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A new approach to compute Overlap efficiency in axially pumped Solid State Lasers

Rakesh Kapoor, P. K. Mukhopadhyay, and Jogy George

Open Access Open Access

Abstract

In this work, we are reporting a new approach to compute the overlap efficiency of end pumped solid-state laser systems. Unlike existing methods in which the overlap integral is computed with a linearize approximation near the threshold, in this method the inverse of the overlap integral is computed numerically in the above threshold regime for several values of circulating fields. Now by fitting a linear curve to this data the overlap efficiency is obtained. The effect of the beam quality factor is also taken into account. It is demonstrated that the linearized approximation near the threshold can give rise to 50% error in overlap efficiency. The method was used to estimate the overlap efficiency in different types of axially pumped lasers.

©1999 Optical Society of America

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Figures (4)
Figure 1.
Figure 1. Variation of (a) match function and (b) overlap efficiency with pump-beam waist position with respect to entrance plane of gain medium. The pump beam profile is taken as an elliptic profile. The match function and overlap efficiency is computed with both, the linearize approximation near threshold and the proposed method.

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Figure 2.
Figure 2. Variation of inverse of overlap integral with normalised circulating field intensity in different pump beam conditions. A linear fit to these plots gives the value of constants C and D. Solid lines are the linear fit to the plotted data. z is the distance of pump beam waist from the entrance plane. The pump beam spot in the gain medium increases with z.

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Figure 3.
Figure 3. Variation of match function with mode waist radius ωm 0 for two different values of pump waist sizes. The pump beam profile is taken as an elliptic profile and ωp 0=[ωpx , ωpy ].

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Figure 4.
Figure 4. Variation of overlap efficiency η 0 with pump waist radius for four different values of beam quality factor M 2. (a) The mode waist radius ωm 0=100µm, (b) The mode waist radius ωm 0=300µm, (c) The mode waist radius ωm 0=500µm.

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Equations (27)

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g ( z ) = η q α a I p ( z ) ⁄ I sat ( 1 + 2 I circ ⁄ I sat ) ,
η q ≈ ω m ω p ,
I p ( x , y , z ) = P p A p ( z ) f p ( x , y , z ) Exp ( − α a z ) ,
A p ( x ) = ∬ f p ( x , y , z ) d x d y ,
I circ ( x , y , z ) = P circ A m ( z ) f m ( x , y , z ) ,
Δ P circ = ∭ d I ( x , y , z ) d x d y = 2 ∬ ∫ 0 l g ( x , y , z ) I circ ( x , y , z ) d z d x d y ,
G = 2 ∫ ∫ ∫ g ( x , y , z ) I circ ( x , y , z ) d V P circ
G = 2 η q α a P ′ p ∫ ∫ ∫ Exp ( − α a z ) f p ( x , y , z ) f m ( x , y , z ) A p ( z ) A m ( z ) ( 1 + 2 p ′ circ f m ( x , y , z ) A m ( z ) ) d V ,
P ′ p = P p / I sat
P ′ circ = P circ / I sat = 〈 A m 〉 I ′ circ ,
〈 A m 〉 = 1 l ∫ 0 l ∫ ∫ f m ( x , y , z ) dx dy dz ,
G = 2 α 0 z + ln ( 1 R 1 R 2 ) ≡ L + T 1 + T 2 ,
P out = T 2 P circ ,
G = 2 η q α a P ′ p [ C ( 1 + 2 D I ′ circ ) ] ,
∫ ∫ ∫ Exp ( − α a z ) f p ( x , y , z ) f m ( x , y , z ) A p ( z ) A m ( z ) [ 1 + 2 〈 A m 〉 I ′ circ f m ( x , y , z ) A m ( z ) ] d V = C 1 + 2 D I ′ circ ,
2 η q α a P ′ p [ C ( 1 + 2 D P ′ circ / 〈 A m 〉 ) ] = ( T + L ) ,
P circ = I sat 〈 A m 〉 2 D [ 2 η q α a P ′ P C T + L − 1 ] ,
P out = T 2 η q α a P p C 〈 A m 〉 D ( T + L ) − T 2 I sat 〈 A m 〉 2 D ,
η o = α a C 〈 A m 〉 η a D ,
P out = T 2 ( T + L ) η q η a η o P p − T 2 I sat 〈 A m 〉 2 D ,
P th = I sat ( L + T ) 2 η q α a C ,
m = η o η a η q T 2 ( L + T ) ,
F = 2 α a C 〈 A m 〉 ,
f m ( x , y , z ) = Exp [ − 2 x 2 ω mx 2 ( z ) − 2 y 2 ω my 2 ( z ) ] ,
f p ( x , y , z ) = Exp [ − 2 x 2 ω px 2 ( z ) − 2 y 2 ω py 2 ( z ) ] ,
ω p ( z ) = ω p 0 [ 1 + ( M 2 z ⋋ ⁄ π ω p 0 2 ) 2 ] 1 2 ,
ω 0 θ = M 2 ⋋ π ,
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