Abstract

The k-function of Stavroudis describes a solution of the eikonal equation in a region of constant refractive index. Given the k-function describing the optical field in one region of space, and given a prescribed refractive or reflective boundary, we construct the k-function for the refracted or reflected field. This procedure, which Stavroudis calls refracting the k-function, can be repeated any number of times, and therefore extends the usefulness of the k-function formalism to multielement optical systems. As examples, we present an analytic solution for the k-function, wavefronts, and caustics generated by a biconvex thick lens illuminated by a plane wave propaga ting parallel to the symmetry axis, and numerical results for off-axis plane-wave illumination of a two-mirror telescope.

© 2011 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 109–142.
  2. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 3rd ed. (Pergamon, 1971), pp. 130–136.
  3. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006), pp. 67–146.
    [CrossRef]
  4. K. Schwarzschild, “Untersuchungen zur geometrischen Optik. I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs,” in Astronomische Mittheilungen der Königlichen Sternwarte zu Göttingen (Dieterich’schen University, 1905), pp. 1–28.
  5. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964), pp. 82–116.
  6. A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995), pp. 59–68.
    [CrossRef]
  7. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
    [CrossRef]
  8. J. A. Hoffnagle and D. L. Shealy, “Wavefront generated by reflection of a plane wave from a conic section,” Proc. SPIE 7060, 70600V-1–70600V-9 (2008).
    [CrossRef]
  9. E. Román-Hernández, J. G. Santiago-Santigo, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts and caustics of a spherical wave reflected by an arbitrary smooth surface,” J. Opt. Soc. Am. A 26, 2295–2305 (2009).
    [CrossRef]
  10. A. Aveñdana-Alejo, L. Castañedo, and I. Moreno, “Properties of caustics produced by a positive lens: meridional rays,” J. Opt. Soc. Am. A 27, 2252–2260 (2010).
    [CrossRef]
  11. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), p. 83
  12. O. N. Stavroudis and R. C. Fronczek, “Caustic surfaces and the structure of the geometrical image,” J. Opt. Soc. Am. 66, 795–800 (1976).
    [CrossRef]
  13. O. N. Stavroudis, “The k-function in geometrical optics and its relationship to the archetypal wavefront and the caustic surface,” J. Opt. Soc. Am. A 12, 1010–1016 (1995).
    [CrossRef]
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanney, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed., (Cambridge University, 2003), p. 181, Eq. (5.7.7).
  15. A. Cordero-Dávila and J. Castro-Ramos, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror,” Appl. Opt. 37, 6774–6778 (1998).
    [CrossRef]
  16. G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028(2001).
    [CrossRef]
  17. R. W. Hosken, “Circle of least confusion of a spherical reflector,” Appl. Opt. 46, 3107–3117 (2007).
    [CrossRef] [PubMed]
  18. J. A. Hoffnagle and D. L. Shealy, “Caustic surfaces of a Keplerian two-lens beam shaper,” Proc. SPIE 6663, 666304-1–666304-9 (2007).
    [CrossRef]
  19. J. J. Stamnes, Waves in Focal Region: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, 1986), pp. 163–200.
  20. D. J. Schroeder, Astronomical Optics (Academic, 1987), pp. 15–19.
  21. R. W. Wood, Physical Optics (Optical Society of America, 1988), 3rd ed.,, p. 50.
  22. M.Born and E.Wolf, Principles of Optics, 7th ed., (Cambridge University, 1999), pp. 203–207.

2010

2009

2008

J. A. Hoffnagle and D. L. Shealy, “Wavefront generated by reflection of a plane wave from a conic section,” Proc. SPIE 7060, 70600V-1–70600V-9 (2008).
[CrossRef]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
[CrossRef]

2007

R. W. Hosken, “Circle of least confusion of a spherical reflector,” Appl. Opt. 46, 3107–3117 (2007).
[CrossRef] [PubMed]

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

2001

1998

1995

1976

Aveñdana-Alejo, A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 109–142.

M.Born and E.Wolf, Principles of Optics, 7th ed., (Cambridge University, 1999), pp. 203–207.

Castañedo, L.

Castro-Ramos, J.

Cordero-Dávila, A.

Flanney, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanney, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed., (Cambridge University, 2003), p. 181, Eq. (5.7.7).

Fronczek, R. C.

Hoffnagle, J. A.

J. A. Hoffnagle and D. L. Shealy, “Wavefront generated by reflection of a plane wave from a conic section,” Proc. SPIE 7060, 70600V-1–70600V-9 (2008).
[CrossRef]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
[CrossRef]

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

Hosken, R. W.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 3rd ed. (Pergamon, 1971), pp. 130–136.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 3rd ed. (Pergamon, 1971), pp. 130–136.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964), pp. 82–116.

Moreno, I.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanney, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed., (Cambridge University, 2003), p. 181, Eq. (5.7.7).

Román-Hernández, E.

Santiago-Santigo, J. G.

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, 1987), pp. 15–19.

Schwarzschild, K.

K. Schwarzschild, “Untersuchungen zur geometrischen Optik. I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs,” in Astronomische Mittheilungen der Königlichen Sternwarte zu Göttingen (Dieterich’schen University, 1905), pp. 1–28.

Shealy, D. L.

J. A. Hoffnagle and D. L. Shealy, “Wavefront generated by reflection of a plane wave from a conic section,” Proc. SPIE 7060, 70600V-1–70600V-9 (2008).
[CrossRef]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
[CrossRef]

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

Silva-Ortigoza, G.

Silva-Ortigoza, R.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Region: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, 1986), pp. 163–200.

Stavroudis, O. N.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanney, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed., (Cambridge University, 2003), p. 181, Eq. (5.7.7).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanney, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed., (Cambridge University, 2003), p. 181, Eq. (5.7.7).

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995), pp. 59–68.
[CrossRef]

Wolf, E.

M.Born and E.Wolf, Principles of Optics, 7th ed., (Cambridge University, 1999), pp. 203–207.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 109–142.

Wood, R. W.

R. W. Wood, Physical Optics (Optical Society of America, 1988), 3rd ed.,, p. 50.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. SPIE

J. A. Hoffnagle and D. L. Shealy, “Wavefront generated by reflection of a plane wave from a conic section,” Proc. SPIE 7060, 70600V-1–70600V-9 (2008).
[CrossRef]

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

Other

J. J. Stamnes, Waves in Focal Region: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, 1986), pp. 163–200.

D. J. Schroeder, Astronomical Optics (Academic, 1987), pp. 15–19.

R. W. Wood, Physical Optics (Optical Society of America, 1988), 3rd ed.,, p. 50.

M.Born and E.Wolf, Principles of Optics, 7th ed., (Cambridge University, 1999), pp. 203–207.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 109–142.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 3rd ed. (Pergamon, 1971), pp. 130–136.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006), pp. 67–146.
[CrossRef]

K. Schwarzschild, “Untersuchungen zur geometrischen Optik. I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs,” in Astronomische Mittheilungen der Königlichen Sternwarte zu Göttingen (Dieterich’schen University, 1905), pp. 1–28.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964), pp. 82–116.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995), pp. 59–68.
[CrossRef]

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), p. 83

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanney, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed., (Cambridge University, 2003), p. 181, Eq. (5.7.7).

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Figures (8)

Fig. 1
Fig. 1

Schematic depiction of a collimated beam incident on an optical system with two refracting surfaces.

Fig. 2
Fig. 2

Refraction at a single surface dividing regions with uniform refractive indices n 0 and n 1 . The meaning of the symbols is explained in the text.

Fig. 3
Fig. 3

Geometry of a thick lens illuminated with collimated light. The refracted ray vector, S 1 , gives the direction of light from the first surface to the second surface. The refracted ray vector, S 2 , gives the direction light leaving the second surface.

Fig. 4
Fig. 4

Illustration of a thick lens (black curves joined by horizontal lines) with R 1 = 5 , R 2 = 5 , and d = 3 . The lens is assumed to have an index of refraction of 1.5 and is surrounded by air. The tangential and sagittal caustics formed by a plane wave incident along the symmetry axis of the lens are shown by red curves and line at right.

Fig. 5
Fig. 5

Series of refracted wavefronts with s = 4 , 5 , , 10 from a thick lens (black curves joined by horizontal lines on left side of figure) are shown by blue lines on right side of figure. The tangential and sagittal caustics are shown by red curves from surface of lens converging at point on right.

Fig. 6
Fig. 6

Classical Cassegrain telescope, after Schroeder [20]. The heavy curves P and S are the primary and secondary mirrors, respectively. The focal length of the primary mirror is denoted f p , and the focus of the entire system is at F. The point E is at the edge of the exit pupil.

Fig. 7
Fig. 7

Aberration function, E, for a plane wave that enters the Cassegrain telescope described in the text at an angle of 2.1 mrad . The units of x and y are normalized to the focal length of the primary, and the units of E are waves.

Fig. 8
Fig. 8

Caustic surfaces of a plane wave entering the Cassegrain telescope at angle of 1.0 mrad . The axes are labeled in units normalized to the focal length of the primary mirror.

Equations (81)

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[ Φ ( r ) ] 2 = n 2 ( r ) .
Φ ( r ) = r · S + k ( S ) ,
k ( 1 ) ( S 1 ) = Ω 1 x 1 · N 1 ,
S 1 = S 0 + Ω 1 N 1 .
Ω 1 = 2 cos ( α 1 ) for     reflection ,
Ω 1 = n 1 cos ( β 1 ) n 0 cos ( α 1 ) for     refraction .
Φ ( r ) = r · S 1 + k ( 1 ) ( S 1 ) ( r     in Region     1 ) .
Φ ( r ) = r · S 2 + k ( 2 ) ( S 2 ) ( r     in Region     2 ) ,
k ( 2 ) ( S 2 ) = k ( 1 ) ( S 1 ) + x 2 · ( S 1 S 2 ) ,
k ( 2 ) ( S 2 ) = k ( 1 ) ( S 1 ) Ω 2 x 2 · N 2 ,
W = q S 2 n 2 2 K 2 ,
q = s n 2 k ( 2 ) ( u 2 , v 2 ) + S 2 · K 2 ,
K 2 = ( k u ( 2 ) , k v ( 2 ) , 0 ) ,
ρ ± = 1 n 2 ( q H ± S 2 ) ,
H = ( n 2 2 u 2 2 ) k u u ( 2 ) + ( n 2 2 v 2 2 ) k v v ( 2 ) 2 u 2 v 2 k u v ( 2 ) ,
S 2 = H 2 4 n 2 2 w 2 2 T 2 ,
T 2 = k u u ( 2 ) k v v ( 2 ) [ k u v ( 2 ) ] 2 .
C ± = 1 2 n 2 2 ( H ± S ) S 2 K .
k u ( 2 ) = x 2 + u 2 z 2 / w 2 ,
k v ( 2 ) = y 2 + v 2 z 2 / w 2 .
D = u 2 x 1 v 2 y 1 u 2 y 1 v 2 x 1 ,
f u 2 = 1 D ( f x 1 v 2 y 1 f y 1 v 2 x 1 )
f v 2 = 1 D ( f y 1 u 2 x 1 f x 1 u 2 y 1 ) .
z 1 ( r 1 ) = R 1 [ 1 1 ( r 1 R 1 ) 2 ] ,
z 2 ( r 2 ) = d + R 2 [ 1 1 ( r 2 R 2 ) 2 ] .
S 1 = n 1 2 w 1 2 e r + w 1 e z ,
r 1 ( w 1 ) = R 1 n 1 2 w 1 2 Ω 1 ( w 1 ) ,
Ω 1 2 ( w 1 ) = n 0 2 + n 1 2 2 n 0 w 1 .
x 2 = x 1 + S 1 ,
( w 1 ) = β ( w 1 ) β 2 ( w 1 ) 4 n 1 2 γ ( w 1 ) 2 n 1 2 ,
β ( w 1 ) = 2 R 1 ( n 1 2 w 1 2 ) Ω 1 ( w 1 ) + 2 w 1 { R 1 [ 1 ( w 1 1 ) Ω 1 ( w 1 ) ] d R 2 } ,
γ ( w 1 ) = { R 1 [ 1 ( w 1 1 ) Ω 1 ( w 1 ) ] d R 2 } 2 + R 1 2 ( n 1 2 w 1 2 ) Ω 1 2 ( w 1 ) R 2 2 .
x 2 ( w 1 ) = r 2 ( w 1 ) e r + { d + R 2 [ 1 Θ ( w 1 ) ] } e z ,
N 2 ( w 1 ) = [ r 2 ( w 1 ) R 2 ] e r + Θ ( w 1 ) e z ,
Θ ( w 1 ) 1 [ r 2 ( w 1 ) R 2 ] 2 .
r 2 ( w 1 ) = n 1 2 w 1 2 [ R 1 Ω 1 ( w 1 ) ( w 1 ) ] n 1 2 w 1 2 g ( w 1 ) .
S 2 = S 1 + Ω 2 N 2 ,
Ω 2 ( w 1 ) = n 1 cos α 2 ( w 1 ) + n 2 cos β 2 ( w 1 ) ,
cos α 2 ( w 1 ) = S 1 · N 2 = n 1 2 w 1 2 [ r 2 ( w 1 ) n 1 R 2 ] + w 1 n 1 Θ ( w 1 ) ,
sin α 2 ( w 1 ) = | S 1 × N 2 | n 1 = w 1 r 2 ( w 1 ) n 1 R 2 + n 1 2 w 1 2 n 1 Θ ( w 1 ) ,
n 2 sin β 2 ( w 1 ) = n 1 sin α 2 ( w 1 ) ,
cos β 2 ( w 1 ) = 1 sin 2 β 2 ( w 1 ) .
w 2 = w 1 + Ω 2 ( w 1 ) Θ ( w 1 ) .
k ( 1 ) ( w 1 ) = R 1 [ Ω 1 ( w 1 ) + n 0 w 1 ] ,
k ( 2 ) ( w 1 ) = k ( 1 ) ( w 1 ) + Ω 2 ( w 1 ) [ R 2 ( d + R 2 ) Θ ( w 1 ) ] .
k w ( 2 ) d k ( 2 ) ( w 1 ) d w 2 = 1 [ d w 2 / d w 1 ] [ d k ( 2 ) ( w 1 ) d w 1 ] ,
k w w ( 2 ) = 1 [ d w 2 / d w 1 ] 2 d 2 k ( 2 ) ( w 1 ) d w 1 2 1 ( d w 2 / d w 1 ) 3 [ d k ( 2 ) ( w 1 ) d w 1 ] [ d 2 w 2 d w 1 2 ] .
K 2 ( w 1 ) = [ 0 , 0 , k w ( 2 ) ( w 1 ) ]
T 2 = 0
S = H
H ( w 1 ) = [ n 2 2 w 2 2 ( w 1 ) ] k w w ( 2 ) ( w 1 ) ,
W ( w 1 ) = q ( w 1 ) S 2 ( w 1 ) n 2 2 k w ( 2 ) ( w 1 ) e z
C + ( w 1 ) = [ n 2 2 w 2 2 ( w 1 ) ] k w w ( 2 ) ( w 1 ) S 2 ( w 1 ) k w ( 2 ) ( w 1 ) e z ,
C ( w 1 ) = k w ( 2 ) ( w 1 ) e z ,
q ( w 1 ) = s n 2 k ( 2 ) ( w 1 ) + w 2 ( w 1 ) k w ( 2 ) ( w 1 ) .
z h = z 0 + c h r 2 1 + 1 ( 1 + κ ) c h 2 r 2 ,
z 0 = 1 + 1 c h ( 1 + κ ) .
β = 2 κ c h ( κ + 1 ) 1 .
u 1 = sin θ r cos ϕ 1 + r 2 / 4 [ cos θ ( r / 2 ) sin θ cos ϕ ] ,
v 1 = r sin ϕ 1 + r 2 / 4 [ cos θ ( r / 2 ) sin θ cos ϕ ] ,
w 1 = cos θ + 2 1 + r 2 / 4 [ cos θ ( r / 2 ) sin θ cos ϕ ] .
k ( 1 ) ( u 1 , v 1 ) = ( u 1 + sin θ ) 2 + v 1 1 ( u 1 2 + v 1 2 ) + cos θ .
d w 2 d w 1 = 1 + [ d Ω 2 ( w 1 ) d w 1 ] Θ ( w 1 ) + Ω 2 ( w 1 ) [ d Θ ( w 1 ) d w 1 ] ,
d 2 w 2 d w 1 2 = [ d 2 Ω 2 ( w 1 ) d w 1 2 ] Θ ( w 1 ) + 2 [ d Ω 2 ( w 1 ) d w 1 ] [ d Θ ( w 1 ) d w 1 ] + Ω 2 ( w 1 ) [ d 2 Θ ( w 1 ) d w 1 2 ] .
d k ( 2 ) ( w 1 ) d w 1 = d k ( 1 ) ( w 1 ) d w 1 + [ d Ω 2 ( w 1 ) d w 1 ] [ R 2 ( d + R 2 ) Θ ( w 1 ) ] ( d + R 2 ) Ω 2 ( w 1 ) [ d Θ ( w 1 ) d w 1 ] ,
d 2 k ( 2 ) ( w 1 ) d w 1 2 = d 2 k ( 1 ) ( w 1 ) d w 1 2 + [ d 2 Ω 2 ( w 1 ) d w 1 2 ] [ R 2 ( d + R 2 ) Θ ( w 1 ) ] 2 ( d + R 2 ) [ d Ω 2 ( w 1 ) d w 1 ] [ d Θ ( w 1 ) d w 1 ] ( d + R 2 ) Ω 2 ( w 1 ) [ d 2 Θ ( w 1 ) d w 1 2 ] .
d Θ ( w 1 ) d w 1 = r 2 ( w 1 ) R 2 2 Θ ( w 1 ) [ d r 2 ( w 1 ) d w 1 ] .
d r 2 ( w 1 ) d w 1 = w 1 g ( w 1 ) n 1 2 w 1 2 + n 1 2 w 1 2 [ d g ( w 1 ) d w 1 ] ,
r 2 [ d r 2 d w 1 ] = g ( w ) [ w 1 g ( w 1 ) + ( n 1 2 w 1 2 ) d g ( w 1 ) d w 1 ] g ( w 1 ) F ( w 1 ) .
d Θ ( w 1 ) d w 1 = g ( w 1 ) F ( w 1 ) R 2 2 Θ ( w 1 ) ,
d 2 Θ ( w 1 ) d w 1 2 = { [ d F ( w 1 ) d w 1 ] g ( w 1 ) + F ( w 1 ) [ d g ( w 1 ) d w 1 ] } R 2 2 Θ ( w 1 ) [ g ( w 1 ) F ( w 1 ) ] 2 R 2 4 [ Θ ( w 1 ) ] 3 ,
d g ( w 1 ) d w 1 = R 1 Ω 1 3 ( w 1 ) + d ( w 1 ) d w 1 ,
d 2 g ( w 1 ) d w 1 2 = 3 R 1 Ω 1 5 ( w 1 ) + d 2 ( w 1 ) d w 1 2 ,
d F ( w 1 ) d w 1 = g ( w 1 ) 3 w 1 [ d g ( w 1 ) d w 1 ] + ( n 1 2 w 1 2 ) [ d 2 g ( w 1 ) d w 1 2 ] .
d Ω 2 ( w 1 ) d w 1 = n 1 d cos α 2 ( w 1 ) d w 1 + n 2 d cos β 2 ( w 1 ) d w 1
d 2 Ω 2 ( w 1 ) d w 1 2 = n 1 d 2 cos α 2 ( w 1 ) d w 1 2 + n 2 d 2 cos β 2 ( w 1 ) d w 1 2 ,
cos α 2 ( w 1 ) = ( n 1 2 w 1 2 ) n 1 R 2 g ( w 1 ) + w 1 n 1 Θ ( w 1 ) .
d cos α 2 ( w 1 ) d w 1 = 2 w 1 g ( w 1 ) n 1 R 2 + ( n 1 2 w 1 2 ) n 1 R 2 [ d g ( w 1 ) d w 1 ] + Θ ( w 1 ) n 1 + w 1 n 1 [ d Θ ( w 1 ) d w 1 ] ,
d 2 cos α 2 ( w 1 ) d w 1 2 = 2 g ( w 1 ) n 1 R 2 4 w 1 n 1 R 2 [ d g ( w 1 ) d w 1 ] + ( n 1 2 w 1 2 ) n 1 R 2 [ d 2 g ( w 1 ) d w 1 2 ] + 2 n 1 [ d Θ ( w 1 ) d w 1 ] + w 1 n 1 [ d 2 Θ ( w 1 ) d w 1 2 ] .
d cos β 2 ( w 1 ) d w 1 = n 1 2 cos α 2 ( w 1 ) n 2 2 cos β 2 ( w 1 ) [ d cos α 2 ( w 1 ) d w 1 ] ,
d 2 cos β 2 ( w 1 ) d w 1 2 = [ n 1 2 n 2 2 cos β 2 ( w 1 ) ] { [ d cos α 2 ( w 1 ) d w 1 ] 2 + cos α 2 ( w 1 ) d 2 cos α 2 ( w 1 ) d w 1 2 } n 1 4 cos 2 α 2 ( w 1 ) n 2 4 cos 3 β 2 ( w 1 ) [ d cos α 2 ( w 1 ) d w 1 ] 2 .

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