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Qualitative question
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Chap.5 Optical Resonators Containing Amplifying Media U D • Objectives – Interactions between resonator & amplifying p y g media – Mode behavior – Output coupling • Reading R di – Chapter 5 of Davis • Outline – – – – – Laser oscillation Threshold of laser oscillation R Resonant t ffrequency Multimode oscillations Output power of a laser EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 1 U D Resonator with Amplifying Media Gain medium z 2 k '  k [ 1   ' (  ) / 2 n ] Propagation constant  ( )   k  ' ' ( ) / n 2 Gain coefficient Propagation of plane wave e  jk ' z   e  2 z  j(k   2 z k ' jk  ' ' j   ) z 2 2 2 2 n 2 n EOP 505: Introduction to Lasers  e Copyright © Qiwen Zhan  jk g z 2 U D Optical Oscillation l Gain medium E 0 t1 r2 e E0  j2kgl E 0 t1 r2 e E 0 t1 e E 0 t1  jk g l  jk g l E 0 t1 t 2 e  jk g l ( r2 , t 2 ) ( r1 , t1 ) Total transmitted filed: E t  E 0 t1 t 2 e  jk g l [1  r1 r2 e  j2kgl  ( r1 r2 e  j2kgl ) 2  ...]  jk l t1 t 2 e g  E0  j2kgl 1  r1 r2 e Oscillation: r1 r2 e  j2kgl  1 Finite input leads to infinite output EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 3 Threshold Condition • U D Physically, the oscillation condition corresponds to a condition that the amplitude after a round trip remains the same, which means the gain per round trip is equal to the loss per round trip trip. Meanwhile Meanwhile, the phase shift after a round trip is multiple of 2. These leads to the threshold condition for oscillation. r1 r2 e  j2kgl 1   ( t  )2l  2 2  1,  t iis threshold th h ld gain i  r1 r2 e  k ' (k  ) 2 l  2 m  , m is an integer 2 2n  EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 4 U D Threshold Population Inversion Amplitude condition: t    1 1 ln r1 r2   t l   l  ln R1 R 2 l 2 Gain per pass Loss per pass g2 c2 A21 g2 2 A21  ( )  ( N2  N1) 2 g(v0 , v)  ( N2  N1) g(v0 , v) g1 8v g1 8 Threshold population inversion: g2 1 8 8 (  ln r1r2 )  ( N2  N1)t  2 g1 g(v0 , v) A21 l g(v0 , v)2 A21c 0 Homogeneous broadening: Inhomogeneous broadening: (N2  g2  N1 )t  2 g1  A21 (N2  g2 1 N 1 )t  3 g1  A21 EOP 505: Introduction to Lasers Copyright © Qiwen Zhan Photon lifetime in resonator 5 U D Resonant Frequency: Mode Pulling Phase condition: k ' (k  )2l  2 m  , 2 2n  ' ( )  2( 0  )  ' ' ( )  m is an integer 2( 0  )  2( 0  )  ( )  (  ( ) k n2 ) mc ]  [1  m k 2l   ( )c    m  (  0 ) 2 At threshold, ignore distributed loss:    m  ( m  0 ) 1/ 2  EOP 505: Introduction to Lasers t  1 R l Copyright © Qiwen Zhan 6 U D Resonant Frequency: Mode Pulling If one of the resonator modes frequency coincides with the line center of the line shape function, then the oscillation occurs at the line center frequency. Otherwise the actual oscillation occurs at a frequency near m but is slightly Otherwise, shifted towards line center. This is the so-called “mode-pulling”. A t l oscillation Actual ill ti ffrequency c/2nl 0 EOP 505: Introduction to Lasers m m+1 Copyright © Qiwen Zhan  7 Multi Longitudinal Modes Oscillation: Homogeneous () U D () Loss line t Loss line t   () Loss line t Final oscillation frequency  EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 8 Multi Longitudinal Modes Oscillation: Inhomogeneous U D 1 Single mode: 0.9 0.8 0.7 Hole Image hole 0.6 0.5 0.4 0.3 Loss line t 0.2 0.1 0 -5 0 5 1 0 .9 0 .8 0 .7 Lamb Dip 0 .6 0 .5 0 .4 Loss line t 0 .3 0 .2 0 .1 0 -5 EOP 505: Introduction to Lasers 0 5 Copyright © Qiwen Zhan 9 U D Multi Longitudinal Modes Oscillation: Inhomogeneous Multi-mode: 1 0 .9 0 .8 0 .7 Hole 0 .6 0 .5 0 .4 Loss line0.t3 0 .2 0 .1 0 -5 0 5 If the homogeneous broadening is also significant, mode competition also may occur in inhomogeneous broadened laser. EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 10 U D Mode Beating Assume two longitudinal modes: E i  E 1e j t  E 2e j (   )t Signal on square law detector is: i(t )  Ei 2  [ E1e j t  E 2 e j (     ) t ][ E1 e  jt  E 2 e  j (     ) t ] * *  E1  E 2  E1 E 2* e  j  t  E 2 E1* e j  t 2 2  E1  E 2  2 E1 E 2 cos(   t    ) 2 2 Beat signal • • • Optical oscillation is too fast to observe directly Beat signal of optical signal frequencies normally in the RF range, thus can be easily detected If mode pulling is considered considered, due to its nonlinearity nonlinearity, beat signals frequencies will split. EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 11 U D Output Power of Laser T1I4 T2I2 R1, A1 (loss R1 (l att mirror i or other th distributed loss) R2 A2 R2, I-+II+ I4 I1 II+ I2 I3 z Total output is: Ioutt=T T1I4+T2I2 EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 12 Output Power of Laser U D In the laser, intensity at both directions contribute to saturation  (z)   0 1  (I  I) / Is  1 dI    (z)  I 1 dI  1 dI  dz     0  1 dI  I dz I dz      (z)  I  dz d d (II)   0  II  C dz  I3  I2R2   I1  I 4 R1 EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 13 U D Output Power of Laser 1 I  dI dz  dI I      1  (I dI I   d [ln I   ln I 2 I  I1 ln I I   4 I 2 4   2 I I ( s 3  R s (   C I  I s 0 dz ]   0 dz s  I1 C 1 1  (  )   I s I s I 2 I1  I I s 4 3   I    I   I I ) / I  CdI    2 I  I s s   I    0 R 1 R 1 s 1 C 1 1 (  )   I s I 4 I3 0 ( ) 0 l  ln  R 2 )( 1  R 2 (  0 l  ln  R EOP 505: Introduction to Lasers 2 )( 1  R1R 2 0 l l R1R 2 ) R1R 2 ) R1R 2 ) Copyright © Qiwen Zhan 14 U D Output Power of Laser  T1  1  R  T2  1  R 1 2  A1  A2 1. If A1=A2=A I out  I s (1  A  1  R1R 2 ) R1R 2 (  0 l  ln l R1R 2 ) 2. If T1=0, R1=1 I out  I 2 T 2  T 2 I s [ 0 l  1 ln( 1  A 2  T 2 )] 2 A2  T2 3 Symmetric resonator: R1=R2=R 3. R1=R2=R, A1=A2=A A1=A2=A, output from each mirror: I out 2  I s (1  A  R ) (  0 l  ln R ) 2 1  R EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 15 Optimal Coupling U D Consider a symmetric mirror resonator, for maximal output: dI out dR  0  ( 1  A  R )(  0 l  ln R ) (1  A  R ) 1   0 2 R 1 R (1  R ) 1 R 1 A  R R   (  0 l  ln R ) A 1 R T opt 1  A  T opt   [  0 l  ln( 1  A  T opt )] A A  T opt    0 l  ln R 1  A  T opt A  T opt  T opt  (  [  0 l  ( A  T opt )]  0l A  1) A   0 lA  A EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 16 Optimal Coupling U D With the optimal coupling, output power can be calculated: Po  I out Amod e  Amod e I s  Ps  Ps T opt  A T opt T opt  A T opt T opt  A T opt [  0 l  ln( 1  T opt )] [  0 l  ln( 1  T optt )] [  0 l  T opt ]  Ps [  0 l  2 A   0 lA   0 lA  0 lA  A  Ps [  0 l  A  2  0 lA ]  Ps [  0 l   Ps [ g 0  ] A ]2 A ]2 EOP 505: Introduction to Lasers Copyright © Qiwen Zhan 17

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