Pulsed Lasers Revised: 2/12/14 15: , Henry Zmuda Set 5a Pulsed Lasers

Transcription

1 Pulsed Lasers Revised: 2/12/14 15: , Henry Zmuda Set 5a Pulsed Lasers 1

2 Laser Dynamics Puled Lasers More efficient pulsing schemes are based on turning the laser itself on and off by means of an internal modulation process, designed so that energy is stored during the off-time and released during the on-time. Energy may be stored either in the resonator, in the form of light that is periodically permitted to escape, or in the atomic system, in the form of a population inversion that is released periodically by allowing the system to oscillate. These schemes permit short laser pulses to be generated with peak powers far in excess of the constant power deliverable by CW lasers. 2014, Henry Zmuda Set 5a Pulsed Lasers 2

3 Laser Dynamics Puled Lasers Four common methods used for the internal modulation of laser light are: gain switching, Q-switching, cavity dumping, and mode locking. These are considered in turn. 2014, Henry Zmuda Set 5a Pulsed Lasers 3

4 Laser Dynamics Puled Lasers Gain Switching In this rather direct approach, the gain is controlled by turning the laser pump on and off. The pump is well above threshold and almost instantaneously creates a population inversion though the laser oscillation requires time to establish itself. During the on-times, the gain coefficient exceeds the loss coefficient and laser light is produced. Most pulsed semiconductor lasers are gain switched because it is easy to modulate the electric current used for pumping. 2014, Henry Zmuda Set 5a Pulsed Lasers 4

5 Laser Dynamics Gain Switching From: Fundamentals of Photonics, Saleh and Teich, Wiley, p , Henry Zmuda Set 5a Pulsed Lasers 5

6 Laser Dynamics Gain Switching Turning the pumping rate R on and off is equivalent to modulating the small-signal population difference. The following regimes are evident in the previous figure: For t < 0, the population difference lies below the threshold and oscillation cannot occur. The pump is turned on at t = 0, which increases the population from a value below threshold to a value above threshold in step-function fashion. The population difference N(t) begins to increase as a result. 2014, Henry Zmuda Set 5a Pulsed Lasers 6

7 Laser Dynamics Gain Switching As long as N(t) < N th, the normalized photon-number density P is negligible. Then N(t) grows exponentially toward its equilibrium value. Once N(t) crosses the threshold N th, laser oscillation begins and P increases. The population inversion then begins to deplete so that the rate of increase of N(t) slows. As P becomes larger, the depletion becomes more effective so that N(t) begins to decay toward its steady-state value. The pump is then turned off which reduces the population to its initial value and N(t)) decay to the initial values. 2014, Henry Zmuda Set 5a Pulsed Lasers 7

8 Laser Dynamics Gain Switching Encyclopedia of Laser Physics and Technology, , Henry Zmuda Set 5a Pulsed Lasers 8

9 Laser Dynamics Case e) Pulsed excitation or Gain Switching - Analysis Start with a laser at rest (no excitation the pump is a pulse) with zero photons excited by a pump whose amplitude is well above that required to exceed threshold on a steady-state basis. How does the laser amplitude develop with time? Assume that the pulsed pump is large enough to make the relative photon number P grow from an insignificant value (as in case a) to something appreciable during its duration. Recall: w w sat = I I sat = P ( ~ 0.1 for this example) 2014, Henry Zmuda Set 5a Pulsed Lasers 9

10 Laser Dynamics e) Gain Switching Under this assumption, dg 0 dt = a R g 1+ P = a R g This has solution (case a): g( t) = R e 1 e at R e, t >> a or t >> τ 2 This suggests that the gain can perhaps exceed threshold before the photons have time to build up from the spontaneous emission level to a level where they utilize the population inversion. 2014, Henry Zmuda Set 5a Pulsed Lasers 10

11 Laser Dynamics e) Gain Switching Now for the photons, assume that the gain g has reached a value in excess of g th but can be treated as as if it were a constant. If we to include the fact that g is a really a function of P the the equation is nonlinear. dp dt = ( S exp [ g ] 1)P + βg constant g t Neglect this. Why? P t ( [ ] 1)t = δ P o exp S exp g If δp o is small enough so as to keep P small (as in case a) the nonlinearities can be ignored. For this case the exponential factor must be large to make P comparable to unity. 2014, Henry Zmuda Set 5a Pulsed Lasers 11

12 Laser Dynamics e) Gain Switching If δ P o = 10 7, P = 0.1 requires, P = 0.1 = 10 7 exp ( S exp[ g] 1)t exp ( S exp[ g] 1)t = 106 ( S exp[ g] 1)t = ln 10 6 = 13.8 If the gain is just above threshold, for instance if g = 1.1g th and if g th is small (~ 0.1), then the time needed to reach oscillation is: ln( 10 ( S exp[ 6 ) g] 1)Δt = ln( 10 6 ) 13.8 Δt = ~ S exp[ g] 1 exp[ g g th ] g g th 1 = 13.8 = g g th S=exp[ g th ] 13.8 = 13.8 = g th g th 0.1g th 0.01 = , Henry Zmuda Set 5a Pulsed Lasers 12

13 Laser Dynamics e) Gain Switching For this example, it will take a normalized time of 1380 (the number of round trips) before the laser reaches an amplitude that is beginning to be interesting then the nonlinearities kick in (our next step). If the cavity length was 12 inches and n = 1, then τ RT = 1 ns and the buildup time is 1.38 µs. Obviously, the pumping must be maintained for a long enough time to permit this evolution. If we "seed" the cavity with a reasonable value of photonssay from another laser-we can speed up this process and shrink the factor of 1380 down to a smaller value by starting with a larger value of δp o. 2014, Henry Zmuda Set 5a Pulsed Lasers 13

14 Laser Dynamics e) Gain Switching Once P reaches an appreciable level nonlinearities must be considered. Recall the fundamental equation we are dealing with: dp dt = ( Seg 1)P + βg dp dt = ( ( eg t) g th 1)P + βg t dp dt = ( ( eg t) g th 1)P dg dt = a R g 1+ P dg( t) dt = a R e g t 1+ P( t) Neglect. 2014, Henry Zmuda Set 5a Pulsed Lasers 14

15 Laser Dynamics e) Gain Switching These must be solved numerically: dp dt = ( ( eg t) g th 1)P dg t = a R e g t dt 1+ P( t) Normalize the time as: T = t τ p, τ p photon lifetime = τ RT ( 1 S), S = e g th 2014, Henry Zmuda Set 5a Pulsed Lasers 15

16 Laser Dynamics e) Gain Switching Normalization: T = t τ p, τ p = τ RT ( 1 S), S = e g th dp( t τ ) p τ p d t τ p dg( t τ ) p τ p d t τ p dp( T ) dt dg( T ) ( = e g ( t τ p) g th ) 1 P t τ p = a R e g t τ p = τ RT e g T dt = aτ p = b = τ p τ 2 dp T 1+ P( t τ ) p g th 1 1 e g th P T R e g T 1+ P( T ) dt = τ ( p e g ( T ) g th 1)P T dg T dt = aτ p R e g T 1+ P( T ) 2014, Henry Zmuda Set 5a Pulsed Lasers 16

17 Laser Dynamics e) Gain Switching Numerical Solution: Slide57: P b = R b 1 g th R e R th = 4 P CW = 3, b = 0.5 τ 2 = 20τ p g th = 0.1 S = , Henry Zmuda Set 5a Pulsed Lasers 17

18 Laser Dynamics e) Gain Switching For T > 9, the relative photon number P becomes larger than 1 and begin to depress the population inversion (or gain) and becomes larger than the CW value by a factor of ~ 6 at T ~ 12. (The CW value is shown as the dashed line at ~ 19.3.) With the number of photons in the cavity large, stimulated emission reduces the gain and the photon number reaches a maximum when the gain crosses threshold, shown as the dashed line at , Henry Zmuda Set 5a Pulsed Lasers 18

19 Laser Dynamics e) Gain Switching There are still a large number of photons in the cavity and they continue to stimulate the atoms and thus reduce the gain to below the threshold and as a consequence the photon number decreases to below the CW value. This allows the gain to re-establish itself and leads to a secondary peak in the photons. Eventually, both the photons and the gain settle down to their CW values. This large initial pulse is sometimes referred to as a gain switched pulse and occurs because the gain can build up faster than the photons. 2014, Henry Zmuda Set 5a Pulsed Lasers 19

20 Laser Dynamics e) Gain Switching Depending upon the choice of the parameters, we can have one or more significant pulses with the initial one being intense. If we could build up the inversion to a larger value than was found here, for example by making the initial seed of the photons even smaller, we would expect a larger initial pulse. Techniques for accomplishing this and modifications to our model are covered in the next on Q switching. 2014, Henry Zmuda Set 5a Pulsed Lasers 20

21 Laser Dynamics Q Switching Giant Pulses Basic Pulse Forming Network: RF Radar Pulse Generator Camera Flash Circuit Electronic auto ignition etc. R C Pump V Battery Switch Q Switch R L Cavity Impedance Level Shift 2014, Henry Zmuda Set 5a Pulsed Lasers 21

22 Laser Dynamics Q Switching Giant Pulses For circuits such as this the peak power delivered to the load can be many times the average power extracted from the source. Similar ideas apply for the giant pulse operation of a laser. Energy can be stored for future use by creating a population inversion. Obviously the spontaneous emission represents a drain on the stored energy, just as in leakage through the capacitor. 2014, Henry Zmuda Set 5a Pulsed Lasers 22

23 Laser Dynamics Q Switching Giant Pulses Here however, spontaneous emission causes another difficulty; it is amplified by the population inversion, and if the round trip gain exceeds 1, it will build up to a steady state value whose intensity is limited by the rate at which energy can be pumped into the system. This must be stopped from happening. To do this, the laser is prevented from oscillating by making the loss per pass very high while pumping the system. If amplified spontaneous emission can be prevented from saturating the active medium with a single-pass gain length, then considerable energy can be stored in the population difference N 2 N , Henry Zmuda Set 5a Pulsed Lasers 23

24 Laser Dynamics Q Switching Giant Pulses This stored energy can be extracted by suddenly lowering the loss. Then the gain greatly exceeds the loss and the intensity rapidly builds up from spontaneous noise, reaching a level where further growth is impossible (i.e., when the gain per pass equals the loss per pass). The result in an intense pulse. The intensity at which further growth is impossible represents the energy stored in the initial population inversion, not the intensity given by the CW oscillation condition. As we will see, the peak intensity can be many times the CW level. 2014, Henry Zmuda Set 5a Pulsed Lasers 24

25 Laser Dynamics Q Switching Giant Pulses Consider a simple laser where a shutter has been added to spoil the cavity Q. T g T g I Gain I R + R 1 2 Shutter Pump We continually pump energy into the population inversion until some sort of equilibrium is reached between the pump and the spontaneous decay processes of the system. This initial population inversion may be many times that required for CW oscillation in the absence of the shutter. 2014, Henry Zmuda Set 5a Pulsed Lasers 25

26 Laser Dynamics Q Switching Giant Pulses Consider a simple laser where a shutter has been added to spoil the cavity Q. T g T g I Gain I R + R 1 2 Shutter opens at t = 0 Pump The shutter opens at t = 0, and we (almost instantly) have a system that is far above threshold, so the spontaneous emission along the axis of the cavity is greatly amplified and builds up to a value sufficiently strong to start depleting the population inversion. All this can happen very quickly. 2014, Henry Zmuda Set 5a Pulsed Lasers 26

27 Laser Dynamics Q Switching Giant Pulses Suppose that the net single-pass gain is 5: R 1 R 2 e γ o = 5 After only five round trips the photon flux would be amplified by a factor of 5 10 = , thus we can neglect any pumping that occurs after t = 0. Because of this large increase in photon flux, the population inversion will become depleted as the photon number increases. Note that the optical power reaches a peak when the inversion crosses the threshold value. Shutter opens at t = , Henry Zmuda Set 5a Pulsed Lasers 27

28 Laser Dynamics Q Switching The math Time dynamics: In one round trip the number of photons in the cavity will increase by a factor of: e 2 ( N 2 N 1 )σ g as well as decrease due to attenuation and transmissivity with a factor of: R 1 R 2 T s 2 T g 2 e 2α s s = e 2g th Where g th is the threshold gain. T s T g T g σ, n g I α s,n s Gain Section Length g I R + R 1 2 Shutter Thus: ΔN p Δt Pump = e 2g th ( e2 N 2 N 1 )σ g 1 N 2 dn p τ RT dt This approximation for the derivative requires that the change in N p is small in a round-trip time interval 2014, Henry Zmuda Set 5a Pulsed Lasers 28

29 Laser Dynamics Q Switching The math Let = N 2 ( t) N 1 ( t) g t σ g be the line-integrated gain, hence dn p dt = e 2g th e2g 1 τ RT N p Also recall the photon lifetime for the passive cavity: τ p = τ RT 1 e 2g th Thus dn p dt = e 2g th e2g 1 1 e 2g th N p τ p ( = e2 g gth ) 1 1 e 2g th N p τ p 2014, Henry Zmuda Set 5a Pulsed Lasers 29

30 Laser Dynamics Q Switching The math Since the variation in N p is small over one round-trip, dn p dt ( = e2 g g ) th 1 1 e 2g th N p τ p 1+ 2 ( g g th ) g th e N p τ p = g g th g th N p τ p = N p τ p g 1 g th, = N 2 ( t) N 1 ( t) g t σ g τ p = τ RT 1 e 2g th τ RT 2g th, = N 2 ( t) N 1 ( t) n t A g Total number of inverted atoms interacting with an optical mode of cross sectional area A This all gives, dn p dt = N p τ p n 1 n th (very intuitive) 2014, Henry Zmuda Set 5a Pulsed Lasers 30

31 Laser Dynamics Q Switching The math Now study the time dynamics. Recall that while stimulated emission increases the photon count it also depletes the inversion with a 2-for1 split. Usually the time scale for the build-up or decay of the photons is on the order of a few photon lifetimes, and this time is much shorter than the lifetime of state 2 atoms and/or the characteristic time for pumping. Because of this we can examine the change in populations caused by stimulated emission only ignoring other causes. For stimulated emission we have Similarly, dn 2 dt dn 1 dt = σ I + + I hf = + σ ( I + + I ) hf ( N 2 N 1 ) ( N 2 N 1 ) 2014, Henry Zmuda Set 5a Pulsed Lasers 31

32 Laser Dynamics Q Switching The math Subtract, d( N 2 N 1 ) dt = n t d A( N 2 N 1 ) g dt dn( t) dt Number of photons in cavity: = 2 σ ( I + + I ) hf ( N 2 N 1 ) ( = 2 I + + I )A = g t σ g ( N 2 N 1 ) hf ( = 2 I + + I )A hf g( t) N p = I τ RT 2 Area 1 hf 2014, Henry Zmuda Set 5a Pulsed Lasers 32

33 Laser Dynamics Q Switching The math N p = I τ RT 2 Area 1 hf dn( t) dt ( = 2 I + + I ) A τ RT hf 2 g( t) 2 τ RT N p dn( t) dt dn( t) dt = 2N p g( t) 2, τ RT = 4N p g( t) = 2 N p τ p n t n th τ p τ RT 2g th 1 = 2 N p τ p 2g th τ p g( t) g th = 2 N p τ p n( t) n th 2014, Henry Zmuda Set 5a Pulsed Lasers 33

34 Laser Dynamics Q Switching The math dn( t) dt dn t τ p dt dn( T ) dt = 2 N p τ p and from Slide 107, n( t) n th = 2N p n t = 2N p n th dn t d t τ p Remember, N p is the total number of photons in the cavity while n(t) is the total number of inverted atoms interacting with an optical mode of cross sectional area A. n( T ), T = t τ p n th dn p dt = N p = 2N p n 1 n th n( t) n th 2014, Henry Zmuda Set 5a Pulsed Lasers 34

35 Laser Dynamics Q Switching The math dn( T ) dt dn p dt = N p = 2N p These equations are nonlinear and require a numerical solution. Let s take a different approach. n( T ) n th n 1 n th 2014, Henry Zmuda Set 5a Pulsed Lasers 35

36 Laser Dynamics Q Switching The math The chain rule: Integrate: dn p dt = dn p dn 2N p dn p dn = 1 2 dt dn T = 2 N p dn dt n( T ) n th n n th dn p dn = N p n th n n = N p 1 n th n 1 n th n 1 n th = 1 2 n th n 1 N p max dn p = N p initial 0 n th n initial 1 2 n th n 1 dn 2014, Henry Zmuda Set 5a Pulsed Lasers 36

37 Laser Dynamics Q Switching The math Integrate: N p max dn p = N p initial 0 n th n initial 1 2 n th n 1 dn N p max = 1 2 n th n initial n th = n initial n th 2 n 1 dn = n th n initial 2 n th 2 ln n initial + n th 2 lnn n th n initial Now if the cavity contains N p t photons, then it stores an energy of hfn p ( t) joules, and that power is being lost through the various losses in the cavity, Q-switch, mirrors, interfaces, etc. n th 2014, Henry Zmuda Set 5a Pulsed Lasers 37

38 Laser Dynamics Q Switching The math The output power equals the stored energy times the fraction lost per round-trip divided by the time for a round-trip. P out = hfn p hfn p fraction lost through coupling per round trip ( t) τ RT ( t) coupling loss per round trip total loss per round trip coupling efficiency η coupling < 1 total loss per round trip τ RT ( photon lifetime) 1 If the output mirror is R 2, then (recall the photon survival factor S) P out = hfn p ( t) 1 R 2 1 = N p ( t) hfη coupling P max out = N hfη max coupling p 1 S τ p τ p τ p η coupling We ve determined the maximum output power. 2014, Henry Zmuda Set 5a Pulsed Lasers 38

39 Laser Dynamics Q Switching The math Let us now estimate the pulse width. 0 = n final n initial 1 2 n th n 1 dn N p final 0 0 = dn p = N p initial 0 n final n initial 1 2 n th n 1 dn = n initial n final 2 n th 2 ln n initial n final 2014, Henry Zmuda Set 5a Pulsed Lasers 39

40 Laser Dynamics Q Switching The math Numerical solution: 0 = n i n f 2 ln n i n f n th 2 ln n i = n n i f n th n f n f = n i exp n n i f n th 2014, Henry Zmuda Set 5a Pulsed Lasers 40

41 Laser Dynamics Q Switching The math How much of the initial inversion is converted to photons? The fraction: η external = n i n f n i The total energy generated in the form of photons is this fraction times the maximum available. From Slide 107: but: n i = N 2i n f = N 2 f N 2i N 2 f N 1i N 1f A g A g A = N = N g p 1 f N 1i A g 2014, Henry Zmuda Set 5a Pulsed Lasers 41

42 Laser Dynamics Q Switching The math Subtracting: n f n i = N 2 f N 1f = N 1f N 1i A N g 2 i N 1i ( N 2i N ) 2 f A g A g = 2N p N p = n i n f 2 Photon Energy: using: η external = n n i f n i Output Energy: W p = hfn p = hf n i n f 2 W out = hfη external η coupling n i = hfη external n i 2014, Henry Zmuda Set 5a Pulsed Lasers 42

43 Laser Dynamics Q Switching The math A reasonable estimate for the pulse width is: Δt W out P = hfη externalη coupling n i max out N hfη max coupling p τ p n = η external τ i p N = η N 2i N 1i max externalτ p p N p max A g This number can be quite small. 2014, Henry Zmuda Set 5a Pulsed Lasers 43

44 Laser Dynamics Q Switching Example A Ruby Laser T a = 0.98 T b = cm A = 0.8cm 2 2cm 10 cm R 1 = 0.99 n s = 2.7 Shutter α s = 0.1 cm 1 (when open) n g = 1.78 σ = cm 2 T c = 0.96 T d = 0.95 λ = 694.3µm Assume pumping to four times threshold and equal degeneracy states. n i g = 4 1 = 1 n Round-Trip Gain: th g 2 R 2 = = R 1 R 2 ( T a T b T c T d ) 2 e 2α s s e 2γ th g 2014, Henry Zmuda Set 5a Pulsed Lasers 44

45 Laser Dynamics Q Switching Example A Ruby Laser Solve for the threshold gain & inversion: 1 = R 1 R 2 ( T a T b T c T d ) 2 e 2α s s e 2γ th g 1 R 1 R 2 ( T a T b T c T d ) = 2 e2γ th g 2αs s 1 ln 2 g R 1 R T a T b T c T d + α s s = γ th g γ th = 1 1 ln 2 g R 1 ( T a T b T c T d ) 2 + α s 1 s + ln 1 g 2 g R 2 Cavity Losses Shutter Loss output coupling loss = γ th = ln ( 0.98) ( 0.97) ( 0.96) ( 0.95) = = 4.60 cm 1 γ th = ( N 2 N 1 ) th σ ( N 2 N 1 ) th = γ th σ = ln cm cm 2 = cm , Henry Zmuda Set 5a Pulsed Lasers 45

46 Laser Dynamics Q Switching Example A Ruby Laser Number of inverted atoms at threshold in the cavity: n th = ( N 2 N 1 ) th A g = ( 0.8) ( 10) = Initial inversion: n i = 4n th = Round-Trip time: 1 c = τ RT = 2 air + n s s + n g g Photon lifetime (passive cavity): 1 ( 2 + ( 1.78)10) = 1.75ns τ p = 1 R 1 R 2 ( T T T T a b c d ) 2 e 2α s s ( 2) = e τ RT 1.75 = τ p = 2.91ns 2014, Henry Zmuda Set 5a Pulsed Lasers 46

47 Laser Dynamics Q Switching Example A Ruby Laser Maximum (Peak) Power: P max out = N hfη max coupling p τ p = = watts S = R 1 R 2 η coupling = 1 R 2 1 S N p max = n initial n th 2 = 4n th n th 2 ( T a T b T c T d ) 2 e 2α s s = 0.595e 2 = = n th 2 ln n initial n th n th 2 ln 4n th n th = 3 2 ln 4 2 = n th = n th = , Henry Zmuda Set 5a Pulsed Lasers 47

48 Laser Dynamics Q Switching Example A Ruby Laser Output Energy: W out = hf η external 0.98 viamathcad = P max out = watts η couplingn i Approximate Pulse Width: Δt = W out P = 0.55 max out = 14.1ns = 0.55 joule 2014, Henry Zmuda Set 5a Pulsed Lasers 48

49 Laser Dynamics Q Switching Example A Ruby Laser More exact results require a numerical solution of the nonlinear differential equations. ΔT = N p initial = n i n th 10 4 Less initial photons means that more time is required to establish the pulse. N p initial = n i n th 10 3 Note that as the inversion ratio is increased, the time required to develop a more intense pulse is decreased. This is because: 1. With a higher inversion, the gain is higher, and thus the rate of growth is faster. 2. A higher inversion ratio means more atoms in the upper state which in turn will contribute more spontaneous power to seed the stimulated emission. 2014, Henry Zmuda Set 5a Pulsed Lasers 49

50 Laser Dynamics Q Switching Example A Ruby Laser Less initial photons means that more time is required to establish the pulse. How much more? If the initial number of photons is decreased by a factor of, say 10, the medium must use time ΔT to amplify the smaller number back to the original value. At these small photon numbers we can neglect eh depletion of the inversion and N p grows according to (slide 111): dn p dt = N p n 1 n th N p ( T ) = N p 0 e + n 1 n th T 2014, Henry Zmuda Set 5a Pulsed Lasers 50

51 Laser Dynamics Q Switching Example A Ruby Laser N p ( T 1 ) N p ( T 2 ) = 0 = N p1 0 N p 1 0 N p2 0 ΔT = = e+ n 1 n th n 1 n th T 2 T 1 1 e + ln N p 1 0 N p2 0 n 1 n th T 1 N p2 ( 0)e + n 1 n th T 2 ΔT = ( 6 1) 1 ln[ 10] = , Henry Zmuda Set 5a Pulsed Lasers 51

52 Laser Dynamics Q Switching Example A Ruby Laser We cannot increase the initial inversion too much and obtain arbitrarily high peak powers. Two reason why: 1. As the initial inversion is increased, the spontaneous emission rate increases proportionately, which is then amplified by the remaining section of the gain medium. Since the single-pass gain is very high (because of the high n i ), the amplified spontaneous emission may be sufficient to saturate the gain section. Indeed, this presents a fundamental limit on the amount of energy that can be stored in the population inversion in any laser. 2. The amplified spontaneous emission may be sufficient to damage the switch or at least change its characteristics. 2014, Henry Zmuda Set 5a Pulsed Lasers 52

53 Laser Dynamics NEXT Mode Locking 2014, Henry Zmuda Set 5a Pulsed Lasers 53