Fundamentals of Photonics – Saleh and Teich 1991 and 2006 Nonlinear Optics – R.W. Royd Principles of Nonlinear Optics – Butcher and D. Cotter Applied Nonlinear Optics – Zernike and Midwinter Handbook of Nonlinear optics – Sutherland 2003
Great web sites – used lots of figures/slides from these
What are nonlinear optical effects? Example: Sending infrared light into a crystal yielded this display of green light (second-harmonic generation):
Nonlinear optics allows us to change the colour (frequency) of a light beam, to change its shape in space and time, and to create ultrashort laser pulses, the shortest events ever made by Man. NL is key element for optical data processing
First demonstration of second-harmonic generation P.A. Franken (1961)
The second-harmonic beam was very weak because the process was not phase-matched.
First demonstration of second-harmonic generation The actual published results…
Why don't we see nonlinear optical effects in our daily life?
1. Intensities of daily life are too weak. Sunlight 102 W/m2 Nonlinear effects ~ 1013 W/m2 2. Normal light sources are incoherent. 3. Some NLO effects require specific crystal symmetries 4. “Phase-matching” is required, and it doesn't usually happen on its own.
What are examples of NLO effects?
Linear optics: shine light of a given frequency on a material and will find light of same frequency at output with perhaps some absorption NLO Examples: the medium adds and subtracts light
SHG Medium
ω (IR)
2ω (Green)
ω1
SHG Medium
ω1+ω2
ω2 THG Medium
ω (IR)
3ω (UV)
Can get sum 1064 nm + 532 nm => 355nm and difference 500nm-700nm=1500nm
Difference-frequency generation ω1
ω2 = ω3 − ω1 Green=UV-Blue
ω3
ω1
ω1
ω3
ω2
Optical Parametric Amplification (OPA) Input high power signal at high frequency ω3 and a weaker signal at ω1 can achieve amplification of the signal at ω1 and generate signal at ω2
And Frequency Sum
ω2 = ω3 + ω1
Non co-linear – third order Spatial effects: Third order example: two different input beams, whose frequencies can be different. So in addition to generating the third harmonic of each input beam, the medium will generate interesting sum and difference frequencies with spatial separation.
ω2
THG medium
Signal #1
2ω1 +ω2 2ω2 +ω1
ω1 Signal #2
Self-diffraction … but the frequencies don’t have to be different to generate new optical fields propagating in different directions…
Signal #1
ω
Nonlinear medium
ω
ω Signal #2
ω
Self-focusing Consequence of OPTICAL KERR EFFECT Collimated beam of light propagating through a NLO medium is brought to a focus: the light causes the medium to act like a lens
Nonlinearity key element for optical switches and optical bistability Optical logic gates, flip-flops
Phase conjugation
Reflection of a plane wave from an ordinary and phase conjugate mirror
Reflection of a spherical wave from an ordinary and phase conjugate mirror
Phase conjugate mirror: Light reflected behaves as if time reversal occurring
Questions Many different types of NLO effects... Why do nonlinear-optical effects occur? How can we describe them? How can we use them? Second Order Effects:
Second-harmonic generation Sum- and difference-frequency generation Autocorrelation
Third Order Non-linear Effects:
Frequency generation, Nonlinear refractive index, Selfphase modulation, Phase conjugation… Consequences and Applications
Nonlinear Response Generic example: describe response with a polynomial
2
3
4
R = ζ 1S + ζ 2 S + ζ 3 S + ζ 4 S + ... Example:
R = ζ 1S + ζ 3 S 3 S = V cos ωt R = ζ 1V cos ωt + ζ 3V 3 cos3 ωt cos3 ωt = 34 cos ωt + 14 cos 3ωt R = (ζ 1V + 34 ζ 3V 3 ) cos ωt + 14 ζ 3V 3 cos 3ωt Small cubic nonlinearity gives rise to modified response at ω and generates a new frequency component at 3ω
Representation of Nonlinearity Linear
R = ξ1S
S R
R
S
s Non-Linear
R
S
R = ξ1S − ξ 2 S 2 S
R
The Fourier components R
The same frequency as the stimulus
Double the frequency of the stimulus
A DC component
Describing light interacting with matter r r r r r ∂B ∇× E = − Maxwell’s equations ∇ ⋅ E = 0 ∂t in a medium: r r r r r r ∂D ∇⋅B = 0 ∇ × B = µ0 J + µ0 ∂t These equations reduce to the wave equation: 2 2 1 E P ∂ ∂ 2 ∇ E − 2 2 = µ0 2 c ∂t ∂t
The
“Inhomogeneous Wave Equation”
induced polarization, P, is what contains the effect of the medium on the EM wave propagation And the form of P determines the solution
Linear optics For low light intensity, the polarization is proportional to the incident field:
c = ε 0 χX Optical polarization of dielectric crystals – mostly due to outer loosely bound valence electrons displaced by the optical electric field. Separation of charges gives rise to a dipole moment P(t)=-Nex(t) Polarization = dipole moment per unit volume Polarization is alternating with the same frequency as the applied E field. Electron oscillates about the equilibrium position – oscillating dipole is a source of EM radiation.
+
-
E
Solving the wave equation in the presence of linear induced polarization For low irradiances, the polarization is proportional to the incident field:
c = ε 0 χX In this simple (and most common) case, the wave equation becomes:
∂2E 1 ∂2E ∂2P − 2 2 = µ0 2 2 ∂z c ∂t ∂t ∂2E 1+ χ ∂2E 1 − 2 = 0; ε 0 µ 0 = 2 2 2 ∂z c ∂t c 1 1+ χ c = ; n = = 1+ χ 2 2 v c v This equation has the solution: X ( z, t ) ∝ E cos(ωt − k z) 0 where ω = v k and v = c /n and n = (1+χ)1/2 The induced polarization, at the same frequency as the incident field and only changes the refractive index. FAMILIAR If only the polarization contained other frequencies…
Lorentz Model Lorentz
model – analogous to a mass on a spring
Electron of mass, m, and charge, e, is attached to the ion by a spring.
r r ∑ F = ma Frestoring + Fdamping + Fapplied = ma dU Frestoring = − dx 2 2 3 4 1 1 1 U = 2 mω0 x + 3 mζ 2 x + 4 mζ 3 x + ...
Cont… U = 12 mω02 x 2 + 13 mζ 2 x 3 + 14 mζ 3 x 4 + ... Frestoring
P(t)=-Nex(t) x(t) is small, harmonic potential regime 2 restoring 0
