Physics of 4 Wave Physics

1. 4 Wave Mixing
Erich Wanzek

2. Non Linear Optics
• The optical response of a dielectric optical medium is described by
the relationship between the polarization density vector and electric
field vector at each point in an optical medium.
• A material is a nonlinear material if the relationship between the
polarization density vector and electric field in a medium is nonlinear.

4. Nonlinear Wave Equation
• The propagation of an
electromagnetic wave in a
nonlinear medium is described by
Maxwell’s equations for a dielectric
homogeneous, isotropic medium.
• The wave equation in a nonlinear
dielectric medium is than described
by the nonlinear wave equation
• the inhomogeneous driving/source
term of the differential equation
allows for the radiation of
electromagnetic fields with
frequencies that were not present
in the incident electromagnetic
optical wave
𝛻2
ℇ −
1
𝑐2
𝜕2
ℇ
𝜕𝑡2
= 𝜇0
𝜕2
𝑃 𝑁𝐿
𝜕𝑡2
𝛻2ℇ −
1
𝑐2
𝜕2
ℇ
𝜕𝑡2
= S

5. 4 wave mixing
• Four-wave Mixing (FWM) is a third order nonlinear optical effect with
respect to the polarizability of the material. A third order nonlinear process
is described by the third term
• Four wave mixing is a frequency matching condition in which two optical
electromagnetic waves that have two different frequencies interact to form
another two optical waves of different frequencies.
• Four-wave mixing can occur when two different frequency components of
an electromagnetic wave propagate together in a nonlinear medium.
• With two frequency components, a refractive index modulation at the
frequency difference between the two occurs and this creates two
additional frequency components.

6. 4 wave mixing
• A superposition of three
frequency component
electromagnetic waves in a
medium can be expressed
as
𝐸(𝑡) = 𝐸 𝜔1 𝑒 𝑗𝜔1 𝑡
+ 𝐸 𝜔2 𝑒 𝑗𝜔2 𝑡
+ 𝐸(𝜔3)𝑒 𝑗𝜔3 𝑡
𝐸 𝑡 =
𝑞=−
+1,−
+2,−
+3
3
1
2
𝐸(𝜔 𝑞)𝑒 𝑗𝜔 𝑞 𝑡

7. 4-wave mixing
𝐸 𝑡 =
𝑞=−
+1,−
+2,−
+3
3
1
2
𝐸(𝜔 𝑞)𝑒 𝑗𝜔 𝑞 𝑡
𝜀0 𝜒 3 𝐸3
𝑃 3 =
1
8
𝜒 3
𝑞=−
+1,−
+2,−
+3
3
𝐸(𝜔 𝑞 𝐸(𝜔 𝑟)𝐸(𝜔𝑙)𝑒 𝑗(𝜔 𝑞+𝜔 𝑟+𝜔 𝑙)𝑡

8. 4 wave mixing
• oscillating fields of the waves can be
mixed in six different permutations by
the relation
• Frequency coupling condition
• phase-matching condition for a four
wave-mixing
𝜔1 + 𝜔2 = 𝜔3 + 𝜔4
𝒌1 + 𝒌2, = 𝒌3 + 𝒌4
𝑃 3
𝜔1 + 𝜔1 − 𝜔3 = 6𝜒 3
𝐸(𝜔1)𝐸(𝜔2)𝐸(3

9. Optical Phase Conjugation(OPC)
• Real time holography
• Phase conjugation mirrors

10. Phase conjugation mirrors

11. Optical Parametric Amplification (OPA)