Abstract

A thorough and general geometrical optics analysis of a slab-shaped laser gain medium is presented. The length and thickness ratio is critical if one is to achieve the maximum utilization of absorbed pump power by the laser light in such a medium; e.g., the fill factor inside the slab is to be maximized. We point out that the conditions for a fill factor equal to 1, laser light entering and exiting parallel to the length of the slab, and Brewster angle incidence on the entrance and exit faces cannot all be satisfied at the same time. Deformed slabs are also studied. Deformation along the width direction of the largest surfaces is shown to significantly reduce the fill factor that is possible.

© 2007 Optical Society of America

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References

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  1. J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser. 1. Theory," IEEE J Quantum Electron. 20, 289-301 (1984).
    [CrossRef]
  2. T. J. Kane, J. M. Eggleston, and R. L. Byer, "The slab geometry laser 2. Thermal effects in a finite slab," IEEE J. Quantum Electron. 21, 1195-1210 (1985).
    [CrossRef]
  3. Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
    [CrossRef]
  4. Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
    [CrossRef]
  5. J. Eicher, N. Hodgson, and H. Weber, "Output power and efficiencies of slab laser systems," J. Appl. Phys. 66, 4608-4613 (1989).
    [CrossRef]
  6. J. M. Eggleston, L. M. Frantz, and H. Injeyan, "Derivation of the Frantz-Nodvik equation for zig-zag optical path, slab geometry laser amplifiers," IEEE J. Quantum Electron. 25, 1855-1862 (1989).
    [CrossRef]
  7. Breault Research Organization Tucson, Arizona, asap Optical Modeling Software (2002).

2004

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
[CrossRef]

2002

Breault Research Organization Tucson, Arizona, asap Optical Modeling Software (2002).

1989

J. Eicher, N. Hodgson, and H. Weber, "Output power and efficiencies of slab laser systems," J. Appl. Phys. 66, 4608-4613 (1989).
[CrossRef]

J. M. Eggleston, L. M. Frantz, and H. Injeyan, "Derivation of the Frantz-Nodvik equation for zig-zag optical path, slab geometry laser amplifiers," IEEE J. Quantum Electron. 25, 1855-1862 (1989).
[CrossRef]

1985

T. J. Kane, J. M. Eggleston, and R. L. Byer, "The slab geometry laser 2. Thermal effects in a finite slab," IEEE J. Quantum Electron. 21, 1195-1210 (1985).
[CrossRef]

1984

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser. 1. Theory," IEEE J Quantum Electron. 20, 289-301 (1984).
[CrossRef]

Bass, M.

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
[CrossRef]

Byer, R. L.

T. J. Kane, J. M. Eggleston, and R. L. Byer, "The slab geometry laser 2. Thermal effects in a finite slab," IEEE J. Quantum Electron. 21, 1195-1210 (1985).
[CrossRef]

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser. 1. Theory," IEEE J Quantum Electron. 20, 289-301 (1984).
[CrossRef]

Chen, B.

Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

Chen, Y.

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
[CrossRef]

Eggleston, J. M.

J. M. Eggleston, L. M. Frantz, and H. Injeyan, "Derivation of the Frantz-Nodvik equation for zig-zag optical path, slab geometry laser amplifiers," IEEE J. Quantum Electron. 25, 1855-1862 (1989).
[CrossRef]

T. J. Kane, J. M. Eggleston, and R. L. Byer, "The slab geometry laser 2. Thermal effects in a finite slab," IEEE J. Quantum Electron. 21, 1195-1210 (1985).
[CrossRef]

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser. 1. Theory," IEEE J Quantum Electron. 20, 289-301 (1984).
[CrossRef]

Eicher, J.

J. Eicher, N. Hodgson, and H. Weber, "Output power and efficiencies of slab laser systems," J. Appl. Phys. 66, 4608-4613 (1989).
[CrossRef]

Frantz, L. M.

J. M. Eggleston, L. M. Frantz, and H. Injeyan, "Derivation of the Frantz-Nodvik equation for zig-zag optical path, slab geometry laser amplifiers," IEEE J. Quantum Electron. 25, 1855-1862 (1989).
[CrossRef]

Hodgson, N.

J. Eicher, N. Hodgson, and H. Weber, "Output power and efficiencies of slab laser systems," J. Appl. Phys. 66, 4608-4613 (1989).
[CrossRef]

Injeyan, H.

J. M. Eggleston, L. M. Frantz, and H. Injeyan, "Derivation of the Frantz-Nodvik equation for zig-zag optical path, slab geometry laser amplifiers," IEEE J. Quantum Electron. 25, 1855-1862 (1989).
[CrossRef]

Kane, T. J.

T. J. Kane, J. M. Eggleston, and R. L. Byer, "The slab geometry laser 2. Thermal effects in a finite slab," IEEE J. Quantum Electron. 21, 1195-1210 (1985).
[CrossRef]

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser. 1. Theory," IEEE J Quantum Electron. 20, 289-301 (1984).
[CrossRef]

Kar, A.

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

Kuhn, K.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser. 1. Theory," IEEE J Quantum Electron. 20, 289-301 (1984).
[CrossRef]

Patel, M. K. R.

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
[CrossRef]

Unternahrer, J.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser. 1. Theory," IEEE J Quantum Electron. 20, 289-301 (1984).
[CrossRef]

Weber, H.

J. Eicher, N. Hodgson, and H. Weber, "Output power and efficiencies of slab laser systems," J. Appl. Phys. 66, 4608-4613 (1989).
[CrossRef]

IEEE J Quantum Electron.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, "The slab geometry laser. 1. Theory," IEEE J Quantum Electron. 20, 289-301 (1984).
[CrossRef]

IEEE J. Quantum Electron.

T. J. Kane, J. M. Eggleston, and R. L. Byer, "The slab geometry laser 2. Thermal effects in a finite slab," IEEE J. Quantum Electron. 21, 1195-1210 (1985).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
[CrossRef]

J. M. Eggleston, L. M. Frantz, and H. Injeyan, "Derivation of the Frantz-Nodvik equation for zig-zag optical path, slab geometry laser amplifiers," IEEE J. Quantum Electron. 25, 1855-1862 (1989).
[CrossRef]

J. Appl. Phys.

J. Eicher, N. Hodgson, and H. Weber, "Output power and efficiencies of slab laser systems," J. Appl. Phys. 66, 4608-4613 (1989).
[CrossRef]

Other

Breault Research Organization Tucson, Arizona, asap Optical Modeling Software (2002).

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Figures (18)

Fig. 1
Fig. 1

Slab laser with angled ends showing the parameters used in this analysis. The angled ends may be at Brewster's angle and are in contact with air. The large faces are in contact with coolant assumed to be water.

Fig. 2
Fig. 2

Slab unfolded into a stack of slabs to simplify the analysis.

Fig. 3
Fig. 3

Condition to satisfy FF = 1 is when the ray that passes through point B will also pass through E 1 , which is the image point of E.

Fig. 4
Fig. 4

Sketch showing different optical paths for different parts of a laser beam in zigzag propagation in a nonideal slab-shaped gain medium.

Fig. 5
Fig. 5

General condition to achieve FF = 1 occurs when the length of the rectangular part of the slab is a half-integer multiple of the length of a bounce, Λ.

Fig. 6
Fig. 6

Reflection angle versus apex angle for different ARs using Eq. (9). -.-.-. Cases where the laser beam enters and exits parallel to the length dimension of the slab given by Eq. (18). ················· Cases where the laser beam enters and exits the slab at Brewster's angle given by Eqs. (16) and (15). ----- Cases where the laser beam incident angle ( φ ) is equal to 90° given by Snell's law. The TIR condition shown here is for water coolant on the large surfaces of the Nd:YAG. β max indicates the limitation of the largest reflection angle, β, which can be achieved. It corresponds to cases where the incident angle, φ = 90 ° . Only the triangular region enclosed by the vertical line identifying the TIR angle, the line for FF = 1, and the line for β max offers acceptable solutions if an uncoated Nd:YAG slab is used.

Fig. 7
Fig. 7

Reflection angle versus LTR at different bounce numbers N for an uncoated Nd:YAG slab with apex angle α = 30.61 ° . Solid curves indicate parallel angled end slab designs, and dashed curves represent antiparallel angled end slab designs. The region between TIR and β max corresponds to the bold thick dashed line, the acceptable solution range, in Fig. 6. Any given LTR can have several solutions that yield FF = 1 and are the preferred solutions. The plot was created using Eqs. (6), (9), and (20).

Fig. 8
Fig. 8

AR versus reflection angle ( β ) for N = 30 in an ideal Nd:YAG slab. LTR is 49.02. There are eight solutions ( N = 30 , 32 ⁡, … ,  44 ) satisfying both FF = 1 and TIR conditions. The solid line is the upper aperture, the dashed line is the lower aperture, and the bold dashed line is the total usage of the aperture. This figure was obtained by ray tracing and solving Eq. (8) numerically.

Fig. 9
Fig. 9

FF versus reflection angle ( β ) for N = 30 in an ideal Nd:YAG slab. The solid line is the upper aperture, the dashed line is the lower aperture, and the bold dashed line is the total usage of the aperture. This figure was obtained by ray tracing and solving Eq. (6) numerically.

Fig. 10
Fig. 10

Beam size in the air versus reflection angle ( β ) for N = 30 in an ideal Nd:YAG slab. No solution exists for reflection angle β > β max = 64.22 ° , which corresponds to 90° incident angle, φ. This figure was obtained by ray tracing and solving Eqs. (7) and (8) numerically.

Fig. 11
Fig. 11

Normalized beam size in the air versus incident angle on the Brewster end face of an ideal Nd:YAG slab with N = 30, t = 2   mm . This figure is the same as Fig. 8, but the horizontal axis is changed to incident angle φ by using Snell's law. When φ approaches 90° the beam size in the air becomes much smaller than the beam size in the slab.

Fig. 12
Fig. 12

Calculated incidence angle tolerance versus bounce number for FF = 1, AR = 1 in an ideal Nd:YAG slab. The larger the bounce number, the higher the required incident angle precision. The results were obtained by ray tracing.

Fig. 13
Fig. 13

(a) Apex angle of the slab that satisfies FF = 1 and AR = 1 as a function of the gain medium index of refraction n obtained from Eq. (19). (b) The incident angle, φ, is the Brewster angle for different gain medium index of refraction n obtained from Brewster condition. (c) The incident angle, φ, equals 90 ° α , the apex angle, such that light enters and exits parallel to the length of the slab. When the slab material has index of refraction n = tan 60 ° = 1.732 , the slab simultaneously satisfies all the conditions: FF = 1, AR = 1, Brewster incident angle, and light entering and exiting parallel to the length of the slab.

Fig. 14
Fig. 14

Calculate FF versus AR for an ideal Nd:YAG slab. Limiting the beam size reduces the AR. However, the FF does not decrease as fast as the AR so that most of the absorbed pump power can still be extracted from the slab. This curve was obtained from Eqs. (6) and (8).

Fig. 15
Fig. 15

Comparison of asap (ray tracing) and mathcad (analytic) simulations of FF and AR for N = 30 in an ideal Nd:YAG slab. (a) The position that the laser light enters and exits parallel to the length of the slab. (b) The desired case where FF = 1, AR = 1. AR obtained by mathcad (analytic) using Eq. (8). ·················· FF obtained by mathcad (analytic) using Eq. 6. ⧫ AR obtained by asap (ray tracing). ☐ FF obtained by asap (ray tracing).

Fig. 16
Fig. 16

Sketch of the possible surface deformation of an edge pumped slab laser. The entrance surface has deformation in only one direction. The largest surface may be deformed in two directions.

Fig. 17
Fig. 17

Calculated AR and FF versus surface deformation displacement for the different deformation directions shown in Fig. 15 for N = 30 in an ideal Nd:YAG slab. The most sensitive displacement is along the width direction of the largest surfaces. A 1   μm displacement can result in A R 0.94 but F F 0.99 FF. ----- AR. ☐ Displacement along the slab width direction on the largest surfaces. • Displacement along the slab length direction on the largest surfaces. ◊ Displacement on the entrance and exit surfaces.

Fig. 18
Fig. 18

Optical path difference at the output surface of a 98   mm × 20   mm × 2   mm   ( L × W × T ) Yb:YAG slab edge pumped with total power of 12.24   kW centered at 941   nm with linewidth 3   nm . The heat removal rate assumed was 30   kW / m 2 on the two largest surfaces.

Tables (1)

Tables Icon

Table 1 Parameters of an Ideal Nd:YAG Slab Cooled by Water with N = 30, t = 2

Equations (113)

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φ 1
n c o o l a n t
n a i r
n c r y s t a l
a 0
β = 90 ° tan 1 [ t ( 1 + N ) tan α t + L tan α ] ,
β 90 ° α
F N / 2
F F = 1 ( t a 0 cos φ 1 2 sin β ) ( Λ a 0 cos φ 1 cos β ) Λ t .
Λ = 2 t tan β ,
φ 1 = β α ,
F F = 1 1 2 t 2 [ 2 t sin α a 0 cos ( β α ) ] [ 2 t sin β a 0 cos ( β α ) ] .
F F = 1
F F = 1
a 0 = 2 t sin β cos ( β α ) .
E 1
A R = a 0 sin ( α ) t .
F F = 1
A R = 1
F F = 1
A R F F = 1 = 2 sin ( α ) sin ( β ) cos ( β α ) = 1 cos ( β + α ) cos ( β α ) .
β sin 1 ( n c o o l a n t n c r y s t a l ) β T I R .
β φ 1 , T I R + α = sin 1 n a i r n c r y s t a l + α β max .
( n = 1.82 )
( n = 1.33 )
β T I R = 46.95 °
β max = 33.33 ° + α
S D R = 4 t a 0 sin β ( sin β + cos β ) cos ( β α ) cos β 4 t 2 a 0 2 1 + cos 2 β cos 2 ( β α ) + 4 t a 0 sin β cos ( β α ) 2.
F F 1
F F = 1
A R = 1
a 0 = t sin α ,
α + β = 90 ° .
β α = sin 1 ( n a i r n c r y s t a l sin φ ) .
φ B r e w s t e r = tan 1 ( n c r y s t a l n a i r ) .
ζ = 90 ° φ α .
ζ = 0
α + φ = 90 ° .
n a i r = 1
n c r y s t a l
α = 30 °
β = φ = 60 °
n c r y s t a l = 1.732
φ B r e w s t e r = 61.22 °
α i d e a l = φ B r e w s t e r 2 = 30.61 ° ,
β = 59.53 °
1.83 °
ζ = 0
α + β
α + β < 90 °
α + β > 90 °
α + β < 90 °
A R < 1
α + β > 90 °
F F < 1
α + β < 90 °
Λ / 2
L T R α + β < 90 ° = L t = N tan β ,
α + β < 90 °
α + β > 90 °
Λ / 2
a s
E H = Λ 2 a s = Λ 2 t tan α = t ( tan β 1 tan α ) .
L T R α + β > 90 ° = ( N 1 ) tan β 1 tan α ,
E 1
F N / 2
R N / 2
E 1
G N / 2
α + β
F F < 1
α + β > 90 °
β max
φ = 90 °
β max
ζ = 0 °
α = 30.61 °
β max
2   mm
100   mm
a 0
a 0
a 0
β max = 64.22 °
1.5
E 1
0.1   μm
6   μm
β = β m a x
E 1
( φ )
β max
φ = 90 °
β max
α = 30.61 °
β max
( β )
( N = 30
32 ⁡, … ,  44 )
( β )
( β )
β > β max = 64.22 °
t = 2   mm
90 ° α
n = tan 60 ° = 1.732
1   μm
A R 0.94
F F 0.99
98   mm × 20   mm × 2   mm   ( L × W × T )
12.24   kW
941   nm
3   nm
30   kW / m 2

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