Chapter 8_L20. Relaxation oscillation

Transcription

1 Chater 8_L Relaxation oscillation

2 Carrier number (1 7 ) Photon number (1 4 ) Relaxation oscillation 3 Relaxation oscillation: when the laser undergoes an external erturbation, the laser s couled carrier and hoton will mutually oscillates around their steady-state values. Eventually, the laser will reach the steady state in some time. The laser is erturbed by a ste-like current at time t= ns, the carrier and hoton exhibits damed relaxation oscillations ( 阻尼振荡 ) N 3 th Time (ns) Current Carrier Photon 8 6 4

3 Carrier number (1 7 ) Photon number (1 4 ) Relaxation oscillation 4 T Time (ns) Ste1 ( T 1 ) Carriers increase immediately with the ste-like current, and reach the threshold at a delay time T 1, which is determined by the effective carrier lifetime. Photons begin to increase quickly starting from the delay time T 1.

4 Carrier number (1 7 ) Photon number (1 4 ) Relaxation oscillation 5 T 1 T Time (ns) Ste (T 1 T ) On one hand, the carriers remain increase due to the um current. On the other hand, uolation inversion N>N th leads to the increase of hoton number. The increased hoton number induces carrier saturation. Finally, the carrier number reaches maximum at a certain hoton number at T.

5 Carrier number (1 7 ) Photon number (1 4 ) Relaxation oscillation 6 T 1 T T Time (ns) Ste3 (T T 3 ) The uolation inversion N>N th, so the hoton number remains increase, leading to stronger carrier saturation effect. Therefore, the carrier number decreases. When the carrier number reaches the threshold, the hoton number reaches the maximum at T 3.

6 Carrier number (1 7 ) Photon number (1 4 ) Relaxation oscillation 7 T 1 T T 3 T Time (ns) Ste4 (T 3 T 4 ) The hotons remain reduces the carrier number due to the saturation effect, and making N<N th. On the other hand, because it is below threshold, the hoton undergoes loss leading to the hoton number decrease. The saturation effect becomes weak, and finally is balanced by the um rocess at T 4.

7 Carrier number (1 7 ) Photon number (1 4 ) Relaxation oscillation 8 T 1 T T 3 T 4 T Time (ns) Ste5 (T 4 T 5 ) The hoton number remains decreasing since below threshold. The um rocess accumulates carriers, and again reaches the threshold at T 5. The question is what is the relaxation eriod? How long it is needed to reach the steady state?

8 Carrier number (1 7 ) Photon number (1 4 ) Small-signal (linearized) analysis 9 When the ertubation induced carrier and hoton variation is much smaller than their steady-state values, it is the small-signal erturbation Current Time (ns) Carrier Photon 6 4

9 Small-signal (linearized) analysis 1 The rate equations Steady-state solution Small-signal erturbation dn dt N R vggnp s dnp NP N =vggnp dt The rate equations s R N N N N g N vggnp s N N P vggn P s d N R R vg N N N N dt d N dt P =v R ( t) R R ( t) N( t) N N( t) N ( t) N N ( t) N( t) N ; N ( t) N N N N N N N s s

10 Small-signal (linearized) analysis 11 The rate equations dn N N R v N N R -v N N N N - v N N dt g g g s s dn N N P N N = v N N v N N N N v N N dt g g g s s Ignore high-order terms, we get the linearized equations Linearized rate equations dn dt dn dt P N R vg N N N N N N =vg N N N N Above threshold, vg N s Linearized rate equations 1 s Linearized rate equations dn dt dn dt P N N R vg N N =v N N N g s s

11 Carrier number (1 7 ) Photon number (1 4 ) Small-signal (linearized) analysis 1 The solution of the linearized rate equation is of the form. N( t) N X ex t sin( Rt) N ( t) N X ex t cos( Rt) R is the relaxation resonance frequency, determining the oscillation eriod. is the daming factor, determining the oscillation decay seed Time (ns) 6 4 Linearized rate equations dn N N R vg N N dt dn dt P N =vg N N s s

12 Small-signal (linearized) analysis 13 The carrier and hoton variations are in the form R ( t) r ex( jt) N( t) nex( jt) N ( t) n ex( jt) Linearized rate equations n nex( jt) j nex( jt) rex( jt) vg N nex( jt) ex( jt) nex( jt) j npex( jt)=vg N nex( jt) s s Linearized rate equations n n j n r vg N n j n N n n P=vg s s Linearized rate equations dn N N R vg N N dt dn dt P N =vg N N s s

13 Small-signal (linearized) analysis 14 The linearized equation in the matrix form 1 1 jvgn s n r n P vgn j s The determinant of the matrix j n nn n n nn R n j j j The resonance frequency R n n 1 vg N The daming factor v nn N g 1 s s

14 Small-signal (linearized) analysis 15 The solutions of the linearized equations 1 1 r n j 1 j vg N r 1 s np vg N s n r n r P 1 j 1 v N g s Examles 8.1, jvgn s n r n P vgn j s

15 Chater 8_L1 16 Q switching technique Electro-otic Q switch Acousto-otic Q switch Saturable-absorber Q switch

16 Photon Gain (Carrier) Loss (Q) Pum Q switching 17 Q switching is a technique for obtaining energetic short (but not ultrashort) ulses from a laser by modulating the intracavity losses and thus the Q factor of the laser resonator. The technique is mainly alied for the generation of nanosecond ulses of high energy and eak ower with solid-state bulk lasers. Time Time g th Time Time

17 Q switching 18 T 1 T Ste1 ( T 1 ) The cavity loss is at a high level, the lasing canot occur. So the um accumulates carriers in the gain medium. The maximum stored carrier energy (amount) is limited by the carrier lifetime (sontaneous emission T s ). Therefore, this energy storing eriod is limited as T 1 <T s, usually on the order of ms (.1 ms 1. ms). Ste (T 1 T ) The cavity loss is suddenly reduced to a low level at T 1, then the carrier oulation is much larger than the threshold value. So the ower of the laser radiation build u very quickly in the cavity. When the ower becomes strong, the gain reduces due to the saturation effect. The eak of the laser ulse is reached when the carrier oulation reduces to the threshold. Below threshold, the ulse ower decays, while it remains deletes the carrier oulations.

18 Q switching 19 The laser ulse duration is usually on the order of ns ( 1. ns 5 ns). The ulse energy ranges from mj u to kj, thereby the ulse eak ower ( W) is much higher than the CW laser ower with the same um. The ulse reetition rate is usually in the range of 1. khz 1 khz. Q switching methods include active Q switching using electrootic Q-switches or acousto-otic Q-switches, and assive Q switching using saturable absorber (otical nonlinearity).

19 Electro-otic Q switching Electro-otic effect (Pockels effect): For some nonlinear crystals such as KD * P, LiNbO 3, when a voltage is alied, it will exhibit birefrigence effect. The birefrigence effect slits the electric field E into an ordinary wave (o-wave < or s>, Ex) and an extrodinary wave (e-wave <s or >, Ey), which are erendicular to each other. The o-wave and the e-wave have different refractive indices n o and n e. The two refractive indices change with the alied voltage. When the field roagates along the otical axis of the crystal, o-wave and e-wave will be on the same direction. The device with Pockels effect is called Pockels cell. When an linearly olarized electric field asses through the cell with 45 to the birefrigence axises, the two wave comonents will have a hase difference: k n n L eo e o

20 Electro-otic Q switching 1 If the hase difference is 9, the olarization of the electric field will become circular. If the hase difference is 18, the electric field is still linearly olarized, but the direction is erendicular to the original direction. Y(e) Y(e) Y(e) X (o) X (o) X (o) Linear E-field 9 hase shift 18 hase shift

21 Electro-otic Q switching In Q switching, the Pockels cell is oerated to exhibit a 9 hase shift as the following exerimental setu. medium 9 Polarizer Pockels cell Closed Q switch The required volatage to roduce 9 hase shift is called «quarter-wave volage», which may range between 1 kv and 5 kv. The switch of the voltage must be fast (tyically < ns), smaller than the laser ulse build-u time.

22 Acousto-otic Q switching 3 When a traveling acoustic wave is alied to some crystal like quartz, the strain induced by the acoustic wave results in local changes of the material refractive index through the hotoelastic effect. This eriodic change of refractive index acts as a hase grating with eriod equal to the acoustic wavelength, amlitude roortional to the sound amlitude, and which is traveling at the sound velocity in the medium (traveling-wave hase grating). Its effect is to diffract a fraction of the incident beam out of the direction.

23 Acousto-otic Q switching 4 The requirements of the oeration (Bragg diffraction) L I out out nra ; in ; in L Iin sin a The voltage alited to the AO switch is around 1 V. The AO switch induced loss is smaller than that by the EO switch, so can not be used to very high energy laser ulse generation. L Examles 8.3 is the hase shift of the light due to the hase grating

24 Saturable-absorber Q switch 5 The absorber can be treated as a two-level system with most carriers in the ground state, and with a large absortion cross section (small saturation intensity, I s ), such as Cr 4+ :YAG. The absorber exhibits a high loss (absortion) when the incident light is weak, while becomes almost transarent (no absortion) when the light is strong (>I s ). = 1 I / Is hv Is = a s

25 Saturable-absorber Q switch 6 Photon T1 T I S T3 Loss (threshold) Ste1 ( T 1 ) The aborber has a high loss due to the absortion. The laser is below threshold. At T1, the um increases the gain to the threshold, and the laser starts to lasing. Gain Time

26 Saturable-absorber Q switch 7 Photon T1 T I S T3 Loss (threshold) Gain Time Ste (T 1 T ) The hoton reduces the absortion coefficient due to the absortion saturation effect. At T, the hoton intensity reaches the absorber saturation intensity, and the absorber becomes almost transarent with very low loss. Thus, the gain is far beyond the threshold.

27 Saturable-absorber Q switch 8 Photon T1 T I S T3 Loss (threshold) Ste3 (T T 3 ) Since the gain is far beyond the threshold, the hoton intensity increases raidly. The gain reduces due to the gain saturation effect. The hoton reaches the eak when the gain reduces to be the same as the loss. Gain Time

28 Saturable-absorber Q switch 9 Comared with active Q switching, assive Q switching is simle and cost-effective (eliminating the modulator and its electronics), and is suitable for very high ulse reetition rates. However, the ulse energies are tyically lower. It may also be a disadvantage that the ulse energy and duration are often more or less indeendent of the um ower, which only determines the ulse reetition rate.

29 Chater 8_L 3 Q-switching laser ulse characteristics

30 Q switching characteristics 31 τ in The maximum time τ in for restoring carrier oulation (high loss) must be smaller than the carrier lifetime of the uer lasing state τ s, or else carriers will be killed by the sontaneous emission. Therefore, the laser ulse reetition rate (1/ τ in ) is usually from 1 khz to u to 1 khz. The questions for the laser ulse: What is the ulse eak ower? What is the ulse energy? What is the ulse duration? What is the ulse buid u time?

31 Pulse eak ower 3 When the Q switch changes from closed to oen at, the initial conditions: in The hoton density: N Pi ~ The carrier density: N i During N dn dt ( t) [, in ] N R s, the rate equations t N( t) N R ex R s f s s in N N R ex R s N i f s s During [, the laser ulse duration is on the order in, f i] of ns, it is so fast that the um and the sontaneous emission have little imact on the carrier oulation during this eriod, and thus can be ignored. i dn dt dn dt P v =v g g gn gn P P N P

32 Pulse eak ower 33 At the eak of the laser ulse, fi The carrier density dn dt N P eak NP =vggnp 1 v g N th The hoton density dn dn P 1 = 1 v g g dnp Nth = 1 dn N Ni NP Nth ln N Ni N eak N i N i N N ln 1 P th N th N th The eak ower eak gm eak P N h V v out The eak ower increases with the value N i /N th. Enlarge the difference between high loss and low loss. Enhance the um ower Longer uer-level carrier lifetime P dn dt dn dt P v =v g g gn gn P P N P

33 N N N total P i f Pulse energy 34 Every hoton is generated from the dro of one carrier oulation, so the total hoton density of the laser ulse inside the cavity is The total hoton energy inside the cavity is E N hv N N hv E i with E N hv ; E N hv total P i f E f i i f f E i is the stored energy in the material before the lasing ulse E f is the residual energy in the material after the lasing ulse The outut ulse energy E E E m m out E i T T The energy utilization factor E Ei Ef E E E 1 N i N f i i

34 Pulse energy 35 When the carrier reaches N f, the hoton is zero. N i NPf Nth ln N f Ni N f N f N N th f ln 1 Ni Ni Ni in Ni Rs N f Rs ex s The energy utilization factor increases with the ratio N i /N th. N N f i E N / N N / N i th i th

35 Pulse duration 36 The ulse duration is aroximately the ulse energy divided by the eak ower E E N h V m m out E i T T eak N i N i P N ln 1 out th hv vgm Nth Nth t E / P d out E eak out Ni / Nth N / N ln N / N 1 i th i th The ulse duration deends only on the hoton lifetime and the ratio N i /N th. The laser ulse usually travels several round in the cavity. (See the argument on. 33)

36 Pulse build-u time 37 The ulse build-u time is usually defined as the time needed for the laser ower reaches 1% of the eak ower starting from the Q switching. N i The ulse build-u time is usually defined as the time needed for the laser ower reaches 1% of the eak ower starting from the Q switching. dn dt P =v g gn P N 1 NP( t) NPi ex vgg t P Examles vgg vg Ni v N g th 1 N N i th N N i th 1 1 N i t NP( t) NPi ex 1 Nth 1 eak N i t NP 1ex 1 1 Nth 1 eak bd ln NP N / N 1 1 i th

37 Cavity duming technique 38 In the Q switching technique, the medium restores energy when the cavity Q is low. Once the cavity Q is switched to a high level, the laser ulse will build u in the cavity, meanwhile the ulse is outut from the cavity. The laser ulse will oscillates several round tris inside the cavity before the end, so the ulse duration is usually tens of nanosecond. Q Restoring energy in medium Lasing ulse outut t

38 Cavity duming technique 39 In the cavity duming technique, the medium restores energy when the cavity has a low Q, once the Q value is switched to a high level, the laser oscillates in the cavity but no outut (%). When the Q value is switched back to the low level, the hotons are dumed out (T=1%) of the cavity within only one round tri, so the ulse duration is only several nanosecond. Q Restoring energy in medium Restoring energy in hoton inside the cavity Outut (dum out) laser ulse t

39 Cavity duming technique 4 s-wave -wave Polarizer (Bruster) Gain medium R1=1% QWP (9 hase shift) Pockels cell (9 hase shift) R=1% When the Pockels cell is off (no voltage). The lasing olarization is s-wave, totally reflected by the Bruster olarizer to the quarter-wave late (QWP), a round ass through the QWP induces 18 hase shift, so the olarization becomes -wave and the outut from the olarizer is 1%. The cavity has a low Q to restore energy in the medium. When the Pockels cell is on. The s-wave double ass the QWP and the Pockesl cell, which induces 36 hase shift, thus the olarization is still s-wave and the outut from the olarizer is %. The cavity has a high Q to restore energy in the hoton. When the Pockels cell is off. The outut is 1%, the laser ulse is dumed out in one round tri.

40 Chater 8_L3 41 Mode locking technique

41 Power Mutimode laser 4 Consider a laser with N longitudinal modes, the mode amlitude is A, the mode sacing is v, the mode (oscillating) bandwidth is v, determined by the gain L Nv bandwidth. The hase of each mode varies randomly. v Nv L A v Frequency Consider 3 modes E ( t) A ex j t j ( t) E ( t) A ex j( ) t j ( t) 1 1 E ( t) A ex j( ) t j ( t) 1 1 The total electric field is E( t) E ( t) E ( t) E ( t) 1 1

42 The total ower is (roortional to) E( t) E( t) E *( t) Multimode laser 43 3A A cos( t ) cos( t ) cos( t ) Since the hases vary randomly, the total light ower shows a random time behavior. The waveform is eriodic with a eriod 1/ v Each light ulse duration of the random waveform roughly equals to v 1/ L The average value is the sum of each mode ower, roortional to N*A. The ulse duration is on the order of icoseconds or less, which can not be detected by the PD. Thus, the monitored value is an time averaged value.

43 Mode-locked laser 44 If the hase of each mode has a definite hase relation (eg. difference), rather than varies randomly. The multimode laser is then mode-locked laser. The henomenon is called mode locking, what locked is the hase. l l 1 ; l - n, - n 1... n -1, n The central mode frequency is w, and the hase is set to be. The total electric field is n E( t) A ex j( l) t jl ln n Aex jl( t ) ex j t ln A( t)ex j t Time series of three locked modes Therefore, E(t) can be reresented by a sinusoidal carrier wave at the center mode frequency w, whose amlitude is time deendent.

44 The time deendent amlitude n A( t) A ex jl( t ) ln A( t ') A ex jlt ' = A n ln t ' sin (n 1) t ' sin Mode-locked laser 45 Introduce a new time reference t' t At () A The laser roduces a train of evenly saced light ulses due to the interaction of hase locked modes. The ulse maxima occur at those times for which the denominator becomes zero. t ' Therefore, the successive ulse time interval is m 1 nl r v c

45 Mode-locked laser 46 The first maximum occurs at t =, using the aroximation sin x~x A() (n 1) A The eak ower is roortional to A () (n 1) A The first zero occurs at a time t such that (n 1) t ' The ulse width (FWHM) is aroximately equal to t 1 t ' (n1) v L At () A

46 Mode-locked laser 47 The field comonents can be reresented by vectors in the comlex lane, the l-th comonent corresonds to a comlex vector of amlitude A, which rotates at the angular velocity l, refer to argument on. 34. A( t ') A( t ') A ( t ') A ( t ') A ( t ') 1 1 Max At ( ') A 1 A A 1 3 A A 1 A 1 t ' t' m t ' t' t t ' ' n 1

47 Mode-locked laser 48 In mode locked laser, the ulse duration is equal to the oscillating bandwidth (mode number), which is determined by the gain width. The ulse duration ranges from a few s down to a few fs. In solid-state and semiconductor lasers, the ulse duration is a few icosecond. In dye or tunable solid-state lasers, the gain linewdith is at least 1 times larger, which leads to a ulse duration of a few femtosecond, such as ~5 fs dye laser, ~7 fs Ti:sahire laser. In gas lasers, the gain linewidth is narrower on the order of a few GHz, leading to a long ulses down to ~1 s. In the mode locked laser, the eak ower of each ulse is roortional to (n+1) A, while a laser with modes of random hase has an average ower roortional to (n+1) A. Therefore, mode-locking laser roduces high eak ower. Therefore, mode-locked laser with hase-locked modes rovides ulse trains of both short duration and high eak ower. Requirements: mutimode, equal mode sacing, fixed mode hase relation

48 Mode-locked laser 49 The sectral enveloe of lasers are usually of Gaussian distribution due to the inhomo. gain broadening. The amlitude A l of the l-th mode is described as: A L l A ex ln L l reresents the FWHM bandwidth of the sectrum. The time-deendent amlitude of the total electric field is A( t ') A ex( jlt ') l l t A ( t) ex ln That is, the electric field amlitude A(t) in the time domain is the Fourier transform of the sectral amlitude A l in the otical domain. The ulse duration is Each ulse intensity is a Gaussian function of time 4ln.441 v L L

49 Mode-locked laser (time domain) 5 In mode-locked laser, the time interval τ of two consecutive ulses is equal to the cavity round-tri time T rt. This can be achieved by inserting a fast shutter at one end of the cavity. The shutter is eriodically oened with a eriod T rt, and the oening duration is τ. This is the case shown in (a). The laser oscillation behavoir can be visualized as a single ultrashort ulse roagates back and forth within the cavity. When the ulse goes to the mirror, there is one ulse outut. Therefore, τ = T rt. This is refered as fundamental mode-locking. Note that the shutter is able to lock the hase of each mode.

50 Mode-locked laser (time domain) 51 If the shutter is laced at the middle of the cavity, and has an oen eriod of T rt /, two ultrashort ulses resent in the cavity such that the two ulses cross each other when the shutter is oen. Then, mode locking is achieved with ulse interval time T rt /. This is called the second harmonic mode locking. ( 二次谐波锁模 ) If the shuter is lace at one third length of the cavity, and has an oen eriod of T rt /3, three ulses resent in the cavity such that two of them cross each other when the shutter is oen. Then, mode locking is achieven with ulse intercal time T rt /3. This is called the third harmonic mode locking. ( 三次谐波锁模 ) In the ring cavity, the ulse reetition rate only deends on the shutter oen eriod, indeendent of the shutter osition within the cavity. The hase locking condition: Fundamental + l +1 l l l1 Second / l +1 l l l1 Third / 3 l +1 l l l1

51 Chater 8_L4 5 Mode locking methods Active mode locking Passive mode locking

52 Mode-locking methods 53 Like Q-switching techniques, active mode locking requires elements driven by current/voltage sources. Passive mode locking emloys nonlinear otical effects such as saturation of saturable absorber or nonlinear refractive index change of a certain material. Active mode locking has three main tyes: Amlitude mode (AM) locking using an amlitude modulator Frequency mode (FM) locking using a hase modulator Gain modulation by synchronous uming (seldom)

53 AM mode locking 54 If the multimode laser s amlitude is modulated with a small amlitude m at a frequency w m, E ( t) A ex( j t j ) 1 mcos( t) l l l m m m A ex( jlt jl ) 1 ex( jmt) ex( jmt) m m A ex( jl ) ex( jlt) ex j( l m) t ex j( l m) t Therefore, the modulated signal has three comonents: the carrier wave at w l, and two small sidebands (ure sine waves) with frequencies slightly above and below the carrier frequency (w l +w m ) and (w l -w m ). w l (w l +w m ) (w l -w m )

54 AM mode locking 55 If the modulation frequency equals to the mode sacing, the modulation sidebands will coincides with the adjacent mode frequencies of the laser resonator. Thus, the equations for cavity modes become couled, i. e., the field equation of a given cavity mode will contain two contributions arising from the modulation of the adjacent modes. The mode couling mechanism then lock the mode hases. The modulator modulates the cavity loss with a eriod T /, which equals to the cavity round-tri time. The stable steady-state condiction corresonds to light ulses assing through the modulator at the times t min when a minimum loss of the modulator occurs. As such, the ulse will return to the modulator after the time T, when the loss is again at a minimum. m

55 AM mode locking 56 If the ulse reaches the modulator at a time slightly deviates from the the time t min, the leading (shorter) or the trailing (longer) edge of the ulse will suffer more attenuation. In the following round tris, the ulse eak will move closer to the minimum loss time t min. Eventually, the steady-state situation will be reached. Once the ulse eak is located at the time t min, the ulse duration tends to be shortened each time the ulse asses through the modulator, because both the leading and the trailing edge of the ulse are somewhat attenuated while the eak of the ulse is not attenuated. Therefore, the ulse duration tends to zero with rogressive assages through the modulator.

56 AM mode locking 57 On the other hand, a short ulse duration corresonds to a broad srectum, which is hysically limited by the finite gain bandwidth. In the frequency domain, the swings of the ulse sectrum outside the gain bandwidth wont be amlified. This is the fundamental limitation to the ulse sectrum bandwidth, and hence to the ulse duration. In the inhomogeneous broadening medium, the mode oscillating bandwidth is rougly the gain bandwidth, if the laser is umed sufficiently above the threshold. The rimary urose of the modulator is only to lock the hases of the oscillating modes. Assuming the modes enveloe is of Gaussian distribution, the AM modulator is located at one end of the cavity, the modulation frequency is ulse duration is.441/ v * m, then the

57 AM mode locking 58 In the homogeneous broadening medium, the time-varying loss of the AM modulator narrows the ulse duration, and thus broadens the sectrum, while the gain bandwidth limits the ulse duation. The ulse duration is much longer than the inverse of the gain bandwidth. The ulse intensity is also of Gaussian rofile with a ulse duration given by.45 / vv ; v v Therefore, the mode-locking ulse duration of inhomo broadening medium is much shorter than homo broadening medium for a similar broadening linewidth. For a (ulsed) and high gain laser, AM mode locking is achieved by a Pockels cell (EO) amlitude modulator. For a cw and low gain laser, it is achieved by a AO modulator, owing to its lower insertion loss. Here it is standing sound wave in the AO modulator, in comarison with the traveling sound wave in the Q switch. Examles 8.7

58 AM mode locking theory 59 In the time domain, the mode locking ulse reroduces itself after each round tri. Assuming a homogeneous broadening medium with a lifetime much longer than the cavity round tri time. At the gain eak, the saturated gain coefficient g with a light intensity I is g g I I The AM modulator is very thin and is laced at mirror. Thus, a single ulse travels back and forth within the cavity. At any given osition in the cavity, the electric field of the ulse in the time domain is given by E t ( ) ( )e j A t t ( ) 1 / s The sectral amlitude of the ulse is obtained from the Fourier transfrom ( A( ) A( t)e j ) t dt The equation is what is the solution of A(t)? A t A d j ( ( ) ( )e ) t ( )

59 Gain medium contribution 6 For one single ass of the gain medium (length l) from A 1 to A : gl / A A1 ex ex j nrl 1 j( ) / c The ower gain: A A G( ) ex g( ) l 1 with the Lorzentian gain coefficient g g( ) 1 ( ) / The sectral width of the ulse is much narrower than the gain bandwith: ( ) gl A A1 ex 1 ex j nrl gl c

60 Gain medium contribution 61 Then, the hase delay of the ulse after assing the gain medium n l gl c = r The time delay of the ulse is determined by the grou velocity (amlitude enveloe) l d nl gl d = v g d c Grou velocity Only consider the absolute value of the comlex amlitude, the active medium (single ass) contribution: A A 1 ( ) gl ex 1 Therefore, the round-tri ass of the gain medium ( ) A3 A1 1 1 gl d vg = dk Phase velocity v = k

61 Gain medium contribution 6 In the time domain, through the inverse Fourier transform: 3 1 ( ) 1 1 ( ) d A t gl A t dt Then, the round-tri contribution of the gain medium is given by an oerator ˆ 1 1 g d T gl dt

62 Cavity loss contribution 63 The cavity loss arises from the finite mirror reflectivity and internal losses. The single-ass amlitude is L T 4 1 A ( t) ex A ( t) Thus, the round-tri contribution of the cavity loss to the field amlitude is given by an oerator T ˆ ex 1 TL TL

63 AM modulator contribution 64 The time varying loss coefficient of the modulator is characterized by ( t) 1cos( t) md md m Then, the single-ass field amlitude through the modulator is md () tlm A5( t) ex A4( t) Thus, the round-tri contribution of the modulator to the field amlitude is given by an oerator Tˆ md ex md 1 cos( mt) lm 1md lm 1 cos( mt) The ulse asses through the modulator when the modulator loss is zero, e. g. t= Tˆ 1 l md md m ( mt)

64 AM mode locking theory 65 The ulse reroduction after each round tri requires that Tˆ Tˆ Tˆ A( t) A( t) md g md l m d 1 ( mt) 1 TL1 gl1 A( t) A( t) dt gl L t A t d md lm 1 ( ) ( ) T m dt The solution is t At ( ) ex with md lm m gl ˆ Tg 1 gl 1 l ˆ md m md 1 ( mt) T T ˆ ex 1 TL TL d dt

65 AM mode locking theory 66 The ulse duration is given by ln 1/ 1/4 1/ ln gl 1 md lm vmv 1.45 vmv 1/

66 FM (PM) mode locking 67 If the multimode laser s hase is modulated with a small amlitude β at a frequency w m, two sidebands will be formed for each mode as the AM modulation. When the modulation frequency equals to the mode sacing, all the cavity modes are couled (hase locked ) to each other through the sidebands. The laser is then mode-locked. E ( t) A ex( j t j ) ex j sin( t) ex( ) 1 sin( ) l l l m A j t j j t l l m A ex( jlt jl ) 1 ex( jmt) ex( jmt) A ex( jl ) ex( jlt) ex j( l m) t ex j( l m) t The above formula used the first-order aroximation of the Taylor exansion. In truth, there are an infinite number of sidesbands, described by Bessel functions. El ( t) Aex( jlt + jl ) J( ) Jk ( )ex jmt ( 1) Jk ( )ex jmt k1 k1 k

67 FM (PM) mode locking 68 If the multimode laser s hase is modulated with a small amlitude β at a frequency w m, two sidebands will be formed for each mode as the AM modulation. When the modulation frequency equals to the mode sacing, all the cavity modes are couled (hase locked ) to each other through the sidebands. The laser is then mode-locked. In the time domain, two stable mode locking states can occur. One is the ulse series occuring at the minimum of the refractive index (kn r L). The other is at the maximum of the refractive index. (The hase is changed by the refractive index) The instantaneous laser frequency is ( t) cos( t) l l m m Only light ulses assing through the modulator at the refractive index extremes has no frequency shift, while others have somewhat frequency shifts. Since these ulses take the frequency shift each round tri, those will be out of the gain bandwidth and quench. Eventually, only ulses at extremes survive. l () t

68 FM (PM) mode locking 69 FM mode locking usually oerates on one of the extreme series. Even a small erturbation can switch one series to another, thus some tehnique is required to stablize the ulse outut on only one series. For the FM mode locking, a Pockels cell EO hase modulator can be used. Here the light olarization is oriented along one of the birefringence axis, rather than 45 angle offset as in a Q switch. ( t) Lnr ( t)

69 Chater 8_L4 7 Mode locking methods Active mode locking Passive mode locking

70 Passive mode locking methods include Passive mode locking 71 Fast saturable absorber ML: using an absorber with very short uer state lifetime Kerr lens ML: using the self-focusing effect (Kerr effect, nonlinear otical effect) Slow saturable absorber ML: using an absorber with long uer state lifetime together with a fast gain medium. Additive ulse ML (APM): using the self-hase modulation effect (nonlinear otical effect).

71 Fast saturable aborber ML 7 Consider an absorber with a low saturation intensity (large cross section), and with a carrier lifetime much shorter than the duration of the mode locking ulse. The laser is initially oscillating with unlocked modes. Then, there is a random sequence of light ulses. For the ulse with a low eak intensity, it will suffer a large attenuation when it asses the absorber due to its weak saturation. For the ulse with a high eak intensity, it will suffer a reduced attenuation owing to the strong saturation. If certain conditions are met, this ulse can grow faster than others, and after many round tris the mode locking ulses are envetually established. Due to the long lifetime of the gain medium, the gain has little change when the ulse asses. The gain saturation is determined by the average intra-cavity laser ower. The ulse is shortened by the time-varying absorber loss, but broadened by the finite gain bandwidth.

72 Fast saturable absorber ML 73 The steady-state ulse amlitude is described by a hyerbolic secant function E( t) sec h( t / ) sec hx ( )= e x e x The ulse duration is related to the ulse time interval The ulse duration is calculated by.79 gs v ( I / I ) s s I s hv = s where g s and α s are saturated value determined by the average intracavity light intensity.

73 Fast saturable absorber ML 74 The requirements of absorber are firstly a short carrier lifetime ( a few s or shorter to obain a short ulse duration), and secondly a large absortion cross section (1-16 cm or larger to botain a low saturation intensity). The most oular saturable absorbers are dye molecules and semiconductors. The dye absorbers carrier lifetime is tyically tens of s. Thus, the absorber remains saturated for a time roughly equal to this lifetime, and ML ulses shorter than a few s can not be obtained. The semiconductor absorber has mutile mechanisms determining the carrier decay: a) Interband sontaneous emission and nonradiative decay of sub-nanosecond; b) Intraband carrier-carrier scattering of ~.1 s; c) Intraband carrier-honon scattering of ~1. s. The slow mechanism a) leads to a low saturation intensity, while the fast mechanisms b) and c) lead to a short ulse. Examles 8.8

74 The gain oerator Saturable absorber ML theory 75 ˆ Tg 1 gl 1 d dt The cavity loss oerator (without absorber) Tˆ 1T L The absortion coefficient of the absorber sa 1 / sa sa I Is 1 I / I The absorber oerator Tˆ sa ex 1 l sa sa 1 l 1 I / I s l sa sa sa sa s

75 Saturable absorber ML theory 76 The self reroduction requires Tˆ Tˆ Tˆ A( t) A( t) sa g d 1 salsa 1 I / Is 1T L1 gl 1 A( t) A( t) dt d I gl 1 T L salsa 1 A( t) dt I s The solution is A At () cosh( t / ) The ulse reetition time 1/ 1/ g I s sa A The ulse duration 1.76

76 Mode-locked laser characteristics 77

77 Homework 78 Page 37:

78 About the scores 79 Homework: 1 scores Including six chaters (6 scores), and three simulations (Chater =1 score, Chater 7=1.5 score, and Chater 8=1.5 score) Project: 1 scores Including 3-age reort, and oral resentation (English). Exeriment: 1 scores Final exame: 7 scores