. 11, 440 (1972).
    [CrossRef]
  4. H. Wolter, “Estimation of Image Aberrations for X-Ray Telescopes,” Opt. Acta 18, 425 (1971).
    [CrossRef]
  5. W. Werner, “Imaging Properties of Wolter I Type X-Ray Telescopes,” Appl. Opt. 16, 764 (1977).
    [CrossRef] [PubMed]
  6. C. E. Winkler, D. Korsch, “Primary Aberrations for Grazing Incidence,” Appl. Opt. 16, 2464 (1977).
    [CrossRef] [PubMed]
  7. T. T. Saha, “Transverse Ray Aberrations for Wolter Type I Telescopes,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 10 (1986).
  8. T. T. Saha, “Transverse Ray Aberrations for Paraboloid–Hyperboloid Telescopes,” Appl. Opt. 24, 1856 (1985).
    [CrossRef] [PubMed]
  9. W. T. Welford, Aberrations of Symmetrical Optical Systems (Academic, London, 1974).
  10. W. G. Driscoll, Ed., Handbook of Optics (McGraw-Hill, New York, 1978).
  11. J. D. Mangus, J. H. Underwood, “Optical Design of a Glancing Incidence X-Ray Telescope,” Appl. Opt. 8, 95 (1969).
    [CrossRef] [PubMed]
  12. J. D. Mangus, “Optical Design of Glancing Incidence XUV Telescopes,” Appl. Opt. 9, 1019 (1970).
    [CrossRef] [PubMed]
  13. M. C. Hettrick, S. Bowyer, “Grazing Incidence Telescopes: a New Class for Soft X-Ray and EUV Spectroscopy,” Appl. Opt. 23, 3732 (1984).
    [CrossRef] [PubMed]
  14. J. C. Green, S. Bowyer, “Analysis of a New Class of Grazing Incidence Spectroscopic Telescope,” Appl. Opt. 25, 1991 (1986).
    [CrossRef] [PubMed]

1987 (1)

1986 (2)

J. C. Green, S. Bowyer, “Analysis of a New Class of Grazing Incidence Spectroscopic Telescope,” Appl. Opt. 25, 1991 (1986).
[CrossRef] [PubMed]

T. T. Saha, “Transverse Ray Aberrations for Wolter Type I Telescopes,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 10 (1986).

1985 (1)

1984 (1)

1977 (2)

1972 (1)

1971 (1)

H. Wolter, “Estimation of Image Aberrations for X-Ray Telescopes,” Opt. Acta 18, 425 (1971).
[CrossRef]

1970 (1)

1969 (1)

1952 (1)

H. Wolter, “Mirror Systems with Glancing Incidence as Image Producing Optics for X-Rays,” Ann. Phys. 10, 94 (1952).
[CrossRef]

Bowyer, S.

Chase, R. C.

Green, J. C.

Hettrick, M. C.

Korsch, D.

Mangus, J. D.

Saha, T. T.

Underwood, J. H.

Van Speybroeck, L. P.

Welford, W. T.

W. T. Welford, Aberrations of Symmetrical Optical Systems (Academic, London, 1974).

Werner, W.

Winkler, C. E.

Wolter, H.

H. Wolter, “Estimation of Image Aberrations for X-Ray Telescopes,” Opt. Acta 18, 425 (1971).
[CrossRef]

H. Wolter, “Mirror Systems with Glancing Incidence as Image Producing Optics for X-Rays,” Ann. Phys. 10, 94 (1952).
[CrossRef]

Ann. Phys. (1)

H. Wolter, “Mirror Systems with Glancing Incidence as Image Producing Optics for X-Rays,” Ann. Phys. 10, 94 (1952).
[CrossRef]

Appl. Opt. (9)

Opt. Acta (1)

H. Wolter, “Estimation of Image Aberrations for X-Ray Telescopes,” Opt. Acta 18, 425 (1971).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

T. T. Saha, “Transverse Ray Aberrations for Wolter Type I Telescopes,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 10 (1986).

Other (2)

W. T. Welford, Aberrations of Symmetrical Optical Systems (Academic, London, 1974).

W. G. Driscoll, Ed., Handbook of Optics (McGraw-Hill, New York, 1978).

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Figures (4)

Fig. 1
Fig. 1

Cross section of a Wolter type II telescope.

Fig. 2
Fig. 2

Reference sphere and principal ray.

Fig. 3
Fig. 3

Schematic diagram illustrating three ray paths through the telescope.

Fig. 4
Fig. 4

Schematic diagram illustrating three ray paths and the reference sphere.

Tables (3)

Tables Icon

Table I Aberration Coefficientsa

Tables Icon

Table II Design Parameters of Wolter Telescopesa

Tables Icon

Table III Comparison of the rms Spot Radius and rms OPD Values Calculated from Exact Ray Tracing Results and TRA and OPD Polynomials

Equations (62)

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1 / ρ = ( cos θ 1 ) / R 1 ,
1 / q = ( e cos α 1 ) / R 2 ,
1 / r = ( 1 e cos θ ) / R 2 ,
h = 2 f tan ( α / 2 ) ,
z p z d = ( z p 2 h d 2 + 2 η h d cos β d ) 1 / 2 ,
Δ θ = tan δ cos β 1 s 2 + tan 2 δ ( a cos 2 β 1 + b ) ,
Δ β sin θ 1 = tan δ sin β 1 s 2 + tan 2 δ sin β 1 cos β 1 c ,
H x h = q 1 r 1 H x p sin β 1 I 1 1 I 2 ,
H y h = q 1 r 1 H y p cos β 1 I 1 1 I 2 ,
I 1 = ( H x p sin β 1 + H y p cos β 1 ) ( 1 + cos θ 1 cos α 1 ) + H z p sin θ 1 sin α 1 ,
I 2 = 1 r 1 cos α 1 [ ( H x p sin β 1 + H y p cos β 1 ) sin ( θ 1 + α 1 ) H z p cos ( α 1 + θ 1 ) ]
H x p = ρ tan δ sin β cos β ( 1 + cos θ ) ,
H y p = ρ tan δ [ 1 cos 2 β ( 1 + cos θ ) ] ,
H z p = ρ tan δ cos β sin θ .
H + = H y h cos β 1 + H x h sin β 1 ,
H = H y h sin β 1 H x h cos β 1 .
H + = s 1 tan δ cos α 1 { cos β 1 tan δ ( t 1 s 2 c + cos 2 β 1 s 3 T 1 ) + tan 2 δ cos β 1 [ s 2 2 ( 7 t 1 2 4 c 2 1 4 ) + s 2 t 1 2 c 2 + s 2 s 3 ( 1 + 2 t 1 T 1 c ) + s 3 2 T 1 2 ] tan 2 δ cos β 1 sin 2 β 1 s 3 2 T 1 2 } ,
H = s 1 tan δ { sin β 1 tan δ sin β 1 cos β 1 s 3 T 1 + tan 2 δ sin β 1 [ s 2 2 ( t 1 2 4 c 2 1 4 ) + s 2 s 3 ( 1 + t 1 T 1 c ) ] + tan 2 δ sin β 1 cos 2 β 1 s 3 T 1 2 } ,
H h = r d + 1 q 2 u ˆ q 2 ,
h d = ( z p z d ) tan α d ,
Δ α d = tan δ cos β d f ( 1 + t d 2 ) ( 1 h d 1 h 2 d ) ,
Δ β d = tan δ sin β d sin α d f ( 1 + t d 2 ) ( 1 h d 1 h 2 d ) ,
t 1 sin β 1 = t d sin β d ,
t 1 cos β 1 = t d cos β d tan δ f t d ( 1 + t d 2 ) ( 1 h d 1 h 2 d ) ,
t 1 2 = t d 2 tan δ cos β d 2 f t d 2 ( 1 + t d 2 ) ( 1 h d 1 h 2 d ) .
t d = k 2 + k 3 8 + k 5 16 Y 0 cos β d ( k 2 2 z p + 3 k 4 8 z p ) ,
H x h = Y 0 sin 2 β d ( 1 4 k 2 + 1 4 k 4 ) Y 0 2 sin β d [ k ( 1 2 z p + 1 R 1 + 1 R 2 ) + k 3 ( 5 8 z p + 1 R 1 + 1 R 2 + f 2 R 1 R 2 ) + k 5 ( 3 f 8 R 1 R 2 ) + cos 2 β d ( 3 2 z p k 3 ) ] + Y 0 3 sin 2 β d [ k 2 1 2 R 1 R 2 ( f z p R 2 R 1 + R 1 R 2 1 ) + k 4 f 4 R 1 R 2 ( 1 R 2 1 R 1 ) ]
H y h = Y 0 [ 1 + k 2 4 + k 4 8 + cos 2 β d ( k 2 2 + k 4 2 ) ] Y 0 2 cos β d * [ k ( 3 2 z p + 1 R 1 + 1 R 2 ) + k 3 ( 5 8 z p + 1 R 1 + 1 R 2 + f 2 R 1 R 2 ) + k 5 ( 3 f 8 R 1 R 2 ) + ( 1 + cos 2 β d ) ( k 3 3 2 z p ) ] ,
H x h = L q r ( OPD ) x d ,
H y h = L q r ( OPD ) y d ,
A C = ρ cos δ ,
B C = r 1 cos γ,
OPL = D A + A C B C + B F ,
OPL = z cos δ + h cos β sin δ + ρ cos δ + ( q 1 r 1 ) cos γ + q 1 cos γ ( I + I 2 + ) ,
cos γ = 1 tan 2 δ ( s 3 2 / 2 ) ,
I = [ I p 1 sin ( α 1 + θ 1 ) H z p 1 cos ( α 1 + θ 1 ) ] / ( r 1 cos α 1 ) ,
H x p 1 = ρ tan δ [ 1 2 sin 2 β ( 1 + cos θ ) + tan δ sin β sin θ ] ,
H y p 1 = ρ tan δ [ 1 cos 2 β ( 1 + cos θ ) + tan δ cos β sin θ ] ,
H z p 1 = ρ tan δ ( cos β sin θ + tan δ cos θ ) .
OPL = L p + 2 a h + tan δ cos β 1 ( h 1 s 1 tan α 1 ) + tan 2 δ * ( cos 2 β 1 s 1 s 3 T 1 2 L p 2 + s 1 s 2 t 1 T 1 c ρ 1 s 2 a h s 3 2 + s 1 s 3 ) ,
OPL = L p + 2 a h f tan δ cos β d ( k 3 2 + k 5 2 ) + tan 2 δ [ L p 2 R 1 2 f R 1 2 R 2 + k 2 f 2 ( 1 2 z p + 1 2 R 1 + 1 2 R 2 ) + k 4 f 2 ( 5 8 z p + 3 4 R 1 + 3 4 R 2 + 3 f 8 R 1 R 2 ) ] + tan 2 δ cos 2 β d f 2 ( k 2 1 z p + k 4 3 z p ) ,
L q r = z p Y 0 cos β d ( 3 4 k 3 + 5 8 k 5 ) + Y 0 2 [ 1 2 z p + k 2 ( 3 4 z p + 1 R 1 + 1 R 2 ) + k 4 ( 25 32 z p + 1 R 1 + 1 R 2 + f 2 R 1 R 2 ) + cos 2 β ( k 2 3 2 z p + k 4 15 4 z p ) ] .
OPD 1 = Y 0 cos β d ( W 131 h d 3 + W 151 h d 5 ) + y 0 2 [ W 220 h d 2 + W 240 h d 4 + ( W 222 h d 2 + W 242 h d 4 ) cos 2 β d ] .
H x h 1 = z p { Y 0 ( W 131 h d 2 sin 2 β d + 2 W 151 h d 4 sin 2 β d ) + Y 0 2 * sin β d [ 2 W 220 h + 4 W 240 h d 3 + W 242 h d 3 ( 1 + cos 2 β d ) ] } ,
H y h 1 Y 0 = z p { Y 0 [ W 131 h d 2 ( 2 + cos 2 β d ) + W 151 h d 4 ( 3 + 2 cos 2 β d ) ] + Y 0 2 cos β d [ 2 W 220 h d + 2 W 222 h d + 4 W 240 h d 3 + W 242 h d 3 ( 3 + cos 2 β d ) ] } .
OPD = OPD 1 + Y 0 2 W 260 h d 6 + Y 0 3 cos β d ( W 331 h d 3 + W 351 h d 5 ) ,
H x h = H x h 1 z p [ Y 0 2 sin β d 6 W 260 h d 5 + Y 0 3 sin 2 β d × ( W 331 h d 2 + 2 W 351 h d 4 ) ] ,
H y h Y 0 = H y h 1 z p { Y 0 2 cos β d 6 W 260 h d 5 + Y 0 3 [ W 331 h d 2 × ( 2 + cos 2 β d ) + W 351 h d 4 ( 3 + 2 cos 2 β d ) ] } .
h d = z p sin [ 2 arctan ( h 0 2 f ) ] ,
H x h x d = L q r u x q 2 ,
H y h y d = L q r u y q 2 ,
z p z d = L q r u z q 2 ,
H h d sin Δ β d = z p z d u z q 2 ( u y q 2 sin β 1 u x q 2 cos β 1 ) .
H + h d cos β d = z p z d u z q 2 ( u y q 2 cos β 1 + u x q 2 sin β 1 ) .
Δ β d tan α d = H ( 1 z p z d 1 q 1 cos α 1 ) ,
Δ α d = cos 2 α d H + ( 1 z p z d 1 q 1 cos α 1 ) .
Δ β d = tan δ sin β d q d ρ d r d ( 1 h d 1 q d sin α d ) ,
Δ α d = tan δ cos β d q d ρ d r d ( 1 h d 1 q d sin α d ) .
L q r = G F = [ ( H x h x d ) 2 + ( H y h y d ) 2 + ( z p z d ) 2 ] 1 / 2 .
L q r = z p ( 1 + 2 f z p 2 h d tan δ cos β d 2 h d z p 2 H d + + R 2 z p 2 ) 1 / 2 ,
H d + = H x h sin β d + H y h cos β d .
L q r = z p Y 0 cos β d ( 3 k 3 4 + 5 k 5 8 ) + Y 0 2 [ 1 2 z p + k 2 ( 3 4 z p + 1 R 1 + 1 R 2 ) + k 4 ( 25 32 z p + 1 R 1 + 1 R 2 + f 2 R 1 R 2 ) + cos 2 β ( k 2 3 2 z p + k 4 15 4 z ρ ) ] ,

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