# Nonlinear Optical Modulators and Switches

This is a continuation from the previous tutorial - optical frequency converters.

In a nonlinear optical modulator, the modulation of an optical wave is accomplished through a nonlinear optical process. A nonlinear optical modulator can be based on either self modulation or cross modulation. In the case of self modulation, only one optical beam is present, and the modulation on the beam is a function of the characteristics of the beam itself. In the case of cross modulation, two or more optical beams are present, and the beam of interest is modulated by one or more other beams that carry the modulation signals. In either case, no electric, magnetic, or acoustic field is needed. Therefore, nonlinear optical modulators and switches are also know as all-optical modulators and all-optical switches, respectively.

There are two fundamentally different types of nonlinear optical modulators and switches. One is the dispersive, or refractive, type, which is based on the optical Kerr effect due to optical-field-induced changes in the real part of the permittivity of a material. Another is the absorptive type, which relies on an intensity-dependent absorption coefficient caused by the nonlinear characteristics of the imaginary part of the permittivity of a material.

Kerr lenses

We first consider the simplest case of the optical Kerr effect discussed in the nonlinear optical interactions tutorial in which $$\mathbf{P}^{(3)}$$ is parallel to a linearly polarized optical field $$\mathbf{E}$$ so that the net effect is an intensity-dependent index of refraction given in (9-49) [refer to the nonlinear optical interactions tutorial].

For a plane optical wave, this optical Kerr effect merely causes a uniform intensity-dependent phase shift across the wavefront. Thus the beam remains a plane wave without any change in its spatial intensity distribution.

If an optical beam has a nonuniform intensity distribution, the intensity-dependent index of refraction leads to a nonuniform phase shift across the wavefront as the beam propagates through the nonlinear medium. This beam will then be focused or defocused as a result of distortion in its phase front.

For simplicity, we consider the propagation of a circular beam, which has a transverse spatial intensity distribution of $$I(r)$$. After such a beam propagates through a thin nonlinear medium of a thickness $$l$$, the total intensity-dependent phase shift can be approximated by

$\tag{9-142}\varphi(r)=\frac{\omega}{c}[n_0+n_2I(r)]l$

The intensity-dependent Kerr phase change given by

$\tag{9-143}\varphi_\text{K}(r)=\frac{\omega}{c}n_2lI(r)$

is known as self-phase modulation because it is imposed by an optical beam on itself through the optical Kerr effect.

Recall that the effect of a thin spherical lens of a focal length of $$f$$ is to cause a spatially varying phase shift of

$\tag{9-144}\varphi(r)=-k\frac{r^2}{2f}=-\frac{\omega}{c}\frac{r^2}{2f}$

in an optical wave passing through the lens, where $$r$$ is the transverse radial distance from the center of the lens.

Therefore, if the intensity-dependent phase shift given in (9-142) has a quadratic dependence on the transverse radial coordinate, the optical Kerr effect in the thin nonlinear medium would be equivalent to the effect of a thin lens. A thin nonlinear medium with such a function is called a Kerr lens

In reality, no optical beam has an ideal quadratic spatial intensity distribution. However, if the intensity distribution of a circular beam is approximately quadratic in $$r$$ near the beam center, the effective focal length of the Kerr lens can be given by

$\tag{9-145}\frac{1}{f_\text{K}}=\left.-a\frac{c}{\omega}\frac{\text{d}^2\varphi}{\text{d}r^2}\right|_{r=0}=\left.-an_2l\frac{\text{d}^2I(r)}{\text{d}r^2}\right|_{r=0}$

where $$a$$ is a correction factor to account for the difference between the true beam profile and the ideal quadratic profile.

Using this relation, we find that the effective focal length of the Kerr lens for a circular Gaussian beam with an intensity distribution of $$I(r)=I_0\exp(-2r^2/w^2)$$ is

$\tag{9-146}f_\text{K}=\frac{w^2}{4an_2lI_0}=\frac{\pi{w}^4}{8an_2lP}$

where $$w$$ is the beam radius at the location of the Kerr medium, $$I_0$$ is the intensity at the beam center, and $$P$$ is the power of the beam.

For a circular Gaussian beam, $$a=1.723$$, and the thin-lens condition for (9-146) to be valid is $$l\lt{z}_\text{R}=\pi{n}w_0^2/\lambda$$.

Note that $$n_2$$ can be either positive or negative because $$\chi^{(3)'}$$ can be either positive or negative. Therefore, a Kerr lens can either focus or defocus a beam, depending on the sign of its effective focal length.

Most applications of Kerr lenses are based on the fact that the effective focal length $$f_\text{K}$$ of a thin Kerr lens is inversely proportional to the peak intensity $$I_0$$ of an optical beam. As a result of this characteristic, the divergence of the beam after passing through a Kerr lens is a function of the intensity of the beam. In addition, the beam divergence also depends on the sign of $$n_2$$ and the location of the Kerr lens with respect to the beam waist, as illustrated in Figure 9-22 below. Figure 9-22. Nonlinear refraction caused by a Kerr lens as a function of beam intensity and the location of the Kerr lens with respect to the beam waist for (a), (b), and (c) $$f_\text{K}\gt0$$, and for (d), (e), and (f) $$f_\text{K}\lt0$$. The solid curves are the propagation lines of a Gaussian beam at a low intensity for which the effect of the Kerr lens is negligible, and the dashed curves are those at a high intensity for which the effect of the Kerr lens is significant.

A Kerr lens is often used as an optical power limiter for the protection of a sensitive optical detector. In this application, the action of the Kerr lens is to increase the beam divergence as the input intensity of a beam is increased, thereby increasing the spread and reducing the intensity of the beam at the surface of the detector.

As demonstrated in Figure 9-22(a), (b), (e) and (f), with proper arrangement, either a Kerr lens of a positive effective focal length, $$f_\text{K}\gt0$$, or one with a negative effective focal length, $$f_\text{K}\lt0$$, can be used for this purpose.

When a Kerr lens in such an arrangement is used as an optical power limiter, only a fraction of the diverging optical beam within a finite central cross-sectional area that is defined either by the area of a small detector or by a hole in a beam block is allowed to reach the detector.

Because the divergence of the beam increases with its intensity, the optical power passing through the finite area to be received by the detector will saturate at a certain level as the input power of the beam continues to increase.

Without the Kerr lens, the beam divergence does not change with its intensity. Then, the optical power received by the detector increases linearly with the input power of the beam without a limit until the detector is damaged even if the detector has a very small area to intercept only a tiny fraction of the beam.

A Kerr lens can also be used as a passive optical switch or an optical thresholding device. For this purpose, an arrangement, such as that shown in Figure 9-22(c) or (d), that leads to a reduction in beam divergence with an increase in input beam intensity is used. Similarly to the setup of a power limiter, only a portion of the beam within a finite central area of the beam cross section is allowed to pass. However, instead of a saturation, the optical power passing through this area increases nonlinearly with the input power of the beam. This behavior can be used to provide a nonlinear feedback to an optical system or to switch on an optical device at a certain threshold. It has been used as the passive mode locker in a technique known as Kerr-lens mode locking for the generation of ultrashort laser pulses.

Example 9-15

A Ti : sapphire laser generates a train of laser pulses of wavelength $$\lambda=780\text{ nm}$$ and pulsewidth $$\Delta{t}_\text{ps}=100\text{ fs}$$ at a repetition rate of $$f_\text{ps}=100\text{ MHz}$$. A beam of such pulses at an average power of $$\bar{P}=50\text{ mW}$$ is focused tightly on a thin silica plate of thickness $$l=1\text{ mm}$$. The nonlinear response time of silica is much faster than $$100\text{ fs}$$ so that the optical Kerr effect can be considered instantaneous in response to the temporal variation of each pulse. Silica has a linear refractive index of $$n_0=1.4537$$ at $$\lambda=780\text{ nm}$$ and, according to Example 9-5 [refer to the nonlinear optical interactions tutorial], a nonlinear refractive index of $$n_2=2.4\times10^{-20}\text{ m}^2\text{W}^{-1}$$. If the laser beam is focused with its waist on the silica plate as tightly as allowed by the thin-lens condition, what is the effective focal length of the Kerr lens caused by self-phase modulation at the peaks of the optical pulses?

The peak power of the pulses is

$P_\text{pk}=\frac{\bar{P}}{f_\text{ps}\Delta{t}_\text{ps}}=\frac{50\times10^{-3}}{100\times10^6\times100\times10^{-15}}\text{W}=5\text{ kW}$

The thin-lens condition, $$l\lt{z}_\text{R}=\pi{n}w_0^2/\lambda$$, requires that

$w_0\gt\left(\frac{l\lambda}{\pi{n}}\right)^{1/2}=\left(\frac{1\times10^{-3}\times780\times10^{-9}}{\pi\times1.4537}\right)^{1/2}\text{m}=13\text{ μm}$

By focusing the beam to the limit of $$w_0=13\text{ μm}$$ allowed by the thin-lens condition and by placing the beam waist on the silica plate, we have the following Kerr focal length at the peak of each pulse using (9-146):

$f_\text{K}=\frac{\pi{w}_0^4}{8an_2lP_\text{pk}}=\frac{\pi\times(13\times10^{-6})^4}{8\times1.723\times2.4\times10^{-20}\times1\times10^{-3}\times5\times10^3}\text{m}=5.42\text{ cm}$

Note that this is the Kerr focal length only at the temporal peak of each pulse. Because $$f_\text{K}$$ is inversely proportional to the optical power and because the nonlinear refractive response of silica is much faster than the $$100\text{ fs}$$ duration of each pulse, we can easily see that the value of $$f_\text{K}$$ varies in time through the duration of a pulse. As a consequence of this temporally varying $$f_\text{K}$$, the divergence of the pulse after the silica plate is a function of time over the pulse duration. Kerr-lens mode locking of lasers takes advantage of this interesting phenomenon.

Polarization and amplitude modulators

The optical-field-induced birefringence of the optical Kerr effects can be used for polarization modulation of an optical wave. Such polarization modulation can be either self induced in a one-beam interaction or cross induced in a two-beam interaction. For simplicity, we consider the interactions in an isotropic medium. The same principle applies to nonlinear optical polarization modulators using anisotropic crystals.

We have already seen from the discussions on Kerr lenses that in a one-beam interaction in an isotropic medium, the induced $$\mathbf{P}^{(3)}$$ and the optical field $$\mathbf{E}$$ have the same polarization state if the optical field is linearly polarized. This is also true for a circularly polarized optical field. Therefore, the optical Kerr effect does not change the polarization state of a linearly or circularly polarized optical wave that propagates alone in an isotropic medium.

The situation is different for an elliptically polarized optical wave in a one-beam interaction, as well as for a linearly or circularly polarized optical wave in a two-beam interaction. In a one-beam interaction with an elliptically polarized optical wave, the polarization state of the induced $$\mathbf{P}^{(3)}$$ is different from that of the optical field $$\mathbf{E}$$, causing the polarization of the optical field to change. The result is a phenomenon known as ellipse rotation because the axes of the ellipse defined by the tip of the elliptically polarized optical field continue to rotate in space as the wave propagates through the nonlinear meduim.

In the interaction of two linearly polarized optical waves, polarization modulation on one wave by the other through the optical Kerr effect is possible if the polarizations of the two waves are neither parallel nor orthogonal to each other. The optical beam being modulated is called the signal or the probe, and that creating the modulation is called the pump.

In an isotropic medium, the coordinate axes can be chosen arbitrarily. With a signal beam at a frequency $$\omega$$ and a pump beam at a frequency $$\omega'$$, we choose the $$xy$$ plane to be that defined by the two linearly polarized field vectors $$\mathbf{E}(\omega)$$ and $$\mathbf{E}(\omega')$$ and the $$y$$ axis to be in the direction of $$\mathbf{E}(\omega')$$, as shown in Figure 9-23(a) below. While the signal beam propagates in the $$z$$ direction, the pump beam propagates in a direction within the $$zx$$ plane that may or may not be collinear with the propagation direction of the signal beam, as also shown in Figure 9-23(a). Figure 9-23. (a) Nonlinear optical polarization modulator and (b) nonlinear optical amplitude modulator.

The optical-field-induced birefringence seen by the signal beam is described by $$\Delta\epsilon_{ij}(\omega,\mathbf{E})$$ given in (9-47) [refer to the nonlinear optical interactions tutorial]. In a practical application, the intensity of the signal beam is much lower than that of the pump beam: $$I(\omega)\ll{I}(\omega')$$. Therefore, the first term on the right-hand side of (9-47), which accounts for the self modulation of the signal beam, can be neglected in comparison to the second term, which accounts for the cross modulation on the signal by the pump. With $$\mathbf{E}(\omega')\parallel\hat{y}$$, we then have

$\tag{9-147}\Delta\epsilon_{xx}\approx6\epsilon_0\chi^{(3)}_{1122}|E(\omega')|^2=\frac{3\chi_{1122}^{(3)}}{cn_0}I(\omega')$

$\tag{9-148}\Delta\epsilon_{yy}\approx6\epsilon_0\chi^{(3)}_{1111}|E(\omega')|^2=\frac{3\chi_{1111}^{(3)}}{cn_0}I(\omega')$

This optical-field-induced birefringence leads to the following intensity-dependent indices of refraction:

$\tag{9-149}n_x=n_0+\frac{3\chi_{1122}^{(3)}}{2c\epsilon_0n_0^2}I(\omega')$

$\tag{9-150}n_y=n_0+\frac{3\chi_{1111}^{(3)}}{2c\epsilon_0n_0^2}I(\omega')$

If the signal beam has a field of $$\mathbf{E}(\omega)=(\hat{x}\mathcal{E}_x+\hat{y}\mathcal{E}_y)\text{e}^{-\text{i}\omega{t}}$$ at the input surface of the nonlinear medium that has a thickness of $$l$$, its field at the output is

$\tag{9-151}\mathbf{E}(\omega)=(\hat{x}\mathcal{E}_x+\hat{y}\mathcal{E}_y\text{e}^{\text{i}\Delta\varphi})\text{e}^{\text{i}k^xl-\text{i}\omega{t}}$

where $$k^x=n_x\omega/c$$ and

$\tag{9-152}\Delta\varphi=\frac{3\pi(\chi_{1111}^{(3)}-\chi_{1122}^{(3)})l}{c\epsilon_0n_0^2\lambda}I(\omega')$

is the phase retardation between the $$x$$ and $$y$$ components of the signal field. Because this phase retardation is linearly proportional to the pump intensity, the polarization state of the signal beam at the output can be modulated by varying the pump intensity if $$\mathbf{E}(\omega)$$ is neither parallel nor perpendicular to $$\mathbf{E}(\omega')$$ so that both $$\mathcal{E}_x$$ and $$\mathcal{E}_y$$ have nonvanishing values.

In comparison to the electro-optic polarization modulators discussed in the electro-optic modulators tutorial, the only difference is that the nonlinear optical polarization modulators discussed here are controlled by a pump optical beam rather than by a voltage. Other than this difference, these two types of polarization modulators have the same function and serve the same purpose.

As seen in the electro-optic modulators tutorial, an amplitude modulator can be easily constructed by placing a polarization modulator between two polarizers. This approach is also applicable to the construction of a nonlinear optical amplitude modulator using a nonlinear optical polarization modulator, as illustrated in Figure 9-23(b) above. A nonlinear optical amplitude modulator and an electro-optic amplitude modulator have the same transmission characteristics, which are discussed in the electro-optic modulators tutorial, if they are set up in the same manner.

When an ultrashort optical pulse is used as the pump beam, a nonlinear optical amplitude modulator can function as a fast optical gate, or a fast all-optic switch, for switching the signal beam within a very short time.

Saturable absorbers

A saturable absorber has an absorption coefficient that decreases with increasing light intensity, such as that characterized by (9-51) with $$\chi^{(1)''}\gt0$$ and $$\chi^{(3)''}\lt0$$ [refer to the nonlinear optical in interactions tutorial]. Note, however, that the relation in (9-51) is rooted in the power series expansion of (9-1) [refer to the optical nonlinearity tutorial]. Because absorption saturation necessarily occurs at a resonant transition between two energy levels, the perturbation approach taken for power series expansion is not valid at sufficiently high intensities. Instead, a full analysis of the resonant absorption has to be carried out. Such an analysis results in an intensity-dependent absorption coefficient characterized by the relation:

$\tag{9-153}\alpha=\frac{\alpha_0}{1+I/I_\text{sat}}$

where $$\alpha_0$$ is the unsaturated absorption coefficient and $$I_\text{sat}$$ is the known as the saturation intensity.

Note: the absorption coefficient described by (9-153) is that for a homogeneously broadened medium. For an inhomogeneously broadened medium, the relation is $$\alpha=\alpha_0/(1+I/I_\text{sat})^{1/2}$$.

The saturation intensity is a characteristic of the resonant transition that is responsible for the absorption under consideration. For $$I\lt{I}_\text{sat}$$, the relation in (9-153) can be expanded:

$\tag{9-154}\alpha=\alpha_0\left[1-\frac{I}{I_\text{sat}}+\left(\frac{I}{I_\text{sat}}\right)^2-\left(\frac{I}{I_\text{sat}}\right)^3+\cdots\right]$

Only when $$I\ll{I}_\text{sat}$$ can $$\alpha$$ be accurately approximated by the first two terms of this expansion, resulting in a linear dependence on $$I$$ like the relation in (9-51) [refer to the nonlinear optical interactions tutorial].

In general, the relation in (9-153) has to be used because the light intensity encountered in a practical device that uses a saturable absorber can easily be comparable to or higher than $$I_\text{sat}$$.

The propagation of an optical wave through a saturable absorber that has an absorption coefficient given in (9-153) is described by

$\tag{9-155}\frac{\text{d}I}{\text{d}z}=-\frac{\alpha_0}{1+I/I_\text{sat}}I$

This equation can be integrated to obtain the following relation:

$\tag{9-156}I(z)\text{e}^{I(z)/I_\text{sat}}=I(0)\text{e}^{I(0)/I_\text{sat}}\text{e}^{-\alpha_0z}$

where $$I(0)$$ is the input light intensity at $$z=0$$.

The transmittance of an optical wave through a saturable absorber of a thickness of $$l$$ is $$T=I_\text{out}/I_\text{in}=I(l)/I(0)$$, which can be calculated by numerically solving (9-156). It is plotted in Figure 9-24 below as a function of the input light intensity, normalized to the saturation intensity, for a few difference values of $$\alpha_0l$$ represented in terms of $$T_0=\text{e}^{-\alpha_0l}$$. Figure 9-24. Transmittance of an optical wave through a saturable absorber that has a thickness $$l$$ and an unsaturated absorption coefficient $$\alpha_0$$ as a function of the input light intensity normalized to the saturation intensity. The curves are plotted for different values of $$\alpha_0l$$ in terms of $$T_0=\text{e}^{-\alpha_0l}$$.

As Figure 9-24 shows, the optical transmittance through a saturable absorber increases nonlinearly as the input intensity is increased and approaches unity at high input intensities.

In a particular application of a saturable absorber, the value of $$\alpha_0l$$ has to be properly chosen for a desired difference between the maximum transmittance at high intensities and the minimum transmittance at low intensities.

A saturable absorber can be used as a spatial light filter, which blocks low-intensity stray light or background optical noise but transmits a high-intensity signal beam. It can be used as an optical discriminator, which transmits optical pulses of intensities above a certain threshold and suppresses those below. A saturable absorber is also commonly used as a passive $$Q$$ switch in a $$Q$$-switched laser or as a passive mode locker in a mode-locked laser for the generation of very short laser pulses. The saturable absorber in this kind of application functions as a passive optical switch in the time domain. It is switched open by the rising intensity of a laser pulse and closes through its own relaxation after the passing of the pulse. Therefore, the relaxation time of a saturable absorber is also an important factor to be considered in its application as a $$Q$$ switch or a mode locker.

The next part continues with the bistable optical devices tutorial.