Spacially resolved coupled mode analysis for TMI threshold powers in quantum and Rayleigh scattering limits

Marc D. Mermelstein

doi:10.1364/AO.420076

Applied Optics
Vol. 60,
Issue 16,
pp. 4901-4915
(2021)
•https://doi.org/10.1364/AO.420076

Spacially resolved coupled mode analysis for TMI threshold powers in quantum and Rayleigh scattering limits

Marc D. Mermelstein

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Marc D. Mermelstein, "Spacially resolved coupled mode analysis for TMI threshold powers in quantum and Rayleigh scattering limits," Appl. Opt. 60, 4901-4915 (2021)

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Table of Contents Category
- Lasers, Optical Amplifiers, and Laser Optics
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About this Article
History
- Original Manuscript: January 20, 2021
- Revised Manuscript: April 13, 2021
- Manuscript Accepted: May 10, 2021
- Published: June 1, 2021

Abstract

A 3D spatially resolved coupled mode and perturbation analysis for the transverse mode instability (TMI) threshold powers in Yb-doped fiber amplifiers is presented in this paper. Threshold powers are computed in the quantum and thermal Rayleigh scattering limits and are compared with those calculated by other coupled mode analyses. Quantum-limited threshold powers are found to be more than three times greater than those calculated with coupled-mode analyses that use uniform and/or average gain approximations. The analysis presented here includes pump depletion, gain saturation, and transverse hole burning. Simulations are applied to co-, cnt-, and bidirectionally pump amplifier configurations. The appearance of TMI is attributed to the formation of a dynamic thermal grating, which enables the exchange of optical power between the fundamental mode (FM) and higher-order mode (HOM). The sole approximation made is that the power in the HOM is much less than that in the FM. A distributed thermal Rayleigh scattering model is introduced that includes a ray-optic representation of the fiber mode structure that relates the Rayleigh power captured by the HOM to the waveguide structure. The location and strength of the thermal gratings are identified to assist in the application of mitigation techniques.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Core radius	$a$	12.5 µm
Cladding radius	$b$	200 µm
Numerical aperture	NA	0.06
Signal wavelength	$λ_{s}$	1064 nm
Pump wavelength	$λ_{p}$	975 nm
Beat length	$Λ$	3.0 mm
Thermal conductivity	$κ$	1.38 W/m-deg
Density	$ρ$	$2200 {k g / m}^{3}$
Heat capacity	$C_{P}$	700 J/kg-deg
Thermal diffusivity	$D$	$8.96 \times 10^{- 7} m^{2} / s$
Thermo-optic coef.	dn/dT	$1.1 \times 10^{5} {1 /}^{\circ} C$
Rayleigh ratio	$R_{t h}$	$5.9 \times 10^{- 9} 1 / m - s r$
Capture fraction	$s_{12}$	$8.4 \times 10^{- 5}$
Gain medium parameters at 1064 nm signal, 975 nm pump
Yb ion density	$n_{0}$	$4.1 \times 10^{25} m^{- 3}$
Pump emission cross sec.	$σ_{e p}$	$2.58 \times 10^{- 24} m^{2}$
Pump abs. cross section	$σ_{a p}$	$2.58 \times 10^{- 24} m^{2}$
Signal emission cross sec.	$σ_{e s}$	$2.30 \times 10^{- 25} m^{2}$
Signal abs. cross section	$σ_{a s}$	$1.69 \times 10^{- 27} m^{2}$
Spont. emission lifetime	$τ$	0.84 ms
Signal saturation intensity	$I_{s}^{s a t} = \frac{ℏ \cdot ω_{s}}{(σ_{s a} + σ_{s e}) \cdot τ}$	$9.4 \times 10^{8} W / m^{2}$
Pump saturation int.	$I_{p}^{s a t} = \frac{ℏ \cdot ω_{p}}{(σ_{p a} + σ_{p e}) \cdot τ}$	$4.7 \times 10^{7} W / m^{2}$
Signal trans rate int.	$I_{s a} = \frac{ℏ \cdot ω_{s}}{σ_{s a} \cdot τ}$	$1.3 \times 10^{11} W / m^{2}$
Signal trans rate int.	$I_{s e} = \frac{ℏ \cdot ω_{s}}{σ_{s e} \cdot τ}$	$9.5 \times 10^{8} W / m^{2}$
Pump trans rate int.	$I_{p a} = \frac{ℏ \cdot ω_{p}}{σ_{p a} \cdot τ}$	$9.4 \times 10^{7} W / m^{2}$
Pump trans rate int.	$I_{p e} = \frac{ℏ \cdot ω_{p}}{σ_{p e} \cdot τ}$	$9.4 \times 10^{7} W / m^{2}$
Constants
Planck’s constant	$ℏ$	$1.05 \times 10^{- 34} J - s$
Boltzmann constant	$k_{B}$	$1.38 \times 10^{- 23} J / d e g$

	Hansen [2]	MDM	Hansen [2]	MDM	Dong [4]	MDM	Tao [8]	MDM
$n_{0}$ [ $m^{- 3}$ ]	(4e25)	4e25	(4e25)	4e25	(4e25)	4e25	3.5e25	3.5e25
$d_{c o r e}$ [µm]	20	20	40	40	30	30	21	21
$d_{c l a d}$ [µm]	(400)	400	(400)	400	400	400	400	400
NA	0.05	0.05	0.03	0.03	0.06	0.06	0.064	0.064
$V$	3.0	3.0	3.5	3.5	5.3	5.3	4.0	4.0
$λ_{S}$ [nm]	1030	1030	1030	1030	1060	1060	1064	1064
$λ_{P}$ [nm]	976	976	976	976	976	976	915	915
$P_{01}$ [W]	1	20	1	20	7.5	7.5	20	20
$P_{11}$ [W]	Q	4e-15	Q	8e-16	7.5e-15	1.5e-15	Q	3e-15
$G$ [dB]	27.0	25	26	20.5	16.3	21.5	23.7	22.4
$γ_{max}$ [1/m-W]	—	0.50	—	0.61	—	0.450	—	0.69
$χ_{max}$ [1/W]	0.07	0.06	0.08	0.065	0.10	0.09	—	0.12
$f_{max}$ [kHz]	5.5	5.8	1.5	1.5	3.4	2.9	—	5.0
$ξ$	0.05	0.06	0.05	0.05	0.01	0.01	0.05	0.06
$P_{t h}$ [kW]	0.44	6.30	0.40	2.33	0.33	1.05	4.70	4.55

Core radius	$a$	12.5 µm
Cladding radius	$b$	200 µm
Numerical aperture	NA	0.06
Signal wavelength	$λ_{s}$	1064 nm
Pump wavelength	$λ_{p}$	975 nm
Beat length	$Λ$	3.0 mm
Thermal conductivity	$κ$	1.38 W/m-deg
Density	$ρ$	$2200 {k g / m}^{3}$
Heat capacity	$C_{P}$	700 J/kg-deg
Thermal diffusivity	$D$	$8.96 \times 10^{- 7} m^{2} / s$
Thermo-optic coef.	dn/dT	$1.1 \times 10^{5} {1 /}^{\circ} C$
Rayleigh ratio	$R_{t h}$	$5.9 \times 10^{- 9} 1 / m - s r$
Capture fraction	$s_{12}$	$8.4 \times 10^{- 5}$
Gain medium parameters at 1064 nm signal, 975 nm pump
Yb ion density	$n_{0}$	$4.1 \times 10^{25} m^{- 3}$
Pump emission cross sec.	$σ_{e p}$	$2.58 \times 10^{- 24} m^{2}$
Pump abs. cross section	$σ_{a p}$	$2.58 \times 10^{- 24} m^{2}$
Signal emission cross sec.	$σ_{e s}$	$2.30 \times 10^{- 25} m^{2}$
Signal abs. cross section	$σ_{a s}$	$1.69 \times 10^{- 27} m^{2}$
Spont. emission lifetime	$τ$	0.84 ms
Signal saturation intensity	$I_{s}^{s a t} = \frac{ℏ \cdot ω_{s}}{(σ_{s a} + σ_{s e}) \cdot τ}$	$9.4 \times 10^{8} W / m^{2}$
Pump saturation int.	$I_{p}^{s a t} = \frac{ℏ \cdot ω_{p}}{(σ_{p a} + σ_{p e}) \cdot τ}$	$4.7 \times 10^{7} W / m^{2}$
Signal trans rate int.	$I_{s a} = \frac{ℏ \cdot ω_{s}}{σ_{s a} \cdot τ}$	$1.3 \times 10^{11} W / m^{2}$
Signal trans rate int.	$I_{s e} = \frac{ℏ \cdot ω_{s}}{σ_{s e} \cdot τ}$	$9.5 \times 10^{8} W / m^{2}$
Pump trans rate int.	$I_{p a} = \frac{ℏ \cdot ω_{p}}{σ_{p a} \cdot τ}$	$9.4 \times 10^{7} W / m^{2}$
Pump trans rate int.	$I_{p e} = \frac{ℏ \cdot ω_{p}}{σ_{p e} \cdot τ}$	$9.4 \times 10^{7} W / m^{2}$
Constants
Planck’s constant	$ℏ$	$1.05 \times 10^{- 34} J - s$
Boltzmann constant	$k_{B}$	$1.38 \times 10^{- 23} J / d e g$

	Hansen [2]	MDM	Hansen [2]	MDM	Dong [4]	MDM	Tao [8]	MDM
$n_{0}$ [ $m^{- 3}$ ]	(4e25)	4e25	(4e25)	4e25	(4e25)	4e25	3.5e25	3.5e25
$d_{c o r e}$ [µm]	20	20	40	40	30	30	21	21
$d_{c l a d}$ [µm]	(400)	400	(400)	400	400	400	400	400
NA	0.05	0.05	0.03	0.03	0.06	0.06	0.064	0.064
$V$	3.0	3.0	3.5	3.5	5.3	5.3	4.0	4.0
$λ_{S}$ [nm]	1030	1030	1030	1030	1060	1060	1064	1064
$λ_{P}$ [nm]	976	976	976	976	976	976	915	915
$P_{01}$ [W]	1	20	1	20	7.5	7.5	20	20
$P_{11}$ [W]	Q	4e-15	Q	8e-16	7.5e-15	1.5e-15	Q	3e-15
$G$ [dB]	27.0	25	26	20.5	16.3	21.5	23.7	22.4
$γ_{max}$ [1/m-W]	—	0.50	—	0.61	—	0.450	—	0.69
$χ_{max}$ [1/W]	0.07	0.06	0.08	0.065	0.10	0.09	—	0.12
$f_{max}$ [kHz]	5.5	5.8	1.5	1.5	3.4	2.9	—	5.0
$ξ$	0.05	0.06	0.05	0.05	0.01	0.01	0.05	0.06
$P_{t h}$ [kW]	0.44	6.30	0.40	2.33	0.33	1.05	4.70	4.55

Abstract

Data Availability

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Tables (2)

Equations (68)

Applied Optics