Nonlinear Vibrational Microscopy

Transcription

1 Nonlinear Vibrational Microscopy Wolfgang Langbein School of Physics and Astronomy, Cardiff University

2 Vibrational Modes In molecules, the energy of the electronic state and the Coulomb repulsion between the nuclei create a potential W for the motion of the nuclei (Born-Oppenheimer approximation) The derivatives of the energy W versus the nuclei coordinates relative to their equilibrium position create forces linear with their displacement, and diagonalizing the set of equations of motion results in a set of vibrational eigenmodes with different eigenfrequencies and reduced masses. In the following, the coordinate of one eigenmode is called Q and its reduced mass m. ɺɺ γɺ Q + Q + Ω Q = F( t) m These modes couple to the environment, resulting in frequency shifts and line broadening. This coupling is specific to each mode. We use here a Markovian model, (negligible memory time of environment), yielding an exponential decay characterized by γ

3 The Sound of Molecules Sound slow-motion 1 Billion to one (1 second vibrations in 30 years audio) Water (H O) 118 THz 115 THz 49 THz Methane (CH 4 ) 95 THz 9 THz 47 THz 41 THz Complex molecule

4 Typical Vibrational Frequencies Vibrational frequencies are dominated by local chemical structure ( neighbour or next neighbour coupling gives the approximate frequency) They allow to identify chemical content of material Vibrational frequency (cm -1 )

5 The number of vibrational modes given by total degrees of freedom minus translation and rotation 3N 6 for N atoms (3N 5 for linear arrangement) Harmonic oscillation of eigenmodes Vibrational modes Optical transition selection rules related to symmetry Rule of thumb: symmetric=raman active, asymmetric=ir active ω = ( m1 + m ) k m m 1 CO Raman: 1335 cm 1 IR: 349 cm 1 H O Raman + IR: 3657 cm 1 Raman + IR: 3756 cm 1 IR: 667 cm 1 Raman + IR : 1594 cm 1

6 IR absorption and Raman spectra Vibrational modes have different relative weights in IR absorption and Raman spectra due to the different selection rules

7 Basic mechanism of Raman scattering The oscillating electric field of the excitation light E( t) Q( t) Classically, the bond length oscillates at the vibrational frequency (Spontaneous Raman scattering is due to zeropoint motion ) The molecular polarizability changes with bond length induced dipole moment ( ) 0 Q p ( t ) = α E ( t ) α + α Q ( t ) E ( t ) Q = 0 Higher orders in Q neglected Q α Q= 0 α R Q( t) = Q cosω t 0 R E( t) = E cosωt 0 ( ) p( t) = α + α Q cosω t E cosωt STOKES ANTI-STOKES cosω t cosωt = cos ω ω t + cos ω + ω t 0 R 0 R 0 ( ) ( ) R R R Equal amplitudes

8 Energy level diagram for Raman E 0 & E 1 : Electronic levels (transition corresponds typically to UV light in biology) incident photon has energy E L molecule gains energy ħω R energy E 1 E E 0 0 virtual state +ħω 0 Q Vibrational level spacing R ħωr Q scattered photon has energy E = E ħω S L R (Stokes Raman) creation of multiple vibrational quanta ES = EL nħωr, n =,3,4.. possible, but typically weak

9 Thermal (Boltzmann) occupation of vibrational levels Thermal Level Occupation E ρ exp kbt kbt I E 0 + ħω E 0 creates unbalance between Stokes and anti-stokes sidebands I Stokes Anti-Stokes ħω R = exp kbt k B T at room temperature is about 00/cm. Thermal occupation of vibrational levels above 500/cm is negligible Dominantly Stokes scattering in the characteristic ( /cm) and C-H,O-H ( /cm) range

10 Flourescence & Raman Vibrational thermalization with environment (molecule motion transfers to surrounding) Surrounding thermal bath E 1 ~ ev photon absorption Vibrational Thermalization Raman Fluorescence photon emission E 0 ~0.1eV picoseconds nanoseconds time Raman: Emission before relaxation of excited state vibration : energy shift represents ground state vibration Fluorescence: Emission after thermalization of excited state vibration: energy shift represents combination of many vibrational modes

11 Non-resonant Raman Avoid flourescence which leads to background & photodamage (molecule stays a long time in excited state, and can use its energy for destructive actions) Use non-resonant excitation with E k T B ~ 6meV~00/cm E 1 E ~ exp kbt E virtual state 1 E Stokes (=real state at wrong energy) ( ) I E 4 S E 0 picoseconds nanoseconds time

12 The Raman cross section Rate of Raman scattering I L : Laser intensity (Power per area) ν I dσ L R R = dω ħωl dω σ R Differential cross-section per solid angle dω dσ R = dω ( ω ω ) ( 4πε ) 4 n L R R 4 0 nl c α R Typical crosssection 10-9 cm per mode (diameter 0.03fm, 100 times smaller than C nucleus) Quantum-mechanical expression for Raman polarizability : Sum over intermediate states j f µ ε R j j µ ε L i f µ ε L j j µ ε R i αr = + ( ) ( ) j ħ ω ji ωl + i Γ ji ħ ω jf + ωr + i Γ jf µ Dipole operator R,L ε Polarization of Laser & Raman

13 Biological Raman Spectra Biophysics J. 84, 3968 (003), Biopolymers 7, 1 (003) Raman spectrum of the nucleus of a single fixed peripheral blood lymphocyte (cw 30mW, 60s)

14 Confocal Raman Micro-spectrometer Spatial resolution, smallest detectable object: ~ µm 3, or about g 633, 514 & 488 nm, 30 mw Back-thinned cooled CCD Piezo-electric microscope stage, 3 nm resolution 60X water & 100X air objectives (courtesy of Max Diem, Northwestern)

15 Nucleus of a cell in buffer mw CW 60x water immersion objective, NA = 1 3 sec (courtesy of Max Diem, Northwestern)

16 Raman imaging Raman spectroscopy is chemically sensitive and label-free Water is transparent in the visible and near IR and gives a weak Raman background: non-resonant Raman spectroscopy is good for deep penetration and in vivo study Raman spectroscopy can be integrated with confocal microscopy for high spatial resolution BUT Spontaneous non-resonant Raman signal is weak: Cross-section typically times lower than fluorescence 10 5 to 10 6 times less intense than Rayleigh scattering of single molecules Raman images typically require high laser power (>10mW) and long acquisition times (>30min)

17 Enhancing Raman Resonant Raman: If the virtual state is close to a real molecular state, the scattering cross section can be greatly enhanced (up to a factor 10 6 ). Can detect concentrations as low as 10-8 M but need UV excitation: cannot be used for deep penetration and gives rise to autofluorescence background. Non-resonant Raman can be strongly (up to ) enhanced by attaching molecules to metallic nanoparticles (e.g. Ag, Au nm). Surface-enhanced Raman scattering (SERS) can be used for single molecule detection but require a special sample preparation and are not suited for imaging Nonlinear Coherent Raman Scattering: a coherent nonlinear laser technique that typically offers efficiencies 10 5 higher than those of non-resonant spontaneous Raman

18 Spontaneous Raman Scattering One excitation field with frequency ν L and intensity I L The total signal is the incoherent superposition of fields generated by an ensemble of N oscillators ( ) I E E E E E E Raman 1 + = 1 + t 1 + t t = 0 t I R ~ N I L N: number of molecules in field

19 Create Coherent Vibrations with Light Green light has a frequency of 600THz, 10 times higher than molecular vibrations Use interference of two light waves to drive vibration by the difference in frequency field amplitude 990Hz 1000Hz time Hz (10Hz difference) Hz (1 Hz difference)

20 Nonlinear Coherent Raman Scattering (CRS) Two optical fields with frequency ν P and ν S excite coherent molecular vibration at ν P - ν S ( ) ( ) CRS 1 + = 1 = 4 1 t t I E E E E t ( ) ICARS N IP IPIS excitation Square in N: Large signals for high densities of molecules

21 Nonlinear Raman Scattering: Equation of Motion E( t) Q ɺɺ α + Q ɺ + Ω Q = F m = E t m R γ p ( ) The intensity ~E is proportional to the driving force The Greens function for this equation of motion is The energy of the dipole moment in the external field is: 1 1 Wp = pe = αe The derivative of the energy with respect to the coordinate defines a force: ( ) ( γ ) G t = Ω Θ t Ω t t 1 ( ) d ( )sin d exp t α R ( ) = ( ') ( ') ' Q t G t t E t dt m 1 F = W = E α p Q p R This vibration gives rise to the coherent Raman scattering polarization Ω = d p Ω γ = α QE R

22 Nonlinear Raman Scattering: Frequency picture The Greens function is varying only slowly compared to the light field oscillation, so that the vibration is driven by the envelope modulation of the light field. In frequency space, the Greens function has a nearly lorentzian shape Assume that the incident field is the superposition of two fields with different frequencies ω P >ω S : ( ) p,s E = R E + E E = E e ω P E E * P S The driving force has then the two components, and P S plus components at the sum frequency and at zero frequency which we neglect p S = α QE P,S P0,S0 * E E The induced dipole moment has frequency components at Frequency ω P ω S ω P ω S ω S - ω P Q ( ) 1 i t G( ω) = G( t) e ω dt = ω Ω iγ Name CARS SRL SRG CSRS Coherent anti-stokes Raman Scattering Stimulated Raman Loss (imaginary part is leading to loss) Stimulated Raman Gain (imaginary part is leading to gain) Coherent Stokes Raman Scattering i t Amplitude E P E S E S E P E P E S E S E P

23 SRL, SRG and Susceptibility Note that the SRL field has the same amplitude as CSRS, and the SRG field as CARS The induced polarization of SRL and SRG has the same frequency as one of the exciting fields. We can therefore interpret it as a change of the linear susceptibility for the field E P or E S repectively, which is given by G( ω) SRG I( G( ω) ) 1 ( ω ω ) Ω P S SRL R( G( ω) ) The real part of G is an intensity-dependent refractive index (Kerr effect), resulting in non-resonant background at low frequencies

24 Coherent Vibrationally Resonant Signals Excitation creates coherence between ψ 0 and ψ1 ψ = ψ + ψ 1 i Rt a 0 a 1e ω a E E P typically S a 1 CARS SRG SRL CSRS E 0 ψ ψ 1 0 EP ES Real intermediate state: Resonance & Memory

25 Coherent Non-Resonant Signals E 1 Non-resonant CARS Non-resonant SRL (Kerr) Non-resonant SRG (Kerr) Non-resonant CSRS E 0 ψ ψ 1 0 Virtual intermediate states: Instantaneous response

26 Nonlinear Raman Scattering: What to detect Since all the third-order Raman fields have similar amplitude, it is a matter of experimental convenience what to detect. Detection of the field in amplitude and phase gives the maximum possible information, but needs an interferometric detection scheme CARS & CSRS: free from pump-background CSRS frequency has incoherent Raman and Flourescence background CSRS has the smaller frequency, which can be problematic for detectors CARS : Non-resonant background interferes. Solutions are: in multiplex detection, use spectral shape to determine non-resonant background and CARS field separately Use propagation effects (epi-detection, wide-field CARS) Use vectorial field effects : background and resonances may have different polarization suppress background by analyzer Probe the difference between two frequencies by temporal modulation (FM-, Differential-, Dual- CARS) heterodyne detection measuring the amplitude and phase can distinguish resonant and nonresonant components (Heterodyne CARS)

27 Nonlinear Raman Scattering: What to detect SRG & SRL: Excitation field background, typically more than signal field. Excitation field background has a given phase relation to the signal ( linear susceptibility ), and provides a phase-stable homodyne-field. The intensity change is proportional to the imaginary part of the Greens function, and is thus proportional to the spontaneous Raman intensity. Incoherent background negligible due to large heterodyne gain Problem: a change of the intensity by 10-5 is also affected by classical laser noise and spatial sample transmission. Solution: Use a modulation technique (amplitude modulate the non-detected excitation field). Adds to experimental complexity The signal is created only half of the time The modulation frequency is limiting the detection bandwidth

28 Maximum signal and sensitivity We estimate the maximum signal achievable using the stimulated Raman process and a quantum-electrodynamics picture with photons The spontaneous Raman process creates both a vibrational quantum and a photon from the vacuum state. Photons are bosons, such that the transition rate is proportional to the final state occupation (Vibrational quanta are often significantly anharmonic - no bosons in general) Consider a single vibrational mode of a single molecule in the focus volume Spontaneous Raman: # Spontaneous Raman photons σ n = n R A Sp # pump photons p F Raman cross-section for the mode Lateral Focus area Number of stimulated photons Stimulated Raman: nst = nspnm Number of photons in Raman field mode Assume that the pulse spectrum is equal to the Raman Gain linewidth of, with a fourier-limited pulse duration τ = π resulting in perfect temporal mode overlap. Consider a single pulse. Estimated using Boyd, Nonlinear optics, chapter 10.

29 Maximum signal and sensitivity n m = Am n A π F S Single mode Area #photons in Stokes pulse Stokes wavelength A m λ = Focus Area Stimulated Photons per pulse: λ σ n = n n 4π A R St p s F One stimulated photon per mode saturates the process: n n n n A λ F St = 1 sat = p s = 4π σ R Numerical example: σ R =10-30 cm, τ=1ps, λ=1µm, A F =(0.5 µm) Number of photons in pump and Stokes pulse at saturation n 11 sat 10 Corresponds to a pulse energy of 0nJ, average power at 10MHz repetition rate 0.W Estimated using Boyd, Nonlinear optics, chapter 10.

30 Maximum signal and sensitivity Transfer efficiency is per mode & molecule at saturation Spontaneous Raman: 10-1 per mode & molecule Enhancement by photons in Stokes Raman Mode Saturation also implies a coherent driving of the vibration into the excited state, which corresponds to a pulse area of π in the Bloch-picture of a two-level system. The pulse area and thus the coherence in the system is proportional to the pump * - Stokes interference E P E S The estimated pulse energy at saturation is times larger than observed damage thresholds CARS experiments are thus normally in the small perturbation limit with a coherence of 1-10% (off-diagonal density matrix element), where coherent quantum control techniques (Rabi-oscillations, adiabatic passage, etc) are not applicable. 1 At the damage threshold, npm = nsm = 10 nsat 10 - of the modes are excited, corresponding to a temperature increase of 40K for a mode of 3000cm -1

31 Shot-noise limited Signal to Noise Shot noise: Noise in the detected photocurrent due to the randomness of photon detection in a coherent field. The Poisson statistics of the detection events results for an average of n detected photons in a standard deviation of n The shot-noise limited signal to noise ratio in SRG detection for a single pulse and single mode is SNR n n n n n = St P S n = using n p s St = n S sat nsat np ns For N m modes and N L sequential pulses: SNR = Nm NL n sat Using n = n = 10 n and : SNR N N n = = Pm Sm sat sat 10 m L 10 Need 10 6 vibrational modes in the focal volume for single pulse detection

32 Signal to Noise: CARS versus SRG and RS for CARS, the field is equal to SRG, nst = ESRGES = ESRG ns so that we can estimate the number of CARS photons for one mode n n n n St P S CARS = ESRG = = 4 ns nsat For N m modes and N L sequential pulses: N = N N n CARS m L CARS SNR= N = N N n n P S CARS m L nsat equal to SRG CARS versus spontaneous Raman scattering (RS) N N CARS Sp λ = N n n σ m P S R 4π AF At saturation = N m At damage threshold λ 4π A At saturation CARS and RS from a single mode are similar for a diffraction limited focus F = 10 N m λ 4π A F

33 Relation with concentrations Focal volume (0.5µm) femtoliter (fl). 1 Molar 10 8 molecules in the focal volume Each molecule can have similar vibrational modes which are spectrally overlapping (e.g. CH in lipids, similar modes) At the assumed damage threshold (0mW at 10MHz, 0.1 of saturation) we estimate: spontaneous Raman photons per pulse and mode, so per second and mode 3 photon/s for 10 µm solution: Practical detection limit is mm CARS photons per pulse and mode, scaling quadratically with the number of modes. 1 photon/second for 10 3 modes in the focal volume, or 100 µm for 1 mode per molecule, or 1-10 µm for lipids photons/second for 1-10mM lipids, good for high-speed imaging Note that all these estimates scale with the assumption for the saturation intensity, the pulse repetition rate, and the Raman crossection (which can be enhanced close to electronic resonance by 4 orders of magnitude)

34 Additional limits and noise in real detectors Detection quantum efficiency QE, the probability that an emitted photon is detected (normally that means absorbed). Typical detector values: Photomultipliers: , determined by photocathode material Photodiodes, CCD cameras: , determined by surface reflection & structuring, and absorption depth compared to intrinsic region Any additional optical elements give additional factors to the detection QE Spectrometers: Mirrors: Bandpass filters: Objectives Transmission, Overlap between angular collection range and signal emission range Excess noise factor f e, due to randomness of avalanche amplification in photomultipliers, avalanche photodiodes, and EM-CCDs. Typical value of. Can be avoided using photon counting or saturated amplification in PMTs (limits the maximum detectable photon flux). Thermal charge generation (dark noise), n d electrons, linear with integration time. Cosmic rays (mainly in scientific CCDs, which have a large active material volume and a very sensitive current amplifier (read noise 3-0 electrons)

35 Additional noise in real detectors Johnson Nyquist noise in current amplifiers (thermal noise in resistors) I q RMS RMS 4kBT kbt = f = R R t = B k T t R t f = Integration time 1 Bandwidth Number of electrons For T=300K, R=1MΩ, t=1µs: 570 electrons Requires a detector capacity C< t/r=1pf For T=300K, R=50Ω, t=1µs: electrons σ = J q RMS e Total detector noise in electrons: σ = σ + f ( n QE + n ) J e P d SNR n QE P = < σ J + fe( npqe + nd ) n P

36 The coherent process is time-ordered Excitation pulse sequence first excitation of the vibrational coherence Q by the intensity, then probing of the coherence Q by the field t αr ( ) = ( ) ( ') ( ') ' p t E t G t t E t dt m The optimum excitation can be defined to produce the maximum amount of useful information for a given photodamage The information is in the system susceptibility χ S, which in general contains a weighted sum of all system vibrations and a non-resonant electronic part α χ ( t) G ( t) χ δ ( t) R S = j + e0 j mj Due to the resonant nature of G, information about individual modes is localized in frequency domain. For single channel detection it is therefore advantageous to measure a localized range in frequency domain as compared to the time domain. Note that the non-resonant electronic background is localized in time-domain.

37 Excitation pulse sequence: Single frequency For time-overlapping unchirped pulses much longer than the memory time (dephasing time) of the excitation, the time-ordering effect is not significant, and the scattered field is determined by the frequency domain representation α χ ω χ ω γ Q S( ) = e0 + Ω j j j m j ( i ) This is the picture which is mostly used. However, time-ordering makes a difference as typically pulses of length similar to the memory time are used. 1 Consider Gaussian pulses of duration and equal linear chirp ( G ) ( G ) E ( t) = α E exp ( t / τ ) + it( ω + βt) 1/ P β P0 P E ( t + t ) = α E exp ( t / τ ) + it( ω + βt) 1/ S 0 β S0 S τ G = α τ Stokes delay Conserves pulse energy I( t) dt τ G0 E0 independent of width β G0 αβ β = + τ β 1 G W.Langbein et al, J. Raman Spectr. 40, 800 (009)

38 Wigner distribution Time-frequency representation Integrating over one of the coordinates gives the intensity versus the other. * I(, t) = E( t ) E ( t + ) e iτω d ω τ τ τ I( t) = I( ω, t) dω I( ω) = I( ω, t) dt Unchirped degenerate CARS ω CARS ω Chirped degenerate CARS CARS ω P Pump ω P Pump ω S Stokes ω S Stokes t t There are more versions of such spectrograms used in signal processing

39 Example of pulse interference 1ps FWHM Gaussian pulses, Pump at 800nm, Stokes at 1037nm (difference 845cm -1 ), equal Amplitudes ( R ( E )) P + Es 4 E P + E s # # # #10-1 t

40 Example of pulse interference Interference at ω P ω S * ( P S ) R E E 1 1.# # # # 10-1 t 1 1ps FWHM Gaussian pulses, Pump at 800nm, Stokes at 1037nm (difference 845cm -1 ), equal Amplitudes

41 Single frequency excitation: driving force ( ) αq EP0E S t t t F( t) = exp + itω D m α β τ G τ G Centred at the instantaneous frequency difference (IFD) ω = ω ω + βt D P S 0 Power spectrum of the driving force ( ) αq πτ G0 t ω ω 0 D τ G SF( ω) = F( ω) = EP0ES 0 exp m τ G 4 The drive spectral intensity at resonance is independent of the pulse width for constant pulse energy The drive spectral width is inversely proportional to the pulse width, 1ps FWHM pulses correspond to 1cm -1 spectral width Shorter drive pulses excite a wider vibrational frequency range without reduction of the drive amplitude. This allows for multiplex excitation without power penalty.

42 Single frequency excitation: vibrational response The vibrational response Q(t) for these driving pulses can be calculated analytically for a resonance of exponential and Gaussian dephasing. Here we show exponential dephasing. detuning ω δ τ G T =τ G Q r T = Q r Delayed response of vibration Readout by the pump determines the measurement time of the vibrational response and leads to a spectral broadening. When driven non-resonantly, the vibration responds instantaneously Larger signal for Pump after Stokes due to causality and probing of vibration by pump (in CARS) 0 0 t 0 =-τ G t 0 =0 t 0 =τ G Q r ε P for T =τ G Q r ε P for T = Q r for T =τ G Q r for T = SC0( ω ) = Q( t) EP( t) dt W.Langbein et al, J. Raman Spectr. 40, 800 (009) ε S ε P time (t/τ G )

43 Optimum delay t 0m Maximum CARS signal SC0( ω ) = Q( t) EP( t) dt is created if the Stokes arrives before the Pump, at t 0m <0 CARS S C0 (µ τ G0 3 ) 0.7 T /τ G = delay t 0 /τ G W.Langbein et al, J. Raman Spectr. 40, 800 (009)

44 Choice of pulse duration Key parameter is the vibrational coherence time T 10 0 CARS intensity at optimum delay decreases with τ G Selectivity ρ (ratio of resonant to nonresonant CARS) increases with τ G S C0 (t 0 ')/S C ρ(t 0 ')/ρ Optimum delay relative to τ G and resulting signal enhancement κ compared to zero delay decreases with τ G /T κ -t 0 '/τ G τ G /T

45 Non-resonant Background: CARS intensity lineshape αr χs( t) G( t) χe0δ ( t) m χ = ω = e0 1 R 1 α χ ω χ ω ω γ m = + R ( ) S( ) = e0 + R i χ( ω) γ = 0.05 α R = m 0.0, 0.05, 0.15 α R = γ = m χ( ω) , 0.05,

46 Single frequency picosecond CRS setup Evans & Xie, /annurev.anchem

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48 SRL Microscopy /science picosecond OPO pulses, Stokes modulated at 1.7MHz, shot Noise 170us pixel time ~10-5 Pump modulation detected by Lock-In 8 10 Hz

49 Video Rate SRL Microscopy /science Higher efficiency detection of backscattered light by photodetector in front of objective Faster modulation (0MHz), 100ns pixel dwell time Tissue (higher signal, higher damage threshold)

50 DOI: /NMETH.1556 Monitoring of Lipid metabolism (SRS) Lipid staining dyes do not give a good representation of lipid distribution