Abstract

The general 4×4 ray matrix of a grating is presented. This matrix suits the cases of transmission and reflection by a chirped grating with curved lines lying on a curved interface. The matrix presented applies to the general oblique incidence of the optical axis and therefore to nonorthogonal cases and to conical diffraction.

© 2008 Optical Society of America

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References

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  1. G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart and Winston, 1975).
  2. D. C. O'Shea, Elements of Modern Optical Design (Wiley, 1985).
  3. S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168-1177 (1970).
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  4. H. Noda, T. Namioka, and M. Seya, “Ray tracing through holographic gratings,” J. Opt. Soc. Am. 64, 1037-1042 (1974).
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  5. C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham) 33, 820-829 (1994).
    [CrossRef]
  6. A. E. Siegman, Lasers (University Science Books, 1986).
  7. A. E. Siegman, “ABCD-matrix elements for a curved diffraction grating,” J. Opt. Soc. Am. A 2, 1793 (1985).
    [CrossRef]
  8. A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. 271, 327-333 (2007).
    [CrossRef]
  9. M. C. Hettrick, “Surface normal rotation: a new technique for grazing-incidence monochromators,” Appl. Opt. 31, 7174-7178 (1992).
    [CrossRef] [PubMed]
  10. M. Koike and T. Namioka, “Grazing-incidence Monk-Gillieson monochromator based on surface normal rotation of a varied-line-spacing grating,” Appl. Opt. 41, 245-257 (2002).
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  11. G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022-1035 (1972).
    [CrossRef]
  12. D. Phillion, “Huygens integral transformation for a 4×4 ray matrix,” http://www.osti.gov/energycitations/purl.cover.jps?purl=/15014361-TKGOi2/native/.
  13. N. Abramson, “Principle of least wave change,” J. Opt. Soc. Am. A 6, 627-629 (1989).
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  14. M. de Labachelerie and G. Passedat, “Mode-hop suppression of Littrow grating-tuned lasers,” Appl. Opt. 32, 269-274 (1993).
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  15. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]

2007 (1)

A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. 271, 327-333 (2007).
[CrossRef]

2002 (1)

1994 (1)

C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham) 33, 820-829 (1994).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

1989 (1)

1986 (1)

A. E. Siegman, Lasers (University Science Books, 1986).

1985 (2)

1975 (1)

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart and Winston, 1975).

1974 (1)

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022-1035 (1972).
[CrossRef]

1970 (1)

Abramson, N.

April, A.

A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. 271, 327-333 (2007).
[CrossRef]

Collins, S. A.

de Labachelerie, M.

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022-1035 (1972).
[CrossRef]

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart and Winston, 1975).

Hettrick, M. C.

Koike, M.

McCarthy, N.

A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. 271, 327-333 (2007).
[CrossRef]

McKinney, W. R.

C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham) 33, 820-829 (1994).
[CrossRef]

Namioka, T.

Noda, H.

O'Shea, D. C.

D. C. O'Shea, Elements of Modern Optical Design (Wiley, 1985).

Palmer, C.

C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham) 33, 820-829 (1994).
[CrossRef]

Passedat, G.

Phillion, D.

D. Phillion, “Huygens integral transformation for a 4×4 ray matrix,” http://www.osti.gov/energycitations/purl.cover.jps?purl=/15014361-TKGOi2/native/.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Seya, M.

Siegman, A. E.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. 271, 327-333 (2007).
[CrossRef]

Opt. Eng. (Bellingham) (1)

C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham) 33, 820-829 (1994).
[CrossRef]

Proc. IEEE (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022-1035 (1972).
[CrossRef]

Other (5)

D. Phillion, “Huygens integral transformation for a 4×4 ray matrix,” http://www.osti.gov/energycitations/purl.cover.jps?purl=/15014361-TKGOi2/native/.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, 1986).

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart and Winston, 1975).

D. C. O'Shea, Elements of Modern Optical Design (Wiley, 1985).

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Figures (6)

Fig. 1
Fig. 1

Paraxial parameters in the input plane U and the output plane V (projections in the y U z U and y V z V planes).

Fig. 2
Fig. 2

Real ray path passing through a spherical boundary at normal incidence (projections in the y z plane).

Fig. 3
Fig. 3

Representation of the paraxial ray path, in three parts (projections in the y z plane).

Fig. 4
Fig. 4

Tridimensional geometry (a) for oblique incidence, with angles β i and ϵ i , and (b) for transmission, with angles β m and ϵ m . The real ray issued from point S i hits the interface at point P z and travels up to point S m . Point P ( x , y ) is the projection of point P z in the x y plane.

Fig. 5
Fig. 5

Representation of the paraxial ray path at oblique incidence (projections in the y U z U and y V z V planes).

Fig. 6
Fig. 6

Positions of (a) the paraxial point P U in the input plane U (oblique incidence) and (b) the paraxial point P V in the output plane V (transmission). The projections of the analyzed paraxial ray are shown, and the bold line segments are those shown in Fig. 4.

Equations (112)

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x n x d x d z = n x tan θ x n x θ x ,
y n y d y d z = n y tan θ y n y θ y .
[ x V x V y V y V ] = [ A xx B xx A xy B xy C xx D xx C xy D xy A yx B yx A yy B yy C yx D yx C yy D yy ] [ x U x U y U y U ] .
[ x V x V ] = [ A xx B xx C xx D xx ] [ x U x U ] ,
[ y V y V ] = [ A yy B yy C yy D yy ] [ y U y U ] .
[ ρ V ρ V ] = [ A B C D ] [ ρ U ρ U ] ,
A = [ A xx A xy A yx A yy ] ,
B = [ B xx B xy B yx B yy ] ,
C = [ C xx C xy C yx C yy ] ,
D = [ D xx D xy D yx D yy ] .
M = [ A B C D ] .
[ A B C D ] 1 = [ D T B T C T A T ] .
[ A B C D ] [ D T B T C T A T ] = [ I 0 0 I ] ,
[ D T B T C T A T ] [ A B C D ] = [ I 0 0 I ] ,
A D T B C T = I ,
B A T A B T = 0 ,
C D T D C T = 0 ,
D A T C B T = I ,
D T A B T C = I ,
D T B B T D = 0 ,
A T C C T A = 0 ,
A T D C T B = I ,
A xx B yx + A xy B yy A yx B xx A yy B xy = 0 ,
B xx D xy + B yx D yy B xy D xx B yy D yx = 0 ,
C yx D xx + C yy D xy C xx D yx C xy D yy = 0 ,
A xy C xx + A yy C yx A xx C xy A yx C yy = 0 ,
A xx D xx + A xy D xy B xx C xx B xy C xy = 1 ,
A xx D yx + A xy D yy B xx C yx B xy C yy = 0 ,
A yx D xx + A yy D xy B yx C xx B yy C xy = 0 ,
A yx D yx + A yy D yy B yx C yx B yy C yy = 1 ,
A xx D xx + A yx D yx B xx C xx B yx C yx = 1 ,
A xy D xx + A yy D yx B xx C xy B yx C yy = 0 ,
A xx D xy + A yx D yy B xy C xx B yy C yx = 0 ,
A xy D xy + A yy D yy B xy C xy B yy C yy = 1 .
A xx D xx B xx C xx = 1 ,
A yy D yy B yy C yy = 1 .
W mat = W axial + W quad ,
W quad = 1 2 ( ρ U T B 1 A ρ U 2 ρ U T B 1 ρ V + ρ V T D B 1 ρ V ) .
( B 1 ) T = C D B 1 A .
ρ V = A ρ U + B ρ U .
ρ U = D T ρ V B T ρ V .
W quad = 1 2 [ ρ U T B 1 A ρ U ρ U T B 1 ρ V ρ V T ( B 1 ) T ρ U + ρ V T D B 1 ρ V ] = 1 2 [ ρ V T C ρ U + ( ρ V T D B 1 ρ U T B 1 ) ( ρ V A ρ U ) ] = 1 2 ( ρ V T C ρ U + ρ V T B ρ U ) .
z ( x , y ) = R [ 1 ( 1 x 2 y 2 R 2 ) 1 2 ] x 2 + y 2 2 R .
x i ( x , y ) = x L i r i + z ( x , y ) ,
y i ( x , y ) = y L i r i + z ( x , y ) ,
x o ( x , y ) = x L o r o z ( x , y ) ,
y o ( x , y ) = y L o r o z ( x , y ) .
W real ( x , y ) = n in Q i P z ¯ + n out P z Q o ¯ ,
Q i P z ¯ = { [ x x i ( x , y ) ] 2 + [ y y i ( x , y ) ] 2 + [ r i L i + z ( x , y ) ] 2 } 1 2 ,
P z Q o ¯ = { [ x o ( x , y ) x ] 2 + [ y o ( x , y ) y ] 2 + [ r o L o z ( x , y ) ] 2 } 1 2 .
W real ( x , y ) ( r i L i ) n in + ( r o L o ) n out + x 2 + y 2 2 ( n in r i + n out r o + n in n out R n in L i r i 2 n out L o r o 2 ) ,
ϕ = n in r i + n out r o + n in n out R = 0 .
ρ i = L i r i ρ U ,
ρ o = L o r o ρ V ,
ρ U = n in r i ρ U ,
ρ V = n out r o ρ V .
[ ρ U ρ U ] = [ I I ( r i L i ) n in 0 I ] [ ρ i ρ i ] ,
ρ U = ρ i + r i L i n in ρ i .
[ ρ V ρ V ] = M [ ρ U ρ U ] = [ A B C D ] [ ρ U ρ U ] .
ρ V = A ρ U + B ρ U ,
ρ U = D T ρ V B T ρ V .
[ ρ o ρ o ] = [ I I ( r o L o ) n out 0 I ] [ ρ V ρ V ] ,
ρ V = ρ o r o L o n out ρ o .
W quad = W 1 + W 2 + W 3 ,
W 1 = 1 2 ( r i L i n in ρ U T ρ i ) = 1 2 ( r i L i n in ρ i T ρ U ) ,
W 2 = 1 2 ( ρ V T C ρ U + ρ V T B ρ U ) ,
W 3 = 1 2 ( r o L o n out ρ o T ρ V ) = 1 2 ( r o L o n out ρ V T ρ o ) .
W quad = 1 2 [ ρ V T ( n out r o A + n in n out r i r o B + C + n in r i D ) ρ U n in L i r i 2 ρ U T ρ U n out L o r o 2 ρ V T ρ V ] .
W mat ( x , y ) = W axial + Φ UV ( x , y ) x 2 + y 2 2 ( n in L i r i 2 + n out L o r o 2 ) ,
Φ UV ( x , y ) = x 2 2 ( n out r o A xx + n in n out r i r o B xx + C xx + n in r i D xx ) + y 2 2 ( n out r o A yy + n in n out r i r o B yy + C yy + n in r i D yy ) + x y 2 [ ( n out r o ) ( A xy + A yx ) + ( n in n out r i r o ) ( B xy + B yx ) + ( C xy + C yx ) + ( n in r i ) ( D xy + D yx ) ] .
W axial = ( r i L i ) n in + ( r o L o ) n out ,
M = [ 1 0 0 0 0 1 0 0 ( n in n out ) R 0 1 0 0 ( n in n out ) R 0 1 ] .
z ̂ U = x ̂ sin β i cos ϵ i + y ̂ sin ϵ i + z ̂ cos β i cos ϵ i .
z ̂ V = x ̂ sin β m cos ϵ m + y ̂ sin ϵ m + z ̂ cos β m cos ϵ m .
W real ( x , y ) = n in S i P z ¯ + n out P z S m ¯ + m λ 0 N ( x , y ) ,
S i P z ¯ = { ( r i sin β i cos ϵ i + x ) 2 + ( r i sin ϵ i + y ) 2 + [ r i cos β i cos ϵ i + z ( x , y ) ] 2 } 1 2 ,
P z S m ¯ = { ( r m sin β m cos ϵ m x ) 2 + ( r m sin ϵ m y ) 2 + [ r m cos β m cos ϵ m z ( x , y ) ] 2 } 1 2 .
W real ( x , y ) W ( 0 , 0 ) + F ( x , y ) ,
W ( 0 , 0 ) = n in r i + n out r m + m λ 0 N O ,
F ( x , y ) = x ( n in sin β i cos ϵ i n out sin β m cos ϵ m + m λ 0 N O x ) + y ( n in sin ϵ i n out sin ϵ m + m λ 0 N O y ) + x 2 2 [ n in r i ( 1 sin 2 β i cos 2 ϵ i ) + n out r m ( 1 sin 2 β m cos 2 ϵ m ) + Ω O xx ] + x y ( n in r i sin β i sin ϵ i cos ϵ i n out r m sin β m sin ϵ m cos ϵ m + Ω O xy ) + y 2 2 ( n in r i cos 2 ϵ i + n out r m cos 2 ϵ m + Ω O yy ) ,
Ω ( x , y ) = z ( x , y ) [ n in cos β i cos ϵ i n out cos β m cos ϵ m ] + m λ 0 N ( x , y )
z ( x , y ) = x 2 2 R x + y 2 2 R y ,
x U ( x ) = x cos β i ,
x V ( x ) = x cos β m ,
y U ( x , y ) = y cos ϵ i x sin β i sin ϵ i ,
y V ( x , y ) = y cos ϵ m x sin β m sin ϵ m .
W mat ( x , y ) = W axial + Φ U V ( x , y ) ,
Φ U V ( x , y ) = 1 2 [ ρ V T ( n out r m A + n in n out r i r m B + C + n in r i D ) ρ U ] = x 2 2 ( ϕ xx cos β i cos β m ϕ yx cos β i sin β m sin ϵ m ϕ xy sin β i sin ϵ i cos β m + ϕ yy sin β i sin ϵ i sin β m sin ϵ m ) + x y 2 [ ϕ yx cos β i cos ϵ m + ϕ xy cos ϵ i cos β m ϕ yy ( cos ϵ i sin β m sin ϵ m + sin β i sin ϵ i cos ϵ m ) ] + y 2 2 ϕ yy cos ϵ i cos ϵ m ,
ϕ xx = n out r m A xx + n in n out r i r m B xx + C xx + n in r i D xx
W ( 0 , 0 ) + F ( x , y ) = W axial + Φ U V ( x , y ) .
W axial = W ( 0 , 0 ) = n in r i + n out r m + m λ 0 N O .
sin ϵ m = n in sin ϵ i + m λ 0 N O y n out ,
sin β m = n in cos ϵ i sin β i + m λ 0 N O x n out cos ϵ m .
A xx = cos β m cos β i ,
A xy = 0 ,
A yx = tan β i tan ϵ i cos ϵ m sin β m sin ϵ m cos β i ,
A yy = cos ϵ m cos ϵ i ,
B xx = 0 ,
B xy = 0 ,
B yx = 0 ,
B yy = 0 ,
C xx = Ω O yy tan β i tan ϵ i tan β m tan ϵ m + Ω O xx cos β i cos β m + Ω O xy ( tan β i tan ϵ i cos β m + tan β m tan ϵ m cos β i )
C xy = Ω O xy cos β m cos ϵ i + Ω O yy tan β m tan ϵ m cos ϵ i ,
C yx = Ω O xy cos β i cos ϵ m + Ω O yy tan β i tan ϵ i cos ϵ m ,
C yy = Ω O yy cos ϵ i cos ϵ m ,
D xx = cos β i cos β m ,
D xy = tan β m tan ϵ m cos ϵ i sin β i sin ϵ i cos β m ,
D yx = 0 ,
D yy = cos ϵ i cos ε m ,
Ω O xx = 1 R x ( n in cos β i cos ϵ i n out cos β m cos ϵ m ) + m λ 0 N O xx ,
Ω O yy = 1 R y ( n in cos β i cos ϵ i n out cos β m cos ϵ m ) + m λ 0 N O yy ,
Ω O xy = m λ 0 N O xy ,

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