Abstract

We present detailed numerical simulations of modal instabilities in high-power Yb-doped fiber amplifiers using a time-dependent temperature solver coupled to the optical fields and population inversion equations. The temperature is computed by solving the heat equation in polar coordinates using a 2D second-order alternating direction implicit method. We show that the higher-order modal content rises dramatically in the vicinity of the threshold and we recover the three power-dependent regions that are characteristic of the transfer of energy. We also investigate the dependence of the threshold on the seed power and the modal content ratio of the seed. The latter has a minimal effect on the threshold while it is shown that for the fiber configuration investigated, the modal instability threshold scales linearly over a wide range with the seed power. In addition, two different gain-tailored core designs are investigated and are shown to have higher thresholds than that of a uniformly doped core. Finally, we show that this full time-dependent model which does not assume a frequency offset between the modes a priori, predicts a reduced threshold when the seed is modulated at the KHz level. This is in agreement with the steady-periodic approach to this phenomenon.

© 2013 OSA

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  1. C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett.36(14), 2686–2688 (2011).
    [CrossRef] [PubMed]
  2. F. J. Kontur, I. Dajani, Y. Lu, and R. J. Knize, “Frequency-doubling of a CW fiber laser using PPKTP, PPMgSLT, and PPMgLN,” Opt. Express15(20), 12882–12889 (2007).
    [CrossRef] [PubMed]
  3. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
    [CrossRef] [PubMed]
  4. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett.36(23), 4572–4574 (2011).
    [CrossRef] [PubMed]
  5. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express20(1), 440–451 (2012).
    [CrossRef] [PubMed]
  6. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19(11), 10180–10192 (2011).
    [CrossRef] [PubMed]
  7. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett.37(12), 2382–2384 (2012).
    [CrossRef] [PubMed]
  8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 1944–1971 (2013).
    [CrossRef] [PubMed]
  9. L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express21(3), 2642–2656 (2013).
    [CrossRef] [PubMed]
  10. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express20(10), 11407–11422 (2012).
    [CrossRef] [PubMed]
  11. H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express20(14), 15710–15722 (2012).
    [CrossRef] [PubMed]
  12. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express19(24), 23965–23980 (2011).
    [CrossRef] [PubMed]
  13. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express20(1), 440–451 (2012).
    [CrossRef] [PubMed]
  14. C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
    [CrossRef]
  15. C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE8237, 82371D, 82371D-10 (2012).
    [CrossRef]
  16. C. Robin and I. Dajani, “Acoustically segmented photonic crystal fiber for single-frequency high-power laser applications,” Opt. Lett.36(14), 2641–2643 (2011).
    [CrossRef] [PubMed]

2013 (2)

2012 (6)

2011 (6)

2010 (1)

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

2007 (1)

Alkeskjold, T. T.

Augst, S. J.

Broeng, J.

Dajani, I.

Dong, L.

Eberhardt, R.

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

Eidam, T.

Fan, T. Y.

Gaida, C.

Goldizen, K. C.

Hansen, K. R.

Jansen, F.

Jauregui, C.

Knize, R. J.

Kontur, F. J.

Lægsgaard, J.

Lanari, A.

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE8237, 82371D, 82371D-10 (2012).
[CrossRef]

Limpert, J.

Lu, Y.

Murphy, D. V.

Otto, H. J.

Otto, H.-J.

Peschel, T.

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

Redmond, S. M.

Rekas, M.

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

Robin, C.

Sanchez, A.

Schmidt, O.

Schreiber, T.

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
[CrossRef] [PubMed]

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

Smith, A. V.

Smith, J. J.

Stutzki, F.

Tsybin, I.

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

Tünnermann, A.

C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express20(1), 440–451 (2012).
[CrossRef] [PubMed]

H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express20(14), 15710–15722 (2012).
[CrossRef] [PubMed]

C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express20(1), 440–451 (2012).
[CrossRef] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
[CrossRef] [PubMed]

F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett.36(23), 4572–4574 (2011).
[CrossRef] [PubMed]

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

Ward, B.

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE8237, 82371D, 82371D-10 (2012).
[CrossRef]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express20(10), 11407–11422 (2012).
[CrossRef] [PubMed]

Wirth, C.

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
[CrossRef] [PubMed]

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

Yu, C. X.

Zeringue, C.

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE8237, 82371D, 82371D-10 (2012).
[CrossRef]

Opt. Express (10)

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 1944–1971 (2013).
[CrossRef] [PubMed]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express21(3), 2642–2656 (2013).
[CrossRef] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express20(10), 11407–11422 (2012).
[CrossRef] [PubMed]

H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express20(14), 15710–15722 (2012).
[CrossRef] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express19(24), 23965–23980 (2011).
[CrossRef] [PubMed]

C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express20(1), 440–451 (2012).
[CrossRef] [PubMed]

F. J. Kontur, I. Dajani, Y. Lu, and R. J. Knize, “Frequency-doubling of a CW fiber laser using PPKTP, PPMgSLT, and PPMgLN,” Opt. Express15(20), 12882–12889 (2007).
[CrossRef] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
[CrossRef] [PubMed]

C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express20(1), 440–451 (2012).
[CrossRef] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19(11), 10180–10192 (2011).
[CrossRef] [PubMed]

Opt. Lett. (4)

Proc. SPIE (2)

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE7580, 75801H, 75801H-6 (2010).
[CrossRef]

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE8237, 82371D, 82371D-10 (2012).
[CrossRef]

Supplementary Material (3)

» Media 1: MOV (3384 KB)     
» Media 2: MOV (3776 KB)     
» Media 3: MOV (3822 KB)     

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Figures (11)

Fig. 1
Fig. 1

a) Temperature difference between fiber core and room temperature, ΔT , along the fiber after 40 ms, 60 ms, and 80 ms at a point away from the center. b) Close-up of part (a) to display the phase shift and oscillations with period 2π/Δβ . The inset reveals the oscillations have a period π/Δβ at the center.

Fig. 2
Fig. 2

Error in temperature versus the number of computational points in log form (Blue) for n = 8, 16 and 32 compared to a line with a slope of −2 (Red).

Fig. 3
Fig. 3

Top picture plots the total intensity of the modes in time and the lower plot displays the fundamental mode power (Blue) and the higher order mode power (Red) in each mode at the output end of the fiber in time for pump power of 720, 820, and 1040 W. The corresponding movies (Media 1, Media 2, and Media 3) show the temporal variation of intensity at the output end as well as temporal variation in modal powers along the propagation direction.

Fig. 4
Fig. 4

Percentage of total power in higher-order mode (LP11) as a function of pump power. Region I displays the “below threshold” region. Region II corresponds to a substantial content of signal power in the higher-order mode. Region III corresponds with the chaotic regime.

Fig. 5
Fig. 5

Threshold as a function of seed power indicating a linear relation over the seed power range investigated. The power injected in the LP11 mode was 5% of the total seed power.

Fig. 6
Fig. 6

The fundamental mode power (Blue) and the higher order mode power (Red) at the output end of the fiber in time for seed power of 30 W whereby only 10-7% of the seed power is injected to the higher-order mode.

Fig. 7
Fig. 7

The core design of the gain-tailored segmented acoustic fiber. There are three acoustic regions. Three segments in the outer ring also does not contain ytterbium in order to suppress the modal instabilities.

Fig. 8
Fig. 8

The core design of the gain tailored fibers that are numerically investigated: a) design A is similar to the gain-tailored SAT fiber whereby Yb is absent in the three white regions b) design B is such that Yb is absent in the outer ring of the core.

Fig. 9
Fig. 9

Higher-order mode content averaged in time between 60 ms and 80 ms as a function of pump power for uniformly Yb-doped core, gain-tailored segmented acoustic design A, and gain-tailored design B whereby ytterbium is absent in annular region near edge of core.

Fig. 10
Fig. 10

Higher-order mode content averaged in time as a function of pump power with modulation frequencies of 1kHz and modulation depth of 1/3000, 1/30000, and 1/300000. The solid lines correspond to best fit to the data.

Fig. 11
Fig. 11

Higher-order mode content averaged in time as a function of pump power with modulation depth 1/3000 and modulation frequencies of 0.5 kHz, 1 kHz, 1.5 kHz, and 2 kHz. The solid lines correspond to best fit to the data.

Tables (1)

Tables Icon

Table 1 Error in temperature for n = 8, 16 and 32

Equations (28)

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2 E ( E ) 1 c 2 2 [ n 2 ( r ,t) E ] t 2 = μ 0 2 P LG t 2 ,
2 E ( 1 n 2 E n 2 ) 1 c 2 2 ( n 2 E ) t 2 = μ 0 2 P LG t 2 .
E x ( r ,t)=(1/2) k A k (z,t) φ k (x,y)e i( β k zωt) +c.c.,
P LG,x =2i ε 0 c k I ω n 0 E x = 2i ε 0 c ω ( σ s (e) N 2 σ S (a) N 1 2 ) n 0 E x ,
k ( 2 β k 2 + n 0 2 ω 2 c 2 ) φ k =0,
2i k β k φ k e i β k z A k z + 2iω n 0 2 c 2 k φ k e i β k z A k t + 2 ω 2 n 0 δn c 2 k A k φ k e i β k z = 2iω n 0 c ( σ s (e) N 2 σ s (a) N 1 2 ) k A k φ k e i β k z .
A j z + n 0 c A j t =i k 0 k κ δn,kj A k exp[ i( β k β j )z ]+ 1 2 k κ LG,kj A k exp[ i( β k β j )z ] ,
κ δn,kj (z,t)= φ j * δn( x,y,z,t ) φ k dxdy κ LG,kj (z,t)= φ j * ( N 2 ( x,y,z,t ) σ s (e) N 1 ( x,y,z,t ) σ s a ) φ k * dxdy .
d A j dz =i k 0 k κ δn,kj (z, t 0 ) A k exp[ i( β k β j )z ]+ 1 2 k κ LG,kj (z, t 0 ) A k exp[ i( β k β j )z ] .
ρC T t = k th ( T r 2 + 1 r T r + 1 r 2 T θ 2 )+Q,
Q=( 1 λ p λ s )[ σ p (a) N 0 ( σ p (a) + σ p (e) ) N 2 ] I p ,
N 2 = σ p (a) I p ω p + σ s (a) I s ω s ( σ p (a) + σ p (e) ) I p ω p +( σ s (a) + σ s (e) ) I s ω s + 1 τ N 0 ,
I s = n 0 2 μ 0 c ( k | A k | 2 | φ j | 2 + kl A k A l * φ k φ l * e iΔ β kl z ),
d I p dz =± ( r core r clad ) 2 ( σ p (e) N ¯ 2 σ p (a) N ¯ 1 ) I p ,
I s = n 0 2c μ 0 [ | A 0 | 2 | φ 0 | 2 + | A 1 | 2 | φ 1 | 2 + A 0 A 1 * φ 0 φ 1 * e iΔ β 01 z + A 0 * A 1 φ 0 * φ 1 e iΔ β 01 z ],
N 2 (x,y,z)= α 1 + η 1 [ A 0 A 1 * φ 0 φ 1 * e iΔ β 01 z + A 0 * A 1 φ 0 * φ 1 e iΔ β 01 z ] α 2 [ 1+ η 2 α 2 ( A 0 A 1 * φ 0 φ 1 * e iΔ β 01 z + A 0 * A 1 φ 0 * φ 1 e iΔ β 01 z ) ] N 0 ,
α 1 (z)= σ p (a) I p ω p + η 1 [ | A 0 | 2 | φ 0 | 2 + | A 1 | 2 | φ 1 | 2 ] α 2 (z)= ( σ p (a) + σ p (e) ) I p ω p + η 2 [ | A 0 | 2 | φ 0 | 2 + | A 1 | 2 | φ 1 | 2 ]+1/τ η 1 = σ s (a) n 0 2c μ 0 ω s η 2 = ( σ s (a) + σ s (e) ) n 0 2c μ 0 ω s .
N 2 = α 1 α 2 [ 1 η 2 α 2 ξ+ ( η 2 α 2 ) 2 ξ 2 + ] N 0 + η 1 α 2 ξ[ 1 η 2 α 2 ξ+ ( η 2 α 2 ) 2 ξ 2 + ] N 0 .
Q N 0 ( 1 λ p λ s )[ σ p (a) ( σ p (a) + σ p (e) ) α 1 α 2 ( σ p (a) + σ p (e) )( α 1 η 2 α 2 2 + η 1 α 2 )ξ ] I p .
κ δn,10 φ 0 φ 1 u(x,y,z)( A 0 A 1 * φ 0 φ 1 e iΔ β 01 z + A 0 * A 1 φ 0 φ 1 e iΔ β 01 z )dxdy,
d A 0 dz =( (i C 0 κ ' δn,10 +ς) | A 1 | 2 + g l,0 2 ) A 0 d A 1 dz =( (i C 0 κ ' δn,10 +ς) | A 0 | 2 + g l,1 2 ) A 1 ,
κ ' δn,10 = u φ 0 2 φ 1 2 dxdy ς= 1 2 ( σ s (e) + σ s (a) ) N 0 ( α 1 η 2 α 2 2 + η 1 α 2 ) φ 0 2 φ 1 2 dxdy g l,0 = N 0 [( σ s (e) + σ s (a) ) α 1 α 2 σ s (a) ] φ 0 2 dxdy g l,1 = N 0 [( σ s (e) + σ s (a) ) α 1 α 2 σ s (a) ] φ 1 2 dxdy .
A 0 (z) A 0 (0)exp( 0 z (i C 0 κ ' δn,10 (z') | A 1 (z') | 2 + g l,0 (z') 2 )dz ' ) A 1 (z) A 1 (0)exp( 0 z (i C 0 κ ' δn,10 (z') | A 0 (z') | 2 + g l,1 (z') 2 )dz' ).
| A 0 (z) | 2 | A 0 (0) | 2 exp( 0 z g l,0 (z')dz ' ) | A 1 (z) | 2 | A 1 (0) | 2 exp( 0 z g l,1 (z')dz' ).
A 0 (z) A 0 (0)exp( 0 z [ i C 0 κ ' δn,10 (z') | A 1 (0) | 2 exp( 0 z' g l,1 (z'')dz '' )+ g l,0 (z') 2 ]dz' ) A 1 (z) A 1 (0)exp( 0 z [ i C 0 κ ' δn,10 (z') | A 0 (0) | 2 exp( 0 z' g l,0 (z'')dz '' )+ g l,1 (z') 2 ]dz' ).
T 1 (r)= T b Q 0 r 2 4k + Q 0 r out 2 2k ln( r out r core )+ Q 0 r core 2 4k 0r r core , 0θ<2π T 2 (r)= T b + Q 0 r out 2 2k ln( r out r ) r core r r out , 0θ<2π
r i =( i 1 2 )Δr i=1,,n+1 θ j =( j1 )Δθ j=1,,m+1
P s = P s 0 [ 1+asin(Ωt) ],

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