Abstract

A simple expression is given for the k-function associated with the general solution of Stavroudis to the eikonal equation for refraction or reflection of a plane wave from an arbitrary surface. Using this result, we specialize the solution to derive analytic expressions for the wavefront and caustic surfaces after refraction of a plane wave from any rotationally symmetric surface. The method is applied to evaluating and comparing the wavefront and caustic surfaces formed both by a planospherical lens and a planoaspheric lens used for laser beam shaping, which provides understanding of how the irradiance is redistributed over a beam as the wavefront folds back on itself within the focal region.

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  2. M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Pure and Applied Mathematics, Interscience Publishers, Wiley, 1965).
  3. S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, 1986).
  4. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH Verlag, 2006).
    [CrossRef]
  5. D. L. Shealy, “Geometrical optics: some applications of the law of intensity,” Proc. SPIE 6289, 62890F-1-16 (2006).
  6. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).
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    [CrossRef]
  8. O. N. Stavroudis, R. C. Fronczek, and R.-S. Chang, “Geometry of the half-symmetric image,” J. Opt. Soc. Am. 68, 739-742 (1978).
    [CrossRef]
  9. O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface,” J. Opt. Soc. Am. A 12, 1010-1016 (1995).
    [CrossRef]
  10. J. A. Hoffnagle and D. L. Shealy, “Caustic surfaces of a keplerian two-lens beam shaper,” Proc. SPIE 6663, 666304-1-9 (2007).
  11. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper,” Proc. SPIE 6668, 666805-1-11 (2007).
  12. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897-909 (1981).
    [CrossRef] [PubMed]
  13. E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry, Mathematical Exposition No. 16 (University of Toronto Press, 1968), p. 87.
  14. G. W. Forbes and M. A. Alonso, “The holy grail of ray-based optical modelling,” Proc. SPIE 4832, 186-197 (2002).
    [CrossRef]
  15. H. Guo and X. Deng, “Differential geometrical methods in the study of optical transmission (scalar theory). I Static transmission case,” J. Opt. Soc. Am. A 12, 600-606 (1995).
    [CrossRef]
  16. L. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  17. H. Römer, Theoretical Optics, An Introduction (Wiley-VCH Verlag, 2005).
  18. J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” US Patent 3,476,463, November 4, 1969.
  19. P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545-3553 (1980).
    [CrossRef] [PubMed]
  20. D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R.Driggers, ed. (Taylor and Francis, 2006). DOI: 10.1081/E-E0E-120029768, ISBN: 0-8247-0940-3 (paper), 0-8247-0939-X (electronic).
  21. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39, 5488-5499 (2000).
    [CrossRef]
  22. J. Hoffnagle and C. Jefferson, “Refractive optical system that converts a laser beam to a collimated flat-top beam,” US Patent 6,295,168, September 25, 2001.
  23. J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
    [CrossRef]

2007

J. A. Hoffnagle and D. L. Shealy, “Caustic surfaces of a keplerian two-lens beam shaper,” Proc. SPIE 6663, 666304-1-9 (2007).

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper,” Proc. SPIE 6668, 666805-1-11 (2007).

2006

D. L. Shealy, “Geometrical optics: some applications of the law of intensity,” Proc. SPIE 6289, 62890F-1-16 (2006).

2003

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
[CrossRef]

2002

G. W. Forbes and M. A. Alonso, “The holy grail of ray-based optical modelling,” Proc. SPIE 4832, 186-197 (2002).
[CrossRef]

2000

1995

1981

1980

1978

1976

Alonso, M. A.

G. W. Forbes and M. A. Alonso, “The holy grail of ray-based optical modelling,” Proc. SPIE 4832, 186-197 (2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Burkhard, D. G.

Chang, R.-S.

Crosignani, B.

S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, 1986).

Deng, X.

DiPorto, P.

S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, 1986).

Forbes, G. W.

G. W. Forbes and M. A. Alonso, “The holy grail of ray-based optical modelling,” Proc. SPIE 4832, 186-197 (2002).
[CrossRef]

Fronczek, R. C.

Guo, H.

Hoffnagle, J.

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
[CrossRef]

J. Hoffnagle and C. Jefferson, “Refractive optical system that converts a laser beam to a collimated flat-top beam,” US Patent 6,295,168, September 25, 2001.

Hoffnagle, J. A.

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper,” Proc. SPIE 6668, 666805-1-11 (2007).

J. A. Hoffnagle and D. L. Shealy, “Caustic surfaces of a keplerian two-lens beam shaper,” Proc. SPIE 6663, 666304-1-9 (2007).

J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39, 5488-5499 (2000).
[CrossRef]

D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R.Driggers, ed. (Taylor and Francis, 2006). DOI: 10.1081/E-E0E-120029768, ISBN: 0-8247-0940-3 (paper), 0-8247-0939-X (electronic).

Jefferson, C.

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
[CrossRef]

J. Hoffnagle and C. Jefferson, “Refractive optical system that converts a laser beam to a collimated flat-top beam,” US Patent 6,295,168, September 25, 2001.

Jefferson, C. M.

Kay, I.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Pure and Applied Mathematics, Interscience Publishers, Wiley, 1965).

Kline, M.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Pure and Applied Mathematics, Interscience Publishers, Wiley, 1965).

Kreuzer, J. L.

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” US Patent 3,476,463, November 4, 1969.

Kreyszig, E.

E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry, Mathematical Exposition No. 16 (University of Toronto Press, 1968), p. 87.

Luneburg, L.

L. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

Rhodes, P. W.

Römer, H.

H. Römer, Theoretical Optics, An Introduction (Wiley-VCH Verlag, 2005).

Shealy, D. L.

J. A. Hoffnagle and D. L. Shealy, “Caustic surfaces of a keplerian two-lens beam shaper,” Proc. SPIE 6663, 666304-1-9 (2007).

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper,” Proc. SPIE 6668, 666805-1-11 (2007).

D. L. Shealy, “Geometrical optics: some applications of the law of intensity,” Proc. SPIE 6289, 62890F-1-16 (2006).

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897-909 (1981).
[CrossRef] [PubMed]

P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545-3553 (1980).
[CrossRef] [PubMed]

D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R.Driggers, ed. (Taylor and Francis, 2006). DOI: 10.1081/E-E0E-120029768, ISBN: 0-8247-0940-3 (paper), 0-8247-0939-X (electronic).

Solimeno, S.

S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, 1986).

Stavroudis, O. N.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
[CrossRef]

Proc. SPIE

G. W. Forbes and M. A. Alonso, “The holy grail of ray-based optical modelling,” Proc. SPIE 4832, 186-197 (2002).
[CrossRef]

D. L. Shealy, “Geometrical optics: some applications of the law of intensity,” Proc. SPIE 6289, 62890F-1-16 (2006).

J. A. Hoffnagle and D. L. Shealy, “Caustic surfaces of a keplerian two-lens beam shaper,” Proc. SPIE 6663, 666304-1-9 (2007).

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper,” Proc. SPIE 6668, 666805-1-11 (2007).

Other

E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry, Mathematical Exposition No. 16 (University of Toronto Press, 1968), p. 87.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Pure and Applied Mathematics, Interscience Publishers, Wiley, 1965).

S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, 1986).

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH Verlag, 2006).
[CrossRef]

L. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

H. Römer, Theoretical Optics, An Introduction (Wiley-VCH Verlag, 2005).

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” US Patent 3,476,463, November 4, 1969.

D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R.Driggers, ed. (Taylor and Francis, 2006). DOI: 10.1081/E-E0E-120029768, ISBN: 0-8247-0940-3 (paper), 0-8247-0939-X (electronic).

J. Hoffnagle and C. Jefferson, “Refractive optical system that converts a laser beam to a collimated flat-top beam,” US Patent 6,295,168, September 25, 2001.

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Figures (9)

Fig. 1
Fig. 1

Refraction of plane wave at a surface. The vector r locates points on the refracted wavefront, when the origin O of the coordinate system is located at the vertex of the lens surface with the optical axis and along refracted ray vector S 2 . The point x is the point of refraction on the lens surface, S 1 is the incident ray vector, and N is the unit normal vector to the refracting surface.

Fig. 2
Fig. 2

Schematic drawing of the refracting surface x showing the incident ray S 1 = n 1 e z , the refracted ray S 2 = n 2 2 w 2 e r + w e z , and the wavefront W when the origin O of the coordinate system is located at the vertex of refracting surface with the optical axis.

Fig. 3
Fig. 3

Geometry of a general refracting surface illuminated with collimated light. The point r 0 indicates the intersection of the incident ray with the reference plane, z = 0 in this case. The point x is the point of refraction, the S 1 is the incident ray vector, S 2 is the refracted ray vector, and the vector r locates a point of a wavefront.

Fig. 4
Fig. 4

Schematic drawing of a concave aspheric mirror x showing the incident ray A 1 = e z and the reflected ray A 2 = sin ( 2 α ) e r + cos ( 2 α ) e z with the origin O of the coordinate system located at the vertex of the mirror with the optical ( z ) axis.

Fig. 5
Fig. 5

(a) Wavefront (solid curves) and caustic (dashed curves) surfaces formed by a spherical surface with n 1 = 1.5 , n 2 = 1 , and unit radius. The paraxial (Gaussian) focus is at z = 2 . The wavefronts are calculated by simple ray tracing and the caustics are calculated by using Eqs. (44a, 44b). (b) (Inset) Expanded view of tangential caustic (dashed curves) and three different, closely spaced wavefronts (solid curves) with the propagation distances of s = 1.495 , 1.5 , 1.507 . The solid circles on the figure indicate approximate locations where three rays from different parts of the input aperture intersect.

Fig. 6
Fig. 6

Scaled drawing of a Keplerian beam shaper that transforms a Gaussian beam with a waist of w 0 = 2.366 mm into a flattened Lorentzian beam with shape and width parameters of q = 15 and R FL = 3.25 mm , and system parameters of n 1 = 1.46071 , n 2 = 1 , t = 150 mm . Plots of selected ray paths of this Keplerian laser beam shaper in which the dark shading of parts of the space between the lenses shows where ray paths converge illustrate visually the spatial distribution of the caustic surfaces.

Fig. 7
Fig. 7

(a) Cross-sectional view of the tangential and sagittal caustic surfaces of a planospherical lens (dashed curve) with the same focal length as the first planoaspheric lens of a Keplerian laser beam shaper (solid curve) described in Subsection 4B. The input of the aperture of the planoaspheric lens is 6 mm , where the z coordinate of the tangential caustic is approximately equal to the paraxial focal length. (b) Three-dimensional view of the tangential caustic surface of the first planoaspheric lens of a Keplerian laser beam shaper with an input aperture radius of 5 mm . The sagittal caustic spike is not visible in this view.

Fig. 8
Fig. 8

Cross-sectional views of refracted wavefronts for s = 0 , 10 , , 150 mm are represented by solid lines. The caustic surfaces of the planoaspheric lens of a Keplerian laser beam shaper are represented by dashed curves.

Fig. 9
Fig. 9

Cross-sectional views with an expanded scale of the wavefronts (solid curves) and the caustic surfaces (dashed lines) of a Keplerian laser beam shaper with an input aperture radius of 7.1 mm with s = ( a ) 60, (b) 70, (c) 80, and (d) 90 mm .

Equations (85)

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u ( r ) = A ( r ) exp [ i k 0 Φ ( r ) ] ,
( Φ x ) 2 + ( Φ y ) 2 + ( Φ z ) 2 = n 2 2 ,
[ A 2 ( r ) Φ ( r ) ] = 0 .
u 2 + v 2 + w 2 = n 2 2
Φ ( r ) = r S 2 ( v , w ) + k ( v , w ) ,
[ k ( v , w ) v ] n 2 2 v 2 w 2 = v x y n 2 2 v 2 w 2 ,
[ k ( v , w ) w ] n 2 2 v 2 w 2 = w x z n 2 2 v 2 w 2 .
Φ ( r ) = n 2 s ,
x n 2 2 v 2 w 2 + y v + z w + k ( v , w ) = n 2 s ,
v x + ( y + k v ) n 2 2 v 2 w 2 = 0 ,
w x + ( z + k w ) n 2 2 v 2 w 2 = 0 ,
W ( v , w : s ) = q S 2 n 2 2 K ,
q = [ n 2 s k ( v , w ) ] + S 2 k ,
K = ( 0 , k v , k w ) .
C ± ( v , w ) = 1 2 n 2 2 ( H ± S ) S 2 K ,
H = ( n 2 2 v 2 ) k v v + ( n 2 2 w 2 ) k w w 2 v w k v w ,
T 2 = k v v k w w k v w 2 ,
S 2 = H 2 4 n 2 2 u 2 T 2 .
W = W r e r + W z e z ,
S 2 = n 2 2 w 2 e r + w e z ,
n 2 s = W r n 2 2 w 2 + W z w + k ( w ) ,
k w = W r w n 2 2 w 2 W z .
Φ = S 1 ( x r 0 ) + S 2 ( r x ) .
Φ = S 1 x + S 2 ( r x ) .
k = x ( S 1 S 2 ) .
S 2 = S 1 + Ω N ,
Ω = ( n 1 cos α + n 2 cos β ) ,
k = Ω x N .
k ( w ) = Ω ( w ) [ r z + z ( r ) 1 + ( z ) 2 ] ,
k ( w ) = r n 2 2 w 2 + ( n 1 w ) z ( r ) .
k w = r w n 2 2 w 2 z ( r ) .
W ( r , ϕ ; s ) = x ( r , ϕ ) + [ n 1 z ( r ) + n 2 s ] S 2 n 2 2 .
k ww = [ ( n 2 2 n 1 w ) ( w n 1 ) n 2 2 w 2 ] ( d r d w ) r n 2 2 [ n 2 2 w 2 ] 3 2 ,
d w d r = z n 2 2 w 2 ( w n 1 ) 2 ( n 1 w n 2 2 ) .
k ww = ( n 1 w n 2 2 ) 2 z ( n 2 2 w 2 ) ( w n 1 ) 3 r n 2 2 ( n 2 2 w 2 ) 3 2 .
C + = e r [ r n 2 2 w 2 ( n 1 w n 2 2 ) 2 z n 2 2 ( w n 1 ) 3 ] + e z [ z ( r ) + w ( n 1 w n 2 2 ) 2 z n 2 2 ( w n 1 ) 3 ] ,
C = e z [ z ( r ) + r w n 2 2 w 2 ] .
C ± = x + r ± S 2 n 2 2 ,
r + = ( n 1 w n 2 2 ) 2 z ( w n 1 ) 3 ,
r = r n 2 2 n 2 2 w 2 .
N = sin α e r + cos α e z .
A 1 = e z .
A 2 = sin ( 2 α ) e r + cos ( 2 α ) e z ,
k ( w ) = r n 2 2 w 2 z ( r ) ( n 2 + w ) ,
k w = r w n 2 2 w 2 z ( r ) ,
k ww = n 2 2 z ( n 2 w ) ( n 2 + w ) 2 r n 2 2 ( n 2 2 w 2 ) 3 2 .
C + = e r [ r n 2 2 w 2 z ( w + n 2 ) ] + e z [ z ( r ) + w z ( w + n 2 ) ] ,
C = e z [ z ( r ) + r w n 2 2 w 2 ] .
lim r 0 C ± e z 2 z ( 0 ) ,
x = R sin α e r + R ( 1 cos α ) e z ,
N = + sin α e r + cos α e z ,
k = Ω R ( 1 cos α ) ,
k ( w ) = R [ Ω ( w ) + n 1 w ] .
d Ω d w = n 1 Ω .
k w = R [ d Ω d w 1 ] = R [ n 1 Ω + 1 ] .
k ww = R [ n 1 Ω 2 d Ω d w ] = n 1 2 R Ω 3 .
C + = R n 1 2 ( n 2 2 w 2 ) n 2 2 Ω 3 S s + R [ n 1 Ω + 1 ] e z ,
C = R [ n 1 Ω + 1 ] e z .
z = z ( r ) .
x ( r , ϕ ) = r e r ( ϕ ) + z ( r ) e z ,
N = z e r + e z 1 + ( z ) 2 .
S 2 = S 1 + Ω N ,
Ω = S 1 N + S 2 N .
Ω = ( n 1 cos α + n 2 cos β ) .
Ω 2 = n 1 2 + n 2 2 2 S 1 S 2 .
Ω 2 = n 1 2 + n 2 2 2 n 1 w .
Ω = 2 n 1 cos α = 2 n 2 cos α .
A 2 = ( N × A 1 ) × N ( N A 1 ) N ,
= A 1 2 ( N A 1 ) N ,
w = n 2 cos θ ,
θ = β α .
cos α = 1 1 + ( z ) 2 ,
sin α = z 1 + ( z ) 2 ,
sin β = γ sin α = γ z 1 + ( z ) 2 ,
cos β = 1 γ 2 sin 2 α = 1 + ( 1 γ 2 ) ( z ) 2 1 + ( z ) 2 ,
sin θ = sin β cos α sin α cos β = n 1 z z n 2 2 + ( n 2 2 n 1 2 ) ( z ) 2 n 2 [ 1 + ( z ) 2 ] ,
cos θ = cos β cos α + sin β sin α = n 1 ( z ) 2 + n 2 2 + ( n 2 2 n 1 2 ) ( z ) 2 n 2 [ 1 + ( z ) 2 ] .
sin α = ± n 2 2 w 2 Ω ,
cos α = w n 1 Ω ,
n 2 sin β = ± n 1 n 2 2 w 2 Ω ,
n 2 cos β = ( n 2 2 n 1 w ) Ω ,
z = ± n 2 2 w 2 ( w n 1 ) ,
d θ d r = d β d r d α d r = [ n 1 cos α n 2 cos β 1 ] d α d r ,
= z Ω cos 2 α n 2 cos β ,
= z ( w n 1 ) 2 n 1 w n 2 2 ,

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