Abstract

The “circular plane-wave” expansion of a spherical wave is newly proposed, and the reconstructed images from volume holograms are studied analytically by applying this method to the expansion of the object wave. The analysis is performed on the weakly coupled amplitude holograms in the Fraunhofer approximation. Both the amplitude and phase of the reconstructed waves are modulated, except the case of exact reconstruction, and are determined by recording and reconstruction parameters and the hologram thickness. The amplitude modulation by the volume hologram of a semitransparent object causes the space-variant or the space-invariant bandpass filtering effect by the spherical-wave or the plane-wave illumination of the object, respectively. The phase modulation causes aberrations and becomes small for a hologram having the large thickness.

© 1976 Optical Society of America

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References

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  1. D. Gabour, Nature (Lond.) 161, 777 (1948).
    [Crossref]
  2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  3. Yu. N. Denisyuk, Opt. Spectrosc. 15, 279 (1963).
  4. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).
    [Crossref] [PubMed]
  5. C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
    [Crossref]
  6. H. Kogelnic, Bell. Syst. Tech. J. 48, 2909 (1969).
    [Crossref]
  7. D. Kermisch, J. Opt. Soc. Am. 59, 1409 (1969).
    [Crossref]
  8. D. Gabour and G. W. Stroke, Proc. R. Soc. A 304, 275 (1968).
    [Crossref]
  9. C. B. Burckhardt, J. Opt. Soc. Am. 57, 601 (1967).
    [Crossref]
  10. N. George and J. W. Matthewa, Appl. Phys. Lett. 9, 212 (1966).
    [Crossref]
  11. A. A. Friesem and J. L. Walker, Appl. Opt. 9, 201 (1970).
    [Crossref] [PubMed]
  12. A. A. Friesem, Appl. Phys. Lett. 7, 102 (1965).
    [Crossref]
  13. M. J. Landry, Appl. Opt. 6, 1947 (1967).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  15. M. R. B. Forshaw, Opt. Commun. 8, 201 (1973).
    [Crossref]
  16. R. Magnusson and T. K. Gaylord, Appl. Opt. 13, 1545 (1974).
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  17. M. R. B. Forshaw, Appl. Opt. 13, 2 (1974).
    [Crossref]
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    [Crossref]
  20. E. Wolf, J. Math. Phys. 11, 2254 (1970).
    [Crossref]
  21. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, England, 1927), p. 397.
  22. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, England, 1922), p. 51.
  23. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 585.
  24. H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1943), pp. 119 and 236.
  25. See Ref. 23, p. 384.
  26. Y. Shono, T. Inuzuka, and T. Hoshino, Appl. Phys. Lett. 22, 299 (1973).
    [Crossref]
  27. Y. Shono and T. Inuzuka, Optik (Stuttg.) 41, 50 (1974).
  28. J. N. Latta, Appl. Opt. 10, 609 (1971).
    [Crossref] [PubMed]
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  30. J. M. Moran, Appl. Opt. 10, 1909 (1971).
    [Crossref] [PubMed]
  31. N. Uchida, J. Opt. Soc. Am. 63, 280 (1973).
    [Crossref]

1974 (3)

1973 (4)

Y. Shono, T. Inuzuka, and T. Hoshino, Appl. Phys. Lett. 22, 299 (1973).
[Crossref]

N. Uchida, J. Opt. Soc. Am. 63, 280 (1973).
[Crossref]

J. M. Moran and I. P. Kaminow, Appl. Opt. 12, 1964 (1973).
[Crossref] [PubMed]

M. R. B. Forshaw, Opt. Commun. 8, 201 (1973).
[Crossref]

1971 (3)

1970 (3)

1969 (3)

V. I. Sukhnov and Yu. N. Denisyuk, Opt. Spectrosc. 21, 62 (1969).

H. Kogelnic, Bell. Syst. Tech. J. 48, 2909 (1969).
[Crossref]

D. Kermisch, J. Opt. Soc. Am. 59, 1409 (1969).
[Crossref]

1968 (1)

D. Gabour and G. W. Stroke, Proc. R. Soc. A 304, 275 (1968).
[Crossref]

1967 (2)

1966 (3)

1965 (1)

A. A. Friesem, Appl. Phys. Lett. 7, 102 (1965).
[Crossref]

1963 (1)

Yu. N. Denisyuk, Opt. Spectrosc. 15, 279 (1963).

1962 (1)

1948 (1)

D. Gabour, Nature (Lond.) 161, 777 (1948).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 585.

Burckhardt, C. B.

Denisyuk, Yu. N.

V. I. Sukhnov and Yu. N. Denisyuk, Opt. Spectrosc. 21, 62 (1969).

Yu. N. Denisyuk, Opt. Spectrosc. 15, 279 (1963).

Doherty, E. T.

Forshaw, M. R. B.

M. R. B. Forshaw, Appl. Opt. 13, 2 (1974).
[Crossref]

M. R. B. Forshaw, Opt. Commun. 8, 201 (1973).
[Crossref]

Friesem, A. A.

Gabour, D.

D. Gabour and G. W. Stroke, Proc. R. Soc. A 304, 275 (1968).
[Crossref]

D. Gabour, Nature (Lond.) 161, 777 (1948).
[Crossref]

Gaylord, T. K.

George, N.

N. George and J. W. Matthewa, Appl. Phys. Lett. 9, 212 (1966).
[Crossref]

Hoshino, T.

Y. Shono, T. Inuzuka, and T. Hoshino, Appl. Phys. Lett. 22, 299 (1973).
[Crossref]

Inuzuka, T.

Y. Shono and T. Inuzuka, Optik (Stuttg.) 41, 50 (1974).

Y. Shono, T. Inuzuka, and T. Hoshino, Appl. Phys. Lett. 22, 299 (1973).
[Crossref]

Kaminow, I. P.

Kermisch, D.

Kogelnic, H.

H. Kogelnic, Bell. Syst. Tech. J. 48, 2909 (1969).
[Crossref]

Kozma, A.

Landry, M. J.

Latta, J. N.

Leith, E. N.

Lin, L. H.

Magnusson, R.

Margenau, H.

H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1943), pp. 119 and 236.

Marks, J.

Massey, N.

Matthewa, J. W.

N. George and J. W. Matthewa, Appl. Phys. Lett. 9, 212 (1966).
[Crossref]

Moran, J. M.

Murphy, G. M.

H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1943), pp. 119 and 236.

Shono, Y.

Y. Shono and T. Inuzuka, Optik (Stuttg.) 41, 50 (1974).

Y. Shono, T. Inuzuka, and T. Hoshino, Appl. Phys. Lett. 22, 299 (1973).
[Crossref]

Stroke, G. W.

D. Gabour and G. W. Stroke, Proc. R. Soc. A 304, 275 (1968).
[Crossref]

Sukhnov, V. I.

V. I. Sukhnov and Yu. N. Denisyuk, Opt. Spectrosc. 21, 62 (1969).

Uchida, N.

Upatnieks, J.

Walker, J. L.

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, England, 1927), p. 397.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, England, 1922), p. 51.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, England, 1927), p. 397.

Wolf, E.

E. Wolf, J. Opt. Soc. Am. 60, 18 (1970).
[Crossref]

E. Wolf, J. Math. Phys. 11, 2254 (1970).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 585.

Appl. Opt. (9)

Appl. Phys. Lett. (3)

Y. Shono, T. Inuzuka, and T. Hoshino, Appl. Phys. Lett. 22, 299 (1973).
[Crossref]

N. George and J. W. Matthewa, Appl. Phys. Lett. 9, 212 (1966).
[Crossref]

A. A. Friesem, Appl. Phys. Lett. 7, 102 (1965).
[Crossref]

Bell. Syst. Tech. J. (1)

H. Kogelnic, Bell. Syst. Tech. J. 48, 2909 (1969).
[Crossref]

J. Math. Phys. (1)

E. Wolf, J. Math. Phys. 11, 2254 (1970).
[Crossref]

J. Opt. Soc. Am. (6)

Nature (Lond.) (1)

D. Gabour, Nature (Lond.) 161, 777 (1948).
[Crossref]

Opt. Commun. (1)

M. R. B. Forshaw, Opt. Commun. 8, 201 (1973).
[Crossref]

Opt. Spectrosc. (2)

V. I. Sukhnov and Yu. N. Denisyuk, Opt. Spectrosc. 21, 62 (1969).

Yu. N. Denisyuk, Opt. Spectrosc. 15, 279 (1963).

Optik (Stuttg.) (1)

Y. Shono and T. Inuzuka, Optik (Stuttg.) 41, 50 (1974).

Proc. R. Soc. A (1)

D. Gabour and G. W. Stroke, Proc. R. Soc. A 304, 275 (1968).
[Crossref]

Other (5)

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, England, 1927), p. 397.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, England, 1922), p. 51.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 585.

H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1943), pp. 119 and 236.

See Ref. 23, p. 384.

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Figures (7)

FIG. 1
FIG. 1

Circular plane-wave expansion of a spherical wave.

FIG. 2
FIG. 2

A geometrical arrangement of recording and reconstruction processes of a volume hologram. Ps(a), a point source of an object-illumination wave; Q(r), an arbitrary point on the object; QC(b), the center of the object; PH(ρ), an arbitrary point in the hologram; PI(R), an arbitrary point in the reconstructed field; Uk, an object-wave component; Ur, a reference wave; Ui, an incident wave for reconstruction; V, reconstructed waves.

FIG. 3
FIG. 3

A geometrical arrangement of experiments.

FIG. 4
FIG. 4

An object pattern and reconstructed images from a volume hologram (t = 2 mm). (a) The object pattern, (b) the virtual image, k i = k r, and plane-eave illumination of the object, (c) the real image where ψi = 6′ and plane-wave illumination of the object, and (e) the real where ψi = 0° and spherical-wave illumination of the object.

FIG. 5
FIG. 5

Bandpass filtering effect where (a) ψi = 0° and plane-wave illumination of the object, and (b) ψi = 6′ and plane-wave illumination of the object. A full line shows the spectra taking the maximum value of the filtering function and the dotted lines show the spectra taking the first minimum (=0) of the filtering function. The spatial-frequency spectra in the region between dotted lines are reconstructed.

FIG. 6
FIG. 6

Space-variant bandpass filtering effect, where ψi =0°, and spherical-wave illumination of the object. An object pattern, the reconstructed real image, and the filtering functions are shown.

FIG. 7
FIG. 7

Reconstructed images from in-line holography. (a) Images from a volume hologram (t = 5 mm); (b) Images from conventional in-line hologram (t = 10 μm).

Tables (1)

Tables Icon

TABLE I Recording and reconstruction parameters of volume holograms.

Equations (96)

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U p ( r ) = h 0 ( k r ) = e i k r / i k r .
r - r cos Δ θ = λ / 2 ,
Δ θ = ( λ / r ) 1 / 2 ,
Δ S = π r λ ,
Δ Ω = π λ / r ,
U Δ S ( r ) = S ( k k × r ) e i k · r ,
S ( k k × r ) = { 1 , | k k × r | ( r λ ) 1 / 2 , 0 , | k k × r | > ( r λ ) 1 / 2 ,
Δ P x · Δ x h ,             P x = k x = k sin ψ ,             Δ x = A B ¯ .
Δ ψ λ / Δ x = 1 2 ( λ / r ) 1 / 2 .
Δ ψ ~ Δ θ .
U Δ S ( r ) = Δ Ω S ( k k × r ) e i k · r d Ω .
U ( r ) = U Δ S ( r ) = S ( k k × r ) e i k · r d Ω ,
U p ( r ) = 0 π 0 2 π S ( k k × r ) exp { i k r [ cos ( φ + φ p ) sin θ sin θ p + cos θ cos θ p ] } sin θ d φ d θ .
U Q ( r ) = 0 π 0 2 π S ( k k × r ) e i k r cos θ sin θ d φ d θ = 0 Δ θ 0 2 π e i k r cos θ sin θ d φ d θ 4 π e i k r i k r .
U ( r ) = e i k r i k r = 1 4 π S ( k k × r ) e i k · r d Ω ,
λ r - b a - b
λ ρ ρ - b .
U 0 ( ρ ) = A K 1 ( α ) f ( r ) e i k ρ - r ρ - r d r ,
f ( r ) = f ( r ) e i k r - a r - a ,
k = 2 π / λ .
K 1 ( α ) = ( 1 / 2 i λ ) ( 1 + cos α ) ,
α = ( b - a , ρ - r ) .
e i k r r = i 4 π S ( k k × r ) e i k · r d k .
U 0 ( ρ ) = i 4 π A f ( r ) K 1 ( α ) S ( k k × [ ρ - r ] ) e i k · [ ρ - r ] d k d r ,
f ( r ) = i 4 π f ( r ) S 0 ( k 0 k × [ r - a ] ) e i k 0 · [ r - a ] d k 0 .
S ( k k × [ ρ - r ] ) = { 1 , | k k × [ ρ - r ] | ( b λ ) 1 / 2 , 0 , | k k × [ ρ - r ] | > ( b λ ) 1 / 2 ,
S ( k 0 k × [ ρ - r ] ) = { 1 , | k 0 k × [ r - a ] | ( b - a λ ) 1 / 2 , 0 , | k 0 k × [ r - a ] | > ( b - a λ ) 1 / 2 ,
Δ S = S ( k k × [ ρ - r ] ) d | k k × [ ρ - r ] | = π b λ ,
Δ S 0 = S ( k 0 k × [ r - a ] ) d | k 0 k × [ r - a ] | = π b - a λ .
K 1 ( α ) = 1 2 i λ ( 1 + [ b - a ] [ ρ - r ] b - a ρ - r ) = 1 2 i λ ( 1 + k [ b - a ] k b - a ) = K 1 ( k ) .
U k ( ρ ) = i 4 π K 1 ( k ) A S ( k k × [ ρ - r ] ) f ( r ) e i k [ ρ - r ] d r
U 0 ( ρ ) = U k ( ρ ) d k .
U k ( ρ ) = i 4 π K 1 ( k ) S ( k k × [ ρ - r * ] ) A Δ S * f ( r ) e i k [ ρ - r ] d r ,
k k × [ ρ - r * ] = 0.
U k ( ρ ) = i 4 π K 1 ( k ) S ( k k × [ ρ - r ] ) A f ( r ) e i k [ ρ - r ] d r - i 4 π K 1 ( k ) S ( k k × [ ρ - r ] ) A - ( A Δ S * ) f ( r ) e i k [ ρ - r ] d r .
Δ S * = π b λ A ~ b - r max 2 ,
Δ S * A ,
U k ( ρ ) i 4 π K 1 ( k ) S ( k k × [ ρ - b ] ) e i k ρ A f ( r ) e - i k · r d r
H ( ρ ) = U 0 ( ρ ) + U r ( ρ ) 2 = U 0 ( ρ ) 2 + U r ( ρ ) 2 + U 0 ( ρ ) U r * ( ρ ) + U 0 * ( ρ ) U r ( ρ ) ,
U r ( ρ ) = e i k r ρ ,
H ( p ) = U 0 ( ρ ) U r * ( ρ ) + U 0 * ( ρ ) U r ( ρ ) = n = ± 1 i 4 π K 1 ( k ) S ( k k × [ ρ - b ] ) e i n [ k - k r ] ρ × A f n ( r ) e - i n kr d r d k ,
f n ( r ) = i 4 π f ( r ) S 0 ( k 0 k × [ r - a ] ) e i n k 0 [ r - a ] d k 0 .
U i ( ρ ) = e i k i ρ ,
V ( R ) = K 2 ( β ) H ( ρ ) e i k i ρ e i k d R - ρ R - ρ d ρ ,
K 2 ( β ) = 1 2 i λ ( 1 + cos β ) 1 2 i λ ( 1 + k i k d k 2 ) = K 2 ( k d ) .
V ( R ) = i 4 π S d ( k d k × R ) K 2 ( k d ) H ( ρ ) e i k i ρ e i k d [ R - ρ ] d ρ d k d ,
S d ( k d k × R ) = { 1 , | k d k × R | ( R λ ) 1 / 2 , 0 , | k d k × R | > ( R λ ) 1 / 2 ,
R b p .
V ( R ) = n = ± 1 - S d ( ( k d / k ) × R ) K 2 ( k d ) 16 π 2 e i k d R × K 1 ( k ) S ( k k × [ ρ - b ] ) exp { i ( n k - n k r + k i - k d ) ρ } × A f n ( r ) e - i n kr d r d ρ d k d k d .
V k ( R ) = s ( k k × [ ρ - b ] ) exp { i ( n k - n k r + k i - k d ) ρ } d ρ .
| k k × [ ρ k - b ] | = 0 ,
V k ( R ) = B t exp { i ( n k - n k r + k i - k d ) [ ρ k + ρ ] } d ρ ,
B ~ [ ( Δ S ) 1 / 2 + A ] 2 .
V k ( R ) = B t δ ( n k x - n k r x + k i x - k d x ) δ ( n k y - n k r y + k i y - k d y ) × sinc [ 1 2 t ( n k z - n k r z + k i z - k d z ) ] e i ( n k - n k r + k i - k d ) ρ k ,
V n ( R ) = - B t 8 π K 1 ( k ) K 2 ( k - w ) S d ( n [ k - w ] k - w × R ) f ˆ n ( n k ) × sinc [ 1 2 t g ( k ) ] e i g ( k ) R z e i n ( k - w ) R d k ,
w = k r - n k i ,
g ( k ) = n k z - n k r z + k i z - k d z ,
k d x = n ( k x - w x ) ,
k d y = n ( k y - w y ) ,
k d z = m ( k 2 - k d x 2 - k d y 2 ) 1 / 2 ,
m = ± 1 ,
n = ± 1.
m = sign ( k i z )
m = - sign ( k i z )
f ( r ) = f ( r ) e i k 0 r .
f ˆ n ( n k ) = 1 2 π f ( r ) e - i n ( k - k 0 ) r d r = f ˆ ( n s ) ,
k = s + k 0 .
V n ( R ) = - B t 8 π K 1 ( k ) K 2 ( k - w ) S d ( n ( k - w ) k - w × R ) f ˆ ( n s ) × sinc [ 1 2 t g ( k ) ] e - i g ( k ) R z e i n [ k - w ] R d k ,
k 0 - k Q Δ k 0 = k Q ( λ r - a ) 1 / 2 ,
k = s + k Q ,
k Q = k r - a r - a .
- π 1 2 t g ( k ) π ,
g ( k ) = 0 ,
g ( k ) = - m [ k 2 - ( k x - w x ) 2 - ( k y - w y ) 2 ] 1 / 2 + n ( k z - w z ) .
( w x 2 + w z 2 ) k x 2 + 2 w x w y k x k y + ( w y 2 + w z 2 ) k y 2 - w x ( w 2 + k 2 - k 2 ) k x - w y ( w 2 + k 2 - k 2 ) k y + ( w 2 + k 2 - k 2 2 ) 2 - k 2 w z 2 = 0 ,
k x = s x + k x - a x b - a , k y = s y + k y - a y b - a .
k i = n k r ,
g ( k ) 0.
V n ( R ) = - B t 16 π 2 A K 1 2 S d ( n k k × R ) f n ( r ) e - i n k ( R - r ) d r d k = i B t 4 π A K 1 f n ( r ) e i n k R - r R - r d r .
w x k x + w y k y - 1 2 ( w 2 + k 2 - k 2 ) = 0.
( k x - m k w x m k - n k ) 2 + ( k y - m k w y m k - n k ) 2 - k k m k - n k ( n m m k - n k ( w x 2 + w y 2 ) + 2 m k - 2 n k + 2 n w z ) = 0 ,
w x k x + w y k y - 1 2 ( w x 2 + w y 2 ) + k w z = 0 ,
V - 1 ( R ) 2 4 × 10 - 5 V 1 ( R ) max 2 .
( S x / 2 π + 96.3 ) 2 + ( S y / 2 π ) 2 = ( 96.3 ) 2 , ( 92.1 ) 2 ( S x / 2 π + 96.3 ) 2 + ( S y / 2 π ) 2 ( 100.3 ) 2 .
( S x / 2 π + 97.7 ) 2 + ( S y / 2 π ) 2 = ( 95.3 ) 2 , ( 91.1 ) 2 ( S x / 2 π + 97.7 ) 2 + ( S y / 2 π ) 2 ( 99.4 ) 2 .
V 1 ( R ) 2 2 × 10 - 5 V - 1 ( R ) max 2 .
( 0.32 S x / 2 π + x + 30.47 ) 2 + ( 0.32 S y / 2 π + y ) 2 = ( 30.47 ) 2 , ( 29.14 ) 2 ( 0.32 S x / 2 π + x + 30.47 ) 2 + ( 0.32 S y / 2 π + y ) 2 ( 31.74 ) 2 .
( 27.75 ) 2 ( 0.32 S x / 2 π + x + 30.47 ) 2 + ( 0.32 S y / 2 π + y ) 2 ( 29.14 ) 2 , ( 31.74 ) 2 ( 0.32 S x / 2 π + x + 30.47 ) 2 + ( 0.32 S y / 2 π + y ) 2 ( 32.96 ) 2 ,
0.32 S y / 2 π + y = 0 , - 8.9 0.32 S y / 2 π + y 8.9 ,
0.32 S x / 2 π + x = 0 , - 1.33 0.32 S x / 2 π + x 1.27 ,
V 1 ( R ) 2 2 × 10 - 5 V - 1 ( R ) max 2 .
V - 1 ( R ) = - B t 8 π e i k R z K 1 ( s ) K 2 ( s ) S d ( s s × R ) sinc [ 1 2 t g ( k ) ] × e - i g ( k ) R z f ˆ ( - s ) e - i sR d s .
( S x 2 π ) 2 + ( S y 2 π ) 2 = 0 , ( S x 2 π ) 2 + ( S y 2 π ) 2 ( 17.8 ) 2 .
( S x 2 π ) 2 + ( S y 2 π ) 2 = 0 , ( S x 2 π ) 2 + ( S y 2 π ) 2 ( 397 ) 2 .
( 0.025 S x 2 π + x + 30.58 ) 2 + ( 0.025 S y 2 π + y ) 2 = ( 30.58 ) 2 , ( 30.56 ) 2 ( 0.025 S x 2 π + x + 30.58 ) 2 + ( 0.025 S y 2 π + y ) 2 ( 30.60 ) 2 ,
( 0.025 S x / 2 π + x + 30.6 ) 2 + ( 0.025 S y / 2 π + y ) 2 = ( 30.6 ) 2 , ( 0.025 S x / 2 π + x + 30.4 ) 2 + ( 0.025 S y / 2 π + y ) 2 ( 33.8 ) 2 , ( 0.025 S x / 2 π + x + 30.8 ) 2 + ( 0.025 S y / 2 π + y ) 2 ( 26.7 ) 2 .