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The Maker’s fringe technique can be applied to all nonlinear materials and thin films. Hereafter the particular case of the measurement of the second-order nonlinearity in a poled glass is discussed.

The glass plates are rotated along an axis perpendicular to the plane of incidence to perform the MFT measurement (Fig. 4.3). The pump electric-field is p-polarized in order to be co-planar with the frozen-in field E_dc. The direction of E_dc defines the optical axis of the poled glass as well as the group symmetry the poled glass belongs to: (6mm or C_∞mm). A straightforward picture of the principle of operation of the Maker’s fringe technique arises if the second harmonic is considered to be always collinear with the pump (i.e. ϑ2ωwϑω). Such approximation is extremely useful: it gives an intuitive physical description and moreover it fails only for extremely high incident anglesϑ. The approximation is justified because, owing to the low dispersion of silica, the difference between the index of refraction at the two frequencies is typically very small: w0.01. As a result, the difference between the internal angle of propagation for the pump and the second harmonic ϑω−ϑ2ω, given by eqs. (4.1 and 4.2), is always smaller than 0.4◦

J J 2w Ew E2w Poled region X Y Z

Figure 4.3: Geometry of a Maker’s fringe experiment for a thermally poled glass. The pump beam is parallel to the plane of incidence (p-pol), which contains the Y and Z axis. The Z-axis is the optical axis of the poled glass (group symmetry 6mmor C∞mm).

The sample is rotated along the X axis.

−100 −80 −60 −40 −20 0 20 40 60 80 100 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

Angle of incidence ϑ (deg)

ϑω − ϑ2ω −100 −80 −60 −40 −200 0 20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Angle of incidence ϑ (deg)

SH power (a.u.) L 2 /(cos ϑ )[sinc( π L/ (2 l c cos ϑ ))] 2 (a.u.)

Figure 4.4: (left) Plot of the angular difference,ϑω−ϑ2ω, between pump and second

harmonic internal propagation directions, as a function of the angle of incidence ϑ. (right) Theoretical computation of the oscillating function in the Maker Fringe formula (eq. (4.20)) assuming the case of a poled glass. (Red curve) ϑω6=ϑ2ω, (Green curve)

Approximated curve forϑω=ϑ2ω. Note that the green and red curves are overlapped.

The parameters used for the computation are the same as described in Fig. (4.5)

(Fig. (4.4(left))) It is worth noting that for normal incidence (i.e. ϑ= 0) the difference is rigorously zero. Therefore for normal incidence the approximationϑ2ωwϑωis rigorously exact.

A numerical computation of the output of a Maker’s fringe measurement of a poled glass with a uniform nonlinear depletion layer of 10µm is shown in Fig. (4.5). On the same graph, the approximate curve is also shown. No appreciable difference can be observed.

Due to the dispersion, the polarization wave (bound wave) and the second harmonic wave (free wave) travel at different velocities inside the medium. As a result, the intensity I_2ω(L) of the second harmonic along the length of the sample is determined by the

interference between the two waves. Rotation of the sample alters the phase relationship between the bound and the free wave at the output of the sample because of the variation in the optical path length. There will be angles for which the two waves interfere constructively and angles for which there is complete destructive interference. The result of a Maker’s fringe measurement is an oscillating curve obtained by recording the second harmonic power as a function of the incident angle.

The sinc2 _{function, in eq. (4.20), accounts for this oscillating behavior. The angular}

separation between two adjacent peaks of the interference pattern corresponds to a difference in the optical path length equal to one coherence length. According to lc = π/∆k, the argument of the sinc2 _{function in eq. (4.20) can be expressed in term of the}

coherence length as: π_2l L

ccos(ϑ2ω). When (L/cosϑ2ω)lcthe sinc

2 _{tends to one and the}

second harmonic efficiency scales with the square of the interaction length (L/cosϑ2ω). This condition holds for small angles of incidence and for thin nonlinear samples. The poled glass, havinglc= 24µm and a nonlinear region of w10µm is a typical case. The sinc2 _{function, plotted in Fig. (4.4(right)), is close to one for all the incident angles.}

∆k00_{represents the projection of the vector∆k=k}

2ω−2kωalong the second harmonic direction and it was defined in eq. (4.21): Owing to the low chromatic dispersion of silica glass, the dependence of ∆k00 _{on the angle of incidence is limited. Whenϑ}

2ω is assumed to be equal toϑ_ω, ∆k00 _{reduces to the well known modulus of the wave-vector mismatch}

|∆k|= ∆k= 4π

λ (n2ω−nω). (4.23)

As ∆k can be considered constant for all ϑ, the major contribution to the oscillations in the MFT pattern is given by the variation of the path travelled by the light in the material: L/cosϑ2ω. The wave-vector mismatch ∆k accounts for the dispersion of the medium.

In Fig. (4.4(right)) the variation between the exact and the approximated oscillating function can be observed. The computed Maker’s fringe curve is also shown.

The envelope of the MFT pattern (eq. 4.20) reflects the angular dependence of the Fresnel transmission losses T (eq. 4.19), the beam correction factor a (eq. 4.18) and of the effective nonlinear coefficient d_eff (eq. 4.22).

In the case of the poled glass with the 10µm depletion layer depicted in Fig. (4.5), the path travelled by the light in the nonlinear region is smaller than the coherence length for all the incident angles. The maximum angle of refraction in silica is in fact 43.6◦_{, hence the maximum path travelled by the light in the nonlinear region is 10µm}

/cos(43.6) = 13.8µm < lc = 24µm. As long as no coherence length is reached in the Maker Fringe measurement, we would expect the second harmonic to grow with the

angle according to (L/cosϑ_2ω)2_{. The maxima and the subsequent drops in Fig. (4.5) are}

due to the Fresnel transmission losses of the pump and of the second harmonic beam at the sample interfaces. The situation is clarified in Fig. (4.6) where the transmittance coefficient for the second harmonic and the Fresnel loss correction factor to the Maker Fringe formula (T2ω_×(Tω₎2_{) are plotted versus the angle of incidence.}

From symmetry considerations poled glass can be considered as belonging to the group symmetry 6mm or C_∞mm. According to the Kleinman symmetry condition [5], the nonlinear tensor for a poled glass is given by:

   0 0 0 0 d₃₁ 0 0 0 0 d31 0 0 d31 d31 d33 0 0 0    (4.24) withd33= 3d31 anddij = (1/2)χ(2)_ij .

The angular dependence ofdeff is obtained from eq. 4.22:

d_eff = 2

3 d33(sinϑωcosϑωcosϑ2ω) + 1

3 d33(cosϑω)

2_sinϑ

2ω+d33(sinϑω)2sinϑ2ω (4.25) The simpler case ofϑ2ω wϑω, hence,

deff wd33sinϑω, (4.26)

provides a better understanding of the role played by this parameter in the Maker’s fringe output pattern. At normal incidence the nonlinear optical coefficient causes the second harmonic intensity to be zero. A plot of d2_eff is shown in Fig. (4.7(left)).

Finally, the beam correction factor (eq. 4.18)versus the angle of incidence is plotted in Fig. (4.7(right)). The effect of this correction is merely a reshaping of the envelope. The envelope of the MFT pattern is dominated at smallϑby the contribution given by d_eff and at large ϑby the Fresnel transmission losses.