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Extraction characteristics of a dual fiber compound cavity

T. B. Simpson, A. Gavrielides, and P. Peterson

Open Access Open Access

Abstract

We study experimentally the time dependence, steady state behavior and spectra of a dual fiber-laser compound cavity. Theoretically we confirm the CW and spectral characteristics. This particular cavity is formed with two Er-doped fiber amplifiers, each terminated with a fiber Bragg grating, and coupled through a 50/50 coupler to a common feedback and output coupling element. The experiment and theory show that a low Q, high gain symmetric compound cavity extracts nearly 4 times the power of a component resonator. This extraction is maintained even when there is significant difference in the optical pathlengths of the two component elements. Further, our measurements and theory show that the longitudinal modes of the coupled cavity are distinct from the modes of the component cavities and that the coherence is formed on a mode-by-mode basis using these coupled-cavity modes. The time behavior of the compound cavity shows slow fluctuations, on the order of seconds, consistent with perturbations in the laboratory environment.

©2002 Optical Society of America

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Figures (10)
Fig. 1.
Fig. 1. Diagram and nomenclature for the compound cavity.

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Fig. 2.
Fig. 2. Schematic of the fiber compound cavity. Additional lengths of fiber can be inserted between the FBGs and the amplifiers, and polarization control elements between the amplifiers and the 50/50 coupler.

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Fig. 3.
Fig. 3. Time dependence of the compound (black) and component cavities (colored).

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Fig. 4.
Fig. 4. (a) The extraction curves for the component cavities. (b) outcoupled power, left axis, and the power exiting the beam splitter, right axis for the compound cavity. The pump current is in amps.

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Fig. 5.
Fig. 5. Spectra of the component (colored) and the compound cavity (black).

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Fig. 6.
Fig. 6. Frequency offset of the 79 th harmonic for both component lasers and the compound laser.

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Fig. 7.
Fig. 7. Mode beating spectra of the two component cavities and the compound cavity. One of the two components cavities has approximately 5 meters of additional fiber inserted. The spectrum analyzer is operating on its “peak hold” setting and the spectra shown here are accumulated over a period of a couple of minutes with the beating frequencies filling in as the laser randomly changed oscillation modes. Note that the compound cavity only showed strong output on those modes that are simultaneously nearly resonant with the two component cavity modes.

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Fig. 8.
Fig. 8. Simulated extraction curves for the compound cavity. The red line is the outcoupled power and the black curve is power exiting the beam splitter.

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Fig. 9.
Fig. 9. (a) Output spectra just above threshold. (b) Output spectra far above above threshold. The power is normalized to the total incoherent power.

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Fig. 10.
Fig. 10. Power output as a function of δl

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Equations (32)

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d E i dz = ± ( g i 1 + f i 2 + b i 2 + i k i ) E i
F i ( z ) = f i ( z ) exp ⁡ ( i ϕ i ( z ) ) , B i ( z ) = b i ( z ) exp ⁡ ( i β i ( z ) ) .
F 1 ( L ) = itF ( L ) , and F 2 ( L ) = rF ( L ) ,
B ( L ) = − it B 1 ( L ) + r B 2 ( L ) , E BS ( L ) = r B 1 ( L ) − it B 2 ( L ) .
F ( L ) = F ( 0 ) exp ⁡ ( ikL ) , B ( 0 ) = B ( L ) exp ⁡ ( ikL ) ,
r 2 ( sin ⁡ 2 [ Ψ ( 1 + δl ) ] − R 2 t 4 sin ⁡ 2 ( 2 Ψ δl ) ) ( G 2 + ln ⁡ [ − R r 2 sin ⁡ ( 2 Ψ δl ) sin ⁡ [ Ψ ( 1 − δl ) ] ] )
− t 2 ( sin ⁡ 2 [ Ψ ( 1 − δl ) ] − R 2 r 4 sin ⁡ 2 ( 2 Ψ δl ) ) ( G 1 + ln ⁡ [ − R t 2 sin ⁡ ( 2 Ψ δl ) sin ⁡ [ Ψ ( 1 + δl ) ] ] ) = 0 .
Ψ = k ( l 1 + l 2 + 2 L ) , δl = l 2 − l 1 l 1 + l 2 + 2 L , and δ = 2 k ( l 2 − l 1 ) .
b i 2 = G i + ln ⁡ ( β i ) 1 − β i 2 , and f i = β i b i ,
β 1 = R t 2 sin ⁡ ( 2 Ψ δl ) sin ⁡ [ Ψ ( 1 + δl ) ] > 0 , and β 2 = − R r 2 sin ⁡ ( 2 Ψ δ l ) sin ⁡ [ Ψ ( 1 − δ l ) ] > 0 .
I out = ( 1 − R 2 ) < ∣ B ( 0 ) ∣ 2 > = ( 1 − R 2 ) < ∣ B ( L ) ∣ 2 >
= ( 1 − R 2 ) < ∣ − it B 1 ( L ) + r B 2 ( L ) ∣ 2 > .
I BS = < ∣ r B 1 ( L ) − it B 2 ( L ) ∣ 2 > .
I out = ( 1 − R 2 ) ∑ m [ t 2 ( b 1 m ) 2 + r 2 ( b 2 m ) 2 + 2 rt b 1 m b 2 m cos ⁡ ( 2 Ψ δl ) ] ,
I BS = ∑ m [ r 2 ( b 1 m ) 2 + t 2 ( b 2 m ) 2 − 2 rt b 1 m b 2 m cos ⁡ ( 2 Ψ δl ) ] .
I component = ( 1 − R 2 ) t 2 G i + ln ⁡ ( t 2 R ) 1 − t 4 R 2 , i = 1 or 2 ,
Δ ν n = c 2 π ( k n + 1 − k n ) = c 2 ( l 1 + L ) , with b i 2 = G i + ln ⁡ R 1 − R 2
sin ⁡ [ Ψ ( 1 + δl ) ] = − sin ⁡ [ Ψ ( 1 − δl ) ] .
Δ ν n = c 2 π ( k n + 1 − k n ) = c ( l 1 + l 2 + 2 L ) ≈ c 2 ( l 1 + L ) , for l 1 ≈ l 2
d f i dz = g i 1 + f i 2 + b i 2 f i .
β i ( L ) − ϕ i ( L ) = 2 k i l i ,
ln ⁡ b i 2 ( L ) f i 2 ( L ) + [ b i 2 ( L ) − f i 2 ( L ) ] = 2 g i L i ≡ G i .
F 1 ( L ) = itR ( − it B 1 ( L ) + r B 2 ( L ) ) exp ⁡ ( 2 ikL ) ,
F 2 ( L ) = rR ( − it B 1 ( L ) + r B 2 ( L ) ) exp ⁡ ( 2 ikL ) .
f 1 f 2 = t r , and exp ⁡ ( i ( ϕ 1 ( L ) − ϕ 2 ( L ) ) ) = i .
b 1 = C b 2 , where C = − r sin ⁡ [ Ψ ( 1 + δ l ) ] t sin ⁡ [ Ψ ( 1 − δ l ) ] > 0
Ψ = k ( l 1 + l 2 + 2 L ) , δl = l 2 − l 1 l 1 + l 2 + 2 L , and δ = 2 k ( l 2 − l 1 ) .
f 1 = β 1 b 1 , where β 1 = R t 2 sin ⁡ ( 2 Ψ δ l ) sin ⁡ [ Ψ ( 1 + δ l ) ] > 0 ,
f 2 = β 2 b 2 , where β 2 = − R r 2 sin ⁡ ( 2 Ψ δ l ) sin ⁡ [ Ψ ( 1 − δ l ) ] > 0 .
b i 2 = G i + ln ⁡ ( β i ) 1 − β i 2 .
r 2 ( sin ⁡ 2 [ Ψ ( 1 + δl ) ] − R 2 t 4 sin ⁡ 2 ( 2 Ψ δl ) ) ( G 2 + ln ⁡ [ − R r 2 sin ⁡ ( 2 Ψ δl ) sin ⁡ [ Ψ ( 1 − δl ) ] ] )
− t 2 ( sin ⁡ 2 [ Ψ ( 1 − δl ) ] − R 2 r 4 sin ⁡ 2 ( 2 Ψ δl ) ) ( G 1 + ln ⁡ [ − R t 2 sin ⁡ ( 2 Ψ δl ) sin ⁡ [ Ψ ( 1 + δl ) ] ] ) = 0 .
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