레이저의주파수안정화방법및그응용 박상언 ( 한국표준과학연구원, 길이시간센터 )

레이저의주파수안정화방법및그응용 박상언 ( 한국표준과학연구원, 길이시간센터 )

Contents Frequency references Frequency locking methods Basic principle of loop filter Example of lock box circuits Quantifying frequency stability Applications of frequency stabilized laser

Frequency references Atomic spectra Simple vapor cell Cs, Rb, Na. K, etc. Optical frequency standard Sr, Yb, Ca, Ion Hg +, Yb +, Molecular spectra I 2, O 2, 13 C 2 H 2, H 2 O Optical cavity Ultra-high finesse ULE optical cavity

Generation of frequency discriminators DC locking Slope of signal Absorption signal of atom and molecule Transmitted signal of reference optical cavity Phase-sensitive detection Low frequency modulation (< few 100 khz, ω m << Δ ω 0 ) Demodulator: lock-in amplifier High frequency modulation (> 1 MHz, ω m >> Δ ω 0 ) Demodulator: high frequency lock-in amplifier, mixer Frequency offset locking (Phase locking) Stabilization of relative frequency between two lasers S ν 0 ν

DC locking Atom, molecule, optical cavity, etc. Beam splitter Vapor cell Beam splitter Diff. Amp. Error signal Mirror Error signal Resonance frequency 0 ν

Phase sensitive detection Frequency / phase modulation method Direct frequency modulation via cavity length PZT: ECDL, Ti:Sapphire laser. Injection current: diode laser, ECDL Electro-optic modulator: Intra-cavity EOM External frequency / phase modulation Acousto-optic modulator Electro-optic modulator Modulation of reference signal Modulation of magnetic field Zeeman modulation Modulation of optical cavity length

Bandwidth of phase sensitive detection Laser EOM Sample Detector Modulated signal ω m 2ω m 3ω m Current Modulation ω m Phase shifter θ Demodulator LPF Frequency Oscillator ω m Lock-in amplifier LPF 2ω m 3ω m Frequency Error signal Loop filter ω c ω m 2ω m 3ω m ω c «ω m Frequency Bandwidth of of error error signal signal will will be be ω c c which which is is cut-off cut-off frequency of of low low pass pass filter. filter.

Low frequency modulation method ω m << Δ ω 0 Amplitude modulated signal ω m ω m 2ω m Δ ω 0 ω m ω m ω m Frequency modulation of laser Demodulated signal from lock-in amplifier

High frequency modulation method Advantage Ultra-high sensitive detection Rapid measurement of absorption or dispersion of sample Wide bandwidth error signal Noise characteristics of detection system ω m >> Δ ω 0 1/f noise Noise spectrum (log) Corner frequency ~ few MHz Frequency range used in FM spectroscopy Quantum limit of detection system Fourier frequency (log)

Simple model of high FMS (1) Phase modulated optical field is given by E( t) = E exp{ i( ω t +β sin( ω t))} 0 0 m β <<1 where β : FM index, ω m : Modulation frequency, ω 0 - ω m ω 0 ω 0 + ω m After reflecting optical cavity or passing through atomic cell, the optical field is E ( t) = E ΣT ( ) J ( )exp{ i( t n t)} 0 n ω R n β ω + ω 0 m n Tn( ω) = exp( δn iφn) where T n (ω) : Complex transfer function, J n (β) : Bessel function, δ n (ω) : Amplitude attenuation, φ n (ω) : Optical phase shift

Simple model of high FMS (2) Assuming small signal modulation (β <<1), high order terms can be neglected and E R (t) simplifies to t m i e m T t m i e m T t i e T t R E + + + ω ω ω ω ω ω ω ω ω ω β β 0 ) 0 ( 1 0 ) 0 ( 1 0 ) 0 ( 0 ) ( 2 2 The light intensity I(t) impinging onto the photo-detector is approximately m t m t t R E t I ω ω φ φ φ β δ δ β sin cos 2 ) ( ) ( ) 2 ( ) ( 0 1 1 1 1 + + Dispersion Absorption

Demodulation of modulated signal using mixer After mixing the modulated signal with the reference signal with a mixer, the mixer output IF mixer will be ( ) ( ) ( ) ) sin(2 ) sin( ) 0 2 1 1 ( ) cos(2 ) cos( ) 1 1 ( ) cos( )sin 0 2 1 1 ( )cos 1 1 ( ) cos( ) ( θ ω θ φ φ φ β θ ω θ δ δ β θ ω ω φ φ φ β ω δ δ β θ ω + + + + + + + + + = + = t m t m t m t m t m t m t I mixer IF 2ω m terms can be removed by using a low pass filter. ) )sin( 0 2 1 1 ( ) )cos( 1 1 ( θ φ φ φ β θ δ δ β + = mixer IF ) 0 2 1 1 ( ) 1 1 ( 90 0 φ φ φ β δ δ β θ θ + = = mixer IF mixer IF Absorption Dispersion

Frequency modulation (FM) spectroscopy 1.0 0.5 Signal [V] 0.0-0.5-1.0 36 40 44 48 52 56 60 64 Frequency detuning [a. u.]

Modulation transfer spectroscopy (MTS) Pump beam ω 0-2Ω Oscillator Ω Phase shifter ω 0 - Ω ω 0 + Ω ω0 +2Ω EOM or AOM Low pass filter ω 0 Error signal Photo detector Mixer LPF Loop filter ω Vapor cell Modulation of pump beam is transferred to unmodulated probe beam only when sub Doppler condition is satisfied. Doppler signal Advantage Advantage No No residual residual offset offset caused caused by by Doppler Doppler background background signal signal Long-term Long-term stability stability can can be be improved improved Disadvantage Disadvantage Probe beam Laser frequency detuning Signal amplitude Difference Doppler signal With dip Laser frequency detuning Large Large signal signal only only at at closed closed transitions transitions For For Rb Rb D2, D2, Fg=3 Fg=3 Fe=4, Fe=4, Fg=2 Fg=2 Fe=1 Fe=1 For For Cs Cs D2, D2, Fg=4 Fg=4 Fe=5, Fe=5, Fg=3 Fg=3 Fe=2 Fe=2

Typical MTS signal for Cs D2 line Saturated absorption signal 6P 3/2 D2 line 852 nm F 5 4 3 2 4 4 6S 1/2 3 Energy level of Cesium D 2 line MTS signal Frequency detuning (a. u.)

Pound-Drever-Hall (PDH) locking Power reflection factor Phase shift ω 0 - ω m ω 0 + ω m E 0 (t) ω 0 ω EOM Optical cavity LO DBM PD Beat signal LPF ω 0 - ω m ω 0 ω 0 + ω m

Basic schematics of frequency stabilization Frequency reference Atom, molecule, optical cavity,. Laser Servo element D(f ) ν ν 0 Detector Servo signal U R = g(f ) C δν Loop filter Servo control g(f ) Error signal S C δν S ν 0 ν

Model of servo loop (1) ν i : Free-running frequency ν s : Actual frequency ν 0 : Reference frequency Error signal S is proportional to the frequency deviation S C ( ν s ν 0 ) = C δν Δν = ν i ν 0 Error signal is filtered and amplified in a δν = ν s ν 0 servo amplifier ν i ν s ν 0 ν Output of servo signal U R will be U R = g(f ) S = g(f ) C δν The frequency-dependence response of S Servo element: D(f ) ν 0 ν The combined frequency-dependent transfer function of the servo loop is D(f ) g(f ) C

Model of servo loop (2) If the servo loop with negative feedback is closed the servo element has to counteract the frequency deviation δν Then the actual frequency ν s Δν = ν i ν 0 ν s = ν i D(f ) g(f ) C δν δν = ν s ν 0 ν s ν 0 = ν i ν 0 D(f ) g(f ) C δν ν i ν s ν 0 ν δν = Δν D(f ) g(f ) C δν Δν = δν + D(f ) g(f ) C δν = δν [1 + D(f ) g(f ) C] δν = Δν 1+ C D( f ) g( f )

Model of servo loop (3) δν = Δν Δν = 1+ C D( f ) g( f ) 1+ G( f ) Overall gain of loop The frequency deviation Δν of a free-running laser is reduced by the factor of 1 + G(f ) If G(f ) is constant (proportional gain). then δν has finite value, which means the frequency of laser couldn t reach the target frequency ν 0. G(f ) has to be infinite to get a ideal frequency stabilization. δν =0, ν s = ν 0

Model of servo loop (4) Frequency-independent case The overall gain of the loop can be set to be as high as possible Unfortunately most of components (detector, amplifier, servo element) in the loop have frequency-dependent response. detector, amplifier, servo element, cable delay,. Frequency dependent phase shift in the loop Total phase shift of 180 produces positive rather than negative feedback.

Constant gain, frequency-dependent phase Negative feedback Positive feedback Phase angle > 180 Gain > 0 db Gain (db) 10 0 f A Bode plot f A f B f C 0 f B Phase ( o ) -180 Frequency (f) f C

Frequency-dependent gain and phase Gain (db) 10 0 Positive feedback Gain > 1 (0 db ) @ Phase angle < -180 Unstable Phase ( o ) 0-180 Stable Gain < 1 (0 db) @ Phase angle < -180 Frequency (f)

Gain and phase margin, loop bandwidth Positive feedback 10 Gain (db) 0 Gain margin Large gain at low frequencies while Loop bandwidth maintaining 0 sufficient phase Phase ( o ) -180 Phase margin margin at unity gain frequency Frequency (f)

Goal of loop filter Large gain at low frequencies! Maintaining sufficient phase margin at unity gain frequency!

Ideal loop filter for given phase shift Gain (db) 10 0 Gain = Gain = 0 0 Impossible! Phase ( o ) -180 Phase margin Frequency (f)

Realistic approach with integrator Gain = @ f 0 Gain (db) 10 0-90 Gain = 0 @ f Phase ( o ) -180 Phase margin Frequency (f)

esketch PRO v1.56 Simple analog circuit simulator Resistor, capacitor, inductor, OP amp Freeware http://www.schematica.com/esketch_files/esketch.htm

Integrator Gain -20 db/dec Phase lag 90

Differentiator (phase lead) Phase lead

Proportional-Integrator Integrator at low frequency Phase lag 90 at low frequency Phase shift 0 at high frequency

First-order delay system with integrator Inverter 1 st -order delay Integrator

First-order delay system with PI amp Inverter 1 st -order delay PI amp.

PID servo Phase lead

OP amp circuits Vin Vout Vout R 2 = 1 + Vin R1 Vin Vout Vout = Vin

OP amp circuits Vin Vout Vout = R2 Vin R1 V1 V2 V3 Vout R4 R4 Vout = ( V1 + V 2 R3 R1 If R1 = R2 = R3 = R4, + V 3 R4 ), R2 Vout = ( V1+ V 2 + V 3)

Block Diagram of Laser Frequency Lock Box Amplitude FM Input ø Signal Input X5 Prop.&Int. Phase X1 Fast Output Ext. Input X5 PI Signal Offset + - Integrator Fast Gain Slow Gain Int. Scan Generator X1 Slow Output PZT Center + - PZT Span

Circuit Diagram of Laser Frequency Lock Box

Measurement of frequency stability Allan deviation Time domain analysis Power spectral density Fourier domain analysis

Allan variance Allan deviation

Comparison of frequency stability

Power spectral density of laser frequency noise Frequency reference Laser ν ν 0 Detector S Servo signal Servo control Error signal ν 0 ν FFT analyzer

Power spectral density of frequency noise 0.9 SAS Error signal F=4 to F'=4x5 Signal (V) 0.6 0.3 0.0 F=4 to F'=5 Saturated absorption signal and error signal -0.3 11.4 MHz / V -0.6-100 -50 0 50 100 150 200 Frequency detuning (MHz) 0.1 Loose locking 0.01 FFT spectrum of laser noise with and without lock PSD(Vrms/Hz 1/2 ) 1E-3 1E-4 1E-5 1E-6 Tight locking 1E-7 10 100 1000 10000 100000 Frequency (Hz)

Linewidth estimation from PSD (b) (a) S ν = 10 5 Hz 2 /Hz f c (a) (b) fc: Corner frequency Lorentzian linewidth Δν = π Sν (Hz) 300 (khz)

Application of frequency stabilized laser Laser cooling and trapping / atom manipulation Bose-Einstein condensation of atom Atomic frequency standard Atom interferometer Sub-Hz linewidth laser for Optical frequency standard Femtosecond laser optical frequency comb

Laser cooling and trapping

Atomic fountain clock

Atomic gravimeter

Hz-level linewidth laser for optical clock PTB setup and result ULE cavity

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