Optical Equations 3 Optical Model for Optoelectronic Devices

1. Optical Equations 3 Optical Model for Optoelectronic Devices

2. Maxwell’s Equations for Optical Waves in Semiconductors For the given semiconductor material property and driven source, 9 variables can be obtained by the above 9 equations using FDTD. A major draw-back, however, is the huge memory requirement due to the to the number of variables involved and material dispersiveness. Despite this pitfall, FDTD is a most general numerical approach that doesn’t need any pre-assumption to the field.

3. Wave Equations in Semiconductors For the given semiconductor material property and driven source, 6 variables can be obtained by the above 6 equations. The 1st term on the LHS represents the coupling between the 3 field components, which is the field divergence contribution due to the structure nonuniformity. In weakly-guided optoelectronic devices, the structure nonuniformity is weak where the field components are strong (in device cross-sections), whereas the field component is weak where the structure nonuniformity is significant (along the device cavity). Therefore, this term (the coupling between different field components) becomes negligible once the device is weakly-guided.

4. Wave Equations in Semiconductors • Most of our current optoelectronic devices are weakly-guided. Hence the 1st term on the LHS of the general wave equation is negligible. • By neglecting this term, the field and polarization components along each direction (x, y, z) become fully decoupled. • Therefore, we just need to solve 2 unknown variables for at most 3 times in a sequence, rather than to solve 6 unknowns simultaneously. Consider our numerical algorithm is always super-linearly scaled, (e.g., the computational complexity of a full matrix is N2~3,) such arrangement usually saves tremendous amount of time. • For some strongly-guided optoelectronic devices, e.g., the polarization rotator using the asymmetric ridge waveguide design, we cannot throw away this term hence the wave equation must be solved in its original form. Otherwise, no component coupling would appear in the governing equations and we will certainly fail in modeling such device with its operating mechanism established on the coupling between different field components.

5. Wave Equations in Semiconductors In semiconductors, different field components are decoupled as indicated in above wave equation. The only term on the LHS represents the wave diffraction in space. The 1st term on the RHS represents the wave dispersion along with time. These two terms exist even in vacuum, hence they reflect the inherent property of the electromagnetic wave, i.e., the wave propagation is a balance between the wave diffraction in space and wave dispersion along with time.

6. Wave Equations in Semiconductors • The 2nd term on the RHS purely represents the medium effect, under the excitation of the electric field, the medium is polarized and equivalent dipoles are formed which is collectively represented by the material susceptibility ( ). • We can view the material susceptibility as a medium response to the input field, which generally cannot follow the time change of the input field. Therefore, the medium output represented by the polarization vector must be a summation of all the historical medium responses. That’s why the polarization and the field is connected through the equation in an integration form.

7. Wave Equations in Semiconductors • The polarization vector, as a summation of many delayed waves, can be sorted into two groups, an in-phase group and an phase orthogonal (with 90 degree difference) group (taking the phase of the input field as the reference). This is why in frequency domain, the material susceptibility is complex. • The in-phase component of the polarization vector, corresponding to the response of the real part material susceptibility in FD, gives a dispersion effect, which makes different wave component to propagate at different speed. • The orthogonal phase component of the polarization vector, corresponding to the response of the imaginary part material susceptibility in FD, gives an amplification or attenuation effect, which makes different wave component take different amplitude amplification (or attenuation) in its propagation.

8. Wave Equations in Semiconductors • The last term on the RHS represents the source contribution in the host medium, which is independent of the input field hence it shows up in the wave equation in an inhomogeneous manner. • In passive materials with negligible loss, the material susceptibility in FD only has real part. There is no inhomogeneous source contribution either. • In semiconductors, the material susceptibility in FD has both real and imaginary parts; the stimulated emission gain (with sufficient pumping) or stimulated absorption loss (with insufficient pumping) in semiconductor contributes to the imaginary part, whereas the spontaneous emission contributes to the inhomogeneous source.

9. Wave Equations in Semiconductors • A major advantage of working on the wave equation, as compared to working on the original Maxwell’s equations, is that less variables are involved. • However, we have to deal with PDEs with a lifted (2nd) order, which may result in unstable problem. • Naturally we will ask a greedy question: Can we further reduce the order of PDEs without introducing extra unknowns variables? • Mathematically we can prove that it is not possible. • Practically, however, under certain approximation, we managed to reach this goal!

10. Reduced Wave Equations in Time and Space - The Traveling Wave Model A basic assumption: the wave propagates along a dominant direction. Hence the field forms a 1D traveling wave with its propagation constant in its traveling direction much larger than the rests. This assumption is valid in most edge-emitting devices.

11. Reduced Wave Equations in Time and Space - The Traveling Wave Model Due to the system linearity, we just need to study one basic element composed of one polarization, one transverse mode, and one propagation factor with varying envelope, without losing the generality. The analysis of each basic element follows a similar way. A complete solution can therefore be constructed by taking the summation of these basic elements.

12. Reduced Wave Equations in Time and Space - The Traveling Wave Model • Reasons to use the phaser expression for the field: • Sine and cosine functions are not eigen solutions. However, their linear combination, i.e., the exponential function is the eigen solution of any linear, parameter invariant system. • The wave equation is linear and real, (not necessarily parameter invariant though), the orthogonality between the real and imaginary components in its solution remains. For this reason, one don’t need to include the complex conjugate part in solving the linear wave equation. • We can also throw away the factor ½ and take the real part of the phaser expression as our final real-world solution. • This approach is only valid for linear and real wave equation.

13. Reduced Wave Equations in Time and Space - The Traveling Wave Model Following equations are obtained once we substitute the field in its factorized form back into the wave equation: Parameters in these equations are obtained by:

14. Reduced Wave Equations in Time and Space - The Traveling Wave Model The equation for the backward propagating wave can readily be obtained by replacing “z” with “-z”: For any given waveguide structure, once the transverse eigen value problem and the longitudinal traveling wave equations are solved, we find the real-world field as:

15. Reduced Wave Equations in Time and Space - The Traveling Wave Model • Assumptions on the optical waveguide • Wave propagates along the device longitudinal direction only • Wave is fully confined in the cross-sectional area perpendicular to the propagation direction • Wave has discrete optical frequencies with relatively slow-varying envelopes (multiple channel, modulated narrow-band signal) • Assumptions on the material • Material has linear optical property • Material takes no time to respond to any optical wave envelope variation • Other assumptions (removable) • Single operation frequency • Single transverse mode • Single polarization state

16. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide Phase matching condition in periodically corrugated waveguide: Any periodically corrugated shape can be expanded into Fourier series in harmonic orders, i.e., if the period of the corrugation is Λ, we have many sinusoidal gratings with periods Λ/M (M=1,2,3,…) simultaneously.

17. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide Therefore, the phase matching condition is found to be: For waves with their wavelength shorter than the above value, radiation happens due to the “free-running” among multiple reflections. For waves with their wavelength longer than the above value, rapid damping happens due to the destructive addition of multiple reflections. Therefore, if the operating wavelength matches to the Mth order sinusoidal grating, the effect of the (M+1)th, (M+2)th, … sinusoidal gratings is negligible since the reflected waves by these grating orders must decay rapidly. Whereas the effect of the (M-1)th, (M-2)th, … sinusoidal gratings is to couple the forward and backward traveling waves to the radiating waves that leave the waveguide.

18. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide • The effect of the Mth order grating is to couple the forward traveling wave to the backward traveling wave and vice versa. • This effect provides a distributed feedback mechanism hence can be used for optical feedbacks as an alternative approach to the classical lump sum feedback (e.g., the mirrors in FP cavity).

19. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide Enforced by the phase matching condition, we can write the field as: since only following wave components exist: (1) the forward and backward propagating waves with their propagation constants matched by the Mth order grating: and (2) the radiating waves created through the forward and backward propagating waves coupling due to those (M-1)th, (M-2)th, … grating orders, with their propagation constants given by:

20. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide Following equations are obtained once we substitute the field in its factorized form back into the wave equation:

21. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide The forward – backward propagating wave coupling coefficient is: which generally comprises the refractive index coupling, bias induced index change coupling, bias induced gain coupling, and modal loss coupling.

22. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide The propagating – radiating wave coupling coefficients are:

23. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide • Forward – backward wave couplings: • The refractive index coupling: wave interference • The bias induced index change coupling: bias dependent wave interference • The bias induced gain coupling: bias dependent wave amplification • The modal loss coupling: wave attenuation where item 2 and 3 co-exist as they are linked through the linewidth enhancement factor.

24. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide In-phase Gain Coupling Anti-phase Gain Coupling Modal Loss Coupling Index Coupling Quarter Wavelength Phase-Shifted Index Coupling

25. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide • Uniform index coupled DFB – relatively easy fabrication, inherently dual mode operation, HR/AR asymmetrical coating must be incorporated to achieve certain SMSR, half yield on precise lasing wavelength control • Quarter wavelength phase-shifted index coupled DFB – higher cost under current fabrication technology, symmetric AR/AR coating wastes half of the optical power • Multiple phase-shifted index coupled DFB – performance in between

26. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide • In-phase gain coupled DFB – red side wavelength lasing, high SMSR, high immunity to external reflection, poor linearity, uncertain reliability • Anti-phase gain coupled DFB – blue side wavelength lasing, high SMSR, high modulation bandwidth, poor linearity, uncertain reliability • Complex coupled DFB – relative easy to fabricate, compatible performance to above purely gain coupling, poor linearity, uncertain reliability

27. An Application Example of the TWM- Optical Governing Equations for Periodically Corrugated Waveguide • Modal loss coupling – blue side wavelength lasing, high SMSR, high immunity to external reflection, higher threshold, poor linearity, moderate reliability • Other nonuniform gratings • Chirped grating: further narrow bandwidth or box-like bandwidth • Apodized grating: enhance SMSR • Dispersive grating: new scheme to realize the single mode operation

28. Method to Deal with Broadband Devices- Optical Governing Equations for SOA and SLED • Unlike semiconductor lasers where lasing happens in a narrow band hence the material is effectively non-dispersive, SOAs and SLEDs are inherently broadband optoelectronic devices as the amplified spontaneous emission noise always plays an important role, not to mention that there could be multiple channels to be dealt with by SOAs. • Therefore, we have proposed special techniques in dealing with equation (2-9). • DSP convolution method • Effective Bloch equation method • Wavelength slicing method

29. Computation Efficiency Enhancement- Standing Wave Approach • Expand the longitudinal field into quasi-modes and time-dependent amplitude to obtain separate equations in replacing the longitudinal wave equation. • Advantage: the PDEs are converted into a set of ODEs, which greatly speeds up the computation efficiency for the DC and small-signal dynamic analyses • Draw-back: doesn’t apply to non-resonant structures such as SOAs and SLEDs

30. Computation Efficiency Enhancement- Rate Equation Approach • Taking wave any spatial dependence by averaging the field, as such, only the time-dependence is left over; the resulted equation is named as the photon rate equation. • Advantage: only one ODE is left over which further enhances the computation efficiency; consequently, a circuit (i.e., a symbolic) model can be extracted to trace the signal flow in systems or networks. • Draw-back: purely phenomenological model without physics insights; can only be used as simulation tool fordevice users, but not for device makers.

31. Noise Treatment Rescale the traveling wave equation to obtain a power flow equation: The spontaneous emission power can be directly found as: