Abstract

Two methods are described for synthesizing the electrooptic light scanner with an arbitrarily prescribed one-dimensional spatial distribution of the light beam. The electrooptic light scanner is constructed by a series of cascaded stages, each stage containing a natural-birefringent electrooptic crystal with an inclination angle between opposite faces. One method is realized by a set of cascaded stages, each of which contains a polarizer, electrooptic crystal, optical compensator, and analyzer; and the other method by a multistage construction that is inserted between the input and output polarizers, each stage being composed of only an electrooptic crystal and optical compensator. Two examples for each method are given to illustrate the synthesis procedures.

© 1971 Optical Society of America

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References

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  1. S. E. Harris, E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 54, 1267 (1964).
    [CrossRef]
  2. E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 55, 835 (1965).
    [CrossRef]
  3. E. O. Ammann, J. Opt. Soc. Amer. 56, 943 (1966).
    [CrossRef]
  4. E. O. Ammann, J. M. Yarborough, J. Opt. Soc. Amer. 56, 1746 (1966).
    [CrossRef]
  5. M. Okada, S. Ieiri, Tech. Group on Quant. Electron. IECE Japan, paper QE69-16 (1969).
  6. M. Okada, S. Ieiri, Japan. J. Appl. Phys. 9, 153 (1970).
    [CrossRef]
  7. J. W. Evans, J. Opt. Soc. Amer. 39, 229 (1949).
    [CrossRef]

1970

M. Okada, S. Ieiri, Japan. J. Appl. Phys. 9, 153 (1970).
[CrossRef]

1966

E. O. Ammann, J. Opt. Soc. Amer. 56, 943 (1966).
[CrossRef]

E. O. Ammann, J. M. Yarborough, J. Opt. Soc. Amer. 56, 1746 (1966).
[CrossRef]

1965

E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 55, 835 (1965).
[CrossRef]

1964

S. E. Harris, E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 54, 1267 (1964).
[CrossRef]

1949

J. W. Evans, J. Opt. Soc. Amer. 39, 229 (1949).
[CrossRef]

Ammann, E. O.

E. O. Ammann, J. Opt. Soc. Amer. 56, 943 (1966).
[CrossRef]

E. O. Ammann, J. M. Yarborough, J. Opt. Soc. Amer. 56, 1746 (1966).
[CrossRef]

E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 55, 835 (1965).
[CrossRef]

S. E. Harris, E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 54, 1267 (1964).
[CrossRef]

Chang, I. C.

E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 55, 835 (1965).
[CrossRef]

S. E. Harris, E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 54, 1267 (1964).
[CrossRef]

Evans, J. W.

J. W. Evans, J. Opt. Soc. Amer. 39, 229 (1949).
[CrossRef]

Harris, S. E.

S. E. Harris, E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 54, 1267 (1964).
[CrossRef]

Ieiri, S.

M. Okada, S. Ieiri, Japan. J. Appl. Phys. 9, 153 (1970).
[CrossRef]

M. Okada, S. Ieiri, Tech. Group on Quant. Electron. IECE Japan, paper QE69-16 (1969).

Okada, M.

M. Okada, S. Ieiri, Japan. J. Appl. Phys. 9, 153 (1970).
[CrossRef]

M. Okada, S. Ieiri, Tech. Group on Quant. Electron. IECE Japan, paper QE69-16 (1969).

Yarborough, J. M.

E. O. Ammann, J. M. Yarborough, J. Opt. Soc. Amer. 56, 1746 (1966).
[CrossRef]

J. Opt. Soc. Amer.

S. E. Harris, E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 54, 1267 (1964).
[CrossRef]

E. O. Ammann, I. C. Chang, J. Opt. Soc. Amer. 55, 835 (1965).
[CrossRef]

E. O. Ammann, J. Opt. Soc. Amer. 56, 943 (1966).
[CrossRef]

E. O. Ammann, J. M. Yarborough, J. Opt. Soc. Amer. 56, 1746 (1966).
[CrossRef]

J. W. Evans, J. Opt. Soc. Amer. 39, 229 (1949).
[CrossRef]

Japan. J. Appl. Phys.

M. Okada, S. Ieiri, Japan. J. Appl. Phys. 9, 153 (1970).
[CrossRef]

Other

M. Okada, S. Ieiri, Tech. Group on Quant. Electron. IECE Japan, paper QE69-16 (1969).

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Figures (9)

Fig. 1
Fig. 1

Basic configuration of E.O. light scanner. P1 and P2 denote polarizer and analyzer. Input light is a monochromatic plane wave polarized as shown by E.

Fig. 2
Fig. 2

Schematic diagram of three stage E.O. light scanner similar to Lyot filter. M: mirror, T: He-Ne laser tube, R: Rotator of polarization direction, P: polarizer, L: lens, Xk: E.O. crystal of the kth stage, Ak: analyzer of the kth stage, S: screen, S.W.G.: sawtooth-wave generator.

Fig. 3
Fig. 3

Arrangements of the kth stage in the first type of E.O. light scanner. Pk and Pk+1: input and output polarizer, Ck: optical compensator, u: basic line (direction of transmission axis of the input polarizer of the top stage), Tk−1 and Tk: directions of transmission axis of the input and output polarizer respectively, So and Fo: slow and fast axis of E.O. crystal, Sk: slow axis of optical compensator, γk−1 and γk: angles between baseline and Tk−1 and Tk, respectively, φo; angle between u and So, φk: angle between u and Sk.

Fig. 4
Fig. 4

General configuration of the first type of E.O. light scanner with m stages.

Fig. 5
Fig. 5

(a) Arrangements of the kth stage in the second type of E.O. light scanner. The same symbols as in Fig. 3 are used. (b) A general configuration of the second type of E.O. light scanner with m stages.

Fig. 6
Fig. 6

Directions of elements of the kth stage in the first type of light scanner with respect to u.

Fig. 7
Fig. 7

Amplitude and intensity distributions of two-stage E.O. light scanner similar to Lyot filter.

Fig. 8
Fig. 8

Triangular amplitude distribution and its intensity characteristics.

Fig. 9
Fig. 9

Directions of E.O. crystal and optical compensator with respect to u.

Tables (2)

Tables Icon

Table I Four Sets of Ten-stage E.O. Light Scanner with Triangular Spatial Amplitude Distributiona

Tables Icon

Table II Solutions of |D(Y)|2 = 0 and a Pair of Related Roots of X + 1/X = Y in the Example of Triangular Amplitude Distributiona

Equations (91)

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( Δ n ) · a · tan δ = λ ,
ω = 2 π λ ( Δ n ) · h · tan δ + 2 π V V o = 2 π ( h a + V V o ) ,
C ( ω ) = C o + C 1 e - i ω + C 2 e - i 2 ω + + C m - i m ω ,
C ( ω ) = C m ( e - i ω + R 1 e i α 1 ) ( e - i ω + R 2 e i α 2 ) × ( e - i ω + R m e i α m ) ,
E k E k - 1 = [ cos ( γ k - φ k ) sin ( γ k - φ k ) 0 0 ] × [ e - i b k cos ( φ k - φ o ) e - i b k sin ( φ k - φ o ) - sin ( φ k - φ o ) cos ( φ k - φ o ) ] × [ e - i ω cos ( φ o - γ k - 1 ) - sin ( φ o - γ k - 1 ) ] = exp [ - ( i b k ) / 2 ] cos ( φ o - γ k - 1 ) cos ( φ o - γ k ) cos ( b k / 2 ) × [ 1 - i tan b k 2 cos ( φ o + γ k - 2 φ k ) cos ( φ o - γ k ) ] × [ e - i ω + tan ( φ o - γ k - 1 ) + sin ( φ o - γ k - i tan ( b k / 2 ) · sin ( φ o + γ k - 2 φ k ) cos ( φ o - γ k ) - i tan ( b k / 2 ) cos ( φ o + γ k - 2 φ k ) ] , ( k = 1 , 2 , , m ) ,
R k e i α k = tan ( φ o - γ k - 1 ) × sin ( φ o - γ k ) - i tan ( b k / 2 ) sin ( φ o + γ k - 2 φ k ) cos ( φ o - γ k ) - i tan ( b k / 2 ) cos ( φ o + γ k - 2 φ k ) , ( k = 1 , 2 , , m ) ,
cos ( b k / 2 ) cos ( φ o - γ k - 1 ) cos ( φ o - γ k ) | 1 - i tan ( b k / 2 ) × cos ( φ o + γ k - 2 φ k ) cos ( φ o - γ k ) | ,             ( k = 1 , 2 , , m ) ,
arg ( C m ) = - k = 1 m ( 1 2 b k + p k ) ,
tan p k = tan b k 2 cos ( φ o + γ k - 2 φ k ) cos ( φ o - γ k ) ,             ( k = 1 , 2 , , m ) .
γ k - γ k - 1 = β k , φ k - γ k - 1 = θ k , φ o - γ k - 1 = μ k . }
R k e i α k = tan μ k sin ( μ k - β k ) - i tan ( b k / 2 ) sin ( μ k + β k - 2 θ k ) cos ( μ k - β k ) - i tan ( b k / 2 ) cos ( μ k + β - 2 θ k ) ,             ( k = 1 , 2 , , m ) .
tan ( b k / 2 ) = tan μ k sin ( μ k - β k ) - R k cos α k cos ( μ k - β k ) R k sin α k cos ( μ k + β k - 2 θ k ) , = R k sin α k cos ( μ k - β k ) R k cos α k cos ( μ k + β k - 2 θ k ) - tan μ k sin ( μ k + β k - 2 θ k ) , ( k = 1 , 2 , , m ) .
tan ( μ k + β k - 2 θ k ) = R k cot μ k cos α k tan μ k tan ( μ k - β k ) - R k tan μ k tan ( μ k - β k ) - R k cos α k ,             ( k = 1 , 2 , , m ) .
E k E k - 1 = e - i ( b k / 2 + p k ) cos μ k ( 1 + R k 2 cot 2 μ k ) 1 2 ( e - i ω + R k e i α k ) ,             k = 1 , 2 , , m ) .
cos μ k / ( 1 + R k 2 cot 2 μ k ) 1 2 ,
tan 2 μ k tan 2 ( φ o - γ k - 1 ) = R k ,             ( k = 1 , 2 , , m ) .
tan ( φ o + γ k - 2 φ k ) = R k cos ( φ o - γ k - 1 ) × cos α k · tan ( φ o - γ k - 1 ) tan ( φ o - γ k ) - R k tan ( φ o - γ k - 1 ) tan ( φ o - γ k ) - R k cos α k , ( k = 1 , 2 , , m - 1 ) ,
tan ( b k / 2 ) = cos ( φ o - γ k ) cos ( φ o + γ k - 2 φ k ) × tan ( φ o - γ k - 1 ) tan ( φ o - γ k ) - R k cos α k R k sin α k , ( k = 1 , 2 , , m - 1 ) ,
tan p k = tan ( φ o - γ k - 1 ) tan ( φ o - γ k ) - R k cos α k R k sin α k ,             ( k = 1 , 2 , , m - 1 ) ,
b m 2 + p m = - arg ( C m ) + k = 1 m - 1 ( b k 2 + p k ) tan - 1 q ,
x = tan p m ,             y = tan ( b m / 2 ) ,
q = tan ( p m + b m / 2 ) = ( x + y ) / ( 1 - x y ) .
x 2 + R m cos 2 α m · ( x - tan α m ) 2 = y 2 [ 1 + R m cos 2 α m ( 1 + x tan α m ) 2 ] .
tan p m = [ B ± ( B 2 + A 2 ) 1 2 ] / A ,
A = q 2 ( 1 + tan 2 α m + R m ) - R m tan 2 α m , B = R m tan α m ( 1 + q 2 + q tan α m ) - q ( 1 + tan 2 α m + R m ) ,
b m = 2 ( tan - 1 q - p m ) .
tan p m = tan ( φ o - γ m - 1 ) tan ( φ o - γ m ) - R m cos α m R m sin α m ,
tan ( φ o + γ m - 2 φ m ) = R m cot ( φ o - γ m - 1 ) × cos α m · tan ( φ o - γ m - 1 ) · tan ( φ o - γ m ) - R m tan ( φ o - γ m - 1 ) · tan ( φ o - γ m ) - R m cos α m .
C ( ω ) = 1 4 ( 1 + e - i ω ) ( 1 + e - i 2 ω ) = 1 4 ( 1 + e - i ω ) ( i + e - i ω ) ( - i + e - i ω ) .
R 1 = R 2 = 1 ,             α 1 = α 2 = 0.
φ o = ± π 4 ,             γ 1 = 0.
1 = ( 1 2 ) 1 2 - i tan ( b 1 / 2 ) sin ( π / 4 - 2 φ 1 ) ( 1 2 ) 1 2 - i tan ( b 1 / 2 ) cos ( π / 4 - 2 φ 1 ) .
K ( ω ) = 1 2 + 2 π 2 { ( e i ω + e - i ω ) + 1 9 ( e i 3 ω + e - i 3 ω ) + 1 25 ( e i 5 ω + e - i 5 ω ) } ,
C ( ω ) = 2 π 2 ( 1 25 + 1 9 e - i 2 ω + e - i 4 ω + π 2 4 e - i 5 ω + e - i 6 ω + 1 9 e - i 8 ω + 1 25 e - i 10 ω )
R 1 = 1.25697 α 1 = α 10 = 0 R 2 = R 3 = 0.390720 α 2 = - α 3 = α 8 = - α 9 = 0.771253 R 4 = R 5 = 0.420170 α 4 = - α 5 = α 6 = - α 7 = 1.97027 R 6 = R 7 = 2.37999 R 8 = R 9 = 2.55938 R 10 = 0.795566
tan 2 φ 0 = R 1 tan 2 ( φ o - γ k - 1 ) = R k tan 2 ( φ o - γ k ) = R k + 1
C ( ω ) = C 0 + C 1 e - i ω + C 2 e - i 2 ω + + C m e - i m ω ,
ψ o = - φ o , ψ k = φ k - φ o , ( k = 1 , 2 , , m ) ψ a = φ a - φ m
( e - i b k cos ψ k e - i b k sin ψ k - sin ψ k cos ψ k ) ( e - i ω cos ψ k - 1 - e - i ω sin ψ k - 1 sin ψ k - 1 cos ψ k - 1 )
( cos ψ a sin ψ a - sin ψ a cos ψ a ) .
( C D ) = ( cos ψ a sin ψ a - sin ψ a cos ψ a ) ( e - i b m cos ψ m e - i b m sin ψ m - sin ψ m cos ψ m ) × ( e - i ω cos ψ m - 1 - e - i ω sin ψ m - 1 sin ψ m - 1 cos ψ m - 1 ) × ( e - i b k cos ψ k e - i b k sin ψ k - sin ψ k cos ψ k ) ( e - i ω cos ψ k - 1 - e - i ω sin ψ k - 1 sin ψ k - 1 cos ψ k - 1 ) × ( e - i b k - 1 cos ψ k - 1 e - i b k - 1 sin ψ k - 1 - sin ψ k - 1 cos ψ k - 1 ) × ( e - i b 1 cos ψ 1 e - i b 1 sin ψ 1 - sin ψ 1 cos ψ 1 ) × ( e - i ω cos ψ o - e - i ω sin ψ o sin ψ o cos ψ o ) ( 1 0 ) ,
C ( ω ) = k 0 + k 1 e - i w + k 2 e - i 2 ω + + k m e - i m ω
( e - i ω cos ψ k - e - i ω sin ψ k sin ψ k cos ψ k ) ( e - i b k cos ψ k e - i b k sin ψ k - sin ψ k cos ψ k ) = ( ( e - i b k c k 2 + s k 2 ) e - i ω c k s k ( e - i b k - 1 ) e - i ω c k s k ( e - i b k - 1 ) e - i b k s k 2 + c k 2 ) ,
( e - i ω c o - e - ω s o s o c o ) ,
( cos ψ a sin ψ a - sin ψ a cos ψ a ) ( e - i b m cos ψ m e - i b m sin ψ m - sin ψ m cos ψ m ) = ( c a c m e - i b m - s a s m c a s m e - i b m + s a c m - s a c m e - i b m - c a s m - s a s m e - i b m + c a c m )
( C D ) = ( c a c m e - i b m - s a s m c a s m e - i b m + s a c m - s a c m e - i b m - c a s m - s a s m e - i b m + c a c m ) × ( ( c m - 1 2 e - i b m - 1 + s m - 1 2 ) e - i ω s m - 1 c m - 1 ( e - i b m - 1 - 1 ) e - i ω s m - 1 c m - 1 ( e - i b m - 1 - 1 ) s m - 1 2 e - i b m - 1 + c m - 1 2 ) × ( ( c k 2 e - i b k + s k 2 ) e - i ω s k c k ( e - i b k - 1 ) e - i ω s k c k ( e - i b k - 1 ) s k 2 e - i b k + c k 2 ) × ( ( c 1 2 e - i b 1 + s 1 2 ) e - i ω s 1 c 1 ( e - i b 1 - 1 ) e - i ω s 1 c 1 ( e - i b 1 - 1 ) s 1 2 e - i b 1 + c 1 2 ) × ( c o e - i ω - s o e - i ω s o c o ) ( 1 0 ) .
( S 1 1 F o 1 ) = ( c 0 s o )
( S 2 2 S 1 2 F 1 2 F 0 2 ) = ( c 1 2 e - i b 1 + s 1 2 0 0 s 1 c 1 ( e - i b 1 - 1 ) s 1 c 1 ( e - i b 1 - 1 ) 0 0 s 1 2 e - i b 1 + c 1 ) ( S 1 1 F 0 1 )
( S 3 3 S 2 3 S 1 3 F 2 3 F 1 3 F 0 3 ) = ( A 2 0 0 0 0 A 2 B 2 0 0 0 0 B 2 B 2 0 0 0 0 B 2 A 2 0 0 0 0 A 2 ) ( S 2 2 S 1 2 F 1 2 F 0 2 )
( S k + 1 k + 1 S k k + 1 S k - 1 k + 1 · · · S 3 k + 1 S 2 k + 1 S 1 k + 1 F k k + 1 F k - 1 k + 1 · · · F 2 k + 1 F 1 k + 1 F 0 k + 1 ) = ( A k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A k 0 0 0 0 0 B k 0 0 0 0 0 0 0 0 A k 0 0 0 0 0 B k 0 0 0 0 0 0 0 0 0 0 A k 0 0 0 0 0 B k 0 0 0 0 0 0 0 0 A k 0 0 0 0 0 B k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B k B k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B k 0 0 0 0 0 A k 0 0 0 0 0 0 0 0 0 0 0 B k 0 0 0 0 0 A k 0 0 0 0 0 0 0 0 B k 0 0 0 0 0 A k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A k ) ( S k k S k - 1 k S k - 2 k · · · S 2 k S 1 k F k - 1 k F k - 2 k F k - 3 k · · · F 0 k )             ( k = 1 , 2 , , m - 1 )
A k = c k 2 e - i b k + s k 2 , B k = s k c k ( e - i b k - 1 ) , A k = s k 2 e - i b k + c k 2 . }
D ( ω ) 2 = 1 - C ( ω ) 2 .
D ( ω ) = D 0 + D 1 e - i ω + D 2 e - i 2 ω + + D m e - i m ω ,
( S j m F j m ) = ( e i b m c a c m - s a s m - e i b m s a c m - c a s m e i b m c a s m + s a c m - e i b m s a s m + c a c m ) ( C j D j )
( e i b m c a c m - s a s m ) C 0 = ( e i b m s a c m + c a s m ) D 0
( e i b m c a s m + s a c m ) C m = ( e i b m s a s m - c a c m ) D m .
b m = 0 , tan ( ψ a + ψ m ) = tan ( φ a - φ o ) = C 0 / D 0 = - D m / C m b m = π , tan ( ψ a - ψ m ) = C 0 / D 0 = - D m / C m ψ m = ± π / 2 , tan ψ a = C m / D m = - D 0 / C 0 ,             b m : indefinite ψ m = 0 or π , tan ψ a = - D m / C m = C 0 / D 0 ,             b m : indefinite .
S j m = ( D 0 C j - C 0 D j ) / ( C 0 2 + D 0 2 ) 1 2 F j m = ( C 0 C j + D 0 D j ) / ( C 0 2 + D 0 2 ) 1 2 . }
S k + 1 k + 1 F k k + 1 = A k B k = c k 2 e - i b k + s k 2 c k s k ( e - i b k - 1 ) = cos ( b k / 2 ) - i sin ( b k / 2 ) cos 2 ψ k - i sin ( b k / 2 ) sin 2 ψ k ( k = 1 , 2 , , m - 1 )
S 1 k + 1 F 0 k + 1 = B k A k = c k s k ( e - i b k - 1 ) s k 2 e - i b k + c k 2 = - i sin ( b k / 2 ) sin 2 ψ k cos ( b k / 2 ) + i sin ( b k / 2 ) cos 2 ψ k ( k = 1 , 2 , , m - 1 )
b m - 1 = π cot 2 ψ m - 1 = S m m / F m - 1 m = - F 0 m / S 1 m . }
S j k + 1 = A k S j - 1 k + B k F j - 1 k { k = 1 , 2 , , ( m - 1 ) j = 2 , 3 , , k } F j - 1 k + 1 = B k S j - 1 k + A k F j - 1 k .
S j - 1 k = i exp [ ( i b k ) / 2 ] × F k k + 1 F k k + 1 F k k + 1 * F j - 1 k + 1 + S k + 1 k + 1 * S j k + 1 ( F k k + 1 2 + S k + 1 k + 1 2 ) 1 2 F j - 1 k = i exp [ ( i b k ) / 2 ] × F k k + 1 * F k k + 1 F k k + 1 S j k + 1 - S k + 1 k + 1 F j - 1 k + 1 ( F k k + 1 2 + S k + 1 k + 1 2 ) 1 2 } k = 1 , 2 , , m - 1 ; j = 1 , 2 , , k + 1 ,
b m - 2 = π cot 2 ψ m - 2 = S m - 1 m - 1 F m - 2 m - 1 = - F 0 m - 1 S 1 m - 1 }
b k = π cot 2 ψ k = S k + 1 k + 1 F k k + 1 = - F 0 k + 1 S 1 k + 1 } ( k = 1 , 2 , , m - 1 )
( S 0 k S 1 k S j k S k k ) = - F k k + 1 / F k k + 1 ( F k k + 1 2 + S k + 1 k + 1 2 ) 1 2 ( F 0 k + 1 S 1 k + 1 F 1 k + 1 S 2 k + 1 F j k + 1 S j + 1 k + 1 F k k + 1 S k + 1 k + 1 ) ( F k k + 1 S k + 1 k + 1 )             ( k = 1 , 2 , , m - 1 )
( F 0 k F 1 k F j k F k k ) = F k k + 1 / F k k + 1 ( F k k + 1 2 + S k + 1 k + 1 2 ) 1 2 ( F 0 k + 1 S 1 k + 1 F 1 k + 1 S 2 k + 1 F j k + 1 S j + 1 k + 1 F k k + 1 S k + 1 k + 1 )             ( S k + 1 k + 1 - F k k + 1 )             ( k = 1 , 2 , , m - 1 )
tan ψ 0 - tan 0 = F 0 1 / S 1 1
C ( ω ) = 1 4 ( 1 + e - i ω + e - i 2 ω + e - i 3 ω )
D ( Y ) 2 = - 1 16 Y 3 - 1 8 Y 2 + 1.
Y = 2 X = 1 Y = - 2 ( 1 - i ) X = - - i [ 2 ± ( 2 - i ) 1 2 ] Y = - 2 ( 1 + i ) X = - i [ 2 ± ( 2 + i ) 1 2 ] .
d 0 = - [ 2 + 5 - 2 ( 2 + 5 ) 1 2 ] d 0 = - 2 [ 2 + 5 + 2 ( 2 + 5 ) 1 2 ] d 1 = 5 - ( 2 + 5 ) 1 2 - ( 5 - 2 ) 1 2 d 1 = 5 + ( 5 + 2 ) 1 2 + ( 5 - 2 ) 1 2 d 2 = 1 - ( 5 + 2 ) 1 2 + ( 5 - 2 ) 1 2 d 2 = 1 + ( 5 + 2 ) 1 2 - ( 5 - 2 ) 1 2 d 3 = 1 d 3 = 1 ,
D 0 = 0.0865036 D 0 = 0.722513 D 1 = 0.222513 D 1 = ± 0.413496 D 2 = 0.413496 D 2 = ± 0.222513 D 3 = ± 0.722513 D 3 = ± 0.0865036.
tan ( φ a - φ o ) = - 2.89005.
S 1 3 = - 0.128530 F 0 3 = - 0.264543 S 2 3 = - 0.309014 F 1 3 = - 0.309014 S 3 3 = 0.764542 F 2 3 = - 0.371469 ,
tan 2 ψ 2 = tan 2 ( φ 2 - φ o ) = - 0.485857.
S 1 2 = - 0.142891 F 1 2 = 0.412989 S 2 2 = 0.850007 F 0 2 = 0.294114
tan 2 ψ 1 = tan 2 ( φ 1 - φ 0 ) = 0.452188.
S 1 1 = - 0.945024             F 0 1 = 0.326987
tan ψ o = - tan φ o = - 0.346009.
b 1 = b 2 = π ,             b 3 = 0 φ 0 = 19.086 ° , φ 1 = 32.040 ° , φ 2 = 6.131 ° , φ a = - 51.827 ° .
b 1 = b 2 = π ,             b 3 = 0 φ 0 = 70.914 ° , φ 1 = 57.959 ° , φ 2 = 83.869 ° , φ a = 51.827 ° .
D ( Y ) 2 = - 4 π 4 ( Y 10 625 - 8 Y 8 1125 + 782 Y 6 10125 + π 2 Y 5 50 - 104 Y 4 675 - 2 π 2 Y 3 45 + 169 Y 2 225 + 13 π 2 Y 30 ) + 3 4
D 0 = - 0.5531094 × 10 - 2 D 1 = - 0.3519861 × 10 - 1 D 2 = - 0.1314751 D 3 = - 0.2650086 D 4 = - 0.2192773 D 5 = 0.2045062 D 6 = 0.4511530 D 7 = - 0.4567066 D 8 = 0.2646903 D 9 = - 0.7559295 × 10 - 1 D 10 = 0.1187867 × 10 - 1 .
b 10 = 0 , b 9 = b 8 = b 7 = = b 2 = b 1 = π φ 0 = - 34.31 ° φ 1 = φ 9 = - 69.98 ° φ 2 = φ 8 = - 23.74 ° φ 3 = φ 7 = - 5.54 ° φ 4 = φ 6 = - 59.17 ° φ 5 = - 47.19 ° φ a = - 90.00 °
e - i ω = X
D ( X ) 2 = m ( X m + 1 X m ) + m - 1 ( X m - 1 + 1 X m - 1 ) + + 2 ( X 2 + 1 X 2 ) + 1 ( X + 1 X ) + 0 .
Y = X + 1 X ,
D ( Y ) 2 = e m Y m + e m - 1 Y m - 1 + + e 2 Y 2 + e 1 Y + e 0 .
k = 1 m ( X - X k or 1 / X k ) = d m X m + d m - 1 X m - 1 + + d 2 X 2 + d 1 X + d 0 ,
D k = ( 0 d 0 2 + d 1 2 + d 2 2 + + d m - 1 2 + d m 2 ) 1 2 d k             ( k = 1 , 2 , , m ) .

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