Abstract

The coupled differential equations, which govern the evolution of pump and signal power in the gain characterization of Er-doped diffused channel waveguides, involve integrals that depend explicitly on the modal fields at the pump and all signal wavelengths. We use an analytical form of the modal field as an appropriately chosen buried asymmetric Gaussian function centered at the field maximum; this leads to analytical forms of coupled differential equations with no integrals for the calculation of gain characteristics of the amplifying waveguide. Thus, computations are simplified and computation time is also significantly reduced.

© 2009 Optical Society of America

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  1. M. Dinand and W. Sohler, “Theoretical modeling of optical amplification in Er-doped Ti:LiNbO3 waveguide,” IEEE J. Quantum Electron. 30, 1267-1276 (1994).
    [CrossRef]
  2. R. Brinkmann, I. Baumann, M. Dinand, W. Sohler, and H. Suche, “Er-doped single and double pass Ti:LiNbO3 waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2356-2360 (1994).
    [CrossRef]
  3. A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989).
  4. V. T. Gabrielyan, A. A. Kaminski, and L. Li, “Absorption and luminescence spectra and energy levels of Nd3+ and Er3+ ions in LiNbO3 crystals,” Phys. Status Solidi A 3, K37-K42 (1970).
    [CrossRef]
  5. I. Baumann, R. Brinkmann, M. Dinand, W. Sohler, and S. Westenhofer, “Ti:Er:LiNbO3 waveguide laser of optimized efficiency,” IEEE J. Quantum Electron. 32, 1695-1706 (1996).
    [CrossRef]
  6. E. K. Sharma and G. Jain, “Closed form modal field expressions in diffused channel waveguides,” Proc. SPIE 5349, 163-171 (2004).
    [CrossRef]
  7. A. K. Taneja, S. Srivastava, and E. K. Sharma, “Closed form expression for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305-310 (1997).
    [CrossRef]
  8. G. Jain and E. K. Sharma, “Gain calculation in erbium doped LiNbO3 channel waveguides by defining a complex index profile,” Opt. Eng. (Bellingham) 43, 1454-1460 (2004).
    [CrossRef]

2004 (2)

E. K. Sharma and G. Jain, “Closed form modal field expressions in diffused channel waveguides,” Proc. SPIE 5349, 163-171 (2004).
[CrossRef]

G. Jain and E. K. Sharma, “Gain calculation in erbium doped LiNbO3 channel waveguides by defining a complex index profile,” Opt. Eng. (Bellingham) 43, 1454-1460 (2004).
[CrossRef]

1997 (1)

A. K. Taneja, S. Srivastava, and E. K. Sharma, “Closed form expression for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305-310 (1997).
[CrossRef]

1996 (1)

I. Baumann, R. Brinkmann, M. Dinand, W. Sohler, and S. Westenhofer, “Ti:Er:LiNbO3 waveguide laser of optimized efficiency,” IEEE J. Quantum Electron. 32, 1695-1706 (1996).
[CrossRef]

1994 (2)

M. Dinand and W. Sohler, “Theoretical modeling of optical amplification in Er-doped Ti:LiNbO3 waveguide,” IEEE J. Quantum Electron. 30, 1267-1276 (1994).
[CrossRef]

R. Brinkmann, I. Baumann, M. Dinand, W. Sohler, and H. Suche, “Er-doped single and double pass Ti:LiNbO3 waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2356-2360 (1994).
[CrossRef]

1989 (1)

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989).

1970 (1)

V. T. Gabrielyan, A. A. Kaminski, and L. Li, “Absorption and luminescence spectra and energy levels of Nd3+ and Er3+ ions in LiNbO3 crystals,” Phys. Status Solidi A 3, K37-K42 (1970).
[CrossRef]

Baumann, I.

I. Baumann, R. Brinkmann, M. Dinand, W. Sohler, and S. Westenhofer, “Ti:Er:LiNbO3 waveguide laser of optimized efficiency,” IEEE J. Quantum Electron. 32, 1695-1706 (1996).
[CrossRef]

R. Brinkmann, I. Baumann, M. Dinand, W. Sohler, and H. Suche, “Er-doped single and double pass Ti:LiNbO3 waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2356-2360 (1994).
[CrossRef]

Brinkmann, R.

I. Baumann, R. Brinkmann, M. Dinand, W. Sohler, and S. Westenhofer, “Ti:Er:LiNbO3 waveguide laser of optimized efficiency,” IEEE J. Quantum Electron. 32, 1695-1706 (1996).
[CrossRef]

R. Brinkmann, I. Baumann, M. Dinand, W. Sohler, and H. Suche, “Er-doped single and double pass Ti:LiNbO3 waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2356-2360 (1994).
[CrossRef]

Dinand, M.

I. Baumann, R. Brinkmann, M. Dinand, W. Sohler, and S. Westenhofer, “Ti:Er:LiNbO3 waveguide laser of optimized efficiency,” IEEE J. Quantum Electron. 32, 1695-1706 (1996).
[CrossRef]

R. Brinkmann, I. Baumann, M. Dinand, W. Sohler, and H. Suche, “Er-doped single and double pass Ti:LiNbO3 waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2356-2360 (1994).
[CrossRef]

M. Dinand and W. Sohler, “Theoretical modeling of optical amplification in Er-doped Ti:LiNbO3 waveguide,” IEEE J. Quantum Electron. 30, 1267-1276 (1994).
[CrossRef]

Gabrielyan, V. T.

V. T. Gabrielyan, A. A. Kaminski, and L. Li, “Absorption and luminescence spectra and energy levels of Nd3+ and Er3+ ions in LiNbO3 crystals,” Phys. Status Solidi A 3, K37-K42 (1970).
[CrossRef]

Ghatak, A. K.

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989).

Jain, G.

E. K. Sharma and G. Jain, “Closed form modal field expressions in diffused channel waveguides,” Proc. SPIE 5349, 163-171 (2004).
[CrossRef]

G. Jain and E. K. Sharma, “Gain calculation in erbium doped LiNbO3 channel waveguides by defining a complex index profile,” Opt. Eng. (Bellingham) 43, 1454-1460 (2004).
[CrossRef]

Kaminski, A. A.

V. T. Gabrielyan, A. A. Kaminski, and L. Li, “Absorption and luminescence spectra and energy levels of Nd3+ and Er3+ ions in LiNbO3 crystals,” Phys. Status Solidi A 3, K37-K42 (1970).
[CrossRef]

Li, L.

V. T. Gabrielyan, A. A. Kaminski, and L. Li, “Absorption and luminescence spectra and energy levels of Nd3+ and Er3+ ions in LiNbO3 crystals,” Phys. Status Solidi A 3, K37-K42 (1970).
[CrossRef]

Sharma, E. K.

E. K. Sharma and G. Jain, “Closed form modal field expressions in diffused channel waveguides,” Proc. SPIE 5349, 163-171 (2004).
[CrossRef]

G. Jain and E. K. Sharma, “Gain calculation in erbium doped LiNbO3 channel waveguides by defining a complex index profile,” Opt. Eng. (Bellingham) 43, 1454-1460 (2004).
[CrossRef]

A. K. Taneja, S. Srivastava, and E. K. Sharma, “Closed form expression for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305-310 (1997).
[CrossRef]

Sohler, W.

I. Baumann, R. Brinkmann, M. Dinand, W. Sohler, and S. Westenhofer, “Ti:Er:LiNbO3 waveguide laser of optimized efficiency,” IEEE J. Quantum Electron. 32, 1695-1706 (1996).
[CrossRef]

M. Dinand and W. Sohler, “Theoretical modeling of optical amplification in Er-doped Ti:LiNbO3 waveguide,” IEEE J. Quantum Electron. 30, 1267-1276 (1994).
[CrossRef]

R. Brinkmann, I. Baumann, M. Dinand, W. Sohler, and H. Suche, “Er-doped single and double pass Ti:LiNbO3 waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2356-2360 (1994).
[CrossRef]

Srivastava, S.

A. K. Taneja, S. Srivastava, and E. K. Sharma, “Closed form expression for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305-310 (1997).
[CrossRef]

Suche, H.

R. Brinkmann, I. Baumann, M. Dinand, W. Sohler, and H. Suche, “Er-doped single and double pass Ti:LiNbO3 waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2356-2360 (1994).
[CrossRef]

Taneja, A. K.

A. K. Taneja, S. Srivastava, and E. K. Sharma, “Closed form expression for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305-310 (1997).
[CrossRef]

Thyagarajan, K.

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989).

Westenhofer, S.

I. Baumann, R. Brinkmann, M. Dinand, W. Sohler, and S. Westenhofer, “Ti:Er:LiNbO3 waveguide laser of optimized efficiency,” IEEE J. Quantum Electron. 32, 1695-1706 (1996).
[CrossRef]

IEEE J. Quantum Electron. (3)

M. Dinand and W. Sohler, “Theoretical modeling of optical amplification in Er-doped Ti:LiNbO3 waveguide,” IEEE J. Quantum Electron. 30, 1267-1276 (1994).
[CrossRef]

R. Brinkmann, I. Baumann, M. Dinand, W. Sohler, and H. Suche, “Er-doped single and double pass Ti:LiNbO3 waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2356-2360 (1994).
[CrossRef]

I. Baumann, R. Brinkmann, M. Dinand, W. Sohler, and S. Westenhofer, “Ti:Er:LiNbO3 waveguide laser of optimized efficiency,” IEEE J. Quantum Electron. 32, 1695-1706 (1996).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

A. K. Taneja, S. Srivastava, and E. K. Sharma, “Closed form expression for propagation characteristics of diffused planar optical waveguides,” Microwave Opt. Technol. Lett. 15, 305-310 (1997).
[CrossRef]

Opt. Eng. (Bellingham) (1)

G. Jain and E. K. Sharma, “Gain calculation in erbium doped LiNbO3 channel waveguides by defining a complex index profile,” Opt. Eng. (Bellingham) 43, 1454-1460 (2004).
[CrossRef]

Phys. Status Solidi A (1)

V. T. Gabrielyan, A. A. Kaminski, and L. Li, “Absorption and luminescence spectra and energy levels of Nd3+ and Er3+ ions in LiNbO3 crystals,” Phys. Status Solidi A 3, K37-K42 (1970).
[CrossRef]

Proc. SPIE (1)

E. K. Sharma and G. Jain, “Closed form modal field expressions in diffused channel waveguides,” Proc. SPIE 5349, 163-171 (2004).
[CrossRef]

Other (1)

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989).

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Figures (9)

Fig. 1
Fig. 1

(a) Typical refractive index profile of a titanium diffused Li Nb O 3 waveguide. (b) Absorption and emission cross sections of Er : Li Nb O 3 . The solid curve refers to σ a and the dotted curve refers to σ e .

Fig. 2
Fig. 2

(a) Modal field profile X ( x ) at y = 0 . (b) Modal field profile Y ( y ) at x = 0 . The dashed curves show the modal fields as obtained from variational analysis for λ = 1.484 μ m , dark solid curves represent λ = 1.532 μ m , and light solid curves show the modal fields at λ = 1.6 μ m .

Fig. 3
Fig. 3

(a) Modal field profile X ( x ) at y = 0 . (b) Modal field profile Y ( y ) at x = 0 . The dashed curves show the modal fields as obtained from variational analysis for λ = 1.485 μ m , light solid curves show the modal fields at λ = 1.6 μ m , and dark solid curves represent the modal field as represented by approximated Gaussian at λ = 1.532 μ m .

Fig. 4
Fig. 4

(a) Contour plot of the modal field at λ = 1.532 μ m in the x y plane. (b) Contour plot of the modal field at λ = 1.532 μ m in the [ ζ , η ] plane.

Fig. 5
Fig. 5

Calculated gain for different pump powers for different wavelengths, for a 5 cm long waveguide. Signal power is 100 nW . Points correspond to calculations using asymmetric Gaussian field and the continuous curves correspond to the actual three variational modal fields.

Fig. 6
Fig. 6

Gain spectrum of the 5 cm long waveguide for different pump powers, when all the signals, each with 100 nW , are traveling simultaneously. The dotted curves are from the AG field at λ = 1.532 μ m while the solid curves are obtained from actual variational fields at each λ j .

Fig. 7
Fig. 7

Gain spectrum of the 5 cm long waveguide for different signal powers at a pump power of 50 mW . The curves with circles are when each signal is propagating individually while the solid curves are obtained when all the signals are traveling simultaneously. Signal power = ( a ) 1, (b) 10, and (c) 100 μ W , and (d) 1 mW .

Fig. 8
Fig. 8

Gain spectrum for different waveguide lengths and a pump power of 150 mW . Signal powers at (a) 100 nW and (b) 1 mW .

Fig. 9
Fig. 9

Variation of the γ k with z corresponding to the following wavelengths: 1.532 μ m ( k = 49 ) , 1.55 μ m ( k = 67 ) , 1.563 μ m ( k = 80 ) . Pump power = 150 mW , dashed lines corresponding to input signal power = 1 mW at each wavelength and solid curves corresponding to input signal power = 100 nW at each wavelength.

Equations (40)

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n 2 ( x , y ) = n s 2 + 2 n s Δ n exp ( x 2 w 2 ) exp ( y 2 h 2 ) y > 0 ,
= n c 2 y < 0 ,
d I j ( x , y ) d z = [ σ e ( λ j ) N 2 σ a ( λ j ) N 1 ] I j ( x , y ) .
N 2 = 1 + j = 0 N p ̃ j ψ j 2 ( x , y ) 1 + j = 0 N ( η ( λ j ) + 1 ) p ̃ j ψ j 2 ( x , y ) ρ 0 ,
N 1 = 1 + j = 0 N η ( λ j ) p ̃ j ψ j 2 ( x , y ) 1 + j = 0 N ( η ( λ j ) + 1 ) p ̃ j ψ j 2 ( x , y ) ρ 0 ,
d I k ( x , y ) d z = σ a ( λ k ) ρ 0 × j = 1 N ( η k η j ) p ̃ j Ψ j 2 ( x , y ) 1 1 + j = 1 N ( 1 + η j ) p ̃ j Ψ j 2 ( x , y ) I k ( x , y ) .
d P k d z = γ k ( z ) P k ,
γ k ( z ) = σ a ( λ k ) ρ 0 0 Ψ k 2 ( x , y ) × j = 1 N ( η k η j ) p ̃ j ( z ) Ψ j 2 ( x , y ) 1 1 + j = 1 N ( η j + 1 ) p ̃ j ( z ) Ψ j 2 ( x , y ) d x d y .
ψ j ( x , y ) = X j ( x ) Y j ( y ) ,
X ( x ) = 1 w d x exp ( α x 2 x 2 w 2 ) ,
Y ( y ) = 1 h d y ( 1 + γ y y h ) exp ( α y 2 y 2 h 2 ) y > 0 ,
= 1 h d y exp ( γ y y h ) y > 0 ,
d x = 1 α x π 2 , d y = 1 2 γ y + 1 2 α y π 2 + γ y 2 α y 2 + γ y 2 8 α y 3 π 2 .
b = ( n e 2 n s 2 ) 2 n s Δ n = ( r s ) d ,
d = d x d y , r = r x r y , s = p d x 2 γ y + 1 V y 2 s y d x + 1 V x 2 s x d y ,
s x = α x π 2 , s y = 1 8 α y ( π 2 ( 3 γ y 2 + 4 α y 2 ) + 4 γ y α y ) ,
r x = π 1 + 2 α x 2 ,
V x = k 0 w 2 n s Δ n , V y = k 0 h 2 n s Δ n ,
p = ( n s 2 n c 2 ) 2 n s Δ n .
r y = 1 4 ( 1 + 2 α y 2 ) 3 2 [ 2 π ( 1 + 2 α y 2 ) + γ y 2 π + 4 γ y ( 1 + 2 α y 2 ) 1 2 ] .
X G ( x , y ) = ( α x w 2 π ) 1 2 exp ( α x 2 x 2 w 2 ) ,
Y G ( x , y ) = A y exp ( α y 1 2 ( y y 0 ) 2 h 2 ) y < y 0 ,
= A y exp ( α y 2 2 ( y y 0 ) 2 h 2 ) y > y 0 ,
A y = 2 α y 1 α y 2 ( α y 1 + α y 2 ) h π 2
I = 0 Y ( y ) Y G ( y ) d y .
γ k ( z ) = σ a ( λ k ) 0 ρ ( y ) Ψ G 2 ( x , y ) × Ψ G 2 ( x , y ) j = 1 N ( η k η j ) p ̃ j ( z ) 1 1 + Ψ G 2 ( x , y ) j = 1 N ( η j + 1 ) p ̃ j ( z ) d x d y .
γ k ( z ) = σ a ( λ k ) [ y 0 ρ 0 Ψ G 2 ( x , y ) Ψ G 2 ( x , y ) j = 1 N ( η k η j ) p ̃ j 1 1 + Ψ G 2 ( x , y ) j = 1 N ( η j + 1 ) p ̃ j d y + y 0 ρ 0 Ψ G 2 ( x , y ) Ψ G 2 ( x , y ) j = 1 N ( η k η j ) p ̃ j 1 1 + Ψ G 2 ( x , y ) j = 1 N ( η j + 1 ) p ̃ j d y ] d x .
A = A y ( α x w 2 π ) 1 2 .
γ k ( z ) = σ a ( λ k ) d ζ × { 0 A 2 w h ρ ( η ) 2 α x α y 1 e ( ζ 2 + η 2 ) S 2 e ( ζ 2 + η 2 ) 1 + S 1 e ( ζ 2 + η 2 ) d η + 0 A 2 w h ρ ( η ) 2 α x α y 2 e ( ζ 2 + η 2 ) S 2 e ( ζ 2 + η 2 ) 1 + S 1 e ( ζ 2 + η 2 ) d η } ,
S 1 = A 2 j = 1 N ( η j + 1 ) p ̃ j , S 2 = A 2 j = 1 N ( η k η j ) p ̃ j .
γ k ( z ) = σ a ( λ k ) π 0 π 0 A 2 w h ρ 2 2 α x α y 2 exp [ r 2 ] exp [ r 2 ] S 2 1 + exp [ r 2 ] S 1 r d r d θ + π 2 π 0 A 2 w h ρ 1 2 α x α y 1 exp [ r 2 ] exp [ r 2 ] S 2 1 + exp [ r 2 ] S 1 r d r d θ .
γ k ( z ) = σ a ( λ k ) ρ 0 [ S 1 S 2 ( S 2 + S 1 ) ln ( 1 + S 1 ) S 1 2 ] .
d P k d z = σ a ( λ k ) ρ 0 [ S 1 S 2 ( S 2 + S 1 ) ln ( 1 + S 1 ) S 1 2 ] P k ,
d P k d z = ( γ k α ) P k ,
γ k ( z ) = σ a ( λ k ) ρ 0 0 Ψ G 2 ( x , y ) f ( x , y , z ) g ( x , y , z ) d x d y ,
f ( x , y , z ) = Ψ G 2 ( x , y ) ( η k η 1 ) p ̃ 1 ( z ) 1 + Ψ G 2 ( x , y ) j = 2 N ( η k η j ) p ̃ j ( z ) ,
g ( x , y , z ) = 1 + Ψ G 2 ( x , y ) ( η 1 + 1 ) p ̃ 1 ( z ) + Ψ G 2 ( x , y ) j = 2 N ( η j + 1 ) p ̃ j ( z ) .
γ ̃ k ( z ) = σ a ( λ k ) ρ 0 0 Ψ G 2 ( x , y ) f 1 ( x , y , z ) g 1 ( x , y , z ) d x d y ,
f 1 ( x , y , z ) = Ψ G 2 ( x , y ) ( η k η 1 ) p ̃ 1 ( z ) 1 ,
g 1 ( x , y , z ) = 1 + Ψ G 2 ( x , y ) ( η 1 + 1 ) p ̃ 1 ( z ) + Ψ G 2 ( x , y ) ( η k + 1 ) p ̃ k ( z ) .

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