ECE 633 Ridge Waveguide Laser homework Introduction This is a slide from a lecture we will study later on. It is about diode lasers. Although we haven t studied diode lasers, there is one aspect about them that we are studying right now: planar waveguide. Hence, the objectives of this HW is: 1- Reinforce your understanding of waveguide - Apply to a specific problem: semiconductor lasers so that you will get an early introduction to diode lasers that will help later on. Killing birds with one stone, so to speak... The most common structure of diode lasers is edge-emittinge ridge-waveguide laser. It starts out with a crystal wafer that is designed as an optical waveguide. Typical actual structures can be very complex with many layers, but can be optically approximated with a few layers as shown.
ECE633 - homework on ridge waveguide laser_s18_p.nb Figure below shows lateral cross section of the wafer structure. which has 3 layers as shown. The top layer is usually coated with thin Au. However, for simplicity and good approximation, we can just neglect the Au contact layer. Next, a ridge is fabricated by etching down two sides of a stripe as shown below: We can calculate the optical mode of such a waveguide. 1. (60 pts) Planar waveguide 1.1 Basic review: Let the wavelength of the laser be 1.55 m, what is its frequency in unit THz 10 1 Hz? Note that the speed of light (in vacuum of course, unless specified otherwise) is 99.79458 m ps 1. Basic planar modes A wafer has structure as indicated below
ECE633 - homework on ridge waveguide laser_s18_p.nb 3 top cladding 3.4 1.9 core 3.6 0.5 bottom cladding 3.4 infinite How many TE modes does it have? what are the effective indices of the modes, and show the mode profiles 1.3 Mode propagation Let the laser length be 3 mm. Calculate the travel time of the light of each mode from one end of the laser to the other. 1.4 Show the mode intensity - any observation? 1.5 Discussion We will learn later on that only the core region is designed to be active, which means it is the region that emits laser light (not the claddings). It is most desirable to have a single mode centered in the core. What do you think of the mode structure of this wafer? 1.6 Fix the wafer It is not uncommon that when a wafer is fabricated, errors occur in the control of layer thickness and refractive index. For this wafer, parasitic cladding mode is undesirable and there is one way to fix it. You cannot change the layer materials (because the wafer is already grown) and their refractive indices. But if a layer is exposed, it can be etched. Can you think of a way to salvage this wafer? Show your result after fixing.. (30 pts) Modal dispersion A wafer has structure as indicated below top cladding 3.4 1.6 core 3.6 0.5 bottom cladding 3.4 infinite.1 Calculate and plot modal propagation constant vs. Obtain n eff for the lowest order mode for ={1.4,1.45,1.45,1.475,1.5,1.55,1.55,1.575,1.6}, then calculate and plot modal propagation constant vs. frequency in unit of rad THz.. Fit the modal vs. Fit the above data of vs. with this Taylor s series expansion: 0 0 1 0
4 ECE633 - homework on ridge waveguide laser_s18_p.nb using 0 corresponding to =1.5 m. Then, plot the fit curve on top of the data points obtained in.1.3 Fit the modal vs. - quadratic only Replot the fit of. above, but only for the quadratic component, in other words, subtract both data and fit with its linear component and replot..4 Do you know the meaning of and in.? If you don t, it s OK, leave it blank or say I don t know. 3. (30 pts + 10 bonus) Generic waveguide properties Consider this planar waveguide: top cladding glass 1.45 core variable variable bottom cladding glass 1.45 3.1 (10 pts) Number of modes as a function of core thickness Let wavelength be 1.5 m. Let the core index be.15 (zirconia). Vary the core thickness: d={., 4.,6., 8., 10,1.}. Plot the number of modes as a function of core thickness. What do you conclude? 3. (10 pts) Number of modes as a function of core index Let wavelength be 1.5 m. Let the core thickness be 10 m. Vary the core dielectric (not index) core 3., 5., 7., 9., 11., 13.. Plot the number of modes as a function of core dielectric. What do you conclude (approx)? 3.3 (10 pts) Number of modes as a function of k Let the core index be.45 (diamond). Let the core thickness be 10 m. Vary the wave number: k= {0.4, 0.8, 1., 1.6,.}. Plot the number of modes as a function of k. What do you conclude? 3.4 (10 pts bonus) What is the fundamental underlying mathematical relationship about the number of modes vs. core, d core, and k? (this is quite advanced in terms of Math - It s OK if you don t know). 4. (30 pts) TE vs. TM
ECE633 - homework on ridge waveguide laser_s18_p.nb 5 Consider this planar waveguide: top cladding glass 1.45 core diamond.45 1.5 bottom cladding glass 1.45 Let =1.5 m. 4.1 (10 pts) TE vs. TM Efield profile Obtain the E y field mode profiles for TE modes and E x field mode profiles for TM modes. What do you observe the most significant difference between TE and TM and explain. 4. (10 pts) TM Hfield profile Obtain the E x field mode profiles and the associated H y field mode profiles for all TM modes. Compare them, what do you observe and explain. 4.3 (10 pts) Confinement Modal confinement is loosely defined as the fraction of light intensity that is located in the core vs the entire intensity of a mode. Mathematically, conf E x core x E x all x Plot the intensity profiles of the highest order mode of TE and TM and compare them. Which polarization, TE or TM do you think in general has higher modal confinement? 5. (50 pts) Pure-single-mode and multi-mode travel Consider this planar waveguide, let =1.5 m. top cladding glass 1.45 core diamond.45 1.5 bottom cladding glass 1.45 For mode m, the E field can be written as: E m x, z F m x m z t Its intensity profile as function of travel distance z is roughly: P m x, z E m x, z. E m x, z F m x which is constant vs. z. In other words, it doesn t matter where we are along the travel direction, the light intensity profile is the same. However, we can have a wave like this: E x a 0 F 0 x 0 z a 1 F 1 x 1 z... a n F n x n z t
6 ECE633 - homework on ridge waveguide laser_s18_p.nb a 0 F 0 x z v p;0 a 1 F 1 x z v p;1... a n F n x z vp;n t (5.1) where a 0, a 1,.., a n are arbitrary coefficients, and v p;m m is known as the phase velocity of mode m. Note: this has a very important application known as MMI or multi-mode interference in numerous planar waveguide devices. 5.1 Discussion Given Eq. (5.1), what do you think of the light intensity profile as you travel along the z-direction? (just a general discussion). 5. Let a 0 1 and a 1 1. Observe the light. Write what you observe and what you think. Show either E-field or intensity profile to support your discussion. 5.3 Let a 0 1 and a 1. Observe the light. Write what you observe and what you think. Show either E-field or intensity profile to support your discussion. 5.4 Let a 1 1 and a 3 1. Observe the light. Write what you observe and what you think. Show either E-field or intensity profile to support your discussion. 5.5 -coupler Consider this planar WG. Let =1.5 m top cladding 3.4 core 1 3.5 1 core 3.3 0.7 core 3 3.5 1 bottom cladding 3.4 Consider multi-mode a 0 1; a 1 1. Show and discuss its behavior as it travels along the WG. What is its application? 6. Ridge waveguide - the effective index approx method A ridge waveguide is fabricated by etching down two sides of a stripe as shown below:
ECE633 - homework on ridge waveguide laser_s18_p.nb 7 w is the width of the ridge, and d is the etch depth. It turns out that one can use a very good approximation to calculate the ridge waveguide mode by using what is known as effective index method. However, since this HW is already long, we ll save this when we study diode lasers.