Abstract

We derive the relationship between Coddington’s equations and the Gaussian curvature for a stigmatic reflective imaging system. This relationship allows parameterizing off-axis conic optical systems using traditional first-order optics by considering the effective curvature at the center of the off-axis sections. Specifically, we demonstrate parameterizing the system requirements of a 2× achromatic image relay for a terawatt laser system. This system required both collimation (far-field) and pupil imaging (near-field) simultaneously. Long working distances and specific spatial constraints limited the available layout options for the imaging components. By parameterizing these system requirements and packaging constraints, the final specifications could be quickly iterated, while allowing for flexibility in the layout of the system during a multi-year conceptual period.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. M. Rodgers, Proc. SPIE 4832, 33 (2002).
    [Crossref]
  2. D. Reshidko and J. Sasian, Opt. Eng. 57, 1 (2018).
    [Crossref]
  3. S.-W. Bahk, J. Bromage, and J. D. Zuegel, Opt. Lett. 39, 1081 (2014).
    [Crossref]
  4. Mssr. de Berce, Supplement du Journal des Sçavans (1672–1674), pp. 70–71.
  5. A. Baranne and F. Launay, J. Opt. 28, 158 (1997).
    [Crossref]
  6. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).
  7. R. Kingslake, Opt. Photon. News 5(8), 20 (1994).
    [Crossref]
  8. H. Coddington, A Treatise on the Reflexion and Refraction of Light; Being Part I. of a System of Optics (Cambridge University, 1829).
  9. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 191.
  10. S. Krivoshapko and V. N. Ivanov, Encyclopedia of Analytical Surfaces (Springer, 2015).
  11. A. Pressley, Elementary Differential Geometry (Springer, 2001).
  12. D. Y. Wang, D. M. Aikens, and R. E. English, Opt. Eng. 39, 1788 (2000).
    [Crossref]

2018 (1)

D. Reshidko and J. Sasian, Opt. Eng. 57, 1 (2018).
[Crossref]

2014 (1)

2002 (1)

J. M. Rodgers, Proc. SPIE 4832, 33 (2002).
[Crossref]

2000 (1)

D. Y. Wang, D. M. Aikens, and R. E. English, Opt. Eng. 39, 1788 (2000).
[Crossref]

1997 (1)

A. Baranne and F. Launay, J. Opt. 28, 158 (1997).
[Crossref]

1994 (1)

R. Kingslake, Opt. Photon. News 5(8), 20 (1994).
[Crossref]

Aikens, D. M.

D. Y. Wang, D. M. Aikens, and R. E. English, Opt. Eng. 39, 1788 (2000).
[Crossref]

Bahk, S.-W.

Baranne, A.

A. Baranne and F. Launay, J. Opt. 28, 158 (1997).
[Crossref]

Bromage, J.

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Coddington, H.

H. Coddington, A Treatise on the Reflexion and Refraction of Light; Being Part I. of a System of Optics (Cambridge University, 1829).

English, R. E.

D. Y. Wang, D. M. Aikens, and R. E. English, Opt. Eng. 39, 1788 (2000).
[Crossref]

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Ivanov, V. N.

S. Krivoshapko and V. N. Ivanov, Encyclopedia of Analytical Surfaces (Springer, 2015).

Kingslake, R.

R. Kingslake, Opt. Photon. News 5(8), 20 (1994).
[Crossref]

Krivoshapko, S.

S. Krivoshapko and V. N. Ivanov, Encyclopedia of Analytical Surfaces (Springer, 2015).

Launay, F.

A. Baranne and F. Launay, J. Opt. 28, 158 (1997).
[Crossref]

Pressley, A.

A. Pressley, Elementary Differential Geometry (Springer, 2001).

Reshidko, D.

D. Reshidko and J. Sasian, Opt. Eng. 57, 1 (2018).
[Crossref]

Rodgers, J. M.

J. M. Rodgers, Proc. SPIE 4832, 33 (2002).
[Crossref]

Sasian, J.

D. Reshidko and J. Sasian, Opt. Eng. 57, 1 (2018).
[Crossref]

Wang, D. Y.

D. Y. Wang, D. M. Aikens, and R. E. English, Opt. Eng. 39, 1788 (2000).
[Crossref]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 191.

Zuegel, J. D.

J. Opt. (1)

A. Baranne and F. Launay, J. Opt. 28, 158 (1997).
[Crossref]

Opt. Eng. (2)

D. Y. Wang, D. M. Aikens, and R. E. English, Opt. Eng. 39, 1788 (2000).
[Crossref]

D. Reshidko and J. Sasian, Opt. Eng. 57, 1 (2018).
[Crossref]

Opt. Lett. (1)

Opt. Photon. News (1)

R. Kingslake, Opt. Photon. News 5(8), 20 (1994).
[Crossref]

Proc. SPIE (1)

J. M. Rodgers, Proc. SPIE 4832, 33 (2002).
[Crossref]

Other (6)

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

H. Coddington, A Treatise on the Reflexion and Refraction of Light; Being Part I. of a System of Optics (Cambridge University, 1829).

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 191.

S. Krivoshapko and V. N. Ivanov, Encyclopedia of Analytical Surfaces (Springer, 2015).

A. Pressley, Elementary Differential Geometry (Springer, 2001).

Mssr. de Berce, Supplement du Journal des Sçavans (1672–1674), pp. 70–71.

Supplementary Material (2)

NameDescription
» Data File 1       Required off-axis parameters for the long and short image relay. Used to compute the axial parameters given in Data File 2.
» Data File 2       Computed axial parameters corresponding to the required off-axis parameters in Data File 1. These parameters describe the designs in Figs. 4 and 5.

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Figures (5)

Fig. 1.
Fig. 1. Layout of a back-to-back Cassegrain relay with an off-axis chief ray traced through the system. The intersection points of the ray with each n th mirror are labeled P n . The off-axis distances are labeled d n . The vertices are labeled V n , and the stigmatic imaging points of each mirror are labeled S n . The stigmatic imaging points overlap for each successive mirror to stigmatically relay between a collimated object space to a collimated image space.
Fig. 2.
Fig. 2. Geometry of a hyperboloid used as a Cartesian reflector. The incoming ray (solid red) is directed toward the point F , the front focal point of the hyperboloid. The point A is the point of intersection of the ray and the surface. The dashed blue line is the surface normal at A . C is the geometric center of the hyperboloid. R S is the local sagittal RoC.
Fig. 3.
Fig. 3. GCC layout showing locations of the N5 crystal plane and G4 grating plane. The available space for the final design of the AIR is shown. All optics between N5 and the AIR are flat mirrors. All optics between the AIR and G4 are flat optics with regard to imaging.
Fig. 4.
Fig. 4. Layout of the “long” relay, with paraboloids in green and hyperboloids in red. See Data File 1 for off-axis requirements and Data File 2 for the computed axial values.
Fig. 5.
Fig. 5. Layout of the “short” relay, with paraboloids in green and hyperboloids in red. See Data File 1 for off-axis requirements and Data File 2 for the computed axial values.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

n s n s = n cos I n cos I R s ,
n cos 2 I t n cos 2 I t = n cos I n cos I R t ,
1 + 1 = 2 cos I R s ,
1 + 1 = 2 R t cos I .
R s R t = cos 2 I .
1 + 1 = 2 R eff = 2 cos I R s = 2 R t cos I .
R eff = R s cos I = R t cos I .
K G = κ 1 κ 2 = 1 R s R t = cos 2 ( I ) R s 2 = 1 R t 2 cos 2 ( I ) .
R eff = ± 1 K G .
K G = F x x · F y y F x y 2 ( 1 + F x 2 + F y 2 ) 2 ,
z ( r ) = a [ 1 + r 2 a R v 1 ] ,
K G = [ a R v r 2 R v + a ( r 2 + R v 2 ) ] 2 .
R v ( r ; a , R eff ) = a R eff r 2 + ( a R eff r 2 ) 2 4 a 2 r 2 2 a .
R v = r cos I sin η .
R s = r sin ( η ) = r sin ( α β 2 ) .
R v = r cos I sin η = R s cos I = R eff cos 2 I .

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