Nonlinear Optical Microscopy

X

Y

Non-linear?

Actual response can be written as

y = c1 x+ c3 x3 (this is called a cubic distortion)

Assuming the input is a periodic signal x = cos (t)

y=c1 cos(t)+c3[cos (t)]3

Trigonometric identity tells us

[cos (t)]3 = (3/4) cos(t) + (1/4) cos(3t)

The output is thus given by

y=[a1+(3/4)c1] cos(t)-(1/4)c3cos(3t)

Thus a small cubic nonlinearity gives rise to a modified response at w but also generates a new signal at 3w

Nonlinear response

1. Applied field distorts the cloud and displaces the electron

2. Separation of charges gives rise to a dipole moment

3. Dipole moment per unit volume is called the polarisation

P = c1 E ; P is the polarization

c1 is called the linear susceptibility

This describes linear propagation giving rise to speed of propagation through the medium (real part) absorption in the medium (imaginary part)

It can be shown that

C1 = n - 1

where n is the refractive index of the medium

Linear polarization

Nonlinear polarizationA more realistic equation for polarisation is

P = (1) E + (2) E2 + (3) E3 +

where (2), (3) etc are the second and third order nonlinear susceptibilities

Normally,

(3) E3 << (2) E2 << (1) E

Unless, E is very very big.

Symmetry arguments can be used to show that for isotropic materials even order susceptibilities are zero

Typical Nonlinear Optical Phenomena

• Second Order Processes– Second Harmonic Generation– Sum-Frequency Generation

• Third Order Processes– Multi-Photon Absorption*– Stimulated Raman Scattering– Optical Kerr Effect– White Light Generation

Interaction of Light with Matter...3)3(2)2(1)1( EEEP

P = induced polarization,(n) = nth order non-linear susceptibilityE = electric field 

Linear Processes ·    Simple Absorption/Reflection ·   Rayleigh Scattering

(3) << (2)<< (1) (5-7 orders of magnitude per term)

Second Order Processes

·      Second Harmonic Generation*

·      Sum-Frequency Generation

Third Order Processes

·      Multi-Photon Absorption*

·      Stimulated Raman Scattering

·      Optical Kerr Effect

·      White Light Generation

One and two photon absorption physics

Requires high power:Absorption onlyIn focal plane

Greatly Reduces out of plane bleaching

Simultaneous absorptionVirtual State:Very short lifetime ~10-17 s

Goeppart-Mayer, ~1936

e.g. fluorescein

One Photon 2 photon

Absorptionprobability

AbsorptionCoefficient units

(50,000)

(10-16 cm2)

(10-50 cm4s)

10-50 cm4s=1 GM (Goppert-Mayer)

Power (photon)dependence

p P2 (gives rise to sectioning)

Laser Temporaldependence

none 1/

p p2 /

One and 2-photon absorption characteristics

Cannot use cw lasers (Ar+)

Xu and Webb, 1996

Slope of 2 atAll wavelengths:2-photon process

Fluorescein and rhodamine

Power Dependence

2-photon excitation of fluorescein: 3D confinement

Absorption, Fluorescence only in middle at focal point

Compare 1 and 2-pAbsorption1-p excites throughout

Radial PSF Axial PSF

Comparable Lateral and Axial Resolution to confocal

Cro

ss s

ectio

n G

M

Max 820 nmnot 1050 nm

Two-photon Absorption Spectrum

10 SS Nominally forbidden in 2-p

20 SS Nominally forbidden in 1-p:Allowed and stronger in 2-p

Rhodamine Photophysics

10-12 s

1000 nm TPE

500 nm OPE

800 nm TPE

400 nm OPE

10-9 s

S0

S1

S2

800 nm stronger than 1000 nm band

Reverse of 1-photonFor all xanthenes:Fluorescein,rhodamines

All max ~830 nmNot ~1000 nm

1 and 2-photon bands

Same emission spectrumfor 1-p, 2-p excitation

Relaxation is independent ofMode of excitation

Same emission spectrumFor different 2-p wavelengths:750 and 800 nmJust like 1-photon emission

Xu and Webb, 1996

Emission Spectrum

1) Emission spectrum is the same as 1-p

2) Emission quantum yield is the same

3) Fluorescence lifetime is the same

4) Spectral positions nominally scale for the same transition: 2-p is twice 1-p wavelength for

5) Selection rules are often different, especially for xanthenes(fluorescein, rhodamine and derivatives)

Some Generalities about Multi-photon absorption

Non-decanned Detection

White, Biophys J, 1998

Confocal (1-p)<2-p descanned< 2-p direct

2-p direct collects ballistic and scattered photons

X-Zprojection

Non-descanned Detection Increases Sensitivity

White, Biophys J, 1998

1-p

2-p

Improved Imaging Depth Due to Reduced Scattering

All images are descanned

Problems can arise from high peak power giving rise to unwanted non-linear effects  Plasma formation leading to cell destruction (makes holes)  Accidental 3 photon absorption of proteins and nucleic acids (700-800 nm) (abnormal cell division) ~ 10 mW at 1.4 NA is good limit at sample(Scales for lower NA)

Piston, Biophys J. 2000

488 nm 1-photon

Slope=1.2

Bleaching of fluorescein dextran in droplets

710 nm 2-photon

Slope=1.9 (low power)

Piston,2000

NADH=3.65Coumarin=5.1

Indo-1=3.5

Highly nonlinear:Higher order processesExcitation to higher states

Non-linear bleaching (ctd)

For same transition 2-pDoes not bleach moreThan 1-p!

Applications

Autofluorescence of endogenous species in tissues

Need multi-photon excitation, non-descanned detectionFor enough sensitivity: small cross sections and quantum yields

Autofluorescence in Tumors

Mitochondria:NADH, Flavins

NAD not fluorescentNADH emission toMonitor respiration

NADH good diagnosticOf cell metabolism

Small cross sectionQuantum yield ~10%Small delta ~0.1 GMHigh concentration

Need non-descannedDetection to be viable

Imaging Muscle (NADH)With TPE Fluorescence

Low cross section butHigh concentration

Balaban et al

Strata corneum

Keratinocytes

Dermal layer(elastin, collagen)fibers

Human Skin Two-photon imaging

So et alAnn. Rev. BME2000 More versatile than dyes (but weaker)

MPM enabling, very weak in confocal

Multiphoton bleaching

Need 3D treatment, both radial, axial PSF

Two-photon cross section measurement

Xu and Webb, 1996

Measure by fluorescence intensity, need quantum yield(same as 1 photon)

Measure wavelength

Measure pulse widthMeasure power

MeasureFluor.

22

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hcNAPna

Control power