Nonlinear Optics:

A very brief introduction to the

phenomena

Paul Tandy ADVANCED OPTICS (PHYS 545), Spring 2010 Dr. Mendes

INTRO

So far we have considered only the case where the polarization density is proportionate to the electric field. The electric field itself is generally supplied by a laser source.

PE= ε0 χ

Where P is the polarization density, E is the electric field, χ is the susceptibility, and ε 0 is the permittivity of free space. We can imagine that the electric field somehow induces a polarization in the material. As the field increases the polarization also in turn increases. But we can also imagine that this process does not continue indefinitely. As the field grows to be very large the field itself starts to become on the order of the electric fields between the atoms in the material (medium). As this occurs the material itself becomes slightly altered and the behavior is no longer linear in nature but instead starts to become weakly nonlinear. We can represent this as a Taylor series expansion model in the electric field.

∞ n 23 PAEAEAEAE==++∑ n 12 3+... n=1

This model has become so popular that the coefficient scaling has become standardized using the letters d and χ (3) as parameters corresponding the quadratic and cubic terms. Higher order terms would play a role too if the fields were extremely high.

2(3)3 PEdEE=++()εχ0 (2) ( 4χ ) + ...

To get an idea of how large a typical field would have to be before we see a nonlinear effect, let us scale the above equation by the first order term.

Pd24χ (3) =1++EE2 +... εχ000E εχ εχ

When we plot this function for two cases: ADP (non‐centrosymmetric quadratic linearity dominant) and CS2 (centrosymmetric cubic nonlinearity dominant) respectively we see that the nonlinearity in both cases is about of size 1, which is the first order term.

This starts to occur when the electric filed is on the order of the linear term at 10^12 V/m in the case of ADP(left) and 10^12 V/m in the case of carbon disulfide(right).

The respective polarization curves:

ADP (linear), and CS2 (quadratic). These nonlinearities are not seen until the electric field is very large. Until the very strong light sources became available (usually from finely focused laser beams) this effect had pretty much gone unnoticed.

Derivation of a nonlinear wave equation for the electric field

We start our derivation as usual with the standard equation for the arbitrary homogeneous dielectric medium.

1 ∂22EP∂ ∇−2 E =μ ct22∂ 0 ∂ t 2

As noted earlier the polarization density can be expressed as a Taylor series. We truncate it to 3rd order and substitute it into our equation above.

∞ 2 n ∂ AE 22(3)3 2 ∑ n ∂ [εχ EdE++ 2 4χ E + ...] 2 1 ∂ E n=1 ()()0 () ∇−E 22 =μμ002 = 2 ct0 ∂∂ t ∂t

222223 2(1 ∂∂∂EEE3)∂E ∇−Ed22 =εμχ00 2 +24 μ0 2 + χ μ0 2 ct0 ∂∂∂ t t∂ t

22 2211∂∂EE ∇−EE{}222 +εμχ00 =∇− 2 ctc0 ∂∂t

22 1 ∂∂22EE∂2 P ⎛⎞⎛⎛⎞∂∂EE 2⎛⎞∂E⎞ ∇−2 Ed =μμ0 +41⎜⎟⎜EE + +2μχ(3) E+2⎟ 2200 2 ⎜⎟⎜2⎜⎟ 0 2 ⎜⎟⎟ ct∂∂ t⎝⎠⎝ ∂∂ t⎝⎠ t ∂∂t⎝⎠ t⎠ We get back our original equation plus 4 extra terms, but these are not “good” terms in that we can still easily solve the wave equation. We immediately see that the new wave equation is nonlinear in the electric field and in the first derivative of the electric field. We are lucky in the sense that it is not nonlinear in the highest order derivative. The extra term are very small compared to the original terms, which means that we possibly could set up a perturbation scheme and make an approximate analytic approximation. A numerical solution would also be possible using a finite difference or finite element scheme.

Our textbook mentions two other solutions, the Born approximation and coupled wave theory. For the sake of this very brief intro, suffice it to say that indeed we can with some effort solve this approximately. But instead of solving it, let’s look at a few properties of this equation and note some applications that exploit these properties.

Second –order nonlinear optics

Second‐harmonic generation SHG

For the case of the material that has negligible nonlinear terms above second order our wave equation simplifies.

2 1 ∂∂22EE∂2 P ⎛⎞⎛⎞∂E ∇−2 Ed =μμ0 +4 ⎜⎟E + 2200 2 ⎜⎟2⎜⎟ ct∂∂ t⎝⎠ ∂∂ t⎝⎠ t

If we view the equation as having a linear part on the right hand side and nonlinear part on the right hand side lest examine what occurs if the nonlinear part is exposed to a simple plane wave electric field. The nonlinear term is:

2 PdENL = ()2 Et()= Re{( Eω ) ejtω } ⎧⎫1 2 2*⎡⎤jωωtj−−t⎡⎤ j ωωωωtj *tj⎡⎤tj*−t PNL ==2 dEdEeEe 2⎨⎬⎣⎦ ()ωω + () = dEeEe⎣⎦() ωω + ()⎣⎦ EeEe () ωω+ () ⎩⎭4 d 2 =+Ee22()ωωωωjtωωω 2() EeEejt * ()−−jt +() E ()*2 ejt ω 2 ( ) d 2 =+Ee22()ωωjtωω() E () *2 e− jt + 2()*() E ωω E 2 ( ) 2 d ⎛⎞1 ⎡⎤22jtωω *2− jt * = ⎜⎟Ee()ωω++() E () e EE()() ωω 22⎝⎠⎣⎦⎢⎥ =+dEe()Re{()2(2)*ωωωjtω } EE ()() *2jt(2ω) =+dE()()ωω E d Re{() E ω e } = PNL (0,2)ϖ We immediately see that the nonlinear term induces a new frequency harmonic of double the value of the incoming frequency. It also creates a time independent electric field. The first term that is time independent can be viewed as optical rectification, in the sense that it is a constant electric field where none was present before passing through this nonlinear medium. This field is measurable but obviously small. The generation of the second harmonic is called SHG.

Electro‐Optic Effect

One can easily see that if we send different optical signals through this medium we can expect a wide variety of outputs. By the same token, we can send a plane wave mixed with a constant electric field.

2 PdENL = ()2 Et()=+ E (0) Re{( Eω ) ejtω }

2 jtωωjt PdEdEEeENL ==22(0)Re{()}(0)Re{()}() + ωω()+ Ee 2 =+2(0)2(0)Re{()}Re{()}dE( 2 E Eωω ejtωω +() E ejt )

⎛⎞2*jtωω1 2j(2) t =+2dE⎜⎟ (0) 2 E (0) Re{ E (ωωωω ) e } +() E ( ) E ( ) + Re{ E ( ) e } ⎝⎠2 =+dE()22* (0) E ()()ωω E + 4(0) E Re{Ee (ωω )jtωω }+ Re{ E2(2) ( ) ej t }

= PNL (0,ϖϖ ,2 )

Not only do we get a second harmonic in this case but also we get another interesting effect. If the constant electric field is significantly larger than the magnitude of the oscillating electric field, we see that we can control the size of the term EEe(0)Re{ (ω )jωt }as other time varying terms are small. This reduces the equation to a plane wave of the input frequency riding on a constant electric field, but as we change the constant field we can control the size of the time varying field. This is called the electro‐ optic effect or Pockel’s effect. This is an analogy to a load line calculation in amplifier theory.

Three Wave mixing

Continuing with a third and last application. One could imagine what would happen if we had a mixture of two plane waves of different frequency.

2 PdENL = ()2

jtω1 jtω 2 Et()Re{()=+ Eωω12 e } Re{() E e }

222jtωω1 jtωωjt1 jt PNL ==2 dEd 2() Re{() Eeωω12 }+ Re{() Ee }() Re{() Ee ωω 12 }+ Re{() Ee } 22 =++2dEe⎡⎤⎡⎤ Re{ (ωω )jtωω12 } Re{ Ee ( )jt } 2Re{ EeEe ( ωω )jtω 2 }Re{ ( )jtω1 } (⎣⎦⎣⎦12 21)

22 ⎛ 1 jtω1 *2−−jtωω121 jtωω*jt jt 2jtω1 ⎞ =+2()dEeE⎜ ⎡⎤ω1 ()ωωωωω1222eEeEeEeEe++⎡ () () ⎤ + 2Re{() }Re{()1 }⎟ ⎝ 4 ⎣⎦4 ⎣ ⎦ ⎠

⎛⎞EE*2()()Re{()ωω+ Ee ωjt(2ω1 ) } ⎜⎟11 1 *2jt(2ω2 ) =+2()()Re{()}dE⎜⎟ωω22 E + E ω 2 e

⎜⎟jtω 2 jtω1 ⎝⎠+2Re{Ee (ωω21 ) }Re{ Ee ( ) } ⎛⎞ *2jt(2ω1 ) ⎜⎟EE()()Re{()ωω11+ Ee ω 1 } ⎜⎟ *2jt(2ω2 ) =+2()()Re{()}dE⎜⎟ωω22 E + E ω 2 e ⎜⎟1 ⎜+++⎡⎤Ee()ωωjtωω11 E* () e− jt ⎡ E (ωω)()eEejtωω22* − jt⎤ ⎟ ⎝⎠2 ⎣⎦11⎣ 22⎦ ⎛⎞ ⎜⎟ *2jt(2ω1 ) ⎜⎟EE()()Re{()ωω11+ Ee ω 1 }

⎜⎟*2jt(2ω2 ) =+2()()Re{()}dE⎜⎟ωω22 E + E ω 2 e

⎜⎟jtωω12 jt jt ω 1* − jt ω 2 1 ⎡EeEe()ωω12 ()++ EeEe () ω 1 () ω 2 ⎤ ⎜⎟+ ⎢ ⎥ ⎜⎟2 **−−jtωω12 jt jt ω 1*−j ω 2 t ⎝⎠⎣⎢⎥EeEeEeEe()ωω12 ()+ () ω 1 () ω 2⎦ ⎛⎞ ⎜⎟ *2jt(2ω1 ) ⎜⎟EE()()Re{()ωω11+ Ee ω 1 } ⎜⎟ *2jt(2ω2 ) =+2()()Re{()}dE⎜⎟ωω22 E + E ω 2 e ⎜⎟ ⎡⎤jt()ωω12+−* jt()ωω12 ⎜⎟1 EEe()()ωω12 + EE () ω1 () ω 2 e + ⎢⎥ ⎜⎟2 **−−jt()ωω12 *−+jt()ωω12 ⎝⎠⎣⎦⎢⎥++EEe()()ωω12 EEe ()() ω1 ω 2

= PNL (0,2ϖϖωωωω121212 ,2 ,+− , )

Not only do we get frequency doubling of both signals, but we the difference and addition of both frequencies. This is called three‐wave mixing and is the foundation of an optical parametric amplifier. In this case one of the input frequencies is pumped, mixed, then the pump frequency and post additive frequency is filtered, leaving only an amplified input frequency.

Third order analysis for centrosymmetric materials is performed similarly, only the 2nd order term is neglected.

References

• Fundamentals of Photonics Saleh & Teich • High Energy Lasers and their applications Jacobs, Sargent, Scully • Laser Fundamentals Silfvast • Optics and Lasers Young