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CHAPTER 4----LASER Chapter 4 Laser Fundamentals of Photonics 2017/5/24 1 CHAPTER 4----LASER LASERS In 1958 Arthur Schawlow, together with Charles Townes, showed how to extend the principle of the maser to the optical region. He shared the 1981 Nobel Prize with Nicolaas Bloembergen. Maiman demonstrated the first successful operation of the ruby laser in 1960. Fundamentals of Photonics 2017/5/24 2 CHAPTER 4----LASER LASERS an oscillator is an amplifier with positive feedback Two conditions for an oscillation: 1. Gain greater than loss: net gain 2. Phase shift in a round trip is a multiple of 2π Fundamentals of Photonics 2017/5/24 3 CHAPTER 4----LASER Stable condition: gain = loss Gain Loss 0 Power Steady-state power If the initical amplifier gain is greater than the loss, oscillation may initiate. The amplifier then satuates whereupon its gain decreases. A steady-state condition is reached when the gain just equals the loss. An oscillator comprises: ◆ An amplifier with a gain-saturation mechanism ◆ A feedback system ◆ A frequency-selection mechanism ◆ An output coupling scheme Fundamentals of Photonics 2017/5/24 4 CHAPTER 4----LASER Light amplifier with positive feedback Pout gPin Pin Gain medium (e.g. 3level system w population inversion) When the gain exceeds the roundtrip losses, the system goes into oscillation + g + Fundamentals of Photonics 2017/5/24 5 CHAPTER 4----LASER LASERS 点击查看flash动画 Amplified once Initial photon Gain medium (e.g. 3-level system w population inversion) Reflected Output Amplified twice Reflected Amplified Again Light Amplification through Stimualted Emission Radiation Fundamentals of Photonics 2017/5/24 Partially reflecting Mirror 6 CHAPTER 4----LASER Mirror Active medium Partially transmitting mirror d Laser output A laser consists of an optical amplifier (employing an active medium) placed within an optical resonator. The output is extracted through a partially transmitting mirror. Fundamentals of Photonics 2017/5/24 7 CHAPTER 4----LASER Optical amplification and feedback ★ Gain medium The laser amplifier is a distributed-gain device characterized by its gain coefficient 2 0 ( ) N 0 ( ) N 0 g ( ) 8 tsp 5.1-43 Small signal Gain Coefficient 0 ( ) ( ) 1 / s ( ) 5.1-42 Saturated Gain Coefficient Fundamentals of Photonics 2017/5/24 8 CHAPTER 4----LASER 0 ( ) ( ) Phase-shift Coefficient (Lorentzian Lineshape) Figure 5.1-5 Spectral dependence of the gain and phase-shift coefficients for an optical amplifier with Lorentzian lineshape function Fundamentals of Photonics 2017/5/24 9 CHAPTER 4----LASER Optical Feedback-Optical Resonator Feedback and Loss: The optical resonator A Fabry-Perot resonator, comprising two mirrors separated by a distance d, contains the medium (refractive index n). Travel through the medium introduces a phase shift per unit length equal to the wavenumber k 2 c In traveling a round trip through a resonator of length d, the photon-flux density is reduced by the factor R1R2exp(-2asd). The overall loss in one round trip can therefore be described by a total effective distributed loss coefficient ar, where exp(2a r d ) R1R2 exp(2a s d ) Fundamentals of Photonics 2017/5/24 10 CHAPTER 4----LASER Loss coefficient a r a s a m1 a m 2 a m1 1 1 ln 2d R1 am2 1 1 ln 2d R2 a m a m1 a m 2 Photon lifetime 1 1 ln 2d R1 R2 1 p arc ar represents the total loss of energy (or number of photons) per unit length, arc represents the loss of photons per second Fundamentals of Photonics 2017/5/24 11 CHAPTER 4----LASER q q F , q 1, 2,..., F F , F c / 2d F 2 p F ar d F Resonator response c 2d q 1 q q 1 Resonator modes are separated by the frequency F c / 2d and have linewidths F / F 1/ 2 p. Fundamentals of Photonics 2017/5/24 12 CHAPTER 4----LASER Conditions for laser oscillation Gain condition: Laser threshold 0 ( ) a r Threshold Gain Condition N 0 Nt where ar Nt ( ) or Nt 1 c p ( ) 8 tsp 1 Nt 2 c p g ( ) Fundamentals of Photonics 2017/5/24 Threshold Population Difference 13 CHAPTER 4----LASER For a Lorentzian lineshape function, as g ( 0 ) 2 / 2 2 tsp Nt 2 c p If the transition is limited by lifetime broadening with a decay time tsp 2a r 2 Nt 2 2 c p As a numerical example, if 01 mm, p=1 ns, and the refractive index n=1, we obtain Nt=2.1×107 cm-3 Fundamentals of Photonics 2017/5/24 14 CHAPTER 4----LASER Conditions for laser oscillation(2) Phase condition: Laser Frequencies 2kd 2 ( )d 2 q, q 1, 2…… Frequency Pulling c 0 ( ) q 2 or c 0 q ( ) 2 q' q c q 0 q ( q ) 2 ' q Fundamentals of Photonics 2017/5/24 15 CHAPTER 4----LASER q ( q 0 ) ' q Fundamentals of Photonics 2017/5/24 Laser Frequencies 16 CHAPTER 4----LASER The laser oscillation frequencies fall near the coldresonator modes; they are pulled slightly toward the atomic resonance central frequency 0. Fundamentals of Photonics 2017/5/24 17 CHAPTER 4----LASER Characteristics of the laser output Internal Photon-Flux Density Gain Clamping 0 ( ) /[1 / s ( )] a r Fundamentals of Photonics 2017/5/24 18 CHAPTER 4----LASER Laser turn-on Time 0 ( ) Steady state ar Loss coefficient (u) Gain coefficient 0 s ( ) 10 s ( ) 10s ( ) Photon-flux density Determination of the steady-state laser photon-flux density . The smaller the loss, the greater the value of . Fundamentals of Photonics 2017/5/24 19 CHAPTER 4----LASER Steady Photon Density 0 ( ) 1], 0 ( ) a r ar 0, 0 ( ) a r s ( )[ Since 0 ( ) N0 ( ) s ( )( and N0 1), N 0 N t Nt 0, N 0 N t Fundamentals of Photonics a r Nt ( ) 2017/5/24 Steady-State Laser Internal Photon-Flux Density 20 CHAPTER 4----LASER Steady-state values of the population difference N, and the laser internal photon-flux density , as functions of N0 (the population difference in the absence of radiation; N0, increases with the pumping rate R). Output photon-flux density 0 Optical Intensity of Laser Output Fundamentals of Photonics 2017/5/24 T 2 h T I0 2 21 CHAPTER 4----LASER Optimization of the output photon-flux density 1 1 1 ln ln(1 T ) 2d R1 2d From a m1 We obtain 1 a r a s a m 2 ln(1 T ) 2d g0 1 0 sT [ 1], g0 2 0 ( )d , L 2(a s a m 2 )d 2 L ln(1 T ) When, T 1 use the approximation Then ln(1 T ) T Top ( g0 L)1/ 2 L Fundamentals of Photonics 2017/5/24 22 CHAPTER 4----LASER Fundamentals of Photonics 2017/5/24 23 CHAPTER 4----LASER Spectral Distribution Determined both by the atomic lineshape and by the resonant modes M B F Number of Possible Laser Modes Linewidth ? Schawlow-Townes limit Fundamentals of Photonics 2017/5/24 24 CHAPTER 4----LASER 0 ( ) Gain ar Loss B 0 F Resonator modes ……M 1 2 Allowed modes Figure 5.3-3 (a) Laser oscillation can occur only at frequencies for which the gain coefficient is greater than the loss coefficient (stippled region). (b) Oscillation can occur only within of the resonator modal frequencies (which are represented as lines for simplicity of illustration ). Fundamentals of Photonics 2017/5/24 25 CHAPTER 4----LASER Homogeneously Broadened Medium ( ) Fundamentals of Photonics 0 ( ) 1 i 1 j / s ( j ) M 2017/5/24 点击查看flash动画 26 CHAPTER 4----LASER Inhomogeneously Broadened Medium 0 ( ) ( ) 点击查看flash动画 s c 2d ar Frequency q 1 q q 1 (a) (b) Figure 5.3-6 (a) Laser oscillation occurs in an inhomogeneously broadened medium by each mode independently burning a hole in the overrall spectral gain profile. (b) Spectrum of a typical inhomogeneously broadened multimode gas laser. Fundamentals of Photonics 2017/5/24 27 CHAPTER 4----LASER Hole burning in a Doppler-broadened medium 点击查看flash动画 Fundamentals of Photonics 2017/5/24 28 CHAPTER 4----LASER Hole burning in a Doppler-broadened medium 点击查看flash动画 Fundamentals of Photonics 2017/5/24 29 CHAPTER 4----LASER Spatial distribution and polarization Spatial distribution x,y Spherical mirror Spherical mirror Laser intensity The laser output for the (0,0) tansverse mode of a sphericalmirror resonator takes the form of a Gaussian beam. Fundamentals of Photonics 2017/5/24 30 CHAPTER 4----LASER 0,0 1,1 TEM0,0 a11 a00 B11 B00 (1,1) modes (0,0) modes Laser output TEM1,1 Figure 5.3-8 The gains and losses for two transverse modes, say (0,0) and (1,1), usually differ because of their different spatial distributions. Fundamentals of Photonics 2017/5/24 31 CHAPTER 4----LASER Two Issues: Polarization, Unstable Resonators Fundamentals of Photonics 2017/5/24 32 CHAPTER 4----LASER Mode Selection Selection of 1. Laser Line 2. Transverse Mode 3. Polarization 4. Longitudinal Mode Fundamentals of Photonics 2017/5/24 33 CHAPTER 4----LASER High reflectance mirror Output mirror Laser output Active medium Prism Aperture Unwanted line Figure 5.3-9 A paticular atomic line may be selected by the use of a prism placed inside the resonator. A transverse mode may be selected by means of a spatial aperture of carefully chosen shaped and size. Fundamentals of Photonics 2017/5/24 34 CHAPTER 4----LASER Brewster window Active midum Brewster window Polarized laser output qB High reflectance mirror Output mirror Figure 5.3-10 The use of Brewster windows in a gas laser provides a linearly polarized laser beam. Light polarized in the plane of incidence (the TM wave) is transmitted without reflection loss through a window placed at the Brewster angle. The orthogonally polarized (TE) mode suffers reflection loss and therefore does not oscillate. Fundamentals of Photonics 2017/5/24 35 Selection of Longitudinal Mode CHAPTER 4----LASER Etalon High reflectance mirror Active midum d1 d Output mirror Resonator loss c/2d Resonator mdoes Etalon mdoes c/2d1 Laser output Figure 5.3-11 Longitudianl mode selection by the use of an intracavity etalon. Oscillation occurs at frequencies where a mode of the resonator coincides with an etalon mode; both must, of course, lie within the spectral window where the gain of the medium exceeds the loss. Fundamentals of Photonics 2017/5/24 36 CHAPTER 4----LASER Multiple Mirror Resonators (a) (b) (c) Figure 5.3-12 Longitudinal mode selection by use of (a) two coupled resonators (one passive and one active); (b) two coupled active resonators; (c) a coupled resonator-interferometer. Fundamentals of Photonics 2017/5/24 37 CHAPTER 4----LASER Characteristics of Common Lasers Solid State Lasers: Ruby, Nd3+:YAG, Nd3+:Silica, Er3+:Fiber, Yb3+:Fiber Gas Lasers: He-Ne, Ar+; CO2, CO, KF; Liquid Lasers: Dye Plasma X-Ray Lasers Free Electron Lasers Fundamentals of Photonics 2017/5/24 38 CHAPTER 4----LASER Fundamentals of Photonics 2017/5/24 39 CHAPTER 4----LASER Pulsed Lasers Method of pulsing lasers External Modulator or Internal Modulator? Modulator Modulator Average power CW power t t (a) (b) Figure 5.4-1 Comparison of pulsed laser outputs achievable with (a) an external modulator, and (b) an internal modulator 1. Gain switching 2. Q-Switching 3. Cavity Dumping 4.Mode Locking Fundamentals of Photonics 2017/5/24 40 CHAPTER 4----LASER Gain Switching Fundamentals of Photonics 2017/5/24 41 CHAPTER 4----LASER Q- Switching Modulator Loss Gain t Laser output t t Figure 5.4-3 Q-switching. Fundamentals of Photonics 2017/5/24 42 CHAPTER 4----LASER Cavity Dumping Figure 5.4-4 Cavity dumping. Fundamentals of Photonics 2017/5/24 43 CHAPTER 4----LASER Mode locking • Laser modes coupling together • Lock their phases to each other CHAPTER 4----LASER Rate equation for the photon-number density dn n NWi dt p p photon lifetime Wi ( ) cn ( ) From ( ) 1/ c p N t We have Probability density for induced absorption/emission dn n N n dt p Nt p Fundamentals of Photonics 2017/5/24 Photon-Number Rate Equation 45 CHAPTER 4----LASER Rate equation for the Population Difference For a three level system dN 2 N R 2 Wi ( N 2 N1 ) dt tsp Note N1 ( Na N ) / 2, N2 ( Na N ) / 2, N N 2 N1 dN N0 N 2Wi N Then dt tsp tsp Where the small signal population difference Substituting Wi n / N t p We have dN N0 N N n 2 dt tsp tsp Nt p Fundamentals of Photonics 2017/5/24 N 0 2 Rtsp N a Population-difference rate equation (Three-level system) 46 CHAPTER 4----LASER Situation for the gain switching Fundamentals of Photonics 2017/5/24 47 CHAPTER 4----LASER Situation for the Q-switching 点击查看flash动画 Fundamentals of Photonics 2017/5/24 48 CHAPTER 4----LASER Dynamics of a Q-Switching process Dynamics of the Q-switching process Note the time relationship between the photon density and the population inversion variations! Fundamentals of Photonics 2017/5/24 49 CHAPTER 4----LASER Determination of the peak power, energy, width and shape of the optical pulse dn N n ( 1) dt Nt p dN N n 2 dt Nt p Dividing dn 1 N t ( 1) dN 2 N 1 1 n Nt ln( N ) N cons tan t 2 2 Fundamentals of Photonics 2017/5/24 50 CHAPTER 4----LASER Power 1 N 1 n N t ln ( N Ni ) 2 Ni 2 1 c P0 h A0 h cTAn h T Vn 2 2d Peak pulse power np N N N 1 N i (1 t ln t t ) 2 Ni Ni Ni Pp h T c Vn p 2d When Ni>>Nt It is clear np 1 Ni 2 So 1 c Pp h T VNi 2 2d The larger the initial population inversion, the higher the Q-switched pulse peak power. Fundamentals of Photonics 2017/5/24 51 CHAPTER 4----LASER c. Pulse energy: tf Nf c c dt E h T V n(t )dt h T V n(t ) dN t N i i 2d 2d dN Ni dN 1 c E h T VN t p Nf N 2 2d N 1 c E h T VNt p ln i 2 2d Nf The final population difference Nf Ni Ni N f ln Nf Nt 1 c E h T V p ( Ni N f ) 2 2d Fundamentals of Photonics 2017/5/24 52 CHAPTER 4----LASER d. Pulse width: A rough estimation of the pulse width is the ratio of the pulse energy to the peak pulse power. pulse p Ni / Nt N f / Nt Ni / Nt ln( Ni / Nt ) 1 When Ni>>Nth and Ni>>Nf pulse p Fundamentals of Photonics The shorter the photon life time, the shorter the Q-switched pulses. 2017/5/24 53 CHAPTER 4----LASER Fundamentals of Photonics 2017/5/24 54 CHAPTER 4----LASER Techniques for Q-switching 1. Mechanical rotating mirror method: Q-switching principle: rotating the cavity mirror results in the cavity losses high and low, so the Q-switching is obtained. Advantages: simple, inexpensive. Disadvantages: very slow, mechanical vibrations. Fundamentals of Photonics 2017/5/24 55 CHAPTER 4----LASER 2. Electro-optic Q-switching Disadvantages: complicate and expensive Advantages: very fast and stable. Pockels effect: applying electrical field in a uniaxial crystal results in additional birefringence, which changes the polarization of light when passing through it. Q-switching principle: placing an electro-optic crystal between crossed polarizers comprises a Pockels switch. Turning on and off the electrical field results in high and low cavity losses. Fundamentals of Photonics 2017/5/24 56 CHAPTER 4----LASER Electro-optic Q-switch operated at (a) quarter-wave and (b) half-wave retardation voltage Fundamentals of Photonics 2017/5/24 57 CHAPTER 4----LASER 3. Acousto-optic Q-switching Fundamentals of Photonics 2017/5/24 58 CHAPTER 4----LASER Diffracted light Incident light q q L Sound Piezoelectric transducer Transmitted light RF Bragg scattering: due to existence of the acoustic wave, light changes its propagation direction. Q-switching principle: through switching on and off of the acoustic wave the cavity losses is modulated. Advantages: works even for long wavelength lasers. Disadvantages: low modulation depth and slow. Fundamentals of Photonics 2017/5/24 59 CHAPTER 4----LASER 4. Saturable absorber Q-switching What’s a saturable absorber? a a0 1 I Is Absorption coefficient of the material is reversely proportional to the light intensity. Is: saturation intensity. Saturable absorber Q-switching: Insertion a saturable absorber in the laser cavity, the Q-switching will be automatically obtained. Fundamentals of Photonics 2017/5/24 60 CHAPTER 4----LASER General characteristics of laser Q-switching • Pulsed laser output: – Pulse duration – related to the photon lifetime. – Pulse energy - related to the upper level lifetime. • Laser operation mode: – Single or multi-longitudinal modes. • Active verses passive Q-switching methods: – Passive: simple, economic, pulse jitter and intensity fluctuations. – Active: stable pulse energy and repetition, expensive. • Comparison with chopped laser beams: – Energy concentration in time axis. • Function of gain medium – Energy storage Fundamentals of Photonics 2017/5/24 61 CHAPTER 4----LASER Laser mode-locking Aims: 1. Familiarize with the principle of laser mode-locking. 2. Familiarize with different techniques of achieving laser Mode-locking. Outlines: 1. 2. 3. 4. 5. Principle of laser mode-locking. Methods of laser mode-locking. Active mode-locking. Passive mode-locking. Transform-limited pulses. Fundamentals of Photonics 2017/5/24 62 CHAPTER 4----LASER Principle of laser mode-locking 1. Lasing in inhomogeneously broadened lasers: i) Laser gain and spectral hole-burning. ii) Cavity longitudinal mode frequencies. iii) Multi-longitudinal mode operation. Fundamentals of Photonics 2017/5/24 63 CHAPTER 4----LASER 2. Laser multimode operation: Single mode lasers: E (t ) E0 cos0t (t ) N Multimode lasers: E (t ) Ei cos i t i (t ) i 1 Mode-frequency separations: ~ c nL Phase relation between modes: Random and independent! Total laser intensity fluctuates with time ! The mean intensity of a multimode laser remains constant, however, its instant intensity varies with time. Fundamentals of Photonics 2017/5/24 64 CHAPTER 4----LASER 3. Effect of mode-locking: (i) Supposing that the phases of all modes are locked together: i (t ) 0 0 (ii) Supposing that all modes have the same amplitude: Ei E0 purely for the convenience of the mathematical analysis (iii) Under the above two conditions, the total electric field of the multimode laser is: N i i t E (t ) Re Ei e where i 1 c N 1 i 0 i c c 2 L Fundamentals of Photonics 2017/5/24 65 CHAPTER 4----LASER 0 is the frequency of the central mode, N is the number of modes in the laser, c is the mode frequency separation. i is the frequency of the i-th mode. Calculating the summation yields: c t sin N 2 E(t ) E 0 cos 0 t t c sin 2 Note this is the optical field of the total laser Emission ! The optical filed can be thought to consist of a carrier wave of frequency 0 that amplitude modulated by the function sin (Nx ) AN (x ) sin (x ) Fundamentals of Photonics 2017/5/24 66 CHAPTER 4----LASER 4. Characteristics of the mode-locked lasers: The intensity of the laser field is: c t sin 2 N 2 I (t ) E02 c t sin 2 2 The output of a mode-locked laser consists of a series of pulses. The time separation between two pulses is determined by RT and the pulse width of each pulse is tp. Fundamentals of Photonics 2017/5/24 67 CHAPTER 4----LASER 5. Properties of mode-locked pulses: i) The pulse separation RT: c t Sin 2 0 2 RT 2 2L c c c t 2 The round-trip time of the cavity! ii) The peak power: Considering sin a a when a is small, E 2 (0) N 2 E02 N times of the average power. N: number of modes. The more the modes the higher the peak power of the Mode-locked pulses. Fundamentals of Photonics 2017/5/24 68 CHAPTER 4----LASER iii)The individual pulse width: c t sin N 0 2 N a c t p 2 N c 2 1 t p a a Narrower as N increases. t p a: bandwidth of the gain profile. RT N The mode locked pulse width is reversely proportional to the gain band width, so the broader the gain profile, the shorter are the mode locked pulses. Fundamentals of Photonics 2017/5/24 69 CHAPTER 4----LASER Techniques of laser mode-locking Active mode-locking: Actively modulating the gain or loss of a laser cavity in a periodic way, usually at the cavity repetition frequency c/2nL to achieve mode-locking. Amplitude modulation: A modulator with a transmission function of 2t T 1 1 cos RT is inserted in the laser cavity to modulate the light. Where is the modulation strength and < 0.5. Under the influence of the modulation phases of the lasing modes become synchronized and as a consequence become mode-locked. Operation mechanism of the technique: Fundamentals of Photonics 2017/5/24 70 CHAPTER 4----LASER Time domain analysis: Consider the extreme case where a shutter is inside the cavity, and the opens only for a short time every second. Is the cavity round trip time. In this case only a pulse with pulse width narrower than the opening time can survive in the cavity, all the CW type of operation will be blocked by the shutter. To have a pulse moving in the cavity the phase of all lasing modes must be synchronized. The laser modes will arrange themselves to realize such a state. Shutter losses It is a natural competition and the fittest will survive. Fundamentals of Photonics 2017/5/24 71 CHAPTER 4----LASER Frequency domain analysis: Sidebands generation Amplitude modulation Em (t ) m sin (m t m ) Electrical filed of each mode m 0 (1 cos t ) Amplitude of each mode is modulated E m (t ) 0 sin ( m t m ) sin ( m )t m sin ( m )t m 2 2 Sidebands are generated by the modulation m- m Without modulation Fundamentals of Photonics m m+ After amplitude modulation 2017/5/24 72 CHAPTER 4----LASER In the case of a multimode laser As all modes are modulated by the same frequency, the sidebands of one mode will drive its adjacent modes, and as a consequence, all modes will oscillate with locked phase. From both the time domain and the frequency domain analysis it is easy to understand why the modulation frequency must be exactly the cavity longitudinal mode separation frequency. Fundamentals of Photonics 2017/5/24 73 CHAPTER 4----LASER Passive mode-locking: Inserting an appropriately selected saturable absorber inside the laser cavity. Through the mutual interaction between light, saturable absorber and gain medium to automatically achieve mode locking. A typical passive mode locking laser configuration: laser medium saturable absorber Mechanism of the mode-locking: i) Interaction between saturable absorber and laser gain: Survival takes all! ii)Balance between the pulse shortening and pulse broadening: Final pulse width. Fundamentals of Photonics 2017/5/24 74 CHAPTER 4----LASER Transform limited pulses Gaussian pulses: In our analysis we have assumed that En=E0 The real gain line has a Gaussian profile,which results in that the lasing mode amplitudes have a Gaussian distribution. A Gaussian gain line shape function a Gaussian mode-locked pulse intensity variation, namely a Gaussian pulse. 1 2 t E (t ) E0 exp 1 2 ln 2 0 2(ln 2)2 Fundamentals of Photonics 2017/5/24 2 exp (i t ) 0 75 CHAPTER 4----LASER Intensity of the pulse: I (t ) 2 ln 2 0 2 exp 1 2(ln 2)2 2 Gaussian intensity profile! Transform limited pulses: If the product of pulse width and spectral bandwidth of a Gaussian pulse equals 0.441, then the Gaussian pulse is called a transform limited pulse as in this case the pulse width is purely determined by the Fourier transformation of the pulse spectral distribution. t p L 0.441 For transform limited Gaussian pulses! tp: pulse width, L: spectral bandwidth Fundamentals of Photonics 2017/5/24 76 CHAPTER 4----LASER 激光振荡器 返回 CHAPTER 4----LASER 均匀加宽激光器模式竞争 返回 CHAPTER 4----LASER 非均匀加宽激光器模式竞争 返回 CHAPTER 4----LASER 多普勒加宽增益饱和 返回 CHAPTER 4----LASER 兰姆凹陷 返回 CHAPTER 4----LASER 脉冲泵浦调Q 返回