Abstract

A general formalism for the reflection of a strongly focused beam from magneto-optic multilayer thin films is established by using a Fourier-transform technique. It is most useful when the focused spot size is of the order of the wavelength, in which case the paraxial approximation is not valid. A significant asymmetry in the field distribution is found for the reflected beam at normal incidence. This asymmetry can be explained in terms of contributions from plane-wave components with different incidence angles.

© 1988 Optical Society of America

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References

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  1. M. Mansuripur, G. A. N. Connell, W. J. Goodman, “Signal and noise in magneto-optical readout,” J. Appl. Phys. 53, 4485–4495 (1982).
    [CrossRef]
  2. Y. Aoki, T. Ihashi, N. Sato, S. Miyaoka, “A magneto-optic recording system using TbFeCo,” IEEE Trans. Magn. MAG-21, 1624–1628 (1985).
    [CrossRef]
  3. Z.-M. Li, B. T. Sullivan, R. R. Parsons, “Use of the 4 × 4 matrix method in the optics of multilayer magnetooptic recording media,” Appl. Opt. 27, 1334–1338 (1988).
    [CrossRef] [PubMed]
  4. G. J. Sprokel, “Reflectivity, rotation, and ellipticity of magnetooptic film structures,” Appl. Opt. 23, 3983–3989 (1984).
    [CrossRef] [PubMed]
  5. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  6. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  7. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [CrossRef]
  8. P. J. Lin-Chung, S. Teitler, “4 × 4 Matrix formalisms for optics in stratified anisotropic media,” J. Opt. Soc. Am. A 1, 703–705 (1984).
    [CrossRef]
  9. A. E. Siegman, “Optical resonators,” in Lasers: Physics, Systems and Technique, W. J. Firth, R. G. Harrison, eds. (The Scottish University Summer School in Physics, Heriot-Watt University, Edinburgh, 1983), pp. 82–85.

1988

1986

1985

T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

Y. Aoki, T. Ihashi, N. Sato, S. Miyaoka, “A magneto-optic recording system using TbFeCo,” IEEE Trans. Magn. MAG-21, 1624–1628 (1985).
[CrossRef]

1984

1982

M. Mansuripur, G. A. N. Connell, W. J. Goodman, “Signal and noise in magneto-optical readout,” J. Appl. Phys. 53, 4485–4495 (1982).
[CrossRef]

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Aoki, Y.

Y. Aoki, T. Ihashi, N. Sato, S. Miyaoka, “A magneto-optic recording system using TbFeCo,” IEEE Trans. Magn. MAG-21, 1624–1628 (1985).
[CrossRef]

Connell, G. A. N.

M. Mansuripur, G. A. N. Connell, W. J. Goodman, “Signal and noise in magneto-optical readout,” J. Appl. Phys. 53, 4485–4495 (1982).
[CrossRef]

Fukumitsu, O.

Goodman, W. J.

M. Mansuripur, G. A. N. Connell, W. J. Goodman, “Signal and noise in magneto-optical readout,” J. Appl. Phys. 53, 4485–4495 (1982).
[CrossRef]

Ihashi, T.

Y. Aoki, T. Ihashi, N. Sato, S. Miyaoka, “A magneto-optic recording system using TbFeCo,” IEEE Trans. Magn. MAG-21, 1624–1628 (1985).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, Z.-M.

Lin-Chung, P. J.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mansuripur, M.

M. Mansuripur, G. A. N. Connell, W. J. Goodman, “Signal and noise in magneto-optical readout,” J. Appl. Phys. 53, 4485–4495 (1982).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Miyaoka, S.

Y. Aoki, T. Ihashi, N. Sato, S. Miyaoka, “A magneto-optic recording system using TbFeCo,” IEEE Trans. Magn. MAG-21, 1624–1628 (1985).
[CrossRef]

Parsons, R. R.

Sato, N.

Y. Aoki, T. Ihashi, N. Sato, S. Miyaoka, “A magneto-optic recording system using TbFeCo,” IEEE Trans. Magn. MAG-21, 1624–1628 (1985).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Optical resonators,” in Lasers: Physics, Systems and Technique, W. J. Firth, R. G. Harrison, eds. (The Scottish University Summer School in Physics, Heriot-Watt University, Edinburgh, 1983), pp. 82–85.

Sprokel, G. J.

Sullivan, B. T.

Takenaka, T.

Teitler, S.

Yokota, M.

Zauderer, E.

Appl. Opt.

IEEE Trans. Magn.

Y. Aoki, T. Ihashi, N. Sato, S. Miyaoka, “A magneto-optic recording system using TbFeCo,” IEEE Trans. Magn. MAG-21, 1624–1628 (1985).
[CrossRef]

J. Appl. Phys.

M. Mansuripur, G. A. N. Connell, W. J. Goodman, “Signal and noise in magneto-optical readout,” J. Appl. Phys. 53, 4485–4495 (1982).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. A

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Other

A. E. Siegman, “Optical resonators,” in Lasers: Physics, Systems and Technique, W. J. Firth, R. G. Harrison, eds. (The Scottish University Summer School in Physics, Heriot-Watt University, Edinburgh, 1983), pp. 82–85.

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Figures (5)

Fig. 1
Fig. 1

Schematics of the model system and the coordinate system for the beam fields. The incident beam propagates along the +z direction.

Fig. 2
Fig. 2

(a) Contour plot of a Gaussian beam intensity near the focal spot. The beam propagates in the +z direction, and r = (x2 + y2)1/2. The spot size w0 = λ/2. z and r are in units of the wavelength λ. (b) Contour plot of the beam intensity near the focal spot, given by the solution to Eqs. (1) and (2) in the text. The spot size is w0 = λ/2. z and r are in units of the wavelength λ.

Fig. 3
Fig. 3

Coordinate system of p and s light for incident and reflected plane-wave components, respectively. A superscript r denotes parameters for reflected waves.

Fig. 4
Fig. 4

Contour plot of 2000 |Eyr(r, ϕ, zr)|2/|Ex(0, zr)|2 at zr = 5z0. Parameters for the incident beam are described in the text, and those for the multilayer thin films are specified in Table 1. w(z) is the spot size defined in Eq. (26).

Fig. 5
Fig. 5

Spectral range for a focused beam as indicated by the solid and dashed lines. kz and kr are the longitudinal and transverse components of the wave vector, respectively, and k0 = 2π/λ, where λ is the wavelength. θ in the figure is the angle of incidence for a plane-wave component with a wave vector (kr, kz). The kz axis is assumed to be along the film normal.

Tables (1)

Tables Icon

Table 1 Thicknesses and Refractive Indices of the Magneto-Optic Multilayer Used to Calculate the Results Shown in Fig. 4a

Equations (39)

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( 2 + k 0 2 ) E x ( r , z ) = 0 ,
E x ( r , 0 ) = exp ( r 2 / w 0 2 ) ,
E x ( r , z ) = s = 0 p = 0 s m = 0 p ( 1 ) m C p m c p ( 2 s ) f 2 s w 0 2 m × γ m ψ 0 s + p ( 0 ) exp ( i k 0 z ) ,
C p m = p ! ( p m ) ! m ! ,
c 0 ( 0 ) = 1 , c p ( 2 s ) = ( 1 ) s + p ( 2 s ) ! s ( p 1 ) ! ( s p ) ! ( s + p ) ! , s = 1 , 2 , , p = 0 , 1 , , s ,
f = 1 k 0 w 0 , γ = 1 w 0 2 ( 1 + i z / z 0 ) , z 0 = k 0 w 0 2 / 2 .
ψ 0 n ( 0 ) ( r , z ) = j = 0 n ( 1 ) n + j ( n ! ) 2 ( n j ) ! ( j ! ) 2 w 0 2 ( n + 1 ) γ n + j + 1 r 2 j exp ( γ r 2 ) .
U ( k ) = ( 2 π ) 3 / 2 E ( x , y , z ) exp ( i k x x i k y y i k z z ) × d x d y d z ,
r 2 j exp ( γ r 2 i k x x i k y y ) d x d y = exp ( k r 2 / 4 γ ) l = 0 j Q j l γ ( j + l + 1 ) k r 2 l .
q = 1 n ( 1 ) n + q ( n ! ) 2 ( n q ) ! ( q ) 2 ( Q q l ) = δ n , l Q n n ,
U x ( k r , k z ) = s = 0 p = 0 s m = 0 p ( 2 π ) 3 / 2 ( 1 ) m C p m c p ( 2 s ) f 2 s w 0 2 ( s + p m + 1 ) × Q s + p s + p k r 2 ( s + p ) × γ r m exp [ k r 2 / 4 γ i ( k z k 0 ) z ] d z = s = 0 p = 0 s m = 0 p ( 2 π ) 1 / 2 G ( s , p , m ) k r 2 ( s + p ) × exp ( k r 2 w 0 2 / 4 ) exp k b z 0 d m d k b m δ ( k b ) ,
k b = k z + k r 2 2 k 0 k 0 , G ( s , p , m ) = f 2 s C p m c p ( 2 s ) w 0 2 ( s + p + 1 ) z 0 m Q s + p s + p ,
F ( x ) d m d x m δ ( x x o ) d x = ( 1 ) m d m d x m F ( x ) | x = x 0 .
[ U s r ( k r ) U p r ( k r ) ] = exp ( i 2 d k z ) [ r s s ( k ) r s p ( k ) r p s ( k ) r p p ( k ) ] [ U s ( k ) U p ( k ) ] ,
k x U x + k z U z = 0.
U ν r = exp ( i 2 d k z ) U x ( R 1 ν cos 2 ϕ k + R 2 ν sin 2 ϕ k + R 3 ν sin ϕ k cos ϕ k ) , ν = x , y , U z r = exp ( i 2 d k z ) U x ( R 1 z cos ϕ k + R 2 z sin ϕ k ) ,
R 1 x = r p p , R 2 x = r s s , R 3 x = ( r s p / cos θ + r p s cos θ ) , R 1 y = r s p / cos θ , R 2 y = r p s cos θ , R 3 y = r p p r s s , R 1 z = r p p tan θ , R 2 z = r p s sin θ .
tan θ = k r / k z , k 2 = k r 2 + k z 2 .
exp ( k b z 0 ) R j ν [ m k b m δ ( k b ) ] exp [ i ( k z k 0 ) z ] d k z = ( 1 ) m exp [ i ( k 0 k r 2 / 2 k 0 ) z ] W j m ν ( k r ) ,
W j m ν ( k r ) = m k b m { exp [ i k b ( z i z 0 ) ] × R j ν ( k r , k b + k 0 k r 2 / 2 k 0 ) } | k b = 0 .
E ν r ( r , ϕ , z r ) = s = 0 p = 0 s m = 0 p ( 1 ) m 2 π G ( s , p , m ) exp ( i k 0 z ) × 0 k r 2 ( s + p ) exp ( k r 2 / 4 γ ) × { 1 2 ( W 1 m ν + W 2 m ν ) J 0 ( r k r ) [ 1 2 ( W 1 m ν W 2 m ν ) ] × cos 2 ϕ + 1 2 W 3 m ν sin 2 ϕ ] J 2 ( r k r ) } k r d k r , ν = x , y , E z r ( r , ϕ , z r ) = s = 0 p = 0 s m = 0 p ( 1 ) m 2 π G ( s , p , m ) exp ( i k 0 z ) × 0 k r 2 ( s + p ) exp ( k r 2 / 4 γ ) × ( W 1 m z sin ϕ W 2 m z cos ϕ ) i J 1 ( r k r ) k r d k r ,
x = r cos ϕ , y = r sin ϕ ,
z = z r + 2 d , γ = 1 w 0 2 [ 1 + i ( z r + 2 d ) / z 0 ] .
W j m ν ( k r ) [ i ( z + 2 d i z 0 ) ] m R j ν ( k r , k 0 k r 2 / 2 k 0 ) .
( z + 2 d i z 0 ) m g ( ϕ ) 0 k r 2 ( s + p ) exp ( k r 2 / 4 γ ) k r d k r = 1 2 ( 2 i k 0 ) s + p + 1 g ( ϕ ) ( s + p ) ! / ( z + 2 d i z 0 ) s + p m + 1 ,
w ( z ) = w 0 [ 1 + ( z / z 0 ) 2 ] 1 / 2 ,
u 0 = i k x / 2 γ , υ 0 = i k y / 2 γ , u = x + u 0 , υ = y + υ 0 .
I p = r 2 p exp ( γ r 2 i k x x i k y y ) d x d y = exp ( k r 2 / 4 γ ) ( u 2 + υ 2 + u 0 2 + υ 0 2 2 u 0 u 2 υ 0 υ ) p × exp [ γ ( u 2 + υ 2 ) ] d u d υ .
u = V cos ϕ , υ = V sin ϕ , u 0 = V 0 cos ϕ 0 , υ 0 = V 0 sin ϕ 0 ,
I p = exp ( k r 2 / 4 γ ) j m p ! ( 2 ) m ( p j ) ! ( j m ) ! m ! V 2 p 2 j + m × V 0 2 j m cos m ( ϕ ϕ 0 ) exp ( γ V 2 ) d ϕ V d V .
0 2 π cos 2 k ϕ d ϕ = 2 π Γ ( k + 1 / 2 ) / k ! , 0 2 π cos 2 k + 1 ϕ d ϕ = 0.
I p = exp ( k r 2 / 4 γ ) × j k ( 1 ) j k π 2 4 k 2 j p ! ( p j + k ) ! Γ ( k + 1 / 2 ) ( p j ) ! ( j 2 k ) ! ( 2 k ) ! k ! × γ ( p + j k + 1 ) k r 2 ( j k ) .
I p = exp ( k r 2 / 4 γ ) l = 0 p Q p l γ ( p + l + 1 ) k r 2 l ,
Q p l = k ( 1 ) l π 2 2 ( k l ) p ! ( p l ) ! Γ ( k + 1 / 2 ) ( p l k ) ! ( l k ) ! ( 2 k ) ! k ! ,
p ( 1 ) n + p ( n ! ) 2 ( n p ) ! ( p ! ) 2 Q p l = k ( 1 ) n + 1 π 2 2 ( k l ) n ! Γ ( k + 1 / 2 ) ( l k ) ! ( 2 k ) ! k ! × p C n p ( 1 ) p ( p l ) ( p l 1 ) ( p l k + 1 ) .
d k d x k x ι ( 1 x ) n = p C n p ( 1 ) p ( p l ) ( p l 1 ) ( p l k + 1 ) x p l k ,
p C n p ( 1 ) p ( p l ) ( p l 1 ) ( p l k + 1 ) = d k d x k x l ( 1 x ) n | x = 1 = 0 , l < n and k < n .
p = l n ( 1 ) n + p ( n ! ) 2 ( n p ) ! ( p ! ) 2 Q p l = δ n , l Q n n ,
Q n n = ( 1 ) n 2 2 n π .

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