Abstract

Theoretical considerations are presented on characteristics of the optical-alignment technique using dual gratings. The general formulation of the diffraction pattern is derived for superimposition of dual amplitude–phase gratings, as functions of aperture ratios for gratings and separation gaps between dual gratings. Numerical analyses reveal characteristics for precise positioning control or alignment technique, which utilize the detection of the minimum value of the first-order diffraction beam or the detection of the zero cross point of the difference in intensities between +1- and −1-order diffraction light for dual gratings. Alignment errors due to variations in an incident light angle, displacement in detector position, and fabrication errors for the gratings are also discussed.

© 1978 Optical Society of America

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References

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  1. K. Kodate, T. Kamiya, and M. Kamiyama, “Double diffraction in the Fresnel region,” Jpn. J. Appl. Phys. 10, 1040–1045 (1971).
    [Crossref]
  2. K. Kodate, T. Kamiya, and H. Takenada, “Double diffraction of phase gratings in the Fresnel region,” Jpn. J. Appl. Phys. 14, 1323–1334 (1975).
    [Crossref]
  3. Y. Torii and Y. Mizushima, “Optical ultramicrometer technique utilizing a composite diffraction grating,” Opt. Commun. 23, 135–138 (1977).
    [Crossref]
  4. D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett.,  31, 426–428 (1977).
    [Crossref]
  5. J. M. Cowley and A. F. Moodie, “Fourier images: I—the point source,” Proc. Phys. Soc. B 70, 486–496 (1957).
    [Crossref]

1977 (2)

Y. Torii and Y. Mizushima, “Optical ultramicrometer technique utilizing a composite diffraction grating,” Opt. Commun. 23, 135–138 (1977).
[Crossref]

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett.,  31, 426–428 (1977).
[Crossref]

1975 (1)

K. Kodate, T. Kamiya, and H. Takenada, “Double diffraction of phase gratings in the Fresnel region,” Jpn. J. Appl. Phys. 14, 1323–1334 (1975).
[Crossref]

1971 (1)

K. Kodate, T. Kamiya, and M. Kamiyama, “Double diffraction in the Fresnel region,” Jpn. J. Appl. Phys. 10, 1040–1045 (1971).
[Crossref]

1957 (1)

J. M. Cowley and A. F. Moodie, “Fourier images: I—the point source,” Proc. Phys. Soc. B 70, 486–496 (1957).
[Crossref]

Austin, S.

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett.,  31, 426–428 (1977).
[Crossref]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier images: I—the point source,” Proc. Phys. Soc. B 70, 486–496 (1957).
[Crossref]

Flanders, D. C.

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett.,  31, 426–428 (1977).
[Crossref]

Kamiya, T.

K. Kodate, T. Kamiya, and H. Takenada, “Double diffraction of phase gratings in the Fresnel region,” Jpn. J. Appl. Phys. 14, 1323–1334 (1975).
[Crossref]

K. Kodate, T. Kamiya, and M. Kamiyama, “Double diffraction in the Fresnel region,” Jpn. J. Appl. Phys. 10, 1040–1045 (1971).
[Crossref]

Kamiyama, M.

K. Kodate, T. Kamiya, and M. Kamiyama, “Double diffraction in the Fresnel region,” Jpn. J. Appl. Phys. 10, 1040–1045 (1971).
[Crossref]

Kodate, K.

K. Kodate, T. Kamiya, and H. Takenada, “Double diffraction of phase gratings in the Fresnel region,” Jpn. J. Appl. Phys. 14, 1323–1334 (1975).
[Crossref]

K. Kodate, T. Kamiya, and M. Kamiyama, “Double diffraction in the Fresnel region,” Jpn. J. Appl. Phys. 10, 1040–1045 (1971).
[Crossref]

Mizushima, Y.

Y. Torii and Y. Mizushima, “Optical ultramicrometer technique utilizing a composite diffraction grating,” Opt. Commun. 23, 135–138 (1977).
[Crossref]

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “Fourier images: I—the point source,” Proc. Phys. Soc. B 70, 486–496 (1957).
[Crossref]

Smith, H. I.

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett.,  31, 426–428 (1977).
[Crossref]

Takenada, H.

K. Kodate, T. Kamiya, and H. Takenada, “Double diffraction of phase gratings in the Fresnel region,” Jpn. J. Appl. Phys. 14, 1323–1334 (1975).
[Crossref]

Torii, Y.

Y. Torii and Y. Mizushima, “Optical ultramicrometer technique utilizing a composite diffraction grating,” Opt. Commun. 23, 135–138 (1977).
[Crossref]

Appl. Phys. Lett. (1)

D. C. Flanders, H. I. Smith, and S. Austin, “A new interferometric alignment technique,” Appl. Phys. Lett.,  31, 426–428 (1977).
[Crossref]

Jpn. J. Appl. Phys. (2)

K. Kodate, T. Kamiya, and M. Kamiyama, “Double diffraction in the Fresnel region,” Jpn. J. Appl. Phys. 10, 1040–1045 (1971).
[Crossref]

K. Kodate, T. Kamiya, and H. Takenada, “Double diffraction of phase gratings in the Fresnel region,” Jpn. J. Appl. Phys. 14, 1323–1334 (1975).
[Crossref]

Opt. Commun. (1)

Y. Torii and Y. Mizushima, “Optical ultramicrometer technique utilizing a composite diffraction grating,” Opt. Commun. 23, 135–138 (1977).
[Crossref]

Proc. Phys. Soc. B (1)

J. M. Cowley and A. F. Moodie, “Fourier images: I—the point source,” Proc. Phys. Soc. B 70, 486–496 (1957).
[Crossref]

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Figures (12)

FIG. 1
FIG. 1

Optical arrangement for superimposed dual gratings.

FIG. 2
FIG. 2

First-order diffraction-light-intensity curves for dual amplitude gratings for various aperture ratios r and normalized air gaps M(i.e. M = λz/p2, z: air gap), where Y±1 = d/ppMλ−1 sin(θ±1) ± M/2, I0 = z(pL)−2 I, and γ = 0.

FIG. 3
FIG. 3

Plots showing difference in intensities between +1- and −1-order diffraction light for dual amplitude gratings under various aperture ratios r and normalized air gaps M, θ = 0, and γ = 0. d is the relative displacement between dual gratings.

FIG. 4
FIG. 4

nth-order diffraction-light-intensity curves for dual amplitude gratings, where Y = d/ppMλ−1 sin(θ) + nM/2 and γ = 0.

FIG. 5
FIG. 5

First-order diffraction-light-intensity curves for identical dual phase gratings as a function of one period of Y±1.

FIG. 6
FIG. 6

Plots of intensity differences between +1-and −1-order diffraction light for identical dual phase gratings. d is relative displacement between dual gratings.

FIG. 7
FIG. 7

Plots showing first-order diffraction light intensity and difference in intensities between +1- and −1-order diffraction light for dual phase gratings with different phase values.

FIG. 8
FIG. 8

First-order diffraction-light-intensity curves for superimposed phase and amplitude gratings.

FIG. 9
FIG. 9

Plots showing difference in intensities between +1- and −1-order diffraction light for dual amplitude and phase gratings.

FIG. 10
FIG. 10

Plots showing first-order diffraction light intensity and difference in intensities between +1- and −1-order diffraction light as a parameter of deviation in detector position Δω′. I 0 = z P - 2 sin - 2 ( π L Δ ω ) sin 2 ( π Δ ω ) I, and Δ I o = z p - 2 Δ I.

FIG. 11
FIG. 11

Dependence on contrast change for dual amplitude gratings.

FIG. 12
FIG. 12

Dependence on variation in phase value for dual phase gratings.

Tables (3)

Tables Icon

TABLE I General formulas for nth-order diffraction light intensity from dual gratings.

Tables Icon

TABLE II General formulas for ±1-order diffraction light intensity from dual gratings.

Tables Icon

TABLE III General formulas for nth order diffraction light intensity for superimposed dual gratings with no air gap.

Equations (67)

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W 1 ( ξ ) = i λ z exp ( i 2 π z / λ ) × t 1 ( x ) exp [ i 2 π λ ( - x sin θ + ( x - ξ ) 2 2 z ) ] d x ,
W 2 ( ω ) = W 1 ( ξ ) t 2 ( ξ ) exp ( - i 2 π ω ξ ) d ξ ,
W 2 ( ω ) = | 1 z t 1 ( x ) t 2 ( ξ ) exp { i 2 π ( - sin θ λ x - ω ξ + ( x - ξ ) 2 2 z ) } d x d ξ | .
t 2 ( ξ ) = k = - C 2 k exp ( i 2 π k ξ / p )
C 2 k = 1 p - p / 2 p / 2 t 2 ( t ) exp ( - i 2 π k t / p ) d t .
- exp ( i π x 2 / 2 ) d x = 1 + i .
W 2 ( ω ) = 1 z | k = - C 2 k exp ( - i π λ z p 2 ( p ω - k ) 2 ) × t 1 ( x ) exp - i 2 π x p ( p ω - k + p sin θ λ ) d x | .
t 1 ( x ) = l = - L 1 L 2 t 1 p ( x - d + l p )
W 2 ( ω ) = p z | K ( u ) k = - C 2 k T 1 p ( u ) × exp [ - i π ( 2 u d p + M ( p ω - k ) 2 ) ] |
T 1 p ( u ) = 1 p - p / 2 p / 2 t 1 p ( x ) exp ( - i 2 π u x / p ) d x ,
K ( u ) = l = - L 1 L 2 exp ( i 2 π u l ) ,
M = λ z / p 2 ,
u = p ω - k + p sin θ λ ,
| W 2 ( n p - sin θ λ ) | = p L z | k = - C 1 n - k C 2 k × exp { - i π ( n - k ) [ 2 ( d p - p M sin θ λ ) + M ( n - k ) ] } | .
I c ( ω ) = | t 1 ( x ) t 2 ( x ) exp { - i 2 π ( ω + sin θ λ ) x } d x | 2 .
I c ( ω ) = | p K ( u ) k = - C 2 k T 1 p ( u ) exp ( - i 2 π u d / p ) | 2 ,
t i p ( x ) = { a i , ( x r p / 2 ) b i , ( r p / 2 < x p / 2 )
C i k = { b i + ( a i - b i ) Ĉ k , ( k = 0 ) ( a i - b i ) Ĉ k , ( k 0 )
| W 2 ( n p - sin θ λ ) | = p L z [ b 2 ( a 1 - b 1 ) × exp ( - i 2 π n Y ) + b 1 ( a 2 - b 2 ) ] Ĉ n + 2 ( a 1 - b 1 ) ( a 2 - b 2 ) × exp [ - i π n ( Y - n M / 4 ) ] S ( Y ; n ) ,
S ( Y ; n ) = { k = 1 Ĉ k - ( n + 1 ) / 2 Ĉ k + ( n - 1 ) / 2 × exp [ - i π M ( k - ½ ) 2 ] cos [ π ( 2 k - 1 ) Y ] ,             ( for n : odd ) 1 2 Ĉ n / 2 2 + k = 1 Ĉ k - n / 2 Ĉ k + n / 2 × exp ( - i π M k 2 ) cos ( 2 π k Y ) ,             ( for n : even )
| W 2 ( ± 1 p - λ - 1 sin θ ± 1 ) | = p L z { b 2 ( a 1 - b 1 ) exp ( i 2 π Y ± 1 ) + b 1 ( a 2 - b 2 ) } Ĉ ± 1 + 2 ( a 1 - b 1 ) ( a 2 - b 2 ) exp ( i π Y ± 1 ) S ( Y ± 1 ) ,
S ( Y ± 1 ) = k = 1 Ĉ k Ĉ k - 1 exp [ - i π M k ( k - 1 ) ] × cos [ π ( 2 k - 1 ) Y ± 1 ]
Y ± 1 = d / p - p M λ - 1 sin ( θ ± 1 ) ± M / 2 ,
| W 2 ( ± 1 p - λ - 1 sin θ ± 1 ) | = p L π z b 2 ( a 1 - b 1 ) exp ( i 2 π Y ± 1 ) + b 1 ( a 2 - b 2 ) + ( a 1 - b 1 ) ( a 2 - b 2 ) exp ( i π Y ± 1 ) cos ( π Y ± 1 ) = p L 2 π z ( a 1 - b 1 ) ( a 2 + b 2 ) × exp ( i 2 π Y ± 1 ) + ( a 1 + b 1 ) ( a 2 - b 2 ) .
I ( ± 1 / p - λ - 1 sin θ ± 1 ) = ( p L ) 2 z A × exp { - i π ( Ŷ ± 1 - 2 α ) } + B exp { i π ( Ŷ ± 1 + 2 β ) } + k = 1 D ( k ) exp { i 2 δ ( k ) } cos { π ( 2 k - 1 ) Ŷ ± 1 } 2
Ŷ ± 1 = ± ( d / p - p M λ - 1 sin θ ± 1 ) + M / 2 A exp ( i 2 π α ) = Ĉ 1 · b 2 ( a 1 - b 1 ) , B exp ( i 2 π β ) = Ĉ 1 b 1 ( a 2 - b 2 ) , D ( k ) exp { i 2 δ ( k ) } = 2 ( a 1 - b 1 ) ( a 2 - b 2 ) Ĉ k Ĉ k - 1 × exp { - i π M k ( k - 1 ) } ,
Δ I = I ( 1 / p - λ - 1 sin θ + 1 ) - I ( - 1 / p - λ - 1 sin θ - 1 ) = - 4 ( p L ) 2 z { [ R 1 cos ( π 2 ( Ŷ + 1 - Ŷ - 1 ) ) + k = 1 Q 1 ( k ) cos ( π ( k - 1 2 ) ( Ŷ + 1 - Ŷ - 1 ) ) ] × [ R 2 sin ( π 2 ( Ŷ + 1 - Ŷ - 1 ) ) + k = 1 Q 2 ( k ) sin ( π ( k - 1 2 ) ( Ŷ + 1 - Ŷ - 1 ) ) ] - [ R 3 cos ( π 2 ( Ŷ + 1 - Y - 1 ) ) + k = 1 Q 3 ( k ) cos ( π ( k - 1 2 ) ( Ŷ + 1 - Ŷ - 1 ) ) ] × [ R 4 sin ( π 2 ( Ŷ + 1 - Ŷ - 1 ) ) + k = 1 Q 4 ( k ) sin ( π ( k - 1 2 ) ( Ŷ + 1 - Ŷ - 1 ) ) ] } ,
R 1 = A cos ( π 2 ( Ŷ + 1 + Ŷ - 1 - 4 α ) ) + B cos ( π 2 ( Ŷ + 1 + Ŷ - 1 + 4 β ) ) R 2 = A sin ( π 2 ( Ŷ + 1 + Ŷ - 1 - 4 α ) ) + B sin ( π 2 ( Ŷ + 1 + Ŷ - 1 + 4 β ) ) , R 3 = - A sin ( π 2 ( Ŷ + 1 + Ŷ - 1 - 4 α ) ) + B sin ( π 2 ( Ŷ + 1 + Ŷ - 1 + 4 β ) ) , R 4 = - A cos ( π 2 ( Ŷ + 1 + Ŷ - 1 - 4 α ) ) + B cos ( π 2 ( Ŷ + 1 + Ŷ - 1 + 4 β ) ) , Q 1 ( k ) = D ( k ) cos [ 2 π δ ( k ) ] cos [ π ( k - 1 2 ) ( Ŷ + 1 + Ŷ - 1 ) ] , Q 2 ( k ) = D ( k ) cos [ 2 π δ ( k ) ] sin [ π ( k - 1 2 ) ( Ŷ + 1 + Ŷ - 1 ) ] , Q 3 ( k ) = D ( k ) sin [ 2 π δ ( k ) ] cos [ π ( k - 1 2 ) ( Ŷ + 1 + Ŷ - 1 ) ] , Q 4 ( k ) = D ( k ) sin [ 2 π δ ( k ) ] sin [ π ( k - 1 2 ) ( Ŷ + 1 + Ŷ - 1 ) ] .
d = p / 2 + z ( sin θ + 1 + sin θ - 1 )
d = z ( sin θ + 1 + sin θ - 1 )
Δ I = - 1 z ( p L π ) 2 A sin { π M [ 1 - p λ ( sin θ + 1 - sin θ - 1 ) ] - 2 π ϕ } sin ( 2 π d p - p M 2 λ ( sin θ + 1 + sin θ - 1 ) ) ,
A exp ( i 2 π ϕ ) = ( a 1 - b 1 ) ( a 2 + b 2 ) ( a 1 * + b 1 * ) ( a 2 * - b 2 * ) ,
Δ I = - 1 z ( p L π ) 2 A sin [ π ( M - 2 ϕ ) ] sin ( 2 π d / p ) ,
I o ( n p - sin θ λ ) = ( π n p L ) 2 I c ( n p - sin θ λ ) = { 1 4 ( a 1 - b 1 ) ( a 2 e i π n r - b 2 e - i π n r ) e - i 2 π n d / p + ( a 2 - b 2 ) ( b 1 e i π n r - a 1 e - i π n r ) 2 , [ 0 d ( 1 - r ) p ] sin 2 ( π n r ) a 2 ( a 1 - b 1 ) e - i 2 π n d / p + a 1 ( a 2 - b 2 ) 2 , [ ( 1 - r ) p < d < r p ] 1 4 ( a 1 - b 1 ) ( a 2 e - i π n r - b 2 e i π n r ) e - i 2 π n d / p + ( a 2 - b 2 ) ( b 1 e - i π n r - a 1 e i π n r ) 2 , ( r p d < p )
I o ( ± 1 p - λ - 1 sin θ ± 1 ; Y ± 1 , M + l ) = I o ( ± 1 p - λ - 1 sin θ ± 1 ; Y ± 1 , M ) ,
I o ( ± 1 p ; d , M + l ) = I o ( ± 1 p ; d + l p 2 , M ) ,
I o ( ± 1 p ; d , M + 2 l ) = I o ( ± 1 p ; d , M ) ,
I o ( ± 1 p - λ - 1 sin θ ± 1 ; l - Y ± 1 , M ) = I o ( 1 p - λ - 1 sin θ 1 ; Y 1 , M ) ,
Δ I o ( 1 p , - 1 p ; d , M + l ) = Δ I o ( 1 p , - 1 p ; d + l p 2 , M )
I o = z ( p L ) 2 | W 2 ( ± 1 p - λ - 1 sin θ ± 1 ) | 2 ,
I = [ ( p L ) 2 / z ] I 0 ,
Δ I o ( 1 p , - 1 p ; d , M ) = I 0 ( 1 p ; d , M ) - I 0 ( - 1 p ; d , M ) ,
I o ( + 1 / p - λ - 1 sin θ + 1 ; Y + 1 , M ) = I 0 ( - 1 / p - λ - 1 sin θ - 1 ; Y - 1 , M )
I ( ± 1 p - λ - 1 sin θ ± 1 ) = 4 ( p L ) 2 z | ( 1 - γ ) 2 Ĉ ± 1 cos ( π Y ± 1 ) + ( 1 - γ ) k = 1 Ĉ k Ĉ k - 1 exp [ - i π M k ( k - 1 ) ] × cos [ π ( 2 k - 1 ) Y ± 1 ] | 2
I o ( ± 1 p - λ - 1 sin θ ± 1 ; Y ± 1 , M ) = I o ( 1 p - λ - 1 sin θ 1 ; Y 1 , M ) ,
I o ( ± 1 p - λ - 1 sin θ ± 1 ; Y ± 1 , l - M ) = I o ( 1 p - λ - 1 sin θ 1 ; Y 1 , M ) ,
I o ( ± 1 p ; d , l - M ) = I o ( 1 p ; d + l p 2 , M ) ,
I o ( ± 1 p ; d , 2 l - M ) = I o ( 1 p ; d , M )
Δ I o ( 1 p , - 1 p ; d , l - M ) = - Δ I o ( 1 p , - 1 p ; d + l p 2 , M ) ,
Δ I o ( 1 p , - 1 p ; d , 2 l - M ) = - Δ I o ( 1 p , - 1 p ; d , M ) ,
Δ I = - 1 z ( p L / π ) 2 sin ( π M ) sin ( 2 π d / p ) ,
I ( ± 1 p - λ - 1 sin θ ± 1 ) = 16 ( p L ) 2 z sin 2 ( δ 2 ) × | Ĉ ± 1 cos ( π Y ± 1 ) + 2 i exp ( i δ 2 ) sin ( δ 2 ) × k = 1 Ĉ k Ĉ k - 1 exp [ - i π M k ( k - 1 ) ] × cos [ π ( 2 k - 1 ) Y ± 1 ] | 2 ,
I ( ± 1 p - λ - 1 sin θ ± 1 ) = 4 z ( p L π ) 2 sin 2 ( δ ) cos 2 ( π Y ± 1 ) .
Δ I = - 4 z ( p L π ) 2 sin ( δ 1 ) sin ( δ 2 ) sin ( π M ) sin ( 2 π d / p ) ,
I ( ± 1 p - λ - 1 sin θ ± 1 ) = ( p L ) 2 z | Ĉ ± 1 exp ( i π Y ± 1 ) + 4 i sin ( δ 2 ) exp ( i δ 2 ) × k = 1 Ĉ k Ĉ k - 1 exp [ - i π M k ( k - 1 ) ] × cos [ π ( 2 k - 1 ) Y ± 1 ] | 2 .
I ( ± 1 / p - λ - 1 sin θ ± 1 ) = ( 1 / z ) ( p L π ) 2 [ 1 sin ( δ ) sin ( 2 π Y ± 1 ) ] .
Δ I = - 1 2 z ( p L π ) 2 sin ( δ ) cos ( π M ) sin ( 2 π d / p ) .
I ( ω ) = p 2 z | l = - L 1 L 2 exp [ i 2 π l p ( ω - λ - 1 sin θ ± 1 ) ] × k = - C 2 k C 1 p ω - k + p λ - 1 sin ( θ ± 1 ) × exp ( - i 2 π d p ( p ω - k + p λ - 1 sin θ ± 1 ) ) × exp [ - i π M ( p ω - k ) 2 ] | 2 .
θ ± 1 = θ ˆ ± 1 + Δ θ ± 1 ,
= - λ - 1 sin ( θ ± 1 ) ± 1 / p + Δ ω ± 1 ,
- λ - 1 sin ( θ ˆ ± 1 ) ± 1 / p + Δ ω ± 1 ,
Δ ω ± 1 Δ ω ± 1 - λ - 1 Δ θ ± 1 cos ( θ ˆ ± 1 ) ,
I ( ± 1 p - λ - 1 sin ( θ ˆ ± 1 + Δ θ ± 1 ) + Δ ω ± 1 ) = p 2 z ( sin π L Δ ω ± 1 sin π Δ ω ± 1 ) 2 | k = C 2 k C 1 ± 1 - k + p Δ ω ± 1 × exp [ - i 2 π X ± 1 ( ± 1 - k + p Δ ω ± 1 ) ] × exp [ - i π M ( ± 1 - k + p Δ ω ± 1 ) 2 ] | 2
X ± 1 = d / p - p M λ - 1 sin ( θ ˆ ± 1 + Δ θ ± 1 ) .
I ( ± 1 p - λ - 1 sin ( θ ˆ ± 1 + Δ θ ± 1 ) + Δ ω ± 1 ) = p 2 z ( sin π L Δ ω ± 1 sin π Δ ω ± 1 ) 2 | k = - sin ( π k r ) π k × sin [ π ( ± 1 - k + p Δ ω ± 1 ) r ] π ( ± 1 - k + p Δ ω ± 1 ) × exp [ - i 2 π X ± 1 ( ± 1 - k + p Δ ω ± 1 ) ] × exp [ - i π M ( ± 1 - k + p Δ ω ± 1 ) 2 ] | 2 ,
d = p 2 M [ sin ( θ ˆ + 1 + Δ θ + 1 ) + sin ( θ ˆ - 1 + Δ θ - 1 ) ]
d = p / 2 + p 2 M [ sin ( θ ˆ + 1 + Δ θ + 1 ) + sin ( θ ˆ - 1 + Δ θ - 1 ) ] .