Abstract

We calculate the elements of the 4 × 4 matrix that completely describes the geometrical optic properties of a planar surface of an anisotropic crystal as an optical instrument. The linear and angular magnification submatrices clearly demonstrate the lack of rotational symmetry about the principal ray. The results also show that the astigmatism produced by the surface has in general, nonorthogonal axes. The calculations apply to crystals of arbitrary symmetry, surfaces of arbitrary orientation, and rays of arbitrary direction. The results are applicable to sound waves as well as light waves. Byproducts of this calculation are the ratio of solid angles of ray vectors inside and outside the crystals, as well as the demagnification of a source volume. These factors are combined to relate the ratio of scattered to incident power outside the crystal to the corresponding ratio inside the crystal for a general scattering experiment such as Raman or Brillouin scattering.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Szivessy, Handbuch der Physik, Vol. XX (Springer-Verlag, Berlin, 1928), p. 635.
  2. M. Lax and D. F. Nelson, J. Opt. Soc. Am. 65, 668 (1975).
    [Crossref]
  3. R. K. Luneberg, Mathematical Theory of Optics (University of California, Berkeley, 1966), Chap. IV. See also S. Liebes, Am. J. Phys. 37, 932 (1967); A. Arakenzy, ibid. 25, 519 (1957); M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, New York, 1965).
    [Crossref]
  4. M. Lax and D. F. Nelson, in Polaritons, edited by E. Burstein and F. De Martini (Pergamon, New York, 1974), p. 27.
  5. M. Lax and D. F. Nelson, in Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 415.
    [Crossref]
  6. J. J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969).
  7. M. J. Lighthill, Philos. Trans. R. Soc. Lond. Ser. A 252, 397 (1960).
    [Crossref]
  8. M. V. Klein, Optics (Wiley, New York, 1970), p. 100 ff.
  9. M. Lax and D. F. Nelson, Proceedings of the 1975 Moscow Symposium on Light Scattering (Plenum, New York, 1976).

1975 (1)

1960 (1)

M. J. Lighthill, Philos. Trans. R. Soc. Lond. Ser. A 252, 397 (1960).
[Crossref]

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970), p. 100 ff.

Lax, M.

M. Lax and D. F. Nelson, J. Opt. Soc. Am. 65, 668 (1975).
[Crossref]

M. Lax and D. F. Nelson, Proceedings of the 1975 Moscow Symposium on Light Scattering (Plenum, New York, 1976).

M. Lax and D. F. Nelson, in Polaritons, edited by E. Burstein and F. De Martini (Pergamon, New York, 1974), p. 27.

M. Lax and D. F. Nelson, in Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 415.
[Crossref]

Lighthill, M. J.

M. J. Lighthill, Philos. Trans. R. Soc. Lond. Ser. A 252, 397 (1960).
[Crossref]

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (University of California, Berkeley, 1966), Chap. IV. See also S. Liebes, Am. J. Phys. 37, 932 (1967); A. Arakenzy, ibid. 25, 519 (1957); M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, New York, 1965).
[Crossref]

Nelson, D. F.

M. Lax and D. F. Nelson, J. Opt. Soc. Am. 65, 668 (1975).
[Crossref]

M. Lax and D. F. Nelson, in Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 415.
[Crossref]

M. Lax and D. F. Nelson, in Polaritons, edited by E. Burstein and F. De Martini (Pergamon, New York, 1974), p. 27.

M. Lax and D. F. Nelson, Proceedings of the 1975 Moscow Symposium on Light Scattering (Plenum, New York, 1976).

Stoker, J. J.

J. J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969).

Szivessy, G.

G. Szivessy, Handbuch der Physik, Vol. XX (Springer-Verlag, Berlin, 1928), p. 635.

J. Opt. Soc. Am. (1)

Philos. Trans. R. Soc. Lond. Ser. A (1)

M. J. Lighthill, Philos. Trans. R. Soc. Lond. Ser. A 252, 397 (1960).
[Crossref]

Other (7)

M. V. Klein, Optics (Wiley, New York, 1970), p. 100 ff.

M. Lax and D. F. Nelson, Proceedings of the 1975 Moscow Symposium on Light Scattering (Plenum, New York, 1976).

G. Szivessy, Handbuch der Physik, Vol. XX (Springer-Verlag, Berlin, 1928), p. 635.

R. K. Luneberg, Mathematical Theory of Optics (University of California, Berkeley, 1966), Chap. IV. See also S. Liebes, Am. J. Phys. 37, 932 (1967); A. Arakenzy, ibid. 25, 519 (1957); M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, New York, 1965).
[Crossref]

M. Lax and D. F. Nelson, in Polaritons, edited by E. Burstein and F. De Martini (Pergamon, New York, 1974), p. 27.

M. Lax and D. F. Nelson, in Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 415.
[Crossref]

J. J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

FIG. 1
FIG. 1

A typical experimental setup for a Raman scattering experiment which displays the expansion of the solid angle on emerging from the crystal. For a narrow laser beam, the source volume VS = AlS from which scattered light is accepted is limited by the input beam area A and a length lS determined by the field stop.

FIG. 2
FIG. 2

Demagnification corrections when the arrival ray, departure ray, and surface normal are all in one plane. The detector field stop (see Fig. 1) is represented in image space by the knife edges, which accept a dimension lD perpendicular to the beam. The portion lS of the laser beam accepted is determined by the geometrical conditions shown, independent of the orientation of the virtual image lA: lS sinθS/cosβ = l/cosβ = lD/cosα.

FIG. 3
FIG. 3

An object T in the crystal has image L outside. The linear magnification is determined by the requirement that the associated vector M in the crystal surface projects onto T in the object plane and L in the image plane.

FIG. 4
FIG. 4

The source volume VS is limited by the laser beam length lS admitted by the detector field stop, the width wc of the unscattered laser beam and the height hc of the beam. In the less likely, thick beam case, the exit optics limit the height to hs < hc.

FIG. 5
FIG. 5

An angular change δ β ˆ perpendicular to the plane made by the arrival ray and the normal to the crystal surface induces a corresponding perpendicular change δâ outside the crystal or in a second crystal.

FIG. 6
FIG. 6

Solid-angle expansion. The source of illumination S has a virtual image at I1 obtained by tracing two rays at an angle apart in the plane of arrival (resulting in the angle outside the crystal). If instead, a change d β ˆ is made perpendicular to the plane of arrival (shown in Fig. 5), then the virtual image is at I2. Figure 6(a) applies to a general orientation of the crystal axes relative to the crystal surface. It is a fortiori valid for a uniaxial crystal with surface normal in the x direction and the z or crystal axis in the plane of arrival. Figure 6(b) applies to a uniaxial crystal with z axis normal to the crystal face. In this case, the second image I2 is on the axis SP by symmetry because a ray SO′ in direction β must have the same image as SO.

Equations (126)

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

cos β d Ω in r / K = cos α d Ω out ( ω / c ) 2 ,
l / cos β = l D / cos α .
l S = l / sin θ S = l D cos β / ( cos α sin θ S ) .
i t ( n × t ) × t n × t ,             j t n × t n × t ,
T x i t + y j t
i l ( n × l ) × l n × l ,             j l n × l n × l ,
L x i l + y j l
n · t = cos β ,             n × t = sin β ;             n · l = cos α ,             n × l = sin α ,
M = x cos β [ ( n × t ) × n sin β ] + y [ n × t sin β ] ,
M = x cos α [ ( n × l ) × n sin α ] + y [ n × l sin α ] .
[ x / cos α y ] = [ cos ϕ - sin ϕ sin ϕ cos ϕ ] [ x / cos β y ] ,
cos ϕ = j t · j l = ( t · l - cos α cos β ) / sin α sin β ,
sin ϕ [ ( n × l ) · t ] / sin α sin β .
t · [ ( n × l ) × n sin α ] = sin β cos ϕ ,
t · [ n × l sin α ] = sin β sin ϕ .
[ x y ] = M [ x y ] ,
M [ cos β cos ϕ cos α cos β sin ϕ - sin ϕ cos α cos ϕ ] , M - 1 [ cos α cos ϕ cos β - cos α sin ϕ sin ϕ cos β cos ϕ ] .
n × l = n × k = n ( n × s ) ,
k 2 = n 2 ,             s 2 = 1.
cos ϕ = n × t sin β · n × s sin θ = cos δ - cos β cos θ sin β sin θ ,
sin ϕ = ( n × s ) · t sin θ sin β ,
s · n cos θ ,
s · t cos δ .
d x d y = ( cos α / cos β ) d x d y ,
0 x l D .
0 cos ϕ ( x / cos β ) - sin ϕ y l D / cos α .
b = cos θ S t + sin θ S cos ϕ i t + sin θ S sin ϕ j t ,
x = l S b · i t = l S sin θ S cos ϕ ,
y = l S b · j t = l S sin θ S sin ϕ .
l S l D = N cos β sin θ S cos α ,             N 1 cos ϕ cos ϕ - cos β sin ϕ sin ϕ .
P scat / T exit T ent P inc / A c = R V S d Ω in r ,
P scat P inc = R T ent T exit d Ω out ( ω c ) 2 K cos α cos β V S A c .
V S = A c l S .
P scat P inc = R T ent T exit l D d Ω D ( ω / c ) 2 K sin θ S ( cos ϕ cos ϕ - cos β sin ϕ sin ϕ ) ,
V S = ( l S b ) · ( h S h ) × ( w c w ) ,
A c = h c w c h × w .
V S A c = l S h S h c = A S h c ,
A S ( b × h ) · t = d x d y = d x d y ( cos β / cos α ) .
P scat P inc / h c out = R K ( ω c ) 2 T ent T exit A D d Ω D ( b × h ) · t h c out h c ,
h c out / h c = ( cos 2 ϕ i + cos 2 β i sin 2 ϕ i ) - 1 / 2 , cos ϕ i n i × b n i × b · n i × l i n i × l i ,             cos β i b · n .
l 2 = 1 ,             k = n s ,             s 2 = 1.
k - n cos θ n = l - cos α n .
k · n = n cos θ ,             l · n = cos α .
k = l + [ ( n 2 - sin 2 α ) 1 / 2 - cos α ] n .
δ k = δ l + n [ ( 1 - cos α n cos θ ) sin α δ α + δ n cos θ ] .
- sin α δ α = δ l · n .
δ n = s n · δ s = s n · ( 1 - ss ) · δ k / k ,
s j / k i = ( δ i j - s i s j ) / k .
( 1 - ss ) · s n = n ( s - t sec δ ) ,
t v g / v g ,             v g = k ω ( k ) ,             ω ( k ) = c k / n ( s ) ,
δ n = ( s - t sec δ ) · δ k .
δ k = δ E + nF · δ k ,
δ E δ l + n [ 1 - ( cos α / n cos θ ) ] sin α δ α ,
F = ( s - t sec δ ) / cos θ .
δ k = δ E + n F · δ E 1 - F · n .
k · δ E = k · δ l + n cos θ ( 1 - cos α n cos θ ) ( - n · δ l ) = ( k - n cos θ n + cos α n ) · δ l = l · δ l = 0 ,
δ k = δ l + n ( 1 - cos α n cos θ ) sin α δ α - n t cos β · [ δ l + n ( 1 - cos α n cos θ ) sin α δ α ] = δ l - n t · δ l cos β = ( 1 - nt n · t ) · δ l .
δ k · t = 0.
δ k n δ θ i s + n δ θ ˆ j s + L s ,
δ l δ α i l + δ α ˆ j l ,
i s = ( n × s ) × s sin θ ,             j s = n × s sin θ ,
j l = j s = j
n δ θ ˆ = δ α ˆ
n δ θ = ( i s + sin θ t / cos β ) · ( δ α i l + δ α ˆ j ) = [ cos ( θ - α ) + sin θ cos β t · ( n × l ) × l sin α ] δ α + sin θ cos β t · n × l sin α δ α ˆ .
n δ θ = cos α cos δ cos β δ α + sin θ sin β sin ϕ cos β δ α ˆ
[ n δ θ n δ θ ˆ ] = [ cos α cos δ cos β sin θ sin β sin ϕ cos β 0 1 ] [ δ α δ α ˆ ] .
d A k k = cos α cos δ cos β d A l l ,
cos β d A k k cos δ = cos α d A l l 1
d A k k / cos δ = d A t k ,
cos β d A t k
δ k 1 = δ k · i t ,             δ k 2 = δ k · j t ,
[ cos β δ k 1 δ k 2 ] = [ cos ϕ sin ϕ - sin ϕ cos ϕ ] [ cos α δ α δ α ˆ ] .
cos β d k 1 d k 2 = cos α d α d α ˆ ,
t = ω ( k ) / ω ( k ) k D ( ω , k ) / k D ( ω , k ) .
D ( ω , k ) = 0.
δ t = ( 1 - tt ) · k k D ( ω , k ) k D · δ k .
δ t = ( 1 - tt ) · B · ( 1 - tt ) · δ k ,
B k k D ( ω , k ) / k D
D ( ω , k ) 1 2 k · κ · k - 1 2 κ 11 κ 33 ω 2 / c 2 = 0 ,
B = κ / [ k ( s · κ 2 s ) 1 / 2 ] .
K = d Ω r d A k = d t 1 d t 2 d k 1 d k 2 = det [ ( 1 - tt ) · B · ( 1 - tt ) ] ,
K = | B u u B u v B v u B v v | ,
t = u × v
K = i j k l [ ( u i u j B i j ) ( v k v l B k l ) - ( u i v j B i j ) ( v k u l B k l ) ] ,
i j u i u j B i j = i j 2 u i u j B i ; j ,
i j u i v j B i j = i j ( u i v j + u j v i ) B i ; j ,
B i ; j B i j / ( 1 + δ i j )             for i j
0             for i > j
K = i j k l [ 4 u i u j v k v l - ( u i v j + u j v i ) ( u k v l + u l v k ) ] · B i ; j B k ; l = i j k l [ u j v l ( u i v k - u k v i ) + u j v k ( u i v l - u l v i ) + u i v l ( u j v k - u k v j ) + u i v k ( u j v l - u l v j ) ] B i ; j B k ; l .
u i v k - u k v i = b e i k b t b
K = i j k l a i j k l B i j B k l ,
a i j k l = a b 2 t a t b ( e i k a e j l b + e i l a e j k b ) ( 1 + δ i j ) ( 1 + δ k l ) ( 1 + δ ( i j ) ( k l ) ) ,
K = t 1 2 ( B 22 B 33 - B 23 2 ) + t 2 2 ( B 33 B 11 - B 31 2 ) + t 3 2 ( B 11 B 22 - B 12 2 ) + 2 t 2 t 3 ( B 12 B 13 - B 11 B 23 ) + 2 t 3 t 1 ( B 23 B 21 - B 22 B 31 ) + 2 t 1 t 2 ( B 31 B 32 - B 33 B 12 ) ,
K = t 1 2 κ 22 κ 33 k 2 ( s · κ 2 · s ) + cyclic permutations = det κ ( t · κ - 1 · t ) k 2 ( s · κ 2 · s ) = det κ ( s · κ · s ) k 2 ( s · κ 2 · s ) 2 ,
[ x y δ β δ β ˆ ] = [ M 0 0 A ] [ x y δ α δ α ˆ ] ,
[ δ k 1 δ k 2 ] = [ 1 / cos β 0 0 1 ] [ cos ϕ sin ϕ - sin ϕ cos ϕ ] [ cos α 0 0 1 ] [ δ α δ α ˆ ]
[ δ t 1 δ t 2 ] = [ B u u B u v B v u B v v ] [ δ k 1 δ k 2 ]
A = [ B u u B u v B v u B v v ] [ cos α cos ϕ cos β sin ϕ cos β - cos α sin ϕ cos ϕ ] .
d Ω in r d Ω out = δ t 1 δ t 2 δ α δ α ˆ = det A = K cos α cos β ,
x + δ x = x + z δ t 1 , y + δ y = y + z δ t 2 ,
x + δ x = x + z δ α , y + δ y = y + z δ α ˆ .
[ 1 z 1 0 1 ] [ M 0 0 A ] [ 1 z 1 0 1 ] = [ M M z + z A 0 A ] .
t = cos β i + sin β k , u = sin β i - cos β k , v = j
M = [ cos β cos α 0 0 1 ] .
B u v = B v u = 0 ,
B v v = B 22 = 1 n κ 22 ( s 1 2 κ 11 2 + s 3 2 κ 33 2 ) 1 / 2 .
s 1 = cos θ ,             s 3 = sin θ ,             κ 22 = κ 11 ,
κ 33 / κ 11 = tan β / tan θ
B v v = 1 n cos β cos α .
B u u = B 11 sin 2 β + B 33 cos 2 β = B v v sin β cos δ / sin θ .
A = [ cos α cos β B u u 0 0 B v v ] .
z z = - A v v M v v = - 1 n cos β cos θ .
z z = - A u u M u u = - ( cos α cos β ) 2 B u u = - sin β sin α cos 2 α cos θ cos δ cos β ,
d Ω in r d Ω out = d A s cos β / r s 2 d A s cos α / r 1 r 2 .
δ α = x / z ,             δ α ˆ = y / z
M = T · M · S ,             A = T · A · S .
z / z = - A j j / M j j ,             j = 1 , 2
[ x y δ β δ β ˆ ] = [ M M z + z A 0 A ] [ x y δ α δ α ˆ ] ,
M = [ cos β 0 0 1 ] [ cos ϕ sin ϕ - sin ϕ cos ϕ ] [ 1 / cos α 0 0 1 ] .
A = [ B u u B u v B v u B v v ] [ 1 / cos β 0 0 1 ] [ cos ϕ sin ϕ - sin ϕ cos ϕ ] [ cos α 0 0 1 ] ,
d x d y = det M d x d y ,             det M = cos β / cos α ,
d Ω in r d Ω out = δ β δ β ˆ δ α δ α ˆ = det A = cos α cos β K ( ω c ) 2 ,
P scat / P inc = R K N T ent T exit l D d Ω D ( ω / c ) 2 / sin θ S ,
N = 1 / ( cos ϕ cos ϕ - cos β sin ϕ sin ϕ ) ,
R K = ( ω c ) 2 J 8 π 2 n θ cos δ θ n ϕ cos δ ϕ ,
J | e ϕ · V S e - i k ϕ · r P N L ( r , ω ) d 3 r | 2 2 0 2 E θ 2 V S .