SK2811 lab: Statics and dynamics of semiconductor lasers 1/16
Simulation laboration in SK2811
STATICS AND DYNAMICS OF SEMICONDUCTOR LASERS
Richard Schatz KTH 2020
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SK2811 lab: Statics and dynamics of semiconductor lasers 2/16
Contents
1 LaserMatrix installation instructions 3 2 Laser theory 5 3 Optimize the power from a Fabry-Perot-laser 7 4 The DFB-Laser 9 5 Simulations of a Bragg Grating with gain 12 6 Simulation of the the optical spectrum of a DFB 13 7 Analytical solution of the modulation response 14 8 Simulation of the modulation response 16
SK2811 lab: Statics and dynamics of semiconductor lasers 3/16
1 LaserMatrix installation instructions
LaserMatrix is an easy-to-use program, primarily used for the longitudinal design of edge emitter lasers. The software was written for an older generation Macintosh (68k and PPC processor) but works very well using a Macintosh emulator in Windows XP, 7, 8 and 10 (both 32 bit and 64 bit). There are also versions of the emulator that works on new Intel Macintosh, Linux machines. LaserMatrix can be used on Android based phones and Tablet computers using BasiliskII emulator.
Please just follow the instructions below to set up the emulator. If you encounter any problems, don´t hesitate to contact me or consult Installation_ReadMe.txt that you find in the zipped folder. When the emulator works, the installation of LaserMatrix is very simple.
1. Download SheepShaver emulator with MacOS 9.0.4 from here: ftp://81.229.17.37/disk1/pub/LaserMatrixPPCWin.zip
2. Unzip the zipped folder into any directory of your Windows computer.
3a) For Windows 32 bit: Move the file cdenable.sys to "C:\Windows\System32\drivers"
3b) For Windows 64 bit: Move the file cdenable.sys to "C:\Windows\SysWOW64\drivers"
4. Open the folder and run gtk+-2.10.13-setup.exe to install GTK runtime environment. If possible, run the installation with administrator rights (right click on the file and choose "Run as administrator")
5. Start the Macintosh emulator by clicking on SheepShaver.exe. Note: depending on your hard disk it may take up to 30 seconds for the emulator to start.
6. From within the emulator environment, in "Apple Menu/Control Panel/Keyboard" please check that the correct keyboard (e.g. US) is used.
7. There are several options like screen size and memory size in Sheepshaver emulator that can be set using SheepShaverGUI.exe in the installation folder. Within the emulator you can toggle between full screen and window using Ctrl+Enter buttons. In tab "Memory/Misc" the setting "Don´t use CPU when idle" should be set. In the tab "JIT compiler" the setting "Enable JIT compiler" should be set.
8. Install LaserMatrix a) Put the.sit file in a folder on your PC preferably located directly on C:\ root directory (e.g. "C:\MacTemp") b) Start the emulator. If the emulator is correctly setup there should be a "This computer" icon on the desktop to access the PC hard drive.
SK2811 lab: Statics and dynamics of semiconductor lasers 4/16 c) In the emulator environment, transfer the .sit file from C:\MacTemp to the LaserMatrix folder that can be found on the hard disk "PPCAppDisk". Do NOT put the file directly on the desktop in the emulator environment. d) Uncompress the transferred .sit file in the LaserMatrix folder by double clicking on the file
9. Run LaserMatrix from the LaserMatrix folder (located in LaserMatrixPPCAppDisk in the Mac environment). SK2811 lab: Statics and dynamics of semiconductor lasers 5/16
2 Laser theory
The basic structure of a laser is an optical cavity (e.g. two mirrors reflecting light back and forth). To achieve lasing, i.e. oscillation, the light inside the cavity is amplified by stimulated emission. The amplification has to be large enough in order to compensate for the fraction of the photons that leaves the cavity through the end mirrors plus the dissipated photons in the cavity. Stimulated emission in semiconductor materials can be pictured as a stimulated electron- hole recombination by a photon, where the energy difference is emitted as a new photon which is an exact copy of the stimulating photon. The opposite action is called (stimulated) absorption. A net stimulated emission can only be maintained when there is a permanent excess of electrons and holes that are able to recombine. This is called population inversion and can be achieved by injecting a high forward current into the p-n junction of the laser diode. The injection is also called pumping. A change in the number of photons is coupled to a change in the number of electrons and vice versa. This interaction is described by the rate equations, where S presents the number of photons in the lasing mode, and Q the number of carriers (electron-hole pairs). dQ I Q GS (1) dtq sp dS GS r (2) dt i m sp
In equation (1), I is the injected pump current, the carrier lifetime sp expresses the carrier lifetime due to spontaneous recombination, which can be either radiative (spontaneous emission) or non-radiative. Non-radiative recombination transfers the energy of the electron to the semiconductor crystal in form of lattice vibrations (phonons). The concept of a carrier lifetime is a simplification of a more accurate carrier recombination model,
23Q Isp AN BN CN V (3) sp
Here, A is the linear recombination rate, B is the bimolecular recombination rate and C is the Auger recombination rate.The term GS in equation (1) describes the stimulated recombination rate. The same term exists in (2) since each stimulated recombination of a carrier gives a stimulated emitted photon.
The photon gain is often assumed to be a linear function of the number of carriers:
G vg g N G N N N tr (4)
Here, N=Q/V is the carrier density where V is the active layer volume, vg is the group velocity of the light, the confinement factorof the lasing mode in the active layer, and g(N) the active material gain coefficient. GN is called the “differential gain”. The gainThe amount of charge carriers required for transparency, Ntr, depends on the material. SK2811 lab: Statics and dynamics of semiconductor lasers 6/16 Transparency means that the light passes through the material without any loss or amplification.
The spontaneous emission rate, rsp, is the amount of photons per unit time that is generated due to spontaneous emission in the lasing mode. For a laser with low output coupling loss and perfect inversion, rsp=G is valid, but for a real laser rsp is about 2-3 times larger. The last term in equation (1) describes the losses of the photons in the cavity. The photon lifetime τp is defined as:
1 p i m v g i m (5)
Where, i is the internal optical loss per unit length in the laser cavity, m represents the mirror loss, i.e., the light output through the mirror. Observe that the stimulated absorption is included in g(N).
DC operation means no variation with time implying a constant number of photons and electrons. The time derivatives in (1) and (2) become zero. For this case the following expression can be derived from (1)-(4):
r sp imS NNtr (6) GN The spontaneous emission term rsp/S can be neglected under lasing conditions. Further we assume that all the other quantities on the right hand side of equation (5) do not depend on the density of photons, the number of the charge carriers, and the temperature. Provided this, the number of the charge carriers at lasing is independent of the current and corresponds to the value that is reached at the threshold (this is not the whole truth but presents a good approximation within the scope of this course). Nth is the threshold carrier density. The current at threshold becomes Ith =q Nth/sp. Increasing the current above threshold will linearly increase the number of photons generated by stimulated emission.
SK2811 lab: Statics and dynamics of semiconductor lasers 7/16
3 Optimize the power from a Fabry-Perot-laser Normally both facets of a Fabry-Perot laser have the same reflectivity (R≈30%) determined by the refractive index difference between air and the semiconductor material. Only 50% of the optical power is therefore emitted from the front facet. In power demanding applications, e.g., pump-lasers used for pumping fiber amplifiers, one can increase this figure by applying an HR-coating (HR=high reflection) on the back facet and an AR-coating (AR=anti-reflection) on the front facet
1. Assume that the back facet is perfectly HR-coated (R2=0.9999) and that the laser has a maximum allowed injection current Im. Choose current (I) as input parameter in the ProgParameters dialog and use the ThisVsThat function in the LaserMatrix program to find the reflectivity R1 that optimizes the maximum output power Pm. Structure and material parameters are found on next page.
2. Explain, in physical terms, why there exists an optimum value of R1 (What happens when R1=0 or R1=1?)
3. With the optimum value of R1, calculate analytically, using the rate equations, the threshold current, Ith, the differential quantum efficiency, d and the power, Pm.. Check the results with LaserMatrix.
SK2811 lab: Statics and dynamics of semiconductor lasers 8/16 Material and structure parameters Center wavelength 1.49 µm Maximum current: Im=100 mA -1 Internal loss coefficient: i = 40 cm Refractive index: n = 3 Group index: ng = 3 Confinement factor: =0.3 Differential gain: a=2.5.10-16 cm2 . 18 -3 Transp. carrier density: Ntr=1.5 10 cm Laser length: L=300 µm Active layer width: w=2 µm Active layer thickness: d=0.2 µm Spont. carrier lifetime: sp=0.8 ns Internal efficiency: i=100%
Standard rate equations dQ/dt =Iin - Isp - G S dS/dt = (G-i - m ) S + rsp
Q = N V:carrier number, S :photon number
G= vg a(N - Ntr):net stimulated gain rate i=vg i :internal loss rate m=-vg/(2L) ln(R1R2):mirror loss rate Iin:injection current (unit electrons per second) Isp= Q/sp :spontaneous recombination rate P1= F1 m S: output power from facet 1 (unit photons per second). F1 is the fraction of the total power that is emitted from the front facet. Since the back facet has reflectivity R2=1 in our case, all the power is emitted from front facet, i.e., F1=1 SK2811 lab: Statics and dynamics of semiconductor lasers 9/16
4 The DFB-Laser
To minimize the effect of fiber dispersion, it is desirable that lasing occurs only at a single wavelength. The laser should be oscillating in a single mode. The carrier dependent material gain coefficient g(N) is generally also a function of the wavelength. Lasing at a certain wavelength starts with the mode that requires the least number of charge carriers to reach threshold (compared to other possible modes). If the gain function is not very wavelength selective (flat), and if the possible modes are close to each other in wavelength, it happens easily that several modes are lasing simultaneously. Furthermore, the number of injected charge carriers may fluctuate due to noise and intentional modulation. This could also cause multi mode lasing, when the lowest threshold in charge carriers switches between different modes during modulation. The material gain g(N) is only weakly dependent on the wavelength. As the resonances in the FP-laser case are close to each other then the discrimination of the selected lasing mode is not sufficiently good. Hence it usually not single mode. Therefore, in practice and in our experiment a so-called DFB-laser (Distributed Feed Back) is used. DFB-lasers employ wavelength selective gratings as mirrors to reflect the light back and forth. This results in much higher mode selectivity (i.e., the threshold carrier density is reached for one mode while the other modes are still far away from threshold), and thus single mode lasing.
Figure 1. DFB-grating etched at the Semiconductor Laboratory in Kista.
The grating inside the DFB-laser is realized as a corrugation (grating) of the material of the waveguide core region (see Figure 1). The grating represents a periodic change in the refractive index that causes small reflections at each corrugation. In this way, waves are coupled forth and back from many points in the active region, being the reason this laser is called “distributed feedback laser DFB” (in contrast: in the FP laser light is only coupled back due to reflection from the front-end and back-end mirrors). Side mirrors are not needed in the DFB-laser; they are even disturbing since they would form a parasitic FP SK2811 lab: Statics and dynamics of semiconductor lasers 10/16 cavity in addition to the distributed cavity of the grating. Therefore, the end facets of a DFB laser are often anti- reflection coated. The superposition of the many small reflections (Bragg reflections) along the grating shows a very sharp maximum at the so-called Bragg wavelength; which occurs when the wavelength of the guided wave is twice as long as the period of the grating. A complete analysis of a DFB-laser goes beyond the scope of this course and is not necessary to have a complete description in order to understand the principle of the frequency selectivity. Figure 2 shows a structure, which is used for a simplified description of the DFB action.
L2 L1 L2 n1 n2 n1 n2 n1
interface 1 interface 2 interface 3 interface N
Figure 2. Model of a grating
The structure is a sequence of layers with refractive indices alternating between n1 and n2. The different layers have the same optical thickness; L1n1=L2n2, and the difference between n1 and n2 is very small. Assume a planar wave with the amplitude A, and an angular frequency entering from the left. The reflection coefficient at interface 1 is:
n n r 1 2 n1 n2 (6)
The reflected wave has the amplitude Ar. At interface 2 the reflection coefficient is –r. The optical phase shift in each layer is =L1n1 /cL2n2 /c. Assuming that r is very small (|r|<<1; which is the case when n1n2), we can neglect changes of the amplitude of the wave transmitted through each interface (1 r1), and assume also we can neglect all double and higher order reflections (|r|2, |r|3, … are very small). The superposition of the reflected waves including the phase from all interfaces leaving the first interface can then be written as:
Ar Are j2 Are j4 Are j6 . (7)
Dividing by A, the total reflection rtot can be derived from eq. (7):
SK2811 lab: Statics and dynamics of semiconductor lasers 11/16 sin(N) r Nr exp( jN) (8) tot N
L n L n Where 1 1 2 2 and is the deviation from the Bragg angular 2 c c frequency 0. We see that |rtot| is maximal for =0, and has the first zero at N =. Equation (8) can also be seen as the stop band characteristic of a filter, the filter stop band bandwidth decreases with N, i.e. the frequency selectivity increase with N.
L1
rtot
rtot
Interface 1
rtot is the total reflection at the two ‘Interface 1’
Figure 3. Model of the DFB laser cavity
To check the amplitude and phase conditions for lasing in a DFB laser, we describe the total cavity as two such gratings connected via a segment with index n1 and the length L1 (see Figure 3). For lasing, the wave has to be reproduced after a roundtrip. The following amplitude and phase conditions must be fulfilled:
j j j2 2 1e rtot e rtot e rtot 1 (9)
In comparison with the oscillation condition for an FP-laser, rtot corresponds to the mirror amplitude reflectivity and the cavity length is replaced by one layer L1. At the Bragg frequency (=/2, and =0), equation (9) becomes –(Nr)2, and does not fulfil the phase condition. The wavelength has to deviate from the Bragg wavelength in order to achieve lasing. Theoretically, two modes equally spaced on each side of the Bragg wavelength reach threshold simultaneously.1
1Alternatively, the “cavity” can be extended with a quarter of a wavelength in the middle(quarter-wave shifted DFB laser). In this case, the phase condition of equation (9) becomes true at the Bragg wavelength. However, such a structure is more difficult to fabricate.
SK2811 lab: Statics and dynamics of semiconductor lasers 12/16 Far away from the Bragg frequency the grating is not efficient (no reflection). The reflections from the facets (semiconductor-air interface) become noticeable in this case if not anti-reflection coating is applied.
5 Simulations of a Bragg Grating with gain
a) Use the LaserMatrix program to calculate the reflectivity of a Bragg-grating (L = 2, L=240 µm, n = ng = 3.23) and plot the reflectivity, |rg|, as a function of optical -1 -1 -1 -1 wavelength, , for different values of the modal gain, gm=0 cm , 20 cm , 40 cm , 60 cm , 80 cm-1. Use linear y-scale. This is most easily done by entering an AR-coated DFB section of passive material and plotting Matrix Element item |S11|2 (power reflectivity) in the Calculate-menu. Use 100% confinement factor (so that modal gain=material gain) and enter the gain as a negative value of internal waveguide loss (WG Loss). b) Explain the change of the curve as the gain is increased. A clue can be obtained by calculating the effective length (i.e., derivative of the reflection phase with respect to the optical frequency of a Bragg grating without gain. (reflection phase=phase of matrix element S11) c) Iterate manually to find the (lowest) value of gm that gives |rg|=1. At what wavelength, , does this occur. d) Iterate manually to find the (lowest) value of gm that gives |rg|≈∞. At what wavelengths, , does this occur. e) Use LaserMatrix to check the answer in c) by calculating the cavity modes of a phaseshifted DFB laser consisting of two active lossless gratings and a phaseshift inbetween with both facets AR-coated (R=0%). f) Use LaserMatrix to check the answer in d) by calculating the cavity modes of a DFB laser consisting of a single active lossless grating with both facets AR-coated (R=0%).
SK2811 lab: Statics and dynamics of semiconductor lasers 13/16
6 Simulation of the the optical spectrum of a DFB
a) Simulate the spontaneous emission spectrum of the DFB laser using LaserMatrix at different bias currents: 3 mA, 5 mA, 7 mA, 10 mA, 20 mA, 40 mA, 80 mA. You can use the Linear (below TH model) in all simulations. b) Describe qualitatively how the laser spectrum changes when the current is increased: 1. below threshold 2. above threshold c) Discuss the difference of the spectra below and above threshold! d) Explain the "ripples" in the spectrum. SK2811 lab: Statics and dynamics of semiconductor lasers 14/16
7 Analytical solution of the modulation response The usual way to impress a signal on a light source is to change the optical power by some means of the laser. Generally the process to impress the signal on any source is called modulation, in our case intensity modulation. The modulation can be done by an external modulator (i.e. Electro-Absorption type, Mach-Zehnder type) or by a bias current change of the laser, so called direct modulation. DC modulation DC + modulation
Laser External Mod Laser
The advantage of the external modulator scheme is improved modulation performances, e.g. extinction-ratio, chirp, and bandwidth. The disadvantages are higher loss, complexity and cost. Direct modulation is often the easiest and cost effective way of modulation. For many applications where extremely high speed or long links are not required, it is considered as the optimal solution. In this laboratory work only direct modulation is investigated. The simplified rate equations are: dQ I Q GS (1) dtq sp dS GS (2) dt im
G GN (N N 0 ) (3) A numerical solution to the non-linear rate equations for a large signal step is shown in figure 8. The current is a rectangular pulse with amplitude 3Ith with duration 2 ns. The pulse response shows a turn on delay time, rise time, damped oscillations and a turn off time. In this case the turn on delay time is dominating, but it can be eliminated by proper biasing (how?).
30
25 5
)
) 20 4
Ith
mW Oscillations 15 3
≈ exp(-γt/2) 10 Rise time 2
Turn off ≈ 1/f Modulation power ( power Modulation 5 R 1
time current ( Modulation Turn on delay 0 0.0 0.5 1.0 1.5 2.0 2.5 Time (ns)
Figure 4. Rectangular pulse response of a laser. Solid line for the laser response and dashed line for current. SK2811 lab: Statics and dynamics of semiconductor lasers 15/16
The large signal analysis can in practice only be obtained numerically. Simple analytical solutions which give much information of the laser behavior can be obtained by small signal analysis which means that the modulation is so small that the rate equations may be linearised. In the small-signal-analysis a rotating phasor I e jt with a small magnitude in current is superimposed and causes corresponding small changes of the carrier and photon numbers, thus:
jωt jωt jωt I = I0 + I e ; N = N0 + N e ; S = S0 + S e .
Equations 1 and 2 give the following matrix:
j A B NI q / (10) C jS 0
The quotient SC q / H() 22 (11) Ij R is called the transfer function between photon number and current. As the output optical power is proportional to the photon number this transfer function is also proportional to the transfer function of intensity modulation response. Figure 9 shows a typical result. (The real response can deviate because of parasitic effects (roll-off caused by capacitance of metal pads etc, and carrier transport through un-doped layers, which can be modelled by introducing a third pole in the transfer function.)
Response dB 4
2 Frequency GHz 2 4 6 8 10 12 14 -2 -4 -6
-8 -10 Figure 5. Intensity modulation response for different bias currents normalised to the value at 0 GHz. The currents are 1.1 Ith; 2.2 Ith; 3.3 Ith; 4.4 Ith a) Derive the matrix elements A, B and C in Equation 10
2 b) Derive the expressions for R and in Equation 11.
SK2811 lab: Statics and dynamics of semiconductor lasers 16/16 8 Simulation of the modulation response
a) Simulate the same modulation response from 0-20GHz using LaserMatrix as a function of bias current 10-80 mA in 8 steps b) Estimate for each bias, the resonance frequency, the 3dB modulation bandwidth (i.e. frequency the response dropped -3dB from the modulation response at low frequencies) and the relative peak height (the difference in dB of the relaxation peak height compared to modulation response at low frequencies)
Ibias [mA] fR [GHz] f3 dB [GHz] Relative peak height [dB] 10 20 30 40 60 80
c) Is the formula f R const I I th valid?