LASER TECHNOLOGY CW Operation, Q-switching, active Mode-locking

Transcription

1 Chair for laser and X-ray physics E11 Prof. Reinhard Kienberger PD. Hristo Iglev Dr. Wolfram Helml Ultrafast physics II Thursday, 10:15 11:45 h PH II 127, Seminarraum E11 Lecture notes: Ultrakurzzeitphysik 1 (2SWS VO, WS 2017/18) LASER TECHNOLOGY CW Operation, Q-sitching, active Mode-locking I( ) 50 Moden 5 Moden P CW Resonator Umlaufzeit Literature: Solid-State Lasers Walther Koechner, Springer Lasers Anthony E. Siegman, University Science Book Optoelectronics and Photonics S. O. Kasap, Prentice Hall Vorlesungsskripten von Prof. A. Leitenstorfer, Uni Konstanz

2 CW Operation Working principle of a laser: Light Amplification by Stimulated Emission of Radiation - Population Inversion - Stimulated Emission - Resonator à Three-level system: 3> 2> à Four-level system: 3> 2> Source of pump energy Laser active material 1> Laser beam 1> 0> Mirror Semi-transparent mirror Pump: constant operation (continuous) Losses: constant (time-independent) The laser runs at the minimal threshold: è Continuous ave laser g t = - 1 ln( r 1 r 2 ) 2L 3 Pulsed Operation Q-Sitching (Güteschaltung) - optical sitch: changes the quality factor Q of the resonator - active and passive Aim: Creating a single, strong pulse Laser poer Threshold poer S(t) P L (t) Pump poer à P P (t) Inversion Δ N(t) - Until t 0 : optical sitch is open, losses high, quality Q lo, S(t) - large - Inversion (Δ N(t)) is not depleted by stimulated emission - At t 0 : Flipping the optical sitch à S(t) small - Saved inversion energy is depleted in one "giant pulse" P L (t) 4

3 Active: Pockels cell ith polarizers Q-Sitching - Applying voltage U: λ/2 plate à losses high, quality Q lo Voltage U Polarizer Mirror Electrodes Polarizer Mirror Laser Pockels cell Passive: saturable dye à Lo intensity light is absorbed, high intensity light is transmitted I 0 active medium I >> 0 horizontally polarized light saturable absorber: To-level system! saturable dye 5 Laser Modes longitudinal modes Mirror separation = integer multiple of λ/2 discrete values of λ, f optical length of the cavity = n. L; spacing of the longitudinal modes Δf = c / (2 n L) I() n1 n1 r +1 r +1 CC x2 2(n1r CE CE 2n1 r +1 r +1 CC n(ω) is the ) ) E Efrequency-dependent refractive E E index of the cavity medium and L the length of the cavity. In a freely singing oscillator ith several axial modes, the frequencies have a spacing of: % # = 2'())+ = 1 - ()) Round-trip time: - ) = 2+ '()) % = 2+. /0 ()) In a laser oscillator ith only one axial mode and one frequency, resulting line idths can be in the order of magnitude of Δn» 1 Hz. In the visible spectral region, that means a relative bandidth of v/v (This surpasses the accuracy of a Quartz-oscillator by more than a factor of 10 6!) 6

4 Phase-Coherent Generation of Light Waves ith Equidistant Frequency Spacing Oscillating axial resonator modes In a laser many independent frequencies (modes) can oscillate. Losses Gain profile of the medium q-2 q-1 q q+1 q+2 q+3 Possible resonator modes Sum of 10 modes ith the same relative phase Sum of the modes ith random phases 7 Generation of Femtosecond Pulses Laser pulses in the sub-nanosecond regime are generated through modelocking. Therefore, the frequency range of the laser emission has to be sufficiently broad and there has to be a ell-defined phase relation beteen the longitudinal laser modes t The aveform observes constructive interference and therefore has a peak at t = 0. Everyhere else it averages to zero. 8

5 Generation of Ultrashort Laser Pulses Concept of mode-locking Applying the technique of mode-locking, frequency and phase of the axial modes can be linked to each other. Considering N modes: Δω L = 2π ω q = ω 0 +qδω L ; q = N 1 N 1,..., 0,... + φ 2 2 q = φ0 T L or T r is the round-trip group delay in the resonator at the central frequency ω 0. 1 i( kqz qt) i q E( z, t) aq e ω + φ Resulting electric field: = + c. c. 2 q T L 50 Modes 5 Modes P CW Resonator round-trip time 9 Active and Passive Mode-Locking Mode-locking requires periodic amplitude modulation ithin the resonator. E 1 (t) E 3 (t) output Modulator Lasermedium The modulation can be achieved actively by installing an additional modulator, hich is controlled by an electric signal that corresponds to the round-trip time T L. Alternatively, a nonlinear optical component that features loer losses ith rising light intensity may be introduced. These techniques are called active mode-locking and passive mode-locking. 10

6 Ultrashort Pulse Duration à Ultra High Peak Poer As a result a laser pulse is formed, circulating ithin the resonator, hich features a pulse duration a pulse energy and a peak poer T t» p r N W P T p» average r P peak Wp» = NP t p average Using modern, commercial designs, up to N = modes can be phase- and frequency coupled. Thus, the energy circulating ithin the resonator, hich is spread along T r» 10 ns in free singing mode, can be temporally concentrated in τ p»10 fs long pulses. Why 10 ns? Why 1 million modes? Within the same process the natural peak poer is increased, hich no can be one million times the average poer of the laser in CW operation. Therefore, an energy of a fe nano-joule ithin the resonator can result in pulses ith several teraatt to one petaatt ith the help of an amplifier. 11 Active Mode Locking Method: Sinusoidal modulation of the intensity Þ through competition, modes are coupled to modulation-sidebands Þ in best case all modes are being coupled and phase-locked modulator transmission cos( M t) time The time frame of minimal losses in the resonator (at the maximum of the modulator transmission) favours the build-up of a light field. This results in the formation of a single pulse circulating ithin the resonator, alays passing the modulator at its maximum transmission. Hence, the modulation frequency has to match the round-trip time. The Modulation Theorem F { E(t)cos(ω M t) } = Multiplication ith cos( M t) generates side-bands. Here: 100% modulation depth! = 1 2 = 1 2 E(t) " # exp(iω M t) + exp( iω M t) $ % exp( iω t) dt = E(ω ω M ) E(ω +ω M ) 12

7 Active Mode-Locking In the frequency domain a sinusoidal intensity modulation generates side bands of each mode. 0 F {E(t))} F {E(t)cos( M t)} 0 - M M Resonatormodes n - M 0 Frequency n + M pc/l For mode-locking, the generated side band has to feature the same frequency as an adjacent mode. This means: M = 2p/round-trip time = 2p/(2L/c) = pc/l Each mode competes for amplification in the medium ith its adjacent mode. This leads to the assertion of the ave ith higher amplitude and continues for all N modes. 13 Active Mode-Locking: The Electro-Optic Modulator Applying a voltage to the crystal changes its index of refraction and induces birefringence (electro-optic effect). A fe kv turn the crystal into a λ/2 or λ/4-plate. Polarizer For V = 0 the polarization remains untouched "Pockels cell" V For V = V p polarization is rotated by 90. Applying a sinusoidal voltage causes a modulated rotation of the polarization. Using a polarizer a sinusoidal amplitude modulation is realized through here. 14

8 Active Mode-Locking: The Acousto-Optic Modulator An acoustic ave through the crystal causes sinusoidal density modulations of the material. A proportional change of the refractive index is linked to that. At the so generated diffraction grating a small fraction of the light beam is diffracted out of the beam path. Pressure-, density-, and refraction index variation caused by the acoustic ave. initial beam Sound generator transmitted beam Quartz diffracted laser beam (= loss) The diffracted part can amount up to ~70%. Sinusoidal modulation of the grating intensity causes a sinusoidal modulation of the output signal. 15 Theory of Active Mode-Locking E 1 (t) E 3 (t) output 1 Modulator Transmission cos( m t) Modulator Lasermedium E(t) approximated modulator transmission function: T r t 0 Zei t valid if: # # % <<! 2! " # # % = '() 1 2 Δ "- ". (# # % ). D m : modulation depth approximated amplification transmission function: 1 Gain coefficient! % = '() 4 % Ω 3. (- - %). DΩ g valid if: - - % << Ω 3 0 E() Frequency Ω g : effective amplification bandidth g 0 : saturated amplification at ω 0 16

9 Theory of Active Mode-Locking In order to formulate the stationary state of the pulse parameters, several approximations are necessary. Approximation #1: The pulse is Gaussian ith complex temporal pulse parameter: γ =α+ i β! " = 1 2 & ' " )*+ -. / " = 1 2 & /)*+ 3" 4 )*+ -. / " = = 1 2 & /)*+ 5" 4 )*+ -(. / + 7")" Pulse duration: t p = 2ln2 a d 2 Instantaneous frequency: i ( t) = ( 0 t + b t ) = 0 + 2b t d t Approximation #2: Die pulse duration is much shorter than the round-trip time " 9 << ; < = 2=. / Approximation #3: The spectral idth of the pulse is much more narro than the amplification banditdth of the laser medium. 9 << " 9 >> 2= 17 Theory of Active Mode-Locking Approximation #2 ( τ p << T r ) implies that the approximated modulator transmission function can describe the modification of the pulse upon passing through the modulator. Thereby, the laser pulse posses its intensity maximum at the moment of maximum transmission. E 1 (t) E 3 (t) output Modulator Lasermedium Applying the approximated modulator transmission function results in:! " # =! % # & ' # =! ( )*+ - % # " )*+ 1 2 Δ '2 ' " # " =! ( )*+ - " # " This shos that the pulse maintains its Gaussian profile: - " = - % + % " Δ '2 ' " # "..and is shortened doing so!! 18

10 Theory of Active Mode-Locking E 1 (t) Modulator E 3 (t) Lasermedium output The modification of the laser pulse upon pass-through of the laser medium and reflection at the output coupling mirror may be described - in frequency domain - by:! " # # % =! ' # # % ( ) * # # % ( ) +,- ) * # # % =./0 1 % Ω * ' (# # %) ' ) +,- =./0 7 2! ' 8 =! 9 8 ) : 8 =! %./0 ; 9 8 './0 1 2 Δ :# : ' 8 ' =! %./0 ; ' 8 ' d 2 1 dτ 2 2 Δ m ω 2 τ 2 + g 0 m 2 $ & 2& a(τ) % Note that a derivative in the time domain is equivalent to a multiplication ith i(ω ω 0 ) in the frequency domain. 19 Theory of Active Mode-Locking E 1 (t) Modulator E 3 (t) Lasermedium output The modification of the laser pulse upon pass-through of the laser medium and reflection at the output coupling mirror may be described - in frequency domain - by:! " # # % =! ' # # % ( ) * # # % ( ) +,- =! % ( (# # 4/ % ) ' ( % ' ' Ω # # % ' ( 456 : * =! % ( (# # 4/ % ) ' ( % " 2 :! 2 ' (;) =! % ( 4 <= >? > (Fourier transformed) The pulse is lengthened. 1 / " = 1 / ' + 22 % Ω * ' g 0 is the saturated amplification at ω 0 and l represents output coupling and other losses. 20

11 Theory of Active Mode-Locking In the stable state the pulse reproduces itself (temporally) after one complete round-trip in the resonator:! " =! $ 1 = 1 + 2, - /! $ &!! ' $! '! ' Ω ' =! $ + 2, - ' / Ω! $! ' / compare to transmission function:! ' =! $ Δ 12 1 ' 2, - Ω / '! $! ' = 1 2 Δ 12 1 ' Approximation #2 ( τ p << T r =2π/ω m ) requires that the modulator has just slightly modified the pulse shape after one round-trip, i.e.:! '! $ =! 44 Plugged into the above equation this leads to:! ' 44 = 1 Δ 1 2 ' 4, 1 Ω ' / -! 44 = 1 2 Δ 1, Ω / 21 Theory of Active Mode-Locking 2 33 = 1 2 Δ, -. /, Ω 1 If γ ss is real, the pulse does not exhibit chirp (β = 0). For the Gaussian pulse: 2 33 = 1 ln 2 # $ % For the pulse duration (FWHM) of a laser that is actively mode locked through amplitude modulation in stable operation e derive: 2 33 = 1 2 Δ, -. /, Ω 1 1 # $ % = 1 4()2 Δ, -. /, Ω 1 D m : modulation depth g 0 : saturated amplification at ω 0 ω M : modulation frequency Ω g : effective amplification bandidth Depending on the amplification bandidth, the length of the resonator and the modulation depth, actively mode-locked lasers can provide pulses in the range beteen ps. 22

12 Theory of Active Mode-Locking In real lasers e have a small detuning beteen modulation frequency and repetition rate: Δν = ν d m 1 T r δν δ td = Tr ν 1 d 2 m νm A small detuning is not catastrophic. The laser pulse see s a monotonic gradient of the transmission à ill be shifted by δt m back to the transmission maximum (t 0 ) Stable mode-locking regime if: δν d < π ν m 2 n2 Δ m ν 2 2 τ m p Shorter pulses requires more precise matching of the modulation frequency to the repetition rate and stronger modulation amplitudes 23 Passive Mode-Locking Method: Inserting a component that favours high field intensities Þ Þ Strong pulses gro at the expense of eaker ones Finally all energy is combined in one pulse Saturable absorber Losses Gain Gain > Losses time Nonlinear optical component that lets high-intensity light transmit more easily than loer-intensity light. Intensity Time (fs) Round-trips (ns) T = 90% for lo intensities and T 100 % for high intensities k = 1 k = 2 k = 3 k = 7 As a result, ithin the initial intensity noise of the c laser, the highest amplitude gros at the expenses of the others ith every round-trip, finally supressing and eliminating the loer amplitudes. 24

13 Passive Mode-Locking: Saturable Absorber mit! $! =! # = 1 + #/# ()* # ()* = ħ, 2. ) / 0 For a to-level absorber ith an absorption cross section σ a and a lifetime of the excited state given by τ 2 If Δα(τ) in the saturable absorber only changes slightly ithin τ 2 and the intensity is eak I/I sat, than:! = /5 678! $ (1 #/# ()* ) ith # / = <(/) 0 E FGG ß normalized instantaneous poer ß effective cross section of the bunch. The pulse modulates its on amplitude upon the passage through the saturable absorber: < =>* / = <?@ / ABC 1 2! $D ) 1 <?@ 0 E FGG # ()* L a - Length of the absorber This effect is called Self-Amplitude Modulation, SAM. 25 Reformulated: Passive Mode-Locking! "#$ % =! '( % )*+ 1 2 / ! '( $ =! '( % )*+ 9 82: 2 + <! '( 3 SAM coefficient defined as: < = : SAM losses: : = / $ For eak absorption: 9 82: = / ! "#$ % =! '( % )*+ 9 82: 2 + <! '( 3! '( : 2 + <! '( 3 Δ! 82: = : 2 + <! '( 3 a(τ) 26

14 Passive Mode-Locking Passive mode-locking allos for the generation of laser pulses ith durations p < 1 ps! à The spectral idth is getting so large that group delay dispersion à GDD has to be considered à additional modulation!!! Long pulse Irradiance vs. time Spectrum time frequency Short pulse time frequency In the time domain this has the folloing implication (additional modulation): a GDD (τ ) = i D 2 d 2 d t 2 a(τ ) 27 Passive Mode-Locking A derivative in the time domain is equivalent to a multiplication ith i(ω ω 0 ) in the frequency domain. = ABB # # % = = # # % '() C D 2 # # %. Passive ML GDD! " # # % = '() * % Ω ". (# # %).! ABB # # % = '() C D 2 # # %.! 567 = '() 4 89: 2 + < = >?.! 123 = '() 4 2 = 123 # # % = = >? # # % '() C D 2 * %. # #. % + < =. >? + * % 2Ω " : 2 28

15 Passive Mode-Locking i(ω ω 0 ) in the frequency domainà d/dt in the time domain. " &'( ) ) + = ",- ) ) +./ Ω7 ) ) " 7, : + : ;<= 2 For the stationary case after one round-trip: 4 + 2Ω !"!# = 0! 7!? 7 "(?) + 9 "? : + : ;<= 2 "(?) = 0 Main equation of passive mode-locking ith self-amplitude modulation The first term broadens the pulse as a result of the finite amplification bandidth and the GDD, the second term shortens it based on SAM and the final term accounts for amplification and losses. 29 Passive Mode-Locking ith a Fast Saturable Absorber Fast absorber: short lifetime of the excited state à allos for generation of shorter laser pulses High-intensity spikes (i.e., short pulses) see less loss and hence can lase hile lointensity backgrounds (i.e., long pulses) on't. 30

16 Passive Mode-Locking ith a Slo Saturable Absorber What if the absorber responds sloly (more sloly than the pulse)? Then only the leading edge ill experience pulse shortening. Gain saturation shortens the pulses trailing edge. The intense spike uses up the laser gain-medium energy, reducing the gain available for the trailing edge of the pulse (and for later pulses). This is the most common situation, unless the pulse is many ps long. 31 Passive Mode-locking With Saturable Gain and Loss Lasers lase hen the gain exceeds the loss. The combination of saturable absorption and saturable gain yields short pulses even hen the absorber is sloer than the pulse. Due to the GVD in the resonator mirrors the local time-axis of the travelling pulse is mirrored on every next run. This phenomenon compensates the pulse-asymmetry 32

17 The Passively Mode-Locked Dye Laser Gain medium Pump beam Saturabl e absorber Passively mode-locked dye lasers yield pulses as short as a fe hundred fs. They are limited by our ability to saturate the absorber. Some common dyes and their corresponding saturable absorbers 33 The Colliding-Pulse Mode-Locked (CPM) Laser A Sagnac interferometer is ideal for creating colliding pulses. Gain medium Saturable absorber Beamsplitter CPM dye lasers produce even shorter pulses: ~30 fs. Colliding pulses have a higher peak intensity. And higher intensity in the saturable absorber is hat CPM lasers require. Intensity To pulses colliding Single pulse Longitudinal position, z 34

18 Q-Sitched Passive Mode-Locking Q-sitched passive mode-locking is observed hen: (i) The laser pulse does not "clear" the entire gain (lo energy); (ii) Laser medium (gain medium) and absorber are too slo. Stable mode-locking if: Intracavity pulse energy Saturation energy of the absorber x Modulation depth Saturation energy of the gain medium Q-sitched mode-locking is (often) undesirable. It can be suppressed: - by choosing a gain medium ith high laser cross-section - by using a resonator design ith small mode areas - by using a long resonator ith lo losses and high intracavity poer - by inserting a poer-limiting element in the resonator - via optimization of the saturable absorber ( R) - ith electronic feedback techniques 35 Passive Mode-Locking: SESAM SESAM = semiconductor saturable absorber mirror Semiconductor as saturable absorber + multi-layer mirror Solves the "Q-sitching problem for passively mode-locked solid-state lasers Allos high repetition rates (up to 160 GHz) at relatively high pulse energies Frequently a SESAM is used as additional passive mode-locking device à in combination active-passive in Nd:YAG lasers à or passive-passive, in case the pulse energy is too small for proper usage of the main mode-locking device. 36

19 Stable mode-locking (no Q-sitching) if: SESAM: Properties Laser medium ith large transition cross section (fast saturation) Fast Absorbers: Intraband thermalisation 100 fs Interband recombination ns The band gap is tunable over a very broad spectral range: Absorber ith large transition cross section (fast saturation) 37 Passive Mode-Locking: Nonlinear Mirrors DM Nonlinear optical component that lets high-intensity light transmit more easily than loer-intensity light. The reflectivity depends on the intensity à R=R(I 0 ) 2 X (2) NL crystal 2 2 Second-harmonic is reflected by DM and reconverted in the NL crystal. High intensity -> High reflectivity Advantage: X (2) -effects are very fast à very short laser pulses are possible in theory. Disadvantage: The nonlinear cascade requires high light intensity and exact phase matching à orks only ith short crystals and long laser pulses 38