Guided-wave all-optical modulators and switches
This is a continuation from the previous tutorial - guided-wave optical frequency converters.
As discussed in the (bulk medium) nonlinear optical modulators and switches tutorial, an all-optical modulator can be either of refractive type, which utilizes \(\boldsymbol{\chi}^{(3)'}\), or of absorptive type, which utilizes \(\boldsymbol{\chi}^{(3)''}\).
For a guided-wave nonlinear optical device, however, any absorptive loss in the waveguide is detrimental to the device function due to the fact that the primary advantage of using an optical waveguide for the device is the long interaction length possible by the waveguiding effect. Therefore, all practical guided-wave all-optical modulators and switches are of refractive type based on the optical Kerr effect.
The majority of such devices require only one optical frequency for their operation though some involve two or more frequencies at a time.
For a guided-wave all-optical modulator or switch that requires only one frequency at a time for its operation, we have
\[\tag{9-242}\mathbf{P}^{(3)}=3\epsilon_0\sum_{\mu,\xi,\zeta}\boldsymbol{\chi}^{(3)}(\omega=\omega+\omega-\omega)\vdots\hat{\boldsymbol{\mathcal{E}}}_\mu\hat{\boldsymbol{\mathcal{E}}}_\xi\hat{\boldsymbol{\mathcal{E}}}^*_\zeta{A}_\mu{A}_\xi{A}^*_\zeta\text{e}^{\text{i}(\beta_\mu+\beta_\xi-\beta_\zeta)z}\]
According to (9-218) [refer to the nonlinear optical interactions in waveguides tutorial], we have the following general coupled-mode equation for such a device:
\[\tag{9-243}\pm\frac{\text{d}A_\nu}{\text{d}z}=\sum_\mu\text{i}\kappa_{\nu\mu}A_\mu\text{e}^{\text{i}(\beta_\mu-\beta_\nu)z}+\sum_{\mu,\xi,\zeta}\text{i}\omega{C}_{\nu\mu\xi\zeta}A_\mu{A}_\xi{A}^*_\zeta\text{e}^{\text{i}(\beta_\mu+\beta_\xi-\beta_\zeta-\beta_\nu)z}\]
where \(\kappa_{\nu\mu}\) is the linear coupling coefficient defined in the coupled-mode theory tutorial, subject to any modifications such as those caused by the electro-optic, magneto-optic, or acousto-optic effects discussed in the previous tutorials, and \(C_{\nu\mu\xi\zeta}\) is the nonlinear coefficient given by
\[\tag{9-244}C_{\nu\mu\xi\zeta}=3\epsilon_0\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\hat{\boldsymbol{\mathcal{E}}}^*_\nu\cdot\boldsymbol{\chi}^{(3)}\vdots\hat{\boldsymbol{\mathcal{E}}}_\mu\hat{\boldsymbol{\mathcal{E}}}_\xi\hat{\boldsymbol{\mathcal{E}}}^*_\zeta\text{d}x\text{d}y\]
Self-phase modulation
In the simplest situation when a waveguide mode \(\boldsymbol{\mathcal{E}}_\nu\) at a particular optical frequency \(\omega\) is not coupled to any other frequencies or any other modes at the same frequency, (9-243) reduces to
\[\tag{9-245}\frac{\text{d}A_\nu}{\text{d}z}=\text{i}\sigma_{\nu\nu}A_\nu|A_\nu|^2\]
where
\[\tag{9-246}\sigma_{\nu\nu}=\omega{C}_{\nu\nu\nu\nu}=3\omega\epsilon_0\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\hat{\boldsymbol{\mathcal{E}}}^*_\nu\cdot\boldsymbol{\chi}^{(3)}\vdots\hat{\boldsymbol{\mathcal{E}}}_\nu\hat{\boldsymbol{\mathcal{E}}}_\nu\hat{\boldsymbol{\mathcal{E}}}^*_\nu\text{d}x\text{d}y\]
Because \(\boldsymbol{\chi}^{(3)}\) is real for a device based on the purely refractive optical Kerr effect, the nonlinear coefficient \(\sigma_{\nu\nu}\) is also a real quantity. It is then clear from (9-245) that only the phase, but not the magnitude, of \(A_\nu\) varies with \(z\). Therefore, the mode power \(P_\nu=|A_\nu|^2\) is a constant that is independent of \(z\). The solution of (9-245) can be easily obtained:
\[\tag{9-247}A_\nu(z)=A_\nu(0)\exp(\text{i}\sigma_{\nu\nu}P_\nu{z})=A_\nu(0)\exp(\text{i}\beta_\nu^{\text{NL}}z)\]
where \(\beta_\nu^{\text{NL}}=\sigma_{\nu\nu}P_\nu\) is a power-dependent modification on the propagation constant.
Clearly, the consequence of the optical Kerr effect on an individual waveguide mode is an effective propagation constant that is a function of the mode power:
\[\tag{9-248}\beta_\nu^\text{eff}=\beta_\nu+\beta_\nu^\text{NL}=\beta_\nu+\sigma_{\nu\nu}P_\nu\]
where \(\beta_\nu\) is the power-independent linear propagation constant of the mode.
This effect leads to the following self-phase modulation for the mode field over a distance \(l\) in the waveguide:
\[\tag{9-249}\varphi_\nu^\text{NL}=\beta_\nu^\text{NL}l=\sigma_{\nu\nu}P_\nu{l}=\sigma_{\nu\nu}|A_\nu|^2l\]
which is linearly dependent on the mode power.
For a waveguide that is fabricated in an isotropic medium, such as silica glass, \(\chi_{xxxx}^{(3)}=\chi_{yyyy}^{(3)}=\chi_{zzzz}^{(3)}=\chi_{1111}^{(3)}\). Then, \(\sigma_{\nu\nu}\) defined in (9-246) becomes
\[\tag{9-250}\sigma_{\nu\nu}=3\omega\epsilon_0\chi_{1111}^{(3)}\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty|\hat{\boldsymbol{\mathcal{E}}}_\nu|^4\text{d}x\text{d}y\]
It is then convenient to define an effective area for an individual waveguide mode in a third-order nonlinear process as
\[\tag{9-251}\mathcal{A}_\nu^\text{eff}=\frac{\left[\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty|\hat{\boldsymbol{\mathcal{E}}}_\nu|^2\text{d}x\text{d}y\right]^2}{\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty|\hat{\boldsymbol{\mathcal{E}}}_\nu|^4\text{d}x\text{d}y}=\left(\frac{\omega\mu_0}{2\beta_\nu}\right)^2\frac{1}{\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty|\hat{\boldsymbol{\mathcal{E}}}_\nu|^4\text{d}x\text{d}y}\]
where the orthonormality relation given in (44) is used [refer to the wave equations for optical waveguides tutorial].
The orthonormality relation in (44) [refer to the wave equations for optical waveguides tutorial] is strictly accurate for TE modes only. It is used here as an approximation for other types of modes. In a weakly guiding waveguide, it is a good approximation.
Then, by using (9-248), (9-250), and (9-251), we find that the power-dependent effective index of the waveguide mode can be written
\[\tag{9-252}n_\nu^\text{eff}=n_\nu+n_{2\nu}\frac{P_\nu}{\mathcal{A}_\nu^\text{eff}}\]
where \(n_\nu^\text{eff}=c\beta_\nu^\text{eff}/\omega\), \(n_\nu=c\beta_\nu/\omega\), and
\[\tag{9-253}n_{2\nu}=\frac{3\chi_{1111}^{(3)}}{4c\epsilon_0n_\nu^2}\]
Clearly, (9-252) has the form of (9-49) [refer to the nonlinear optical interactions tutorial], and (9-253) has the form of (9-50) [refer to the nonlinear optical interactions tutorial].
Then self-phase modulation for the mode field can be expressed in the form of (9-143) [refer to the nonlinear optical modulators and switches tutorial] as
\[\tag{9-254}\varphi_\nu^\text{NL}=\beta_\nu^\text{NL}l=\frac{\omega}{c}n_{2\nu}\frac{P_\nu}{\mathcal{A}_\nu^\text{eff}}l=\frac{2\pi{n}_{2\nu}}{\lambda}\frac{P_\nu}{\mathcal{A}_\nu^\text{eff}}l\]
Note that (9-253) is valid only for a mode in a waveguide fabricated in a noncrystalline isotropic medium. For a mode in a waveguide based on a crystalline material, such as GaAs or LiNbO3, (9-254) can still be used, but (9-253) is generally not valid because the nonlinear refractive index \(n_{2\nu}\) in this situation is a function of the mode field polarization direction with respect to the principal axes of the crystal.
Two-mode interaction
Guided-wave all-optical modulators and switches function on the same basic principle as guided-wave electro-optic modulators and switches by transforming a differential phase shift between two waveguide modes into an amplitude modulation, except that the required phase shift is controlled by the optical power in the waveguide structure rather than by an externally applied voltage.
There are two basic approaches to transforming a differential phase shift into an amplitude modulation: by interference or by phase-sensitive coupling.
The operation of most devices involves only two modes of either the same waveguide or two separate waveguides. For a device that functions on two waveguide modes \(a\) and \(b\) at the same frequency \(\omega\), the total field is simply \(\mathbf{E}(\mathbf{r},t)=\mathbf{E}(\mathbf{r})\exp(-\text{i}\omega{t})\) with
\[\tag{9-255}\mathbf{E}(\mathbf{r})=A(z)\hat{\boldsymbol{\mathcal{E}}}_a(x,y)\text{e}^{\text{i}\beta_az}+B(z)\hat{\boldsymbol{\mathcal{E}}}_b(x,y)\text{e}^{\text{i}\beta_bz}\]
which has the same form as that given in (25) [refer to the coupled-mode theory tutorial].
By using (9-243) and by keeping only the major nonlinear terms, the coupled-mode equations for such a two-mode device can be written as
\[\tag{9-256}\pm\frac{\text{d}A}{\text{d}z}=\text{i}\kappa_{aa}A+\text{i}\kappa_{ab}B\text{e}^{\text{i}(\beta_b-\beta_a)z}+\text{i}\sigma_{aa}|A|^2A+\text{nonlinear cross terms}\]
\[\tag{9-257}\pm\frac{\text{d}B}{\text{d}z}=\text{i}\kappa_{bb}B+\text{i}\kappa_{ba}A\text{e}^{\text{i}(\beta_a-\beta_b)z}+\text{i}\sigma_{bb}|B|^2B+\text{nonlinear cross terms}\]
where \(\sigma_{aa}=\omega{C}_{aaaa}\) and \(\sigma_{bb}=\omega{C}_{bbbb}\), as defined in (9-246). In general, \(\sigma_{aa}\ne\sigma_{bb}\).
The nonlinear cross terms are those that represent direct nonlinear coupling between the two modes with the nonlinear coefficients \(C_{aaab}\), \(C_{aaba}\), \(C_{abaa}\), \(C_{aabb}\), \(C_{abab}\), \(C_{abba}\), and \(C_{abbb}\) for (9-256) and \(C_{bbba}\), \(C_{bbab}\), \(C_{babb}\), \(C_{bbaa}\), \(C_{baba}\), \(C_{baab}\), and \(C_{baaa}\) for (9-257). Such nonlinear cross terms are generally much smaller than the direct nonlinear terms characterized by \(\sigma_{aa}\) and \(\sigma_{bb}\), which are explicitly expressed in the above coupled equations.
In a device that is based solely on interference, \(\kappa_{ab}=\kappa_{ba}=0\), and the nonlinear cross terms vanish also. Therefore, there is generally no direct power exchange between the two modes. The nonlinear differential phase shift between the two modes controls the interference condition, thus turning an optical-power-dependent phase change into an amplitude modulation or switching.
In a device that is based on coupling, \(\kappa_{ab}\ne0\) and \(\kappa_{ba}\ne0\). The function of modulation or switching is then a result of direct exchange of power between the two modes. In such a device, the power-dependent differential phase shift controls the effective coupling coefficient between the two modes through its influence on the phase matching between them.
Nonlinear optical mode mixers
A nonlinear mode mixer is a simple all-optical switch based on the power-dependent interference effect between two modes in a multimode waveguide, such as the \(\text{TE}_0\) and \(\text{TE}_1\) modes of a slab waveguide.
Two different modes in an unperturbed waveguide are orthogonal to each other in the absence of nonlinear effects. Even when the optical Kerr effect is present, significant direct coupling between them occurs only when the power in the waveguide reaches a critical level.
Below this critical power level, the optical Kerr effect leads to sufficient self-phase modulation in each individual mode but no significant cross-phase modulation or power exchange between the mutually orthogonal modes.
For a nonlinear two-mode mixer operating in this regime, \(\kappa_{aa}=\kappa_{bb}=\kappa_{ab}=\kappa_{ba}=0\), and the nonlinear cross interaction between the two modes can also be neglected. Consequently, both (9-256) and (9-257) reduce to the form of (9-245) with the solution given in (9-247). Therefore, the total field in the two-mode mixer is
\[\tag{9-258}\begin{align}\mathbf{E}(\mathbf{r})&=A(0)\hat{\boldsymbol{\mathcal{E}}}_a(x,y)\text{e}^{\text{i}\beta_a^\text{eff}z}+B(0)\hat{\boldsymbol{\mathcal{E}}}_b(x,y)\text{e}^{\text{i}\beta_b^\text{eff}z}\\&=\left[A(0)\hat{\boldsymbol{\mathcal{E}}}_a(x,y)+B(0)\hat{\boldsymbol{\mathcal{E}}}_b(x,y)\text{e}^{\text{i}(\beta_b^\text{eff}-\beta_a^\text{eff})z}\right]\text{e}^{\text{i}\beta_a^\text{eff}z}\end{align}\]
For a mode mixer of a length \(l\), the total differential phase shift between the two modes over the length of the device is
\[\tag{9-259}\Delta\varphi=(\beta_b^\text{eff}-\beta_a^\text{eff})l=\Delta\varphi_\text{L}+\Delta\varphi_\text{NL}\]
where \(\Delta\varphi_\text{L}=(\beta_b-\beta_a)l\) is the linear differential phase shift due to modal dispersion and \(\Delta\varphi_\text{NL}=(\beta_b^\text{NL}-\beta_a^\text{NL})l\) is the nonlinear differential phase shift due to the difference in the self-phase modulation of the two different modes.
For a given device of a fixed length, the value of \(\Delta\varphi_\text{L}\) is fixed, but that of \(\Delta\varphi_\text{NL}\) varies with the powers in the modes. Therefore, \(\Delta\varphi\) can be controlled by the power coupled into the waveguide.
Even when that power is evenly divided between the two modes, there is still a power-dependent differential phase shift between the two modes because the self-phase modulation expressed in (9-254) for a waveguide mode is also a function of the mode-dependent effective area \(\mathcal{A}_\nu^\text{eff}\).
Figure 9-33 illustrates the principle of a two-mode mixer. In this example, the power launched into the waveguide is equally divided between the two modes so that \(P_a=P_b=P/2\) and \(A(0)=B(0)\) at the input end. Therefore, the total field is asymmetrically distributed with its peak on one side of the waveguide.
The linear differential phase shift in this example is \(\Delta\varphi_\text{L}=2n\pi\), where \(n\) is an integer.
At low power levels when the power-dependent nonlinear differential phase shift \(\Delta\varphi_\text{NL}\) is negligibly small, the field distribution at the output end is the same as that at the input end, as shown in Figure 9-33(a).
At a power level of \(P_\pi\) when \(\Delta\varphi_\text{NL}=\pi\), the total differential phase shift is \(\Delta\varphi=(2n+1)\pi\). Then, at the output end the peak of the total field is switched to the other side of the waveguide, as shown in Figure 9-33(b).
In this manner, a nonlinear mode mixer functions as a power-dependent all-optical switch.
A nonlinear mode mixer can take the form of a two-mode slab waveguide or that of a two-mode channel waveguide. In the latter case, both input and output ends of the mode mixer can be connected to a Y-junction waveguides for all-optical switching of optical power between separate waveguides, as shown in Figure 9-34. Such a device also functions as a nonlinear mode sorter.
All-optical Mach-Zehnder interferometers
A nonlinear mode mixer functions as a nonlinear interferometric device only when the optical power in the waveguide is kept below the critical power level to prevent direct coupling of the modes. This limitation is caused by the fact that the two modes overlap in space while propagating codirectionally. It can be avoided in a nonlinear interferometer that consists of two separate arms such as one in the form of a Mach-Zehnder interferometer as shown in Figure 9-35.
There are a few significant differences between a nonlinear Mach-Zehnder interferometer and a nonlinear mode mixer:
(1) both arms of the interferometer are generally single-mode waveguides;
(2) at any power level, there is no cross modulation between the fields in the two separate arms of the interferometer;
(3) the two fields that are combined at the output end of the interferometer can experience different propagation distances because the lengths of the two arms do not have to be the same.
An all-optical Mach-Zehnder interferometer is based on the same principle as the electro-optic Mach-Zehnder interferometer discussed in the guided-wave electro-optic modulators tutorial except that the differential phase shift \(\Delta\varphi\) between its two arms is controlled by the optical power rather than by an applied electric field. An all-optical Mach-Zehnder interferometer can have a single input channel, as shown in Figure 9-35, or three input channels, as shown in Figure 9-36.
Figure 9-35 shows two possible structures of single-input, all-optical Mach-Zehnder interferometers. The beam-splitting and beam-combining couplers at the input and output ends, respectively, of an all-optical Mach-Zehnder interferometer can be either Y-junction waveguides, as shown in Figure 9-35(a), or directional couplers, as shown in Figure 9-35(b).
The two arms of an all-optical interferometer are not required to be identical. Therefore, in general, the linear differential phase shift is \(\Delta\varphi_\text{L}=\beta_bl_b-\beta_al_a\), and the nonlinear differential phase shift can be expressed as
\[\tag{9-260}\Delta\varphi_\text{NL}=\beta_b^\text{NL}l_b-\beta_a^\text{NL}l_a=\frac{2\pi}{\lambda}\left(n_{2b}\frac{P_b}{\mathcal{A}_b^\text{eff}}l_b-n_{2a}\frac{P_a}{\mathcal{A}_a^\text{eff}}l_a\right)\]
where \(l_a\) and \(l_b\) are the lengths of the two separate arms, respectively.
We see that a power-dependent nonlinear differential phase shift can be obtained only when the two arms are not balanced, due to unbalanced excitation or physical asymmetry between them.
With unbalanced excitation, \(P_a\ne{P}_b\). Physical asymmetry exists when the waveguides that form the two arms have different lengths, \(l_a\ne{l}_b\), or different effective areas, \(\mathcal{A}_a^\text{eff}\ne\mathcal{A}_b^\text{eff}\), or different values of nonlinearity, \(n_{2a}\ne{n}_{2b}\), or any combination of them.
To facilitate the possibility of unbalanced excitation, the Y-junction waveguides or directional couplers used in an all-optical Mach-Zehnder interferometer are not necessarily 3-dB couplers.
For a given device, however, the beam-splitting coupler at the input end and the beam-combining coupler at the output end are usually identical couplers with a fixed power-splitting ratio of \(\xi:(1-\xi)\) between the two arms, as also shown in Figure 9-35.
For an all-optical Mach-Zehnder interferometer that uses such Y-junction waveguides as input and output couplers, the power transmittance is
\[\tag{9-261}T=1-2\xi(1-\xi)(1-\cos\Delta\varphi)\]
where \(\Delta\varphi=\Delta\varphi_\text{L}+\Delta\varphi_\text{NL}\) is the total differential phase shift.
For one that uses such directional couplers as input and output couplers, the power transmittance through the same channel is
\[\tag{9-262}T=1-2\xi(1-\xi)(1+\cos\Delta\varphi)\]
Note that (9-261) reduces to the form of (6-77), and (9-262) reduces to that of (6-79) [refer to the guided-wave electro-optic modulators tutorial], if the two arms of the interferometer are equally excited so that \(\xi=1/2\).
As discussed above, such balanced excitation is feasible only when the two arms of the interferometer are physically asymmetric. Such physical asymmetry leads to a nonvanishing linear differential phase shift, \(\Delta\varphi_\text{L}\ne0\), which acts as a bias phase shift. By properly adjusting the asymmetry between the two arms, the value of this bias phase shift can be chosen for a desired operating point of the device.
Figure 9-36 shows the structure of a three-input, symmetric all-optical Mach-Zehnder interferometer using Y-junction waveguides.
This device consists of three input channels that are fed into a symmetric Mach-Zehnder interferometer with arms of equal lengths. The data signal is sent through the central channel \(c\), and the control signals are fed into either channel \(a\) or \(b\) or both.
The data signal wave is orthogonally polarized with respect to the control signals to avoid interference between them. A polarizer at the output end allows only the polarization of the data signal to pass. Interaction between the data signal and the control signals is through cross-phase modulation only.
Because a data signal sent through channel \(c\) is equally split between the two arms of the interferometer, nonlinear phase shifts caused by self-phase modulation of the data signal in the two arms cancel. The net differential nonlinear phase shift is caused by the cross-phase modulation imposed by any control signals on the data signal.
This differential nonlinear phase shift has exactly the form of (9-260) with \(n_{2a}=n_{2b}=n_2\), \(l_a=l_b=l\), and \(\mathcal{A}_a^\text{eff}=\mathcal{A}_b^\text{eff}=\mathcal{A}_\text{eff}\) for a symmetric Mach-Zehnder interferometer:
\[\tag{9-263}\Delta\varphi_\text{NL}=\frac{2\pi}{\lambda}\frac{n_2l}{A_\text{eff}}(P_b-P_a)\]
where \(n_2\) is the nonlinear refractive index due to cross-phase modulation between orthogonally polarized waves.
Though the two arms of a symmetric Mach-Zehnder interferometer are equal in length, it is still possible to introduce a linear phase difference between them by a bias voltage if the device is fabricated on an electro-optic material.
For a symmetric Mach-Zehnder interferometer using Y-junction waveguides, the transmittance of the data signal is that given in (9-261) with \(\xi=1/2\), which is reduced to the following simple form:
\[\tag{9-264}T=\cos^2\frac{\Delta\varphi}{2}\]
An all-optical interferometer has many useful applications. Like an electro-optic interferometer, it can be used as an amplitude modulator or, when accompanied by a directional coupler instead of a Y-junction waveguide at the output end, as a switch.
Unlike an electro-optic interferometer, however, its function is completely controlled by the input optical power alone. Therefore, there are some unique applications of an all-optical interferometer that are not possible with an electro-optic interferometer.
For instance, with unbalanced excitation in an all-optical interferometer with symmetric arms, it is possible to shape an optical pulse by taking advantage of the fact that the power-dependent transmittance of the device now varies across the envelope of the pulse. Pulse shortening can be achieved if the maximum transmittance occurs at the peak of the pulse while the wings of the pulse have very low transmittance.
All-optical Mach-Zehnder interferometers can be made to perform certain unique functions, such as optical logic, optical sampling, and optical ON-OFF switching.
Example 9-24
A three-input, symmetric all-optical Mach-Zehnder interferometer as shown in figure 9-36 consists of AlGaAs channel waveguides fabricated on a GaAs substrate along the \([110]\) crystal axis on the \((001)\) plane.
The data signal launched into channel \(c\) is a \(\text{TM}\)-like mode polarized in the \([001]\) direction. A control signal is launched into channel \(a\) as a \(\text{TE}\)-like mode polarized in the \([1\bar10]\) direction. No control signal is launched into channel \(b\).
Both data and control signals are at \(\lambda=1.55\text{ μm}\) wavelength. The length of both arms of the interferometer is \(l=2\text{ cm}\), and the effective area of the channel waveguide is \(\mathcal{A}_\text{eff}=6\text{ μm}^2=6\times10^{-12}\text{ m}^2\).
The nonlinear refractive index characterizing cross-phase modulation between TE-like and TM-like modes in this AlGaAs waveguide at \(\lambda=1.55\text{ μm}\) is \(n_2=1.3\times10^{-17}\text{ m}^{2}\text{ W}^{-1}\). No linear bias phase is applied to either arm of the device.
Ignoring all possible linear and nonlinear losses, find the power of the control signal needed for this device to function as an all-optical ON-OFF switch. If the control signal is in the form of an optical pulse of \(\Delta{t}_\text{ps}=1\text{ ps}\) pulsewidth, what is the switching energy of the control pulse?
Solution:
For the device to function as an all-optical ON-OFF switch, both the ON state with a transmittance of \(T=1\) and the OFF state with a transmittance of \(T=0\) have to be accessible by varying the power of the control signal. Because there is no linear phase bias, the total differential phase shift of the device is contributed solely by the nonlinear effect; thus \(\Delta\varphi=\Delta\varphi_\text{NL}\). Because no control signal is launched into channel \(b\), \(P_b=0\). From (9-264), we then find that the minimum nonlinear differential phase shift required for \(T=1\) is \(\Delta\varphi_\text{NL}=0\) and that required for \(T=0\) is \(\Delta\varphi_\text{NL}=-\pi\). Therefore, we find from (9-263) that the ON state can be reached by simply making \(P_a=0\) so that \(\Delta\varphi_\text{NL}=0\). By setting \(\Delta\varphi_\text{NL}=-\pi\) in (9-263), we find the following control signal power required to reach the OFF state:
\[P_a=\frac{\lambda\mathcal{A}_\text{eff}}{2n_2l}=\frac{1.55\times10^{-6}\times6\times10^{-12}}{2\times1.3\times10^{-17}\times2\times10^{-2}}\text{ W}=17.88\text{ W}\]
If the control signal is in the form of an optical pulse of \(\Delta{t}_\text{ps}=1\text{ ps}\) pulsewidth, the device is in the ON state with \(T=1\) in the absence of a control pulse. The device can be switched to the OFF state with a control pulse of a switching energy of \(U_\text{ps}=P_a\Delta{t}_\text{ps}=17.88\text{ pJ}\).
Clearly, a waveguide Mach-Zehnder interferometer operated with a CW beam at 17.88 W is not practical, but it is practical with an ultrashort pulse of 17.88 W peak power such as the one of \(1\text{ ps}\) pulsewidth considered here. For this reason, the control signal for this device is generally in the form of ultrashort laser pulses though the data signal can be of any waveform.
Here we have ignored the losses and dispersion of the waveguide. In reality, the waveguide has both linear losses, mainly from scattering and impurity absorption, and nonlinear losses, from both two-photon and three-photon absorption processes. These losses will increase the switching power of the device while reducing its extinction ratio between ON and OFF states. When the device is operated with short optical pulses, the dispersion of the waveguide can broaden the pulses and introduce an additional linear phase shift in the pulses. The consequences are also an increase in the switching energy and a reduction in the extinction ratio.
Nonlinear optical loop mirrors
A nonlinear optical loop mirror, is a folded Mach-Zehnder interferometer in the so-called Sagnac configuration, as shown in Figure 9-37.
The basic device shown in Figure 9-37(a) consists of a single-mode waveguide loop, such as a single-mode fiber or a single-mode semiconductor waveguide, that is closed with a four-port directional coupler.
The two paths of opposite propagation directions in the loop are equivalent to the two arms of an interferometer. The single coupler, which has a power-splitting ratio of \(\xi:(1-\xi)\), serves as both the power-splitting input coupler and the power-combining output coupler.
An input field is split into two contrapropagating fields that travel through exactly the same loop path but in opposite directions before recombining at the coupler to form the output of the device.
The optical field launched into the device can be a short pulse that has a spatial span much shorter than the loop length. Then interaction between contradirectionally propagating pulses in the loop is negligible so that only the self-phase modulation of each individual pulse needs to be considered.
It can also be a very long pulse or a CW wave that fills up the entire loop. Then the cross-phase modulation between contradirectionally propagating waves needs to be considered as well.
Because of the exact symmetry between the two contradirectional paths, \(\Delta\varphi_\text{L}=0\) irrespective of the operating condition. It can be shown that for both cases discussed here, we have
\[\tag{9-265}\Delta\varphi=\Delta\varphi_\text{NL}=(1-2\xi)\frac{2\pi{n}_2}{\lambda}\frac{P_\text{in}}{\mathcal{A}_\text{eff}}l\]
where \(P_\text{in}\) is the input power launched into the device and \(l\) is the length of the loop.
The transmittance of the device is that given in (9-262) with \(\Delta\varphi=\Delta\varphi_\text{NL}\) given above. The device also has a reflectance \(R=1-T\) back to the original input port.
In the linear regime at low power levels, the device functions as a mirror with \(R=4\xi(1-\xi)\) and \(T=1-4\xi(1-\xi)\).
In the nonlinear regime at high power levels, the device functions as a nonlinear mirror with power-dependent reflectance and transmittance due to the dependence of \(\Delta\varphi_\text{NL}\) on the input power.
Similarly to the Mach-Zehnder interferometer, a nonlinear optical loop mirror can also accept a control signal to switch the data signal. Figure 9-37(b) shows a two-input configuration for such a purpose. More sophisticated configurations are also possible. With a control signal, a nonlinear optical loop mirror can perform such functions as optical switching, sampling, multiplexing, and demultiplexing.
There are several advantages of using the nonlinear optical loop mirror as an all-optical interferometric device over the conventional all-optical Mach-Zehnder interferometer with two separate arms.
Because the two contrapropagating fields in a nonlinear optical loop mirror travel over exactly the same path in opposite directions, they experience exactly the same linear effects, which cancel out when the two fields are combined in returning to the coupler.
Therefore, the device is stable against external perturbations and does not require interferometric alignment. This unique characteristic allows a very long fiber on the order of kilometers to be used for a nonlinear optical loop mirror to function at a low optical power level with sufficient self-phase modulation, making it a truly practical all-optical device.
Because response and relaxation of Kerr nonlinearity in silica fibers are nearly instantaneous, a nonlinear fiber loop mirror is also ideal for many applications that use ultrashort optical pulses. The precise match in length of the contradirectional paths in this device ensures precise coincidence of the returning pulses, which is a daunting task with a conventional interferometer with separate arms considering the fact that the path length can be as long as a few kilometers while the pulses can be shorter than \(1\text{ ps}\).
Example 9-25
A single-input nonlinear optical fiber loop mirror of the configuration shown in Figure 9-37(a) consists of a single-mode fiber that has a loop length of \(l=100\text{ m}\) and an effective cross-sectional area of \(\mathcal{A}_\text{eff}=3\times10^{-11}\text{ m}^2\) for an optical wave at \(\lambda=1.55\text{ μm}\). The self-phase modulation nonlinear refractive index of this fiber is \(n_2=3.2\times10^{-20}\text{ m}^2\text{ W}^{-1}\). At low input power levels, this loop mirror has a transmittance of \(T=25\%\). Find the lowest input power that is required for it to have a transmittance of \(100\%\).
Solution:
With a low-power transmittance of \(T=25\%=1/4\), we find by solving \(T=1-4\xi(1-\xi)=1/4\) that \(\xi=1/4\) for the power-splitting ratio of the coupler in the device.
By plugging \(\xi=1/4\) and \(T=1\) into (9-262), we find that the nonlinear phase shift required for \(T=1\) at a high power level is a solution of the following condition: \(1+\cos\Delta\varphi=0\). Therfore, \(\Delta\varphi=(2n+1)\pi\) for any integer \(n\).
From (9-265), we see that \(P_\text{in}\propto\Delta\varphi\). The lowest required power for \(T=100\%\) can be obtained by plugging \(\Delta\varphi=\pi\) and \(\xi=1/4\) into (9-265) to find that
\[P_\text{in}=\frac{\lambda\mathcal{A}_\text{eff}}{n_2l}=\frac{1.55\times10^{-6}\times3\times10^{-11}}{3.2\times10^{-20}\times100}\text{ W}=14.53\text{ W}\]
This power is too high for this fiber device to be practical if the input is a CW signal. It is not a problem if the input signal consists of very short pulses. For instance, an average power of only 1.453 mW is required if the input signal is made up with pulses of \(1\text{ ps}\) pulsewidth at a repetition rate of 100 MHz. For this reason, nonlinear optical loop mirrors are generally operated with very short laser pulses.
Nonlinear directional couplers
The coupling efficiency of a directional coupler can be varied by varying the phase mismatch or the coupling coefficients between the two waveguides that form the directional coupler.
For an electrically modulated directional coupler discussed in the guided-wave electro-optic modulators tutorial, the coupling coefficient is a function of an externally applied voltage that induces changes in the refractive index of the waveguide material through the Pockels effect.
For an all-optical nonlinear directional coupler based on the optical Kerr effect, the coupling coefficient can be varied by varying the value or the distribution of the optical power launched into the device.
A nonlinear directional coupler can be formed using two parallel waveguides fabricated in such nonlinear crystals as GaAs or LiNbO3, as shown in Figure 9-38(a). It can also be formed using a dual-core optical fiber, as shown in Figure 9-38(b).
The advantage of using a dual-core fiber is that a coupler of a very long interaction length on the order of kilometers can be easily realized to make practical use of the small optical nonlinearity in a fiber.
In the following, we consider for simplicity only symmetric directional couplers in which the two waveguide channels are identical. Asymmetric nonlinear directional couplers have similar characteristics.
For a symmetric nonlinear directional coupler that is formed by two identical single-mode waveguides, we have \(\beta_a=\beta_b\) and \(\kappa_{aa}=\kappa_{bb}\). Therefore, the effective linear propagation constant for each individual waveguide mode is \(\beta=\beta_a+\kappa_{aa}=\beta_b+\kappa_{bb}\). In addition, \(\kappa_{ab}=\kappa^*_{ba}\equiv\kappa\), which is real and positive, and \(\sigma_{aa}=\sigma_{bb}\equiv\sigma\), which is real but can be either positive or negative depending on the sign of \(\chi^{(3)}\) of the Kerr medium. The coupled equations for a symmetric nonlinear directional coupler can then be written as
\[\tag{9-266}\frac{\text{d}\tilde{A}}{\text{d}z}=\text{i}\kappa\tilde{B}+\text{i}\sigma|\tilde{A}|^2\tilde{A}+\text{nonlinear cross terms}\]
\[\tag{9-267}\frac{\text{d}\tilde{B}}{\text{d}z}=\text{i}\kappa\tilde{A}+\text{i}\sigma|\tilde{B}|^2\tilde{B}+\text{nonlinear cross terms}\]
where \(\tilde{A}=A\text{e}^{-\text{i}\kappa_{aa}z}\) and \(\tilde{B}=B\text{e}^{-\text{i}\kappa_{bb}z}\) as defined in (52) [refer to the two-mode coupling tutorial].
The terms characterized by the coupling coefficient \(\kappa\) represent linear coupling between the two modes. The terms characterized by \(\sigma\) contribute to the self-phase modulation of each individual mode. The nonlinear cross terms, which are not explicitly spelled out because of their complexity, contribute to direct nonlinear coupling between the two modes.
In general, the nonlinear cross terms, though not completely negligible, are much smaller than the terms that represent linear coupling and self-phase modulation in each equation.
Indeed, the direct nonlinear coupling contributed by the nonlinear cross terms is not necessary for the functioning of a nonlinear directional coupler. The basic operation principle of a nonlinear directional coupler is that the self-phase modulation of each individual mode creates a power-dependent differential phase shift that leads to a power-dependent phase mismatch between the two modes.
As a consequence, the coupling coefficient would become power dependent even if the only coupling were the linear coupling characterized by the linear coefficient \(\kappa\). The direct nonlinear coupling contributed by the nonlinear cross terms acts as an additional perturbation, which changes the detailed quantitative characteristics of a nonlinear directional coupler. The general characteristics of a nonlinear directional coupler can be fully understood without considering the nonlinear cross terms.
We consider only the simple case when the nonlinear cross terms are neglected. We also assume that the input optical power is initially launched into only waveguide \(a\) so that \(P_a(0)=P_\text{in}\) and \(P_b(0)=0\). Under these assumptions, the coupling efficiency of a nonlinear coupler that has an interaction length \(l\) is found to be
\[\tag{9-268}\eta=\frac{P_b(l)}{P_\text{in}}=\frac{1}{2}[1-\text{cn}(2\kappa{l},m)]=\frac{1}{2}\left[1-\text{cn}(2\kappa{l},\frac{\sigma}{4\kappa}P_\text{in})\right]\]
where
\[\tag{9-269}m=\frac{\sigma}{4\kappa}P_\text{in}=\frac{P_\text{in}}{P_\text{c}}\]
is an index that characterizes the level of the input power with respect to a critical power level \(P_\text{c}\) that is defined as
\[\tag{9-270}P_\text{c}=\frac{4\kappa}{\sigma}=\frac{2\kappa\lambda\mathcal{A}_\text{eff}}{\pi{n}_2}\]
and \(\text{cn}(z,m)\) is a Jacobi elliptic function defined by
\[\tag{9-271}z=\displaystyle\int\limits_x^1\frac{\text{d}t}{(1-t^2)^{1/2}(1-m^2+m^2t^2)^{1/2}}=\text{cn}^{-1}(x,m)\]
For the symmetric coupler under consideration, \(P_a(l)+P_b(l)=P_\text{in}\), and the power transmittance through the input channel is
\[\tag{9-272}T=\frac{P_a(l)}{P_\text{in}}=1-\eta\]
It can be clearly seen from (9-268) that the coupling efficiency of a nonlinear coupler is a function of the input power to the device.
Figure 9-39 shows the coupling efficiency as a function of interaction length \(l\), normalized to the linear coupling length \(l_\text{c}^\text{PM}=\pi/2\kappa\), at various input power levels that are characterized by different values of the index \(m\).
In the limit of very low input powers, \(P_\text{in}\ll{P}_\text{c}\), and \(m\approx0\), the coupling efficiency reduces to that of the phase-matched linear directional coupler, \(\eta=(1-\cos2\kappa{l})/2=\sin^2\kappa{l}\) given in (85) [refer to the two-mode coupling tutorial], because \(\text{cn}(2\kappa{l},0)=\cos(2\kappa{l})\), and the coupling length is just \(l_\text{c}^\text{PM}\).
As the input power increases, a power-dependent phase mismatch between the two waveguide channels is generated by the power-dependent differential phase shift. At relatively low input powers, \(P_\text{in}\lt{P}_\text{c}\) and \(m\lt1\), this power-dependent phase mismatch has the effect of slowing down the power transfer between the two channels. This phase mismatch is reduced as more power is transferred and is later even reversed as more than 50% of the input power is transferred. The nonlinear directional coupler thus acts like a reversed-\(\Delta\beta\) coupler. Complete switching of power with \(\eta=1\) to reach the cross state still occurs, but the coupling length is longer than the linear coupling length and it increases as the input power increases. These effects can be observed from the curve for \(m=0.9\) in Figure 9-39.
At high input powers, \(P_\text{in}\gt{P}_\text{c}\) and \(m\gt1\), the initial phase mismatch is so large that the power transfer never reaches the 50% point for the phase mismatch to be reversed. Therefore, the coupling efficiency oscillates, but \(\eta\lt1/2\) for any device length. The cross state cannot be reached at such high power levels, as can be seen in Figure 9-39 from the curves for \(m=1.1\) and \(m=2\).
At the critical power level, \(P_\text{in}=P_\text{c}\) and \(m=1\), the coupling efficiency stays at \(\eta=1/2\) indefinitely after 50% of the input power is transferred. This state is unstable as any perturbation caused by noise or fluctuations in the input power can tip this balance between the two channels.
Figure 9-40 shows the coupling efficiency as a function of input power \(P_\text{in}\), normalized to the critical power \(P_\text{c}\), for a symmetric nonlinear directional coupler with a fixed length \(l=l_\text{c}^\text{PM}\), known as the half-beat-length coupler, and another with \(l=2l_\text{c}^\text{PM}\), known as the beat-length coupler.
For the half-beat-length coupler, which starts with a linear coupling efficiency of \(\eta=1\) at a very low input power, the coupling efficiency remains high until the input power approaches the level of \(P_\text{c}\) when it drops and remains low for all high power levels above \(P_\text{c}\).
For the beat-length coupler, which starts with a linear coupling efficiency of \(\eta=0\) at a very low power level, only a very narrow power range exists for high coupling efficiencies with a peak value of \(\eta=1\).
In the above discussions, only symmetric directional couplers are considered, and the effect of nonlinear cross terms in (9-266) and (9-267) are ignored. There exists a general analytical solution in the form of elliptic functions for the coupled nonlinear differential equations even when the structure of the coupler is asymmetric and the nonlinear cross terms are considered.
The primary effect of the nonlinear cross terms is to cause a change in the value of \(P_\text{c}\) depending on the strength of the nonlinear cross coupling between the two waveguide modes. For an asymmetric coupler, the initial linear phase mismatch leads to power-dependent characteristics that are nonreciprocal with respect to detuning between the two channels.
Example 9-26
A half-beat-length nonlinear directional coupler of the structure shown in Figure 9-38(a) for a TE-like mode has a length of \(l=1.5\text{ cm}\). It consists of two parallel AlGaAs channel waveguides on a GaAs substrate with the same structural parameters as the AlGaAs waveguides described in Example 9-24. At \(\lambda=1.55\text{ μm}\) wavelength, the nonlinear refractive index characterizing self-phase modulation for the TE-like mode in the waveguide is \(n_2=1.5\times10^{-17}\text{ m}^2\text{ W}^{-1}\). Find the critical power of the device.
Solution:
Because the device is a half-beat-length coupler, we have \(l_\text{c}^\text{PM}=l=1.5\text{ cm}\). Therefore, the coupling coefficient is \(\kappa=\pi/2l_\text{c}^\text{PM}=\pi/2l\). By plugging this relation into (9-270), we find the following critical power:
\[P_\text{c}=\frac{\lambda\mathcal{A}_\text{eff}}{n_2l}=\frac{1.55\times10^{-6}\times6\times10^{-12}}{1.5\times10^{-17}\times1.5\times10^{-2}}\text{ W}=41.3\text{ W}\]
If the device is operated with a short pulse of \(\Delta{t}_\text{ps}=1\text{ ps}\) like that considered in Example 9-24, then the critical pulse energy is \(41.3\text{ pJ}\).
The next tutorial describes optical transitions for laser amplifiers.