Lecture 25
Optical Wave Mixing in Two-Level Systems (1) Saturation Spectroscopy setup:
strong pump weak probe
+
atomic vapor
+
measure transmission of probe wave
One determines how the response of the medium to the probe wave is modified by the presence
Lecture 24
Rabi Oscillations and Dressed Atomic States Intense field - population changes significantly, Stark shift induced by the laser field profoundly modifies the energy-level structure leading to new resonances in the optical susceptibility. No rela
Lecture 23
Optical Bloch Equations
ba (t ) = ba (t ) exp ( it )
ba
= ( u iv )
1 2
It can be shown that the motion equations for and are equivalent to:
u = v u T21 v = u v T21 + Ew w = ( w weq ) T11 Ev
(1)
w = wba
= 2ba 1
Bloch equations consider (in
Lecture 22
Nonlinear Optics in the Two-Level Approximation In simple cases we use power expansion:
b
a
P (t ) = 0 (1) E (t ) + 0 (2) E (t ) 2 + 0 (3) E (t )3 + .
But does not work for all cases, such as Saturable Absorption.
=
0
1+ I Is I = 2n 0 c E 2
0
Lecture 21
Semiconductor Nonlinearities Semiconductors have allowed electronic energy states in broad bands. - The lower mostly filled valence bands. - Upper empty or mostly empty conduction bands. - Bandgap Eg energy is difference between highest valence
Lecture 20
Quantum Mechanical Model of Nonresonant Electronic Nonlinearities:
k j i gn nm ml lhg N (3) P kjih ( ; r , q , p ) = 3 F 0 lmn ( ng )( mg q p ) l g p
(
)
(1)
where = r + q + p
input input and output frequencies. The Cartesian indices are to
Lecture 19
Intensity-Dependent Refractive Index (a)
n = n0 + n2 E 2
time average
(1)
n0 - linear refractive index n2 - intensity dependence of refractive index E (t ) = E ( ) exp ( it ) + c.c.
2 E 2 = 2 E ( )
n = n0 + 2n2 E ( )
2 2
(2) (3) (4)
P NL ( ) =
Lecture 18
Second Harmonic Susceptibility: Resonant Case
Vij = (i | er E | i ) = eE r i* (r ) i (r )d 3 r = 0
(0) 11 = 1
2
i nm = inm nm V , nm nm nm i nm = inm nm (Vnv vm nvVvm ) nm nm v i i i i (1) (0) + i21 + 21 21 = V21 11 V23 31 + 22V21 + 23V31 t
Lecture 17
Density Matrix Calculation of the Linear Susceptibility
(1) nm (t ) = exp ( inm + nm ) t dt t
V (t ) = E (t ) (omit the vector signs) (0) nm =0 for n m E (t ) = E ( ) exp ( i t )
p p
i V (t ), (0) exp ( inm + nm ) t nm (1)
(0) (0) V , (0) = Vn
Lecture 16
Density Matrix Formalism
i s = H s
H = H 0 + V (t )
( s (r , t ) = Cn s ) (t ) un (r ) n
Schrdinger equation for quantum state s Hamiltonian operator
(1)
(2)
H 0 u n ( r ) = En u n ( r )
* um (r ) un (r ) d 3 r = mn
energy eigenfunctions ar
Lecture 15
(1)
= a u (r ) exp ( imt )
(1) mm m
mg E ( p ) 1 a (t ) = exp i (mg p ) t p mg p
(1) n
Linear Susceptibility Linear optical properties of a material system. The expectation value of the electric dipole moment:
p = | |
(19)
(0) (r , t ) = u g
Lecture 14
Quantum-Mechanical Theory of the Nonlinear Optical Susceptibility Using the laws of quantum mechanics explicit expressions for susceptibility motivation: a) functional dependence on material parameters (dipole transitional moments, atomic energ
Lecture 12
Nonlinear Optical Interactions with Focused Gaussian Beams Considered in the approximation in which all interacting waves are infinite plane waves. In practice, the incident radiation is usually focused. Paraxial Wave Equation n 2 2 En 1 2 Pn 2
Lecture 11
If highly reflecting mirrors are placed around the nonlinear medium to form an optical resonator, gain of the parametric amplification process can occur. This is known as a parametric oscillator (PO).
3
1 2
1
Second Harmonic Generation (SHG)
E
Lecture 10
Manley Rowe Relations For lossess media
I i = 2ni 0 cAi Ai*
* dAi dI i dAi* = 2ni 0 c Ai + Ai dz dz dz
1 2 3
(1)
dI1 = 4 0 d eff 1 iA3 A1* A*2 exp(ikz ) + c.c. = 8 0 d eff 1 Im A3 A1* A*2 exp(ikz ) dz dI 2 * = 8 0 d eff 2 Im A3 A1* A2 exp( ik
Lecture 9
Phase-Matching Considerations (PMC) For SFG the intensity of the generated field varies with wave vector mismatch. sin 2 ( kL 2 ) I 3 = I 3 (max) 2 ( k L 2 )
k = 0 is difficult to achieve because the refractive index of materials that are lossle
Lecture 8
The Wave Equation for Nonlinear Optical Media Laser field induces polarization at new frequency components. Generation of new components are described by Maxwells equations.
1
2 1 + 2
3 = 1 + 2
1 + 2
dipole radiation pattern
N atoms: if the rela
Lecture 7
Time-Domain Description of Optical Nonlinearities Frequency-Domain (FD) descriptions of nonlinearities apply to monochromatic input fields. FD relates the nonlinear polarization P( ) to the input field E ( ) Time-Domain (TD) descriptions of nonl
Lecture 6
Symmetry Properties of the Nonlinear Susceptibility
Consider mutual interaction of three waves: 1 , 2 , 3 = 1 + 2
(1 = 3 2 ; 2 = 3 1 )
( 2) Pi (n + m ) = 0 ijk (m + n , n , m ) E j (n ) Ek (m ) jk ( nm )
(1)
Then need to determine six tensors.
(
Lecture 5
Centrosymmetric Media
2 Frestoring = m0 x + mbx3
2 4 U = Fdx = 1 m0 x 2 1 mbx 4 2
(1)
Tensor properties of susceptibility cannot be specified unless internal symmetries of the medium are completely known. Consider isotropic centrosymmetric mater
Lecture 4
Nonlinear Susceptibility of an Anharmonic Oscillator Approach: - Lorentz (harmonic) oscillator: A good approximation of the linear response of an atom to an incident electric field. - Add additional nonlinear restoring force term => Differential
Lecture 3
Definition of Nonlinear Susceptibility Given: Note: Material with dipsersion and/or loss * is complex and frequency dependent. * Prime () in (1) denotes sum over only positive frequencies
E (r , t ) = n E n ( r , t ) + c.c. = n E n (r , t )
'
(1
Lecture 1
Introduction Nonlinear Optics : phenomena that occur as a consequency of the modification of the optical properties of a material system by the presence of intense light. Nonlinear since I n Lasers: Maiman 1960 Basov, Prochorov 1960 Franken 1961
Lecture 26
Self-Focusing of Light
Gaussian beam profile
n = n0 + n2 I
Optical length
Laser beam will tend to be brought to a focus by the action of this lens.
short medium
long medium
(damage to material)
1
Self-Trapping of Light The tendency of the beam