Abstract

Using the formalism of mode overlap, a theoretical analysis of optically pumped fiber laser amplifiers and oscillators is developed. The concept of normalized overlap coefficients is introduced to account for the effects of the transverse structure of the interacting signal and pump modes on the device characteristics. Simple and accurate closed-form expressions are derived for the gain of fiber amplifiers and the threshold and energy conversion efficiency of fiber laser oscillators in terms of the fiber and laser material parameters and the pump and signal modes. When applied to step-index Nd:YAG fiber lasers, this study predicts optimum fundamental mode oscillation in fibers with a V number of 5–25 with submilliwatt thresholds and nearly quantum-limited conversion efficiencies.

© 1985 Optical Society of America

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References

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  1. C. J. Koester, E. Snitzer, “Amplification in a Fiber Laser,” Appl. Opt. 3, 1182 (1964).
    [CrossRef]
  2. J. Stone, C. A. Burrus, “Neodynium-Doped Fiber Lasers: Room Temperature cw Operation with an Injection Laser Pump,” Appl. Opt. 13, 1256 (1974).
    [CrossRef] [PubMed]
  3. H. Injeyan et al., “Amplification of Light Propagating Through a Fiber by Evanescent Wave Coupling,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1981), paper THJ5.
  4. N. Periasamy, Z. Bor, “Distributed Feedback Laser Action in an Optical Fiber,” Opt. Commun. 39, 298 (1981).
    [CrossRef]
  5. J. Stone, C. A. Burrus, “Self-contained LED-Pumped Single-Crystal Nd:YAG Fiber Laser,” Fiber Integrated Opt. 2, 1 (1979).
  6. M. Digonnet, H. J. Shaw, “Diode-Pumped Fiber Laser,” Final Technical Report AFWAL TR-83-1110 (July1983).
  7. J. P. Budin et al., “On the Design of Neodynium Miniature Lasers,” IEEE J. Quantum Electron. QE-14, 831 (1978).
    [CrossRef]
  8. S. R. Chinn et al., “Low-Threshold, Transversely Excited NdP5O14 Laser,” IEEE J. Quantum Electron. QE-11, 747 (1975).
    [CrossRef]
  9. K. Kubodera, K. Otsuka, “Single-Transverse-Mode LiNd-P4O12 Slab Waveguide Laser,” J. Appl. Phys. 50, 653 (1979).
    [CrossRef]
  10. K. Kubodera, K. Otsuka, “Laser Performance of a Glass-Clad LiNdP4O12 Rectangular Waveguide,” J. Appl. Phys. 50, 6707 (1979).
    [CrossRef]
  11. M. Birnbaum et al., “Laser Emission Cross Section of Nd:YAG at 1064 nm,” J. Appl. Phys. 52, 1212 (1981).
    [CrossRef]
  12. H. W. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  13. G. D. Boyd, D. A. Kleinman, “Parametric Interaction of Gaussian Light Beams,” J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]
  14. D. Gloge, “Weakly Guiding Fibers,” Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  15. D. Marcuse, “Quantum Electronics, Principles and Applications,” in Theory of Dielectric Optical Waveguides, (Academic, New York, 1974).
  16. D. Marcuse, “Gaussian Approximation of the Fundamental Modes of Graded-Index Fibers,” J. Opt. Soc. Am. 68, 103 (1978).
    [CrossRef]

1981

N. Periasamy, Z. Bor, “Distributed Feedback Laser Action in an Optical Fiber,” Opt. Commun. 39, 298 (1981).
[CrossRef]

M. Birnbaum et al., “Laser Emission Cross Section of Nd:YAG at 1064 nm,” J. Appl. Phys. 52, 1212 (1981).
[CrossRef]

1979

J. Stone, C. A. Burrus, “Self-contained LED-Pumped Single-Crystal Nd:YAG Fiber Laser,” Fiber Integrated Opt. 2, 1 (1979).

K. Kubodera, K. Otsuka, “Single-Transverse-Mode LiNd-P4O12 Slab Waveguide Laser,” J. Appl. Phys. 50, 653 (1979).
[CrossRef]

K. Kubodera, K. Otsuka, “Laser Performance of a Glass-Clad LiNdP4O12 Rectangular Waveguide,” J. Appl. Phys. 50, 6707 (1979).
[CrossRef]

1978

J. P. Budin et al., “On the Design of Neodynium Miniature Lasers,” IEEE J. Quantum Electron. QE-14, 831 (1978).
[CrossRef]

D. Marcuse, “Gaussian Approximation of the Fundamental Modes of Graded-Index Fibers,” J. Opt. Soc. Am. 68, 103 (1978).
[CrossRef]

1975

S. R. Chinn et al., “Low-Threshold, Transversely Excited NdP5O14 Laser,” IEEE J. Quantum Electron. QE-11, 747 (1975).
[CrossRef]

1974

1971

1968

G. D. Boyd, D. A. Kleinman, “Parametric Interaction of Gaussian Light Beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

1966

1964

Birnbaum, M.

M. Birnbaum et al., “Laser Emission Cross Section of Nd:YAG at 1064 nm,” J. Appl. Phys. 52, 1212 (1981).
[CrossRef]

Bor, Z.

N. Periasamy, Z. Bor, “Distributed Feedback Laser Action in an Optical Fiber,” Opt. Commun. 39, 298 (1981).
[CrossRef]

Boyd, G. D.

G. D. Boyd, D. A. Kleinman, “Parametric Interaction of Gaussian Light Beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Budin, J. P.

J. P. Budin et al., “On the Design of Neodynium Miniature Lasers,” IEEE J. Quantum Electron. QE-14, 831 (1978).
[CrossRef]

Burrus, C. A.

J. Stone, C. A. Burrus, “Self-contained LED-Pumped Single-Crystal Nd:YAG Fiber Laser,” Fiber Integrated Opt. 2, 1 (1979).

J. Stone, C. A. Burrus, “Neodynium-Doped Fiber Lasers: Room Temperature cw Operation with an Injection Laser Pump,” Appl. Opt. 13, 1256 (1974).
[CrossRef] [PubMed]

Chinn, S. R.

S. R. Chinn et al., “Low-Threshold, Transversely Excited NdP5O14 Laser,” IEEE J. Quantum Electron. QE-11, 747 (1975).
[CrossRef]

Digonnet, M.

M. Digonnet, H. J. Shaw, “Diode-Pumped Fiber Laser,” Final Technical Report AFWAL TR-83-1110 (July1983).

Gloge, D.

Injeyan, H.

H. Injeyan et al., “Amplification of Light Propagating Through a Fiber by Evanescent Wave Coupling,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1981), paper THJ5.

Kleinman, D. A.

G. D. Boyd, D. A. Kleinman, “Parametric Interaction of Gaussian Light Beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Koester, C. J.

Kogelnik, H. W.

Kubodera, K.

K. Kubodera, K. Otsuka, “Laser Performance of a Glass-Clad LiNdP4O12 Rectangular Waveguide,” J. Appl. Phys. 50, 6707 (1979).
[CrossRef]

K. Kubodera, K. Otsuka, “Single-Transverse-Mode LiNd-P4O12 Slab Waveguide Laser,” J. Appl. Phys. 50, 653 (1979).
[CrossRef]

Li, T.

Marcuse, D.

D. Marcuse, “Gaussian Approximation of the Fundamental Modes of Graded-Index Fibers,” J. Opt. Soc. Am. 68, 103 (1978).
[CrossRef]

D. Marcuse, “Quantum Electronics, Principles and Applications,” in Theory of Dielectric Optical Waveguides, (Academic, New York, 1974).

Otsuka, K.

K. Kubodera, K. Otsuka, “Laser Performance of a Glass-Clad LiNdP4O12 Rectangular Waveguide,” J. Appl. Phys. 50, 6707 (1979).
[CrossRef]

K. Kubodera, K. Otsuka, “Single-Transverse-Mode LiNd-P4O12 Slab Waveguide Laser,” J. Appl. Phys. 50, 653 (1979).
[CrossRef]

Periasamy, N.

N. Periasamy, Z. Bor, “Distributed Feedback Laser Action in an Optical Fiber,” Opt. Commun. 39, 298 (1981).
[CrossRef]

Shaw, H. J.

M. Digonnet, H. J. Shaw, “Diode-Pumped Fiber Laser,” Final Technical Report AFWAL TR-83-1110 (July1983).

Snitzer, E.

Stone, J.

J. Stone, C. A. Burrus, “Self-contained LED-Pumped Single-Crystal Nd:YAG Fiber Laser,” Fiber Integrated Opt. 2, 1 (1979).

J. Stone, C. A. Burrus, “Neodynium-Doped Fiber Lasers: Room Temperature cw Operation with an Injection Laser Pump,” Appl. Opt. 13, 1256 (1974).
[CrossRef] [PubMed]

Appl. Opt.

Fiber Integrated Opt.

J. Stone, C. A. Burrus, “Self-contained LED-Pumped Single-Crystal Nd:YAG Fiber Laser,” Fiber Integrated Opt. 2, 1 (1979).

IEEE J. Quantum Electron.

J. P. Budin et al., “On the Design of Neodynium Miniature Lasers,” IEEE J. Quantum Electron. QE-14, 831 (1978).
[CrossRef]

S. R. Chinn et al., “Low-Threshold, Transversely Excited NdP5O14 Laser,” IEEE J. Quantum Electron. QE-11, 747 (1975).
[CrossRef]

J. Appl. Phys.

K. Kubodera, K. Otsuka, “Single-Transverse-Mode LiNd-P4O12 Slab Waveguide Laser,” J. Appl. Phys. 50, 653 (1979).
[CrossRef]

K. Kubodera, K. Otsuka, “Laser Performance of a Glass-Clad LiNdP4O12 Rectangular Waveguide,” J. Appl. Phys. 50, 6707 (1979).
[CrossRef]

M. Birnbaum et al., “Laser Emission Cross Section of Nd:YAG at 1064 nm,” J. Appl. Phys. 52, 1212 (1981).
[CrossRef]

G. D. Boyd, D. A. Kleinman, “Parametric Interaction of Gaussian Light Beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

N. Periasamy, Z. Bor, “Distributed Feedback Laser Action in an Optical Fiber,” Opt. Commun. 39, 298 (1981).
[CrossRef]

Other

D. Marcuse, “Quantum Electronics, Principles and Applications,” in Theory of Dielectric Optical Waveguides, (Academic, New York, 1974).

M. Digonnet, H. J. Shaw, “Diode-Pumped Fiber Laser,” Final Technical Report AFWAL TR-83-1110 (July1983).

H. Injeyan et al., “Amplification of Light Propagating Through a Fiber by Evanescent Wave Coupling,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1981), paper THJ5.

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Figures (8)

Fig. 1
Fig. 1

Schematic of the Gaussian pump beam path inside the laser medium.

Fig. 2
Fig. 2

Dependence of the slope efficiency of a free-space Nd: YAG laser at λp = 0.5145 μm (T1 = 1%, δ1 = 5%, Ws = 25 μm): (a) exact solution; (b) approximate solution obtained by replacing Ws and Wp by their respective average value; (c) approximate solution given by Eq. (22).

Fig. 3
Fig. 3

Variation of a few F01nm coefficients with the fiber V number (at the signal frequency) computed with λs = 1.064 μm, λp = 0.5145 μm, n1 = 1.820, and n2 = 1.815.

Fig. 4
Fig. 4

Evolution of the overlap coefficients F01nm for increasing values of the modal number m.

Fig. 5
Fig. 5

Dependence of the threshold of a Nd:YAG fiber laser on the core radius for the LP01/LP01 mode configuration. Parameters are the same as in Fig. 3, with αa = 0.6 cm−1 and l = 1 cm.

Fig. 6
Fig. 6

Exact (solid curve) and approximate (dash curves) theoretical dependence of the output power of a fiber laser oscillator on the absorbed pump power for increasing round-trip cavity loss. Signal and pump modes are LP01; fiber V number is 40.

Fig. 7
Fig. 7

Dependence of the slope efficiency on the fiber V number for a LP01 signal mode and various pump modes. The parameters are the same as in Fig. 3. The dashed curve represents the far-from-cutoff approximation [Eq. (22)] in the case of the fundamental pump mode.

Fig. 8
Fig. 8

Intensity radial profiles of a few LP modes (far from cutoff).

Tables (1)

Tables Icon

Table I Asymptotic Values (V → ∞) of the Modal Overlaps Fνμnm for a Few Low-Order Signal and Pump Modes

Equations (42)

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dn ( x , y , z ) dt = r ( x , y , z ) n ( x , y , z ) τ f c σ n ( x , y , z ) n 1 × j = 1 N S j ( x , y , z ) = 0 ,
d S i dt = c σ n 1 cavity n ( x , y , z ) S i ( x , y , z ) d υ c δ i 2 ln 1 S i = 0 ,
r ( x , y , z ) = R r 0 ( x , y , z ) ,
s i ( x , y , z ) = S i s 0 , i ( x , y , z ) ,
J i ( S 1 , S 2 , , S N ) = δ i 2 l σ τ f R ,
J i ( S 1 , S 2 , , S N ) = cavity s 0 , i ( x , y , z ) r 0 ( x , y , z ) 1 + c σ τ f n 1 j = 1 N S i s 0 , j ( x , y , z ) d υ .
J 1 ( S 1 ) = δ 1 2 l σ τ f R ,
J 1 ( S 1 ) = J 1 ( S 1 , 0 , 0 0 ) = cavity r 0 ( x , y , z ) s 0 , 1 ( x , y , z ) 1 + c σ τ f n 1 S 1 s 0 , 1 ( x , y , z ) d υ .
P abs = P p [ 1 exp ( α a l ) ] .
P th = h ν p σ τ f δ 2 2 l 1 J 1 ( 0 ) ,
J 1 ( 0 ) = J 1 ( 0 , 0 , , 0 ) = cavity r 0 ( x , y , z ) s 0 , 1 ( x , y , z ) d υ .
γ = σ τ f h ν p P abs A p * .
A p * = 1 l J 1 ( 0 ) ,
A p * = A f F ν μ nm ( fiber case ) ,
A p * = A ̅ p F ν μ nm ( unguided case ) ,
γ = γ 0 F ν μ nm ,
γ 0 = σ τ f h ν p P abs A f .
F = n , m ρ nm F μ ν nm ,
P out = T 1 h ν s 2 ln 1 c S 1 ,
1 J 1 ( S 1 ) = 1 J 1 ( 0 ) + c σ τ f n 1 S 1 η p ,
η p = active region r 0 ( x , y , z ) d υ .
s = P out ( P abs P th ) = η p T 1 δ 1 h ν s h ν p .
W p 2 ( z ) = W p 2 [ 1 + ( λ p ( z z p ) π W p 2 n 1 ) 2 ] ,
W ̅ p 2 = 1 l 0 l W p 2 ( z ) dz .
W ̅ p 2 = W p 2 ( 1 + p ) ,
p = 1 3 ( λ p l π n l W p 2 ) 2 .
F 0000 = W ̅ p 2 W ̅ p 2 + W ̅ s 2 ,
A p * = A ̅ p F 0000 = A ̅ p + A ̅ s = π 2 ( W ̅ p 2 + W ̅ s 2 ) .
W p , opt = ( λ p l 3 π n 1 ) 1 / 2 .
F 0100 = W ̅ p 4 ( W ̅ p 2 + W ̅ s 2 ) 2 .
P th ( TEM 10 ) = P th ( TEM 00 ) ( δ 00 δ 10 α 00 ) ,
V = 2 π a λ n 1 2 n 2 2 .
F ν μ nm = F nm ν μ ( V V c )
F ν μ , u = η s ,
P th = h ν p σ τ f δ ν μ 2 A f F ν μ nm .
P th , inc = h ν p σ τ f δ ν μ 2 A f F ν μ nm 1 [ 1 exp ( α a l ) ] ,
s ν μ ( r , ϕ ) = 1 2 E x ( ν , μ ) H y ( ν , μ ) * .
s ν μ ( r , ϕ ) = { c ν μ J ν 2 ( κ r ) { cos 2 ν ϕ sin 2 ν ϕ for r a , c ν μ J ν 2 ( κ r ) K ν 2 ( γ a ) K ν 2 ( γ r ) { cos 2 ν ϕ sin 2 ν ϕ for r a ,
c ν μ = 2 γ s 2 l π e ν V s 2 1 | J ν 1 ( κ s a ) J ν + 1 ( κ s a ) | ,
e ν = { 2 for ν = 0 , 1 for ν 0 .
r nm ( r , ϕ , z ) = { c nm J n 2 ( κ p r ) exp ( α a z ) { cos 2 n ϕ sin 2 n ϕ r a , c nm J n 2 ( κ p a ) K n 2 ( γ p a ) K n 2 ( γ p r ) exp ( α a z ) { cos 2 n ϕ sin 2 n ϕ r a ,
c nm = 2 γ p 2 l π e n V p 2 1 | J n 1 ( κ p a ) J n + 1 ( κ p a ) | 1 [ 1 exp ( α a l ) ] .

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