Abstract

Geometric optics is used to find the shape of the image and the irradiance cast on a spherical target by a circular beam after reflection by a far off-axis paraboloidal mirror. For moderately large f/No. the image is found to be nearly circular with its center shifted slightly from the beam axis. However, the target irradiance can be highly asymmetric unless the beam intensity falls off rapidly with radius. An expansion in powers of inverse f/No. is used to obtain closed form expressions for the image shape and the target irradiance. Numerical studies are carried out for parameters relevant to the design of a laser fusion reactor. Limits are placed on allowable tilt errors by means of a naive analysis of ray aberrations. A useful formula is derived for the perturbed target irradiance under small tilt errors, based on a new expression for the caustic. These simple formulas allow one to carry out detailed design studies without recourse to ray tracing codes.

© 1979 Optical Society of America

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References

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  1. W. H. Reichelt, D. J. Blevins, W. C. Turner, Los Alamos Scientific Lab. Report LA-UR 77-468.
  2. T. T. Saito, Appl. Opt. 14, 1773 (1975).
    [CrossRef] [PubMed]
  3. R. W. Conn et al., Fusion Research Program Report UWFDM-220, U. Wisconsin (December1977).
  4. J. A. Maniscalco, Lawrence Livermore Lab. Report UCRL-78682 (September1976).
  5. H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).
  6. Ref. 3, Sec. V.C.
  7. S. S. Glaros, A. J. Glass, in Sixth Symposium on Engineering Problems of Fusion Research, San Diego, November 1975 (IEEE, New York, 1976), p. 84.
  8. E. S. Bliss et al., Bull. Am. Phys. Soc. 22, 1158 (1977).
  9. T. F. Stratton et al., in Digest of Topical Meeting on Inertial Confinement Fusion, Feb. 1978, San Diego (Optical Society of America, Washington, D. C., 1978), paper P.TuC7-1.
  10. M. J. Monsler, J. A. Maniscalco, Lawrence Livermore Lab. Report UCRL-79990 (November1977).
  11. J. E. Howard, U. Wisconsin FRP Report UWFDM-238 (March1978).
  12. J. Morgan, Introduction to Geometrical and Physical Optics (McGraw-Hill, New York, 1953), p. 420.
  13. H. J. Stalzer, Appl. Opt. 4, 1205 (1965).
    [CrossRef]
  14. E. J. Guay, Mathematics Dept., U. Wisconsin; private communication.

1977 (1)

E. S. Bliss et al., Bull. Am. Phys. Soc. 22, 1158 (1977).

1975 (1)

1965 (1)

Blevins, D. J.

W. H. Reichelt, D. J. Blevins, W. C. Turner, Los Alamos Scientific Lab. Report LA-UR 77-468.

Bliss, E. S.

E. S. Bliss et al., Bull. Am. Phys. Soc. 22, 1158 (1977).

Brueggemann, H. P.

H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).

Conn, R. W.

R. W. Conn et al., Fusion Research Program Report UWFDM-220, U. Wisconsin (December1977).

Glaros, S. S.

S. S. Glaros, A. J. Glass, in Sixth Symposium on Engineering Problems of Fusion Research, San Diego, November 1975 (IEEE, New York, 1976), p. 84.

Glass, A. J.

S. S. Glaros, A. J. Glass, in Sixth Symposium on Engineering Problems of Fusion Research, San Diego, November 1975 (IEEE, New York, 1976), p. 84.

Guay, E. J.

E. J. Guay, Mathematics Dept., U. Wisconsin; private communication.

Howard, J. E.

J. E. Howard, U. Wisconsin FRP Report UWFDM-238 (March1978).

Maniscalco, J. A.

M. J. Monsler, J. A. Maniscalco, Lawrence Livermore Lab. Report UCRL-79990 (November1977).

J. A. Maniscalco, Lawrence Livermore Lab. Report UCRL-78682 (September1976).

Monsler, M. J.

M. J. Monsler, J. A. Maniscalco, Lawrence Livermore Lab. Report UCRL-79990 (November1977).

Morgan, J.

J. Morgan, Introduction to Geometrical and Physical Optics (McGraw-Hill, New York, 1953), p. 420.

Reichelt, W. H.

W. H. Reichelt, D. J. Blevins, W. C. Turner, Los Alamos Scientific Lab. Report LA-UR 77-468.

Saito, T. T.

Stalzer, H. J.

Stratton, T. F.

T. F. Stratton et al., in Digest of Topical Meeting on Inertial Confinement Fusion, Feb. 1978, San Diego (Optical Society of America, Washington, D. C., 1978), paper P.TuC7-1.

Turner, W. C.

W. H. Reichelt, D. J. Blevins, W. C. Turner, Los Alamos Scientific Lab. Report LA-UR 77-468.

Appl. Opt. (2)

Bull. Am. Phys. Soc. (1)

E. S. Bliss et al., Bull. Am. Phys. Soc. 22, 1158 (1977).

Other (11)

T. F. Stratton et al., in Digest of Topical Meeting on Inertial Confinement Fusion, Feb. 1978, San Diego (Optical Society of America, Washington, D. C., 1978), paper P.TuC7-1.

M. J. Monsler, J. A. Maniscalco, Lawrence Livermore Lab. Report UCRL-79990 (November1977).

J. E. Howard, U. Wisconsin FRP Report UWFDM-238 (March1978).

J. Morgan, Introduction to Geometrical and Physical Optics (McGraw-Hill, New York, 1953), p. 420.

R. W. Conn et al., Fusion Research Program Report UWFDM-220, U. Wisconsin (December1977).

J. A. Maniscalco, Lawrence Livermore Lab. Report UCRL-78682 (September1976).

H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).

Ref. 3, Sec. V.C.

S. S. Glaros, A. J. Glass, in Sixth Symposium on Engineering Problems of Fusion Research, San Diego, November 1975 (IEEE, New York, 1976), p. 84.

E. J. Guay, Mathematics Dept., U. Wisconsin; private communication.

W. H. Reichelt, D. J. Blevins, W. C. Turner, Los Alamos Scientific Lab. Report LA-UR 77-468.

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Figures (20)

Fig. 1
Fig. 1

Parabolic mirror imaging geometry. A circular beam is shown incident from the right, parallel to the mirror symmetry axis 00′. The effective focal length of the off-axis mirror is r0 at the principal ray height R0. In practice the parameter a is determined by r0 and the turning angle ϕ0. The effective off-axis f/No. depends on r0 and the beam radius ρB, independent of ϕ0.

Fig. 2
Fig. 2

Image of a circular beam, in a plane normal to r0 at distance r 0 * < r 0 from focus 0′. ρ 1 and ρ 2 are the limits of the image contour in the median plane.

Fig. 3
Fig. 3

Transformation to beam-centered coordinates. The z axis has been rotated through the angle ϕ0 in the median plane. The new z axis z ˜ coincides with the beam axis along r0, while the y axis is unchanged.

Fig. 4
Fig. 4

Image contour for large f/No. To lowest order in the image is a circle of radius , with the beam axis shifted by an amount Δξ in the −ϕ direction.

Fig. 5
Fig. 5

Irradiance profiles for a flat beam profile and focal length r0 = 5ρB. Profiles are shown for turning angles ϕ0 = 45°, 90°, and 135° in the median plane.

Fig. 6
Fig. 6

Irradiance profiles for a Gaussian beam falling off by one e-folding.

Fig. 7
Fig. 7

Irradiance profiles for a Gaussian beam falling off by two e-foldings.

Fig. 8
Fig. 8

Geometry of a ray when the parabolic mirror is tilted through δϕ in the median plane (rotation about the ym axis). Perturbed quantities are denoted by primes.

Fig. 9
Fig. 9

Tilt error criterion. The location of the perturbed ray is limited to the angle αmax measured from the target center.

Fig. 10
Fig. 10

Tilt error geometry for rotation of the mirror through δζ about the z axis, coincident with the beam axis (roll).

Fig. 11
Fig. 11

Tilt error geometry for rotation of the mirror through δη about the xm axis (yaw).

Fig. 12
Fig. 12

Details of rotation about the x axis. The normal n ^ lies in the xm-zm plane, while the perturbed normal n ^ lies in the x m - z m plane.

Fig. 13
Fig. 13

Normalized roll, pitch, and yaw tolerances as a function of turning angle ϕ0.

Fig. 14
Fig. 14

Perturbed target irradiance geometry. Two neighboring rays intersect at point 0″, which lies on the caustic. Points on the caustic are given in polar coordinates (l,χ), as described in the Appendix.

Fig. 15
Fig. 15

Perturbed target irradiance for turning angle ϕ0 = 45° and small tilt errors. Note that the range of β decreases with increasing δϕ. Flat beam profiles are assumed; a Gaussian profile (α = 2) is used in one case to show its strong smoothing effect. The unperturbed irradiance IT varies by 18% for a flat beam profile.

Fig. 16
Fig. 16

Perturbed target irradiance for turning angle ϕ0 = 90° and small tilt errors. In this case IT varies by 49%.

Fig. 17
Fig. 17

Perturbed target irradiance in the vicinity of the caustic, with ϕ0 = 90° and δϕ* = 35.4 μrad. The discontinuities are due to overlapping ray bundles. The spikes at βmax would be strongly modified by diffraction.

Fig. 18
Fig. 18

Mirror-target geometry with the incident ray at a small angle δϕ with the mirror axis. This convention is chosen to agree with the literature on caustics.

Fig. 19
Fig. 19

Full caustic curve for a paraboloidal mirror.

Fig. 20
Fig. 20

Caustic for paraboloidal mirror superimposed upon a 1-mm radius target. The two separate branches correspond to positive and negative tilts using the upper half-mirror alone. Note that since the latus rectum is held constant here, the focal length will vary with ϕ. In the numerical examples in Sec. V, r0 is held constant and a varied.

Equations (72)

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R 2 = 4 a ( a - z )
r = 2 a / ( 1 + cos ϕ ) = a sec 2 ( ϕ / 2 ) ,
R = 2 a sin ϕ 1 + cos ϕ = 2 a tan ( ϕ / 2 ) .
R 2 = R 0 2 + ρ B 2 - 2 R 0 ρ B cos Ψ ,
sin θ = ρ B R sin Ψ .
x ˜ = x cos ϕ 0 - z sin ϕ 0 = r sin ξ cos ( π - Ψ ) , y ˜ = y = r sin ξ sin ( π - Ψ ) z ˜ = x sin ϕ 0 + z cos ϕ 0 = r cos ξ ,
x = r sin ϕ cos θ , y = r sin ϕ sin θ , z = r cos ϕ .
cos ξ = sin ϕ 0 sin ϕ cos θ + cos ϕ 0 cos ϕ ,
tan ψ = sin ϕ sin θ sin ϕ 0 cos ϕ - cos ϕ 0 sin ϕ cos θ ,
sin ξ sin ψ = sin ϕ sin θ .
ρ B / r 0 .
ψ = ψ + 1 2 sin ψ tan ( ϕ 0 / 2 ) + 0 ( 2 ) ,
ξ = + 1 2 2 cos ψ tan ( ϕ 0 / 2 ) + 0 ( 3 )
ξ 2 sin 2 ψ + ( ξ cos ψ - Δ ξ ) 2 = 2 + 0 ( 4 ) ,
Δ ξ 1 2 2 tan ( ϕ 0 / 2 ) .
ξ 1 , 2 = [ 1 ± 1 2 tan ( ϕ 0 / 2 ) ] .
ξ 1 , 2 = [ 1 1 2 tan ( ϕ 0 / 2 ) ] - 1 .
F = 1 2 cot ξ ¯ 1 / 2 ,
tan ξ 1 = ( 1 - / 2 1 - ) ,
tan ξ 2 = ( 1 + / 2 1 + ) ,
I T d A T = I B d A B ,
d A B = R d R d θ ,
d A T = R T 2 sin ϕ d ϕ d θ .
I T = I B ( ρ ) R d R R T 2 sin ϕ d ϕ ,
sin ϕ d R = R d ϕ ,
I T = ( 2 a R T ) 2 I B ( ρ ) ( 1 + cos ϕ ) 2 .
I B ( ρ ) = I 0 exp [ - α ( ρ / ρ B ) 2 ]
I T ( ξ ) [ 1 + 2 t ξ + 1 2 ( 1 + 5 t 2 ) ξ 2 ] exp [ - α ( ξ / ) 2 ( 1 + t ξ ) ] ,
δ z = r sin ( 2 δ ϕ ) sin ϕ 2 r δ ϕ sin ϕ .
sin α = δ z R T sin ϕ 2 r δ ϕ R T .
β = ϕ - α ϕ - α ,             r R T .
δ z = δ z 0 [ 1 + ( 1 - 2 cos ϕ 0 sin ϕ 0 ) Δ ϕ ] ,
δ ϕ R T sin α max 2 r 0 ,
δ ϕ max = R T / 2 r 0 .
δ y R 0 δ ζ = r 0 sin ϕ 0 δ ζ .
δ ζ R T sin α max r 0 sin ϕ 0 .
r ^ = r ^ + 2 cos ( ϕ / 2 ) n ^ .
n ^ = [ sin ( ϕ / 2 ) , δ η cos ( ϕ / 2 ) , cos ( ϕ / 2 ) ] .
r ^ = [ sin ϕ , 2 δ η cos 2 ( ϕ / 2 ) , cos ϕ ] .
δ r m r ( r ^ - r ^ )
δ x m = δ z m = 0 , δ y m = r ( 1 + cos ϕ ) δ η = 2 a δ η .
δ y = R T sin α ,
δ η R T sin α max r 0 ( 1 + cos ϕ 0 ) ,
δ ϕ R T sin α max 2 r 0 , δ ζ = 2 δ ϕ sin ϕ 0 , δ η = 2 δ ϕ 1 + cos ϕ 0 .
I T d A T = I B d A B
d A T = ( R T d β ) ( l 1 sin ϕ ) d θ ,
I T = I B R 2 d ϕ / d β R T l 1 sin 2 ϕ ,
l 1 sec α d ϕ = R T d β ,
l 1 sin ( ϕ - χ ) = R T sin ( β - χ ) .
χ = ( 3 / 2 ) ϕ - π / 2.
I T = I T cos α cos 2 ( ϕ / 2 ) cos 2 ( α + ϕ / 2 ) ,
sin α = 2 r δ ϕ R T = 2 a δ ϕ R T sec 2 ( ϕ / 2 ) .
δ ϕ = R T 2 a cos 3 ( ϕ / 2 ) ,
δ ϕ * R T 2 r 0 cos ( ϕ 0 / 2 ) .
β = ϕ - sin - 1 [ A sec 2 ( ϕ / 2 ) ] ,
A 2 a δ ϕ R T = 2 r 0 δ ϕ R T cos 2 ( ϕ 0 / 2 ) .
cos ( ϕ ˜ / 2 ) = ( δ ϕ δ ϕ * ) 1 / 3 cos ( ϕ 0 / 2 ) .
β max = ( 3 / 2 ) ϕ ˜ - π / 2 ,
ϕ ϕ ˜ ± [ 2 3 sin ϕ ˜ ( β max - β ) ] 1 / 2 ,
R 2 = 4 a z
R - R 0 = - ( z - z 0 ) tan ( ϕ + δ ϕ ) ,
F ( R , z , ϕ , δ ϕ ) = R - 2 tan ( ϕ / 2 ) + [ z - tan 2 ( ϕ / 2 ) ] tan ( ϕ + δ ϕ ) = 0 ,
F / ϕ = 0 ,
cos 2 ( ϕ / 2 ) [ z - tan 2 ( ϕ / 2 ) ] = cos 2 ( ϕ + δ ϕ ) [ 1 + tan ( ϕ / 2 ) tan [ ϕ + δ ϕ ) ] ,
R = 2 tan ( ϕ / 2 ) - [ z - tan 2 ( ϕ / 2 ) ] tan ( ϕ + δ ϕ ) .
27 t ( Δ z - R t ) 2 = ( R + z t ) ( R - 8 t + t Δ z ) 2 ,
R 2 ( 1 - 2 cos ϕ ) 1 + cos ϕ δ ϕ ,
Δ z - 2 sin ϕ ( 1 + 2 cos ϕ ) ( 1 + cos ϕ ) 2 δ ϕ .
Δ z - 3 9 ( 8 - R ) ( 1 + R ) 1 / 2 ,
l = a δ ϕ sec 3 ( ϕ / 2 ) ;
χ = ( 3 / 2 ) ϕ - π / 2 ,
l = a δ ϕ sec 3 ( χ 3 + π 6 ) .

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