2do seminario theoretical background

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Quantum Frequency Conversion

Oscar Adrián Jiménez Gordillo

28/01/2015

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Fundamentals of Nonlinear Optics

• In order to describe precisely what and optical nonlinearity means let us first consider how the dipole moment per unit volume, or polarization P, of a material depends on the strength of an applied optical field E(t).

3


• Nonlinear Polarization– When an external electric field E is applied to a

dielectric medium a polarization P is induced.

– Where ε0 is the permittivity of free space and the are tensors of rank n + 1 which are called the n-th order susceptibilities.

)(...n

(1)

4


• We now write E as an expansion of plane waves with angular frequencies ωn.

• Defining and extending the range of the indices to with the convention of we have

...3,2,1 n

nn

(2)

(3)

5

• When we insert (3) in (1), the first two polarization terms read:

• When electric fields and/or higher-order susceptibilities become large, the nonlinear terms Pi

(NL) = Pi(2) + Pi

(3) + ... come into play. Then, nonlinear optical effects can be observed.

• In this work, we will particularly investigate second-order nonlinear effects.


(4)

(5)

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• Working out the sum in (5) for the simple case of two frequencies ω1 and ω2 produces terms proportional to:– second harmonic generation of ω1 (SHG)

– second harmonic generation of ω2 (SHG)

– sum frequency generation (SFG)

– difference frequency generation (DFG)

– optical rectification (OR)

twie 12

twie 22

twwie )(2 21

twwie )(2 21

0e

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Fundamentals of Nonlinear Optics• In the previous slide we can see that there are 4 non zero

frequency components in the nonlinear polarization. • However, typically no more than one of these frequency

components will be present with any appreciable intensity in the radiation generated by the nonlinear optical interaction.

• The reason for this behavior is that the nonlinear polarization can efficiently produce an output signal only if a certain phase-matching condition is satisfied, and usually this condition cannot be satisfied for more than one frequency component of the nonlinear polarization.

• Operationally, one often chooses which frequency component will be radiated by properly selecting the polarization of the input radiation and the orientation of the nonlinear crystal.

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• For simplicity, we now assume that the χijk are frequency independent.

• Instead of the second-order nonlinear susceptibility (χ(2)

ijk), it is common to use the nonlinear coefficient d ≡ dijk given by the relation dijk = χ(2)

ijk / 2.• Using the Kleinman’s symmetry

(6)

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• And also using this table for contracting the d tensor we can write it in a 3x6 matrix with 10 independent elements.

• Now, the three components of the second-order nonlinear polarization in (5) are then obtained by the multiplication of a 3x6 matrix with a 6-component column vector.

10


• For instance, when we write the polarization in the same way as the electric field in (3)

the frequency component of the nonlinear polarization at ω3 = ω1 + ω2 is given by

(7)

(8)

11

Coupled Mode Equations

• Starting with the inhomogeneous wave equation for a nonlinear, lossless, isotropic and nonmagnetic dieletric.

• For simplicity, we make an ansatz of three scalar electric fields Em (m = 1, 2, 3)

(9)

(10)

12


• The fields in Eq. (10) are oscillating at angular frequencies ω1, ω2 and ω3 = ω1 + ω2 and propagate along the z axis.

• Refractive indices of the medium:• Dispersion relation: • The intensity is given by: (11)

13


• Inserting (10) in (9) we obtain the coupled mode equations:

• Which govern the evolution of the amplitudes Am(z) along the propagation direction.

(12)

(13)

(14)

Are the coupling constants.

14


• The parameter is the wave vector mismatch.

• For our needs it is sufficient to assume that the km are collinear and thus Δk’ is always a scalar.

• The coupled differential equations (12), (13) and (14) are central to all second-order nonlinear interactions. Depending on the actual value of Δk’(k1, k2, k3) the processes that can take place are either SHG, SFG, DFG, or OR.

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Phasematching

• The wave vector mismatch Δk’ is a crucial parameter for the efficiency of nonlinear processes, as the following example for SFG shows.

• For this example we assume that two frequencies ω1 and ω2 are incident in a nonlinear medium and interact to create a third frequency ω3 = ω1 + ω2 via SFG.

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Sum Frequency Generation

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Phasematching

• The amplitude of the sum-frequency (ω3) field at the exit plane of the nonlinear medium is given in this case by integrating Eq. (14) from z = 0 to z = L, yielding

• The intensity of the ω3 wave is given by the magnitude of the time-averaged Poynting vector, which for our definition of field amplitude is given by

(15)

(16)

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Phasematching

• We thus obtain

• The squared modulus that appears in this equation can be expressed as

(17)

(18)

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• Finally, our expression for I3 can be written in terms of the intensities of the incident fields by using Eq. (16) to express |Ai|2 in terms of the intensities, yielding the result

• Note that the effect of wave vector mismatch is included entirely in the factor sinc2(ΔkL/2). This factor, which is known as the phase mismatch factor, is plotted in the next slide.

Phasematching

(19)

20

Phasematching

21

Phasematching

• The sinc2 function is characteristic for second-order nonlinear interactions. It attains its global maximum when the argument vanishes. Therefore, Δk’ = 0 is required to maximize the generated intensity I3(L). This equation is known as the phasematching condition.

• Since the linear momentum for photons is given by p = ħk, Δk’ = 0 represents the conservation of momentum in the photon picture.

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Phasematching

• Also energy conservation must be fullfilled in nonlinear optical processes which is expressed by the relation ω3 = ω1 + ω2.

• In practice it is not trivial to satisfy this to conditions because of material dispersion. With the vacuum wavelength the phasematching condition reads

(20)

(21)

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Phasematching

• Note that n3, n2, and n1 usually also depend on the temperature of the nonlinear material.

• Inspection of (20) and (21) reveals that, in general, conservation of energy and momentum are not satisfied simultaneously and no efficient frequency conversion takes place. Inserting (20) in (21) yields

(22)

2do seminario theoretical background

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