Mathematical Model of a Semiconductor Laser and Thermal Issues (2014F)

The threshold of pump power for a semiconductor is determined using a mathematical model and then compared to experimental results from a real diode laser. Using results of this first experiment a mathematical model predicting the buildup of laser power (based on gain saturation) is developed using a spreadsheet and compared to the Rigrod approach. Finally, the effect of temperature on diode performance is investigated.

- Read chapter 13 from
__FLL__by Csele on Laser Diodes - Download the data sheet for the US Lasers 808-5 diode
- Read the Application Note from ILX Lightwave on the measurement of laser diode threshold current.
- Download the Newport Application Note on Characterizing Diode Lasers which outlines the methods to determine the characteristic temperature of a diode.
- Download the Power Development Simulation (Pass-by-Pass Model, Round-Trip Model) as used in class (Password-protected XLS). This model is designed for a HeNe laser and will be used as a starting point then modified to suit the model for this semiconductor diode laser (Many changes are required including designing for a single-pass, sectioning the amplifier into multiple segments, and careful choice of intensities for the gain saturation equation).
- Download the Power Development Simulation (Pass-by-Pass Model, Single Pass Improvement) as used in class (Password-protected XLS).

A US-Lasers 808-5 diode is installed in the test fixture. Turn on the ILX LDT-5910B Temperature Controller and set the temperature to 20C. Install a fiber such that radiation from the laser diode can be analyzed using the Agilent 86142B OSA (the fiber should be blue - swing it upwards such that the apeture intercepts the infrared beam from the diode).

With the room interlock system enabled, turn ON the ILX LDX-3412 precision current source and increase the current to approximately 30mA. Using an infrared viewer card, align the fiber so that it intercepts the diode output and then align manually such that output appears on the OSA. The laser diode output should appear on the OSA around 808nm and multiple modes may well be visible.

The OSA is seen in this photo along with a sample output from the laser diode. See chapter 13 of __Csele__ to get an idea of where the modes originate. This will allow you to compute the length of the cavity and hence the gain medium, a parameter not found on the data sheet for the diode.

- Set the LDX meter selector to monitor diode current.
- Set the OSA for a center wavelength of 800nm and a span of 10nm to start
- Turn on the output from the ILX current source. Starting at zero mA, increase the current slowly and observe the output of the diode on the OSA. Specifically record the mode spacing (required to determine the length of the diode) at low currents (< 25mA), the threshold current (where a single mode becomes dominant), and the drift of the center wavelength as current increases from threshold to maximum.
- Determine the spacing of the modes - this gives the length of the cavity (see
__Csele__6.3). The width and depth of the active volume of the diode can be found in the datasheet.

- Reduce the current to zero, disconnect the fiber-optic cable from the diode, and point the diode at the sensor of the Ophir Nova power meter
- Set the power meter for a wavelength of 808nm (
**MENU**button, twice, then select**LASER**for 808nm) - Increase laser diode current gradually (in increments of 2mA until the "knee" is reached at which point use increments of 0.5mA to 1mA) collecting (light, current) data points.
- Plot a L/I graph for the diode in the same manner as the application note of the prelab and determine the threshold current using lines of best-fit.
- Knowing the typical voltage across the diode (from calculations of the bandgap voltage) and the current, plot electrical Power IN versus optical Power OUT and determine the slope efficiency of the diode (Power IN being electrical power, Power OUT being optical power).

Repeat the experiment above (Part 2) at various temperatures (10C, 20C, 30C, and 40C), determining the L/I graph at each temperature. This data is required in order to determine the characteristic temperature of the device (see the prelab for details).

- Knowing the reflectivity of the mirrors (from n of the material - use Fresnel equations from chapter 4 of
__Csele__), the absorption of the AlGaInP material (provided below), and the length of the cavity (computed from the measured FSR), compute threshold gain (g_{th})of the laser. Use the threshold gain formula to do this. - From threshold gain, compute threshold inversion ΔN (chapter 5). Cross-section of the material is required (use the pre-lab references). Be mindful of the units for ΔN (volumetric).
- Knowing ΔN, and the lifetime of the ULL, compute the recombination rate, dN
_{ULL}/dt = ΔN/τ. The actual units will be in terms of "electrons per second, per unit volume" (i.e. /m^{3}*s). - Energy in equals Energy Out, so the recombination rate dN
_{ULL}/dt must equal the pumping rate (J/qt) where t is thickness of the recombination layer. Compute current density J (in A/m^{2}) by assuming that each transition of dN_{ULL}/dt represents the energy of one electron charge (q, equal to 1.602*10^{-19}J) , and finally (knowing the emitter size from the datasheet and from the mode spacing) threshold current. Pay attention to units to ensure the final threshold current is really in terms of Amperes, and expect a logical answer of between 10mA and 40mA. Compare to the experimentally-determined value in the lab and describe, using a logical argument, why the experimental threshold might be larger than the experimental. What assumption(s) about the behaviour of semiconductors were made in the above theoretical computations which can possibly lead to a low theoretical threshold current? - From the Power plot (Watts IN versus Watts Out), compute slope efficiency of the laser diode.

You might need a few numbers to get started:

Cross section σ_{0} = 1 * 10^{-19} m^{2}

ULL Lifetime τ = 1 * 10^{-9} s

Index of refraction of AlGaInP = 3.7

Attenuation γ = 25 cm^{-1}

Now, knowing the gain of the device (both threshold gain and gain at a particular drive current) and physical parameters (e.g. length) we may now generate a model to show how power develops in the laser. This model will predict not only output power but also response time, an important consideration when using a laser diode in a communications system.

Develop a spreadsheet model for this diode similar to that demonstrated in class for the HeNe laser. The model must predict the intra-cavity power on each pass through the gain medium (a 'pass' considered in each direction - a round-trip is two passes). Further refinement is required by subdividing the gain medium into smaller lengths (x/2 or x/3, as required ... this will be discussed in lectures). To do this you will need a starting gain (g_{0}): choose a drive current which corresponds to maximum diode output ("Typical drive current" from the datasheet) and determine the gain by scaling ΔN according to the threshold determined in the lab. If, for example, theoretical drive current is twice that of the threshold current, then the inversion is presumed to be twice that of the threshold inversion already determined. From there g_{0} may be determined.

Several parameters must be computed now, including P_{sat}, and several have already been computed or referenced including attenuation, length, and optics parameters.

When the simulation is complete, graph gain and power output vs. pass on the same graph. Determine the time for the laser to reach 90% of the CW output level - this is the risetime of the device for practical purposes.

*To be done individually ...*

- Hand-in a graph of P
_{optical}vs. drive current for the laser diode as observed in the lab for various temperatures. - Show, on the graph above, the threshold current for the device and explain in a paragraph how it was determined (i.e. summarize the method from the ILX application note that you used)
- Calculate the slope efficiency of the diode (you might have to research this). Include a Power IN:OUT plot and show how slope efficiency was computed. Remember that electrical power is calculated as current times voltage (outline
**where**you obtained any constants required such as device voltage). - Calculate the threshold gain of the diode (In the same manner as problem 5, chapter 5 of
__Csele__, but using the device parameters found experimentally in this lab). - Calculate the
**theoretical**threshold current for the diode (show all calculations and formulae used) and compare to the experimentally-determined value.*This question is worth considerably more marks than others in the lab and a large degree of detail is required to show how you developed the model. Show all intermediate steps and equations.* - Calculate the characteristic temperature (T
_{0}) of the diode using the method outlined in the prelab reading (from ILX). Show all calculations required (e.g. ln(J_{th})) as well as the graph (for which the slope of that graph is T_{0}). - Show how parameters used in the simulation model were developed (including P
_{sat}and g_{0}at the rated drive current). - Hand-in a screen shot of the spreadsheet model showing at least the first ten rows of the simulation as well an any anchored parameters at the top. All columns must be shown (this may require more than one screen-shot) and cell references ("ABC" columns and "123" rows) must be visible. The simulation must be on a "per section through the amplifier" basis (with ΔX smaller than the amplifier length), not "round trip" as we have done in class since both the gain, and loss at each cavity mirror, is very large (The entire process will be outlined in lectures before the lab). Calculation of g
_{sat}also requires some forethought since it depends on the intra-cavity beam travelling in**both**directions - this was/will be discussed in lectures (In the simple simulation provided this was done by multiplying the unidirectional intra-cavity power by two ... this cannot be done here). As well, show ALL CALCULATIONS by outlining EACH FORMULA from EACH CELL in the first four rows (simply show these as, for example, "AA6=+AC6*exp($C$7*C6)" on a separate page).*This question is worth considerably more marks than others in the lab and a large degree of detail is required to show how you developed the model. Show all spreadsheet formulae.* - Hand-in a graph (output from the spreadsheet model) showing the dependence of both intra-cavity power and gain with respect to pass # (preferably on the same graph using primary and secondary axes, as per the example simulation provided).
- From the simulation determine the
**time**for the device to reach 90% output levels (i.e. the rise-time of the device) - Outline (showing key mathematical steps) the prediction of output power using the Rigrod model remembering to factor-in attenuation as described in lectures.
- A comparison of
**output power**at optimal drive current as predicted by (i) the pass-by-pass model in this lab, (ii) predictions from the Rigrod model, and (iii) predictions from the simple model to that stated in the datasheet. Explain why the discrepancy might exist.