Abstract

The thermally induced stress birefringence has been known to affect severely the performance of a solid-state-laser system when the beam is linearly polarized. This study analyzes the birefringence as a function of the YAG:Nd rod-axis orientation; this investigation illustrates the angular dependence of birefringence for directions other than the [111] crystal direction and describes the losses for various rod-axis and polarizer orientations.

© 1971 Optical Society of America

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References

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  1. J. D. Foster and L. M. Osterink, J. Appl. Phys. 41, 3656 (1970).
    [CrossRef]
  2. W. Koechner and D. K. Rice, IEEE J. QE-6, 557 (1970).
    [CrossRef]
  3. R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
    [CrossRef]
  4. YAG:Nd Data (Electronics Division, Crystal Products Department, Union Carbide Corporation, San Diego, Calif., 1969).
  5. J. F. Nye, Physical Properties of Crystals (Pergamon, London, 1965).

1970 (2)

J. D. Foster and L. M. Osterink, J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

W. Koechner and D. K. Rice, IEEE J. QE-6, 557 (1970).
[CrossRef]

1967 (1)

R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
[CrossRef]

Dixon, R. W.

R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
[CrossRef]

Foster, J. D.

J. D. Foster and L. M. Osterink, J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

Koechner, W.

W. Koechner and D. K. Rice, IEEE J. QE-6, 557 (1970).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Pergamon, London, 1965).

Osterink, L. M.

J. D. Foster and L. M. Osterink, J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

Rice, D. K.

W. Koechner and D. K. Rice, IEEE J. QE-6, 557 (1970).
[CrossRef]

IEEE J. (1)

W. Koechner and D. K. Rice, IEEE J. QE-6, 557 (1970).
[CrossRef]

J. Appl. Phys. (2)

R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
[CrossRef]

J. D. Foster and L. M. Osterink, J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

Other (2)

YAG:Nd Data (Electronics Division, Crystal Products Department, Union Carbide Corporation, San Diego, Calif., 1969).

J. F. Nye, Physical Properties of Crystals (Pergamon, London, 1965).

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Figures (16)

F. 1
F. 1

Crystal orientation for YAG.

F. 2
F. 2

Relationship of (x″,y″,z″) coordinate system to (x′,y′,z′) system. Description of α.

F. 3
F. 3

Relationship of (x,y,z) coordinate system to (x″,y″,z″) system. Description of β.

F. 4
F. 4

Cylindrical coordinate system in YAG rod.

F. 5
F. 5

Birefringence parameter as a function of Φ for α = 0° and β = 0°, and for α = 0° and β = 90°.

F. 6
F. 6

Birefringence parameter as a function of Φ for α = 0° and β = 45°.

F. 7
F. 7

Birefringence parameter as a function of Φ for α = 45°and β = 0°.

F. 8
F. 8

Birefringence parameter as a function of Φ for α = 45° and β = 54.8°.

F. 9
F. 9

Birefringence parameter as a function of Φ for α = 45° and β = 30°.

F. 10
F. 10

Birefringence parameter as a function of Φ for α = 45°and β = 90°.

F. 11
F. 11

Difference in optical path length ξ as a function of normalized rod radius r/r0 with, growth directions as a parameter.

F. 12
F. 12

YAG:Nd laser resonator with polarizer internal to cavity.

F. 13
F. 13

Analytical equivalent of Fig. 12.

F. 14
F. 14

Relationship of relevant coordinate systems to preferred polarizer direction and rod cross section.

F. 15
F. 15

The ratio of the irradiance of a parallel beam leaving the crystal to that of the irradiance of the incident beam for a crystal between parallel polarizers with growth direction as a parameter.

F. 16
F. 16

Components of beam in a birefringent YAG crystal between parallel polarizers.

Equations (54)

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Δ n τ ρ = ( n 0 3 / 2 ) [ ( B 11 B 22 ) + 4 B 12 2 ] 1 2 .
Δ n τ ρ = Ω S ( r / r 0 ) 2 ,
Ω = f ( ν , p m n , α , β , Φ )
S = α P d / 16 π ( 1 ν ) K L ,
P d = A 0 ( π r 0 2 L ) ,
Ω = Δ n τ ρ / S .
ν = 0.3 n 0 = 1.82 p 11 = 0.0290 p 12 = + 0.0091 p 44 = 0.0615 .
r = 1.2 S Φ = + 1.4 S z = + 1.4 S .
Δ n τ ρ = ( n 0 3 / 6 ) ( p 11 p 12 + 4 p 44 ) ( Φ r ) .
Ω = ( n 0 3 / 3 ) ( p 11 p 12 + 4 p 44 ) ( ν + 1 ) .
Ω = 0.74 .
Ω ( Φ ) = n 0 3 ( ν + 1 ) [ ( p 11 p 12 ) 2 cos 2 2 Φ + 4 p 44 2 sin 2 2 Φ ] 1 2 .
ξ = L Δ n τ ρ / λ ,
ξ = Ω S / 2 π ( r / r 0 ) 2 ,
S = α P d / λ ( 1 ν ) 8 K .
P d = η P in .
ν = 0.3 α = 7.9 × 10 6 / K K = 0.111 W / cm · K .
S = 72 ( P d = 600 W ) S = 4.8 ( P d = 40 W ) .
T = 1 1 4 π Φ = 0 2 π sin 2 2 ( Φ γ ) [ 1 sin 2 Ω ( Φ ) S 2 Ω ( Φ ) S ] d Φ ,
T [ 111 ] = 1 4 [ 3 + sin ( 0.74 S ) / 0.74 S ] ,
r = S [ ( 3 ν 1 ) ν ( 7 1 ) r 2 / r 0 2 ]
Φ = S [ ( 3 ν 1 ) ν ( 5 3 ) r 2 / r 0 2 ]
z = 2 S ( ν 1 ) ( 1 2 r 2 / r 0 2 ) ,
S = α A 0 r 0 2 / ( 1 ν ) 16 K ,
Δ B i j = p ijkl k l ( i , j , k , l = 1 , 2 , 3 ) .
Δ B m = p m n n ( m , n = 1 , 2 6 ) .
p m n ( x , y , z ) = [ p 11 p 12 p 12 0 0 0 p 12 p 11 p 12 0 0 0 p 12 p 12 p 11 0 0 0 0 0 0 p 44 0 0 0 0 0 0 p 44 0 0 0 0 0 0 p 44 ] .
( r , Φ , z ) = [ r 0 0 0 Φ 0 0 0 z ] ,
¨ ( x , y , z ) = T ¨ 1 · ¨ ( r , Φ , z ) · T ¨ ,
T ¨ = R ¨ · Ü
U = [ cos α cos β sin α cos β sin β sin α cos α 0 sin β cos α sin α sin β cos β ]
R = [ cos Φ sin Φ 0 sin Φ cos Φ 0 0 0 1 ] .
( x , y , z ) = [ 11 12 13 12 12 23 13 23 33 ] ,
n ( x , y , z ) = [ 11 22 33 2 23 2 13 2 12 ] .
Δ B m ( x , y , z ) = [ B 11 B 22 B 33 B 23 B 13 B 12 ] ,
Δ B ( x , y , z ) = [ B 11 B 12 B 13 B 12 B 22 B 23 B 13 B 23 B 33 ] .
Δ B ¨ ( x , y , z ) = Ü · Δ B ( x , y , z ) · Ü 1 .
Δ B ( x , y , z ) = [ B 11 B 12 B 13 B 12 B 22 B 23 B 13 B 23 B 33 ] .
( B 0 + B 11 ) x 2 + ( B 0 + B 22 ) y 2 + 2 B 12 x y = 1 ,
( x y ) ( B 0 + B 11 B 12 B 12 B 0 + B 22 ) ( x y ) = 1 .
B ρ = B 0 + 1 2 ( B 11 + B 22 ) + 1 2 [ ( B 11 B 22 ) 2 + 4 B 12 2 ] 1 2
B τ = B 0 + 1 2 ( B 11 + B 22 ) 1 2 [ ( B 11 B 22 ) 2 + 4 B 12 2 ] 1 2 .
Δ n ρ = ( n 0 3 / 2 ) B ρ
Δ n τ = ( n 0 3 / 2 ) B τ
n ρ = n 0 ( n 0 3 / 2 ) B ρ
n τ = n 0 ( n 0 3 / 2 ) B τ .
Δ n τ ρ = ( n 0 3 / 2 ) [ ( B 11 B 22 ) 2 + 4 B 12 2 ] 1 2 .
È in = È 0 cos ω t .
E ρ = E 0 cos θ cos ω t
E τ = E 0 sin θ cos ω t .
E ρ = E 0 cos θ cos ( ω t + φ ρ )
E τ = E 0 sin θ cos ( ω t + φ τ ) ,
( I out I in ) T = 1 π r 0 2 r = 0 r 0 ϕ = 0 2 π [ 1 sin 2 2 θ sin 2 2 π ξ ] rdrd Φ .
T = ( I out I in ) T = 1 1 4 π ϕ = 0 2 π sin 2 2 ( Φ γ ) × [ 1 sin 2 Ω ( Φ ) S 2 Ω ( Φ ) S ] d Φ ,