Abstract

Formulas for the wave aberrations introduced into a beam by a prism or a plane grating are derived from the theory of plane symmetric systems. Emphasis is made on the field-dependent aberrations.

© 2000 Optical Society of America

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References

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  1. R. E. Hopkins, “Mirror and prism systems,” Military Standardization Handbook 141 (Defense Supply Agency, Washington, D.C., 1962).
  2. J. W. Howard, “Formulas for the coma and astigmatism of wedge prisms used in converging light,” Appl. Opt. 24, 4265–4268 (1985).
    [CrossRef] [PubMed]
  3. C. J. Barth, D. Oepts, “Stigmatic and coma-free imaging with a thick prism: comparison of third-order theory and ray-tracing results,” Appl. Opt. 27, 3838–3844 (1988).
    [CrossRef] [PubMed]
  4. W. Mao, Y. Xu, “Distortion of optical wedges with a large angle of incidence in a collimated beam,” Opt. Eng. 33, 580–585 (1999).
    [CrossRef]
  5. J. M. Sasian, “How to approach the design of a bilateral symmetrical optical system,” Opt. Eng. 33, 2045–2061 (1994).
    [CrossRef]
  6. W. T. Welford, “Thin lens aberrations,” Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1989), pp. 234–235.
  7. W. C. Sweatt, “Describing holographic and optical elements as lenses,” J. Opt. Soc. Am. 67, 803–808 (1977).
    [CrossRef]

1999

W. Mao, Y. Xu, “Distortion of optical wedges with a large angle of incidence in a collimated beam,” Opt. Eng. 33, 580–585 (1999).
[CrossRef]

1994

J. M. Sasian, “How to approach the design of a bilateral symmetrical optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

1988

1985

1977

Barth, C. J.

Hopkins, R. E.

R. E. Hopkins, “Mirror and prism systems,” Military Standardization Handbook 141 (Defense Supply Agency, Washington, D.C., 1962).

Howard, J. W.

Mao, W.

W. Mao, Y. Xu, “Distortion of optical wedges with a large angle of incidence in a collimated beam,” Opt. Eng. 33, 580–585 (1999).
[CrossRef]

Oepts, D.

Sasian, J. M.

J. M. Sasian, “How to approach the design of a bilateral symmetrical optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Sweatt, W. C.

Welford, W. T.

W. T. Welford, “Thin lens aberrations,” Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1989), pp. 234–235.

Xu, Y.

W. Mao, Y. Xu, “Distortion of optical wedges with a large angle of incidence in a collimated beam,” Opt. Eng. 33, 580–585 (1999).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

W. Mao, Y. Xu, “Distortion of optical wedges with a large angle of incidence in a collimated beam,” Opt. Eng. 33, 580–585 (1999).
[CrossRef]

J. M. Sasian, “How to approach the design of a bilateral symmetrical optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Other

W. T. Welford, “Thin lens aberrations,” Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1989), pp. 234–235.

R. E. Hopkins, “Mirror and prism systems,” Military Standardization Handbook 141 (Defense Supply Agency, Washington, D.C., 1962).

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Figures (3)

Fig. 1
Fig. 1

(a) Top view of the prism showing the trajectory of the OAR, the apex angle α, the prism thickness t along the OAR, the angle of incidence I 1 of the OAR in the first surface, and the angle of refraction I 2′ of the OAR in the second surface. (b) Side view of the prism showing the marginal and the chief rays, their slope angle u and , and their height y and at the prism middle plane.

Fig. 2
Fig. 2

Spherical mirror (far left) and a plane grating (middle). Note the tilted image plane (far right).

Fig. 3
Fig. 3

Grid distortion for the spherical mirror and the grating configuration. Distortion is indicated by the crosses against the ideal location indicated by the grid corners. (a) As the observation plane is tilted -20°, negative keystone is generated. (b) At the sagittal image plane there is no keystone distortion but a small amount of quadratic distortion I. (c) As the observation plane is tilted +20° from the sagittal one, positive keystone distortion is generated.

Tables (3)

Tables Icon

Table 1 Wave Aberrations and Their Field and Aperture Dependence

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Table 2 Conditions for the Absence of Aberrations

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Table 3 Comparison of Theory and Ray-Tracing Results for a Grating

Equations (20)

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WCte._lateral_chromatic=-1-nn1νsin I1-sin I2y-12sin I1+sin I2utn,WCte._Astigmatism=-121-n2n2 usin2 I1-sin2 I2y-12sin2 I1+sin2 I2utn,WAnamorphism=-1-n2n2 u¯sin2 I1-sin2 I2y-12sin2 I1+sin2 I2utn+121-n2n2 Ψsin2 I1-sin2 I2,WCte._coma=-121-n2n2 u2sin I1-sin I2y-12sin I1+sin I2utn,WLinear_astigmatism=-1-n2n2 uu¯sin I1-sin I2y-12sin I1+sin I2utn,WField_tilt=-121-n2n2 uu¯sin I1-sin I2y-12sin I1+sin I2utn+121-n2n2 uΨsin I1-sin I2,WQuadratic_distortion_I=-121-n2n2 u¯2sin I1-sin I2y-12sin I1+sin I2utn,WQuadratic_distortion_II=-1-n2n2 u¯2sin I1-sin I2y-12sin I1+sin I2utn+1-n2n2 u¯Ψsin I1-sin I2.
WSpherical_aberration=- 18n2-1n3 u4t,  WLinear_coma=4 u¯u WSpherical_aberration,  WQuadratic_astigmatism=4 u¯u2WSpherical_aberration,  WPetzval_field_curvature=0,  WDistortion=4u¯u3WSpherical_aberration,  WLongitudinal_chromatic=-12n-1n21ν u2t,  WLateral_chromatic=-n-1n21ν uu¯t.
Q1=sin I1-sin I2y-12sin I1+sin I2utn,
Q2=sin2 I1-sin2 I2y-12sin2 I1+sin2 I2utn.
y=-12sin I1+sin I2utnsin I1-sin I2  for Q1,
y=-12sin2 I1+sin2 I2utnsin2 I1-sin2 I2  for Q2.
δn-1αn-1nsin I1-sin I2.
WCte._lateral_chromatic=-1-nn1νsin I1-sin I2y,WCte._Astigmatism=-121-n2n2 usin2 I1-sin2 I2y,WAnamorphism=-1-n2n2 u¯sin2 I1-sin2 I2y+121-n2n2 Ψsin2 I1-sin2 I2,WCte._coma=-121-n2n2 u2sin I1-sin I2y,WLinear_astigmatism=-1-n2n2 uu¯sin I1-sin I2y,WField_tilt=-121-n2n2 uu¯sin I1-sin I2y+121-n2n2 uΨsin I1-sin I2,WQuadratic_distortion_I=-121-n2n2 u¯2sin I1-sin I2y,WQuadratic_distortion_II=-1-n2n2 u¯2sin I1-sin I2y+1-n2n2 u¯Ψsin I1-sin I2.
WCte._lateral_chromatic=-λf-λcmd y,WCte._Astigmatism=-12 usin I1+sin I2mλd y,WAnamorphism=-u¯sin I1+sin I2mλd y+12sin I1+sin I2mλd Ψ,WCte._coma=-12 u2mλd y,WLinear_astigmatism=-uu¯ mλd y,WField_tilt=-12 uu¯ mλd y+12 uΨ mλd=-12 u2mλd y¯,WQuadratic_distortion_I=-12 u¯2mλd y,WQuadratic_distortion_II=-u¯2mλd y+u¯Ψ mλd=-u¯uy¯ mλd.
tan θ=i1+nn δ=i δ+iδn,
12Ψu tan θ-u tan θ=WField_tilt.
ΔWQuadratic_distortion_II=-Ψu¯ tan θ-u¯ tan θ.
ΔWQuadratic_distortion_II=-2 u¯u WField_tilt-Ψu¯ uu-u¯tan θ.
WQuadratic_distortion_II=2 u¯u WField_tilt,
WQuadratic_distortion_II+ΔWQuadratic_distortion_II=0.
I1=0.0°,  I2=10.0865°,  u=-0.05,  u¯=0.0875,  y=25 mm,  y¯=43.7443 mm.
Δuny¯=Δu¯ny+ΨΔ1n2,
An-1=cos Icos I-1-12 sin2 IΔ1n2.
JV+JIIIy¯y+JIVy¯y+JIIy¯2y2=-12 n sin IA¯Δu¯ny.
2JIIy¯2y2+JIIIy¯y=-n sin IA¯Δu¯n y+n sin IA¯ΨΔ1n2.

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