Modeling a Tunable Narrow Linewidth Laser

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IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM SWEDEN 2020,

Modeling a Tunable Narrow Linewidth Laser

CHRISTOFFER EJEMYR

KTH ROYAL INSTITUTE OF TECHNOLOGY

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Abstract

In this report a model of a tunable narrow linewidth laser used for telecommunications is presented.

The model uses both theoretical analysis and experimental data to create the mathematical models governing its behaviour and is aimed to be useful in a development environment with requirements on accuracy, efficient implementation and adaptability to future design. Results show that the model presented achieves high accuracy in both optical and electrical measurements. In summary the model could be useful in a development environment with further improvements in adaptability possible in the future.

Keywords: Laser, Tunable, Model, Lasing frequency, DBR

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Abstrakt

I detta arbete presenteras en modell av en avstämbar laser som används för teleoptisk kommunikation.

Modellen baseras p˚a s˚aväl teoretiska modeller som p˚a experimentell data och matematiska uttryck ges för alla delar av modellen. Modellen är tänkt att användas i en utvecklingsmiljö och det ställs därför krav p˚a noggrannhet, beräkningshasighet hos implementationen samt adapterbarhet till framtida liknande lasrar.

Resultaten visar att modellen är noggrann i b˚ade elektrisk och optisk karakteristik. Sammanfattningsvis kan vi konstatera att modellen kan bli användbar i produktutveckling men har möjlighet till förbättring i sin adapterbarhet till ny design.

Nyckelord: Laser, Avst¨ambar, Modell, Lasrande frekvens, DBR

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1 Introduction

1.1 Lasers in a Modern World

Lasers, emitting light only at a narrow range of frequencies in a single direction, are essential to the modern world of today [7]. The industrial and do- mestic applications of lasers are many and wide ranged. Within industry, lasers of high intensity are used to cut through metals and glass with precision better than most mechanical solutions. Within medicine, lasers assist in surgery with precise and safer cuts. In connection to many homes, lasers pro- vide a high speed internet by sending information between countries and continents. The applications all demand different properties and different lasers, but they all make use of the lasers’ great precision.

This paper will focus on a specific type of laser, a tunable narrow linewidth laser (NLL). An NLL is a laser designed to only emit a very narrow range of frequencies. This is crucial when using the NLL in teleoptic communication. When sending information across a wave-guide such as a fiber-optic ca- ble you want to send multiple signals at once, since this would increase the bandwidth of the connection. Differentiating signals from each other can be done by sending the signals at different frequencies. Furthermore a smeared spectra results in an unclear signal and thus information might be lost.

We therefore require a laser with a narrow peak over frequency.

1.2 Background

This thesis project was invented together with the optics company II-VI in pursuit of a more accurate model of their tunable NLL. The general aim when designing the thesis was to model an existing laser with the long term goal that the models would be easily adjustable to help development of similar products in the future.

In current development of a new product the laser goes through many different stages. Beginning with a general design, it goes into a small scale produc- tion for testing. When testing is done the design is altered to better meet the requirements set out in the beginning and so the process repeats until all requirements are met. In this process the testing requires a lot of data extractions and measurement analysis. Having to run the data extraction code on real devices when run for the first time results

in bugs and errors often ending up destroying the first rounds of products.

Having a virtual model of the product would enable the testing team to start testing their code on a model close to what one can expect from the real product. Such a model would reduce the number of errors and thus reduce development time and cost. Therefore the project is to build a model with the general characteristics of the NLL enabling the usual data extractions.

1.3 Aim of the Study

The aim of this thesis is to develop and implement a model of the tunable narrow linewidth laser pro- duced at II-VI, in order to speed up the development process of new similar products.

The requirements on the model are:

• Accuracy: The model must be accurate in frequency, lasing power and electrics such that all measured phenomena are visible and of similar characteristics.

• Speed of calculation: The model should be able to be measured upon at the same rate as a real device.

• Modifiable: It should be possible to change many of the lasers parameters without reim- plementing actual mathematical models, as well as possibly replacing any component in the laser.

2 The Narrow Linewidth Laser

For a more intuitive understanding of the work- ings of the NLL we introduce the device structure and the general idea of how lasing appears. Later, in chapter 3, a more stringent and comprehensive model of the individual components of the NLL will be given.

2.1 Device Structure

The laser to be modeled is shown in a cross section in Figure 1. In principle the laser consists of two Bragg reflectors, one phase tuning section and one gain section, as well as some passive cavities between the former components [3]. As will be clear

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Figure 1: The II-VI tunable narrow linewidth laser in cross section. Illustration from Eriksson et al. [3].

below, the NLL is controlled by temperature mod- ulation and the laser therefore has airgaps beneath all the heated components to insulate and prevent heat leakage.

In addition the NLL has three heating pads located on each reflector and on the phase section. The illustration in Figure 1 also shows a semiconductor optical amplifier (SOA). This will not be included in the model since it in a sense is not part of the laser, but only amplifies the signal after it exits the NLL.

2.2 Principles of Lasing

To make the NLL lase you apply current to the gain section. This in turn emits photons from the medium into the area between the reflectors. As photons pass through or are reflected by each component, they are effected both in amplitude and phase. Representing a wave at a given moment in time and space by an amplitude and phase as E = E0e^iφ⁰ we can define the interaction with each component as a multiplication by a complex transmission coefficient (short. transmission) t = Ae^iφ. As will be seen in chapter 3.3, t might depend on both frequency and temperature as well as material constants such as refractive index and length. We define t₁, t_p, t₂, t_g, t₃ as the transmissions of components between the reflectors in Figure 1 (left to right) and r_r and r_l as the reflections (analogous to transmission) of the rear and lead reflectors respectively. Using this we can define the round trip coefficient, X, as

X = rrrl(t1tpt2tgt3)², (1)

which will be instrumental in the definition of lasing frequency.

The aim of the laser design is to have only a single, or a very narrow spectra of frequencies lase.

Singling out a frequency is achieved with careful design and tuning of the Bragg reflectors as well as by design of the entire device structure. Figure 3 on page 10 reveals the design of the lead and rear reflectors reflectivity. Each reflector has seven peeks in reflectivity at approximately equal distance on the frequency spectra. The figure can be interpreted as showing a plot of |r_r(f )|² and |r_l(f )|² of r_r and r_l from Equation 1. This frequency dependent design makes the round trip amplification, |X(f )|, significantly larger at some frequencies and therefore single out some frequencies to lase.

As well as the reflectors’ reflectivity there is an- other constraint further limiting the possible lasing frequencies. Although having carefully designed reflectors with special reflectivity the NLL is very similar to the principles of a Fabry-Perot laser (based on two mirrors) and must therefore fulfill the phase condition given by Equation 1.6 in [1], although modified for the more complex structure of the NLL. This in principle states that the round trip phase must be such that

arg X = 0. (2)

Intuitively this is understood by the need for constructive interference creating a standing wave in the cavity between the reflectors. This creates a set of allowed frequencies called cavity modes.

These modes, approximately equally spaced over frequency, significantly limit the possible lasing fre-

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quencies.

All of these constraints combined create the spectra of light emitted from the laser with intensities varying over frequency. The lasing frequency is often said to be the strongest of those frequencies.

2.3 Tunable Properties

Explained above are those properties that in an instant defines what frequency is lasing. But as mentioned before, the NLL is tunable. This implies we can change both the frequency and the intensity of the lasing light.

Controlling the NLL is done with electrical connec- tions setting currents and voltages over the various components mentioned in chapter 2.1. We denote the currents Ir, Il, Ip, Ig (rear reflector, lead reflector, phase section and gain section) and the voltages with the same indices. The gain current stimulate the emission of photons in the gain section. This in turn governs the lasing power (explained in further detail in chapter 3.2). The other currents are connected to the corresponding heating pad, which in turn affects the optical material by altering the refractive index of the components.

2.4 Measurements and Testing

The model is supposed to act as a substitute for a real device, and must therefore be able to handle the normal measurements and controls an actual device would in a test environment. When testing; a suite of control settings are set and measurements made for each such operating point. On a real device we are able to

• set currents at each electrical pad,

• measure voltage at each electrical pad,

• measure the lasing frequency and the lasing power.

These expectations set the interface to which our model should adhere, and in extension they define the relations between currents, voltages, tem- peratures and frequencies our mathematical models must describe.

3 Models

3.1 Heaters

To control the lasing frequency of the NLL the phase section and the two reflectors are modulated by temperature pads. These platinum pads are sup- plied with current causing the electrical power to heat the material. Of interest is to model the rela- tionship of current to voltage and current to temperature, which will be used both as a standalone model as well as a part of later presented models.

Heating is generally a much slower process than changes in electrical properties. It might therefore be of interest to model the heating dependent on time. But thanks to the very small scale of the components in the NLL the heating takes place in such a speed that the measuring instruments can not measure the time dependency. We are therefore satisfied by a method giving the stable temperature from a given current.

Let I, V and ∆T denote the current, voltage and temperature (relative to surrounding temperature) of an arbitrary heater and P and R be electrical power and resistance respectively. To derive the relation T (I) and V (I) we use the relations

P = IV = I²R, (3)

R = R0(1 + αT∆T ), (4)

∆T = RthP, (5)

with R_thbeing the thermal impedance, α_T the re- sistivity temperature coefficient and R₀ the restis- tance at ∆T = 0, all material constants. The first equation is the well known Ohm’s law, the second an approximation of resistance in a limited temperature range, and the latter shown in to be a good approximation to experimental data [4] [9]. With some work we then derive the following models,

Vh(Ih) = R0Ih

1 − RthR0αTI_h², (6)

∆Th(Ih) = RthR0I_h²

1 − RthR0αTI_h². (7)

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3.2 Gain Section

The gain section is the active material of the laser emitting photons that in turn goes on to lase. The section is made up of an InGaAsP waveguide common to semiconductor lasers. As with any semiconductor it is possible to model the p and n dop- ing levels and using that to model the macroscopic characteristics such as current versus voltage and current versus lasing power as shown by Coldren, L. A. [2]. In this model we only observe the macroscopic characteristics and using parameters of these to model their general behaviour.

3.2.1 Lasing Power Characteristics

To activate the laser current is pushed through the gain section making it emit photons. Because of losses in the round trip these photons need a certain gain in the gain section to survive repeated round trips and lase. This phenomenon is charac- terized by the PI-curve (lasing power over current).

For this simple model the curve will be character- ized by two parameters, the threshold current, I_th, and the PI-slope, η (sometimes ^∂P_∂I). Using this we define the curve as

P_l(I_g) = max(0, η(I_g− Ith)). (8)

3.2.2 Electrical Characteristics

The electrical characteristics of the gain section are close to the characteristics of a diode, with some modification [2]. The semiconducting material does behave like a diode, but the measurements taken of the gain section include a serial resistance, R_s. We can therefore use the Shockley diode equation to derive an expression of the voltage over applied current.

I(V_d) = I_s∗

e^q(Vd−RsI)^{nkB T} − 1

⇒ (9)

Vd(I) = nkBT q ln I

I_s+ 1

+ RsI, (10)

where Is and n being approximated as constants, kB is Boltzmann’s constant and q the elementary charge. This expression holds for a non-lasing gain section, but when applied with stronger current the energy losses effect the electrical characteristics. We

assume that lasing power is proportional to the loss in electrical power. Therefore our compete model of the electrical characteristics of the gain section is

V_g(I_g) = nkBT q ln Ig

I_s+ 1

+ R_sI_g−∂V

∂PP (I), (11) with^∂V_∂P being constant and P (I) given by Equation 8.

3.3 Lasing Frequency

Since lasing is not a discrete process of one frequency but rather an emission of a complete spectra of light with intensity varying over frequency we need to define the meaning of a single lasing frequency and introduce how to single that frequency out. To calculate the lasing frequency we use the round trip defined in Equation 1 together with the phase condition in Equation 2.

Given a round trip X(f ) we define the lasing frequency, flase, as the frequency where |X(flase)| = sup {|X(f )| where arg X(f ) = 0}. This definition can be understood as the frequency that has the largest amplitude of those fulfilling the phase condition. To calculate this lasing frequency we need the round trip, given by the components’ transmissions and reflections. Below we model these transmissions and reflections depending on frequency and temperature and these models are combined to calculate the lasing frequency.

Following the definition of lasing frequency the models below must only model the relative gain at each frequency and is therefore not bound to model the true gain. The same is true for the phase where only the phase shift modulo 2π must be accurate.

3.3.1 Passive Section

The passive sections of the NLL seen in Figure 1 between the marked sections are non-active regions with no special optical qualities. As any other material the passive regions have a complex optical refractive index, n + iκ, where n is the real refractive index and κ the extinction coefficient. The NLL operates in the regions of approximately 190.5 THz to 197 THz. This very narrow region at high frequency lead to small variations of n and κ [6]. These variations are neglected in the model here, but does make a difference in later calculations.

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As explained above we need only to model the relative transmission gain. We therefore define the gain of the transmission as |tpass(f )| = 1 for all frequencies. Now the model of the phase remains. Given that a specific passive section has the length L and a refractive index n we have that the shift in phase is given by

ϕpass(f ) = 2πLn

λ =2πLnf

c , (12)

with f = _λ^c being the frequency. We therefore have

tpass(f ) = e^iϕ^pass^{(f )} = eⁱ^2πLnf^c . (13)

All the passive sections follow this model, with their respective lengths.

3.3.2 Phase Section

The phase section has the same optical properties as the passive sections. The main difference is that the phase section can be modulated by temperature, as explained in chapter 3.1. This change in temperature does effect the refractive index, n, which in turn effects the phase shift of the transmission coefficient, tp.

Using the same argument as with the passive sections we approximate the transmission gain with

|tp(f )| = 1. Also in parallel to the passive sections we introduce the phase shift through the phase section, ϕp(f ), such that the transmission coefficient is t_p(f ) = e^iϕ^p^{(f )}.

Given the earlier stated fact that the refractive index changes with temperature we assume that ^∂n_∂T is constant over frequency. This assumption leads to a constant change in phase shift for all frequencies given a temperature change ∆T .

∆ϕp(f ) = 2πLf c

∂n

∂T∆T (14)

Since ϕp is linear over frequency we can see this constant shift ∆ϕp as either a vertical shift or a horizontal shift of the linear graph of ϕ(f ). See- ing this as a horizontal shift (a shift in temperature gives a shift in frequency mapping to a given phase shift) will have advantages in implementation seen in chapter 4. Allowing variations in both frequency

and temperature we can, by setting ∆ϕp= 0, derive the shift in frequency given a shift in temperature mapping to a given phase-shift. We have that

ϕp(f ) = 2πLnf

c (15)

which gives

∆ϕ_p(f ) = ∂ϕ_p

∂f ∆f +∂ϕ_p

∂T ∆T = 2πL

c

∂f

∂fn + f∂n

∂f

∆f + ∂f

∂Tn + f∂n

∂T

∆T

= 2πL

c

n + f∂n

∂f

∆f + f∂n

∂T∆T

= 0. (16)

With some calculation we then have

∆f = −f_∂T^∂n ngr

∆T, (17)

where n_gr = n + f^∂n_∂f is the group refractive index of the material. The term −^f

∂n

∂T

ngr varies with frequency, but over the interval of interest this vari- ation is small. Therefore we approximate −^f_n^∂T^∂n

gr =

∂f

∂T = constant.

In summary the transmission coefficient of the phase section is

tp(f, ∆T ) = eⁱ^2πLn^c ^{(f +}^∂T^∂f^{∆T )} (18) with ∆T being the difference of temperature from where n was measured.

3.3.3 Reflectors

The reflectors are specially designed distributed Bragg reflectors (DBR). A DBR is grating made of inter-layered materials of different optical properties. As light passes the grating there is a reflection in each of the boundaries between the layers (see Figure 2). As these reflections interfere with each other, waves of wavelengths not matching the grating structure cancel and only some frequencies are reflected.

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Figure 2: Structure and internal reflections of a distributed Bragg reflector (DBR). Illustation from Coldren [2].

Because of these interfering reflections the complex reflection coefficient, r, will not have |r(f )| = 1, even though as with all the previous sections the DBR’s do not have a frequency dependent gain in the material. Accurate models of the reflectivity spectra directly from design parameters lie beyond the scope of this paper. Instead the goal is, from a given spectra be able to model the shifts due to temperature variations.

Figure 3: Reflectivity spectra of the lead (front) and rear (back) reflectors. Image from Eriksson et al. [3].

Reflectivity gain The reflectivity of the DBR’s (seen in Figure 3) is given to us by the peak frequencies, the amplitude of each peak and the

linewidth of a peak. We define the full-width- at-half-maximum linewidth (FWHM), ∆ν, as the width of a peak at half the amplitude of the peak [8]. Denote the peak frequencies f₁, f₂, . . . , f₇. The respective amplitudes |r(f₁)|, . . . , |r(f₇)| are given.

As a simple model the spectra is modeled as a su- perposition of Gaussian distributions around each peak.

We have

|r(f )| =

7

X

i=1

|r(fi)|e⁻^{(f −fi)}

2

2σ2 (19)

with σ = q ∆ν²

ln 256 to achieve the desired linewidth of the peaks.

The gratings are tunable using temperature modu- lation, just as with the phase section. As mentioned in chapter 3.3.2 the implementation would bene- fit from a model of reflectivity that is shifted over frequency when temperature shifts. As explained above the spectra of reflectivity gain is given by the interactions of many elementary reflections in each boundary between layers in the reflector. We can therefore use, in each of those layers, the argument given in chapter 3.3.2.

Take a peak fn from the reflectivity peaks. This peak is a peak due to the constructive interference between all the reflections appearing in the reflector. When a temperature shift, ∆T , oc- curs the frequency that would experience the same phases when passing trough the reflector shifts. The new frequency f_n + ∆f will therefore experience the same constructive interference at temperature T + ∆T as f_n did for temperature T . As the material of the reflector is very similar to the phase section we assume that the same reasoning given in chapter 3.3.2 is applicable to the reflectors, and therefore say that we model the reflectivity of the gratings with temperature dependence according to

|r(f, ∆T )| = |r(f + ∂f

∂T∆T )| (20)

where _∂T^∂f is the same as in Equation 18. This result is strengthened by the experimental results of H¨aggmark [5].

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Reflectivity phase shift Now having modeled the real part of the complex reflectivity coefficient the phase shift of the reflectors are of interest. As explained on pages 121 - 123 in [2] the reflection in a DBR can be simplified as a total reflection at a certain depth, L_pen. Given such a penetration depth we can use the same argument as with the passive and phase sections, giving the phase shift of the reflection as

ϕr(f ) = 2π · 2Lpenˆnf

c (21)

where L has been exchanged with 2Lpenand n been changed to ˆn being the average refractive index. We introduce the group delay of reflection τ = ^2L^pen_cⁿ^ˆ^gr and use this to differentiate the phase shift over frequency

∆fϕr(f ) = ∂ϕr

∂f ∆f = 4πLpen

c

∂(ˆnf )

∂f ∆f = 4πLpennˆgr

c ∆f ≈ 2πτ ∆f. (22)

Introduced above are three new parameters ˆn, ˆn_gr and L_pen. Instead of specifying all these physical parameters (that might depend on frequency) we have τ and ϕ0 given for each peak from experimental data. Using this we can define a linear relation around each peak frequency. We define the phase shift as a peacewise linear function on the intervals spaced around each peak frequency as

ϕr(f ) =

7

X

i=1

ϕr_i(f ) (23) with

ϕr_i(f ) =

(ϕ0_i+ 2πτi(f − fi) if f ∈ Ii,

0 otherwise. (24)

where Ii = _f

i+f_i−1

2 ,^fⁱ^+f₂ⁱ⁺¹

, i ∈ {1, . . . , 7} with the end regions (I1 and I7) extending to infinity.

This is visualized in Figure 4. The discontinuities in this models can be argued to have little effect on the result thanks to the arguments given in chapter 3.3.

They state that only the frequencies with largest round trip gain have the ability to lase, which by

the model of |r(f )| will not be in the middle between peaks.

1.92 1.94 1.96 1.98

Frequency, f ×10¹⁴

Phaseshift,ϕ(f)

ϕ(f ) ϕ₀

Figure 4: Phase shift over frequency linearized around each given phase shift at reflective peaks.

Now it remains to model the change of phase shift due to a change in temperature. We differentiate ϕrover temperature.

∆Tϕr(f, ∆T ) = ∂

∂T

4πLpennf c

∆T = 4πLpenf

c

∂n

∂T∆T. (25)

Given this differentiation we have that the total shift in phase shift of a reflection peak over temperature shift ∆T is

∆ϕr_peak = ∆Tϕr_peak + ∆fϕr_peak = 0. (26)

This means that the phase shift for a reflection peak does not change with temperature change. There- fore, just as with the reflectivity gain we can shift the phase shift over the frequency spectra when temperature change using ∆f = ^∂f_∂T∆T giving the full expression of the reflection coefficient depending och frequency and temperature as

r(f, ∆T ) = |r(f + ∂f

∂T∆T )|e^iϕ^r^{(f +}^∂T^∂f^{∆T )} (27)

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3.3.4 Gain Section

Experimental data shows that the gain section has a gain bias towards the center of the lasing frequencies. A simple model that seems to fit the experimental data is a parabola. Further we model the phase in the same way as the passive sections, resulting in the transmission of the gain section

tg(f ) = (1 − αg(f − fg))eⁱ^2πLnf^c , (28)

where α_g is a material constant determined experimentally and f_g is the gain peak frequency.

4 Implementation

Explained above are the mathematical models governing how each component effects the model and its output. There are many different approaches to combining these mathematical expressions to ex- tract the voltages, the lasing power and the lasing frequency. As explained in chapter 1.3 one of the core aims of the model is to be modular in the sense of each component being replaceable, changing constants as well as mathematical models.

The implementation of some models such as the IV- relations are straight forward and just a matter of evaluation of an expression. These models are im- plemented as array-methods to enable using built-in or other hardware optimized array-math packages.

Of more interest is the implementation of the round trip calculation and finding the lasing frequency.

This calculation is a lot more complex in the sense that it is not just a simple mathematical expression being evaluated for a set of parameters and vari- ables. Instead it contains multiple different expressions combined to find the one frequency fulfilling the definition set in chapter 3.3. The implementation chosen is explained in the algorithm below.

1. Create an array, freq, with equally spaced frequencies over the relevant frequency band.

The resolution of the array can be chosen ar- bitrarily, but resolution larger than 5GHz is found to have an impact on accuracy. Then create an array, phi, of equal length to freq.

2. Iterate the components simulating a round trip. At each component multiply (in- place) each entity of the array phi with

the corresponding (frequency dependant) transmission/reflection coefficient modeled in chapter 3.3, resulting in the array phi=

X(f, I_r, I_l, I_p), where X is the full round trip coefficient.

3. Find the indices of the elements with the lo- cally largest argument with arguments evaluated on the intervall (0, 2π).

4. Given indices above interpolate the amplitude of phi at which the argument equals zero.

Given such an index i the interpolation can be done using

|ϕi| +(2π − arg ϕi)(|ϕi+1| − |ϕi|) 2π − arg ϕi+1− arg ϕi

, (29)

where ϕj = phi[j].

5. Choose the frequency corresponding to the largest such amplitude.

In addition one other implementation design is key for the speed of the algorithm described above.

Since at every component an array is to be mul- tiplied with an array of coefficients we do not want to regenerate any of the complex arrays as a new measurement is to be taken. Therefore we can use the fact that all tunable properties are shiftable over frequency and generate all the complex coefficient arrays on program start. Changing the temperature of a component is therefore represented by shifting the complex array. We illustrate this with an ex- ample.

We have a frequency-array f of length 10000, cov- ering the frequency band 190.5 THz to 197.5 THz.

We then know that at each component we need to generate an array t with the transmission (or reflection) coefficient corresponding to each frequency. If the component is non-tunable we instantiate t to the correct array and we use the same array on every multiplication. If the component is tunable with temperature we take the maximum tuning temperature ∆Tmax and instantiate the array as a vector of length 10000 + ceil(|^∂f_∂i_∂T^∂f∆Tmax|), where ^∂f_∂i is the change of frequency over index in the array.

We instantiate the array with complex transmission coefficients corresponding to frequencies of f and extending to all indices in t, shifting the indices such that f = 190.5 THz corresponding to index

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i0 = ceil(|^∂f_∂i_∂T^∂f∆Tmax|). Then, when multiplying with t, we use the partial array t[i:i+10000] with i = i0 − round(^∂f_∂i_∂T^∂f∆T |).

An advantage of the general implementation is that every component can model the transmission sep- arately, governed by any number of parameters.

Specifically the implementation of pre-generated arrays has significant effect on computational time, reducing calculation time upwards of 100 times (this result is hardware and software dependent).

5 Results

Results of the model in respect to the aims set out in chapter 1.3 are presented below. The results were generated with parameters given from design spec- ifications as well as from experimental data extractions and manual fitting. Since individual differ- ences between devices are common we have chosen to use data of only a single device. The parameters are fitted to this specific device.

Due to protection of intellectual property some results are presented without axis values. All figures have linear scaled axes, but do not in general origi- nate in the origin.

5.1 Accuracy

We define accuracy of the model as a measurement of how well our model fits the experimental data.

Sometimes the term accuracy can be quantified and be represented by a single value. In this thesis we will not quantify the accuracy of the model since such models often miss what we find most inter- esting. In this case that the model have the same characteristic behavior as the real data, not neces- sarily the same exact values.

5.1.1 Electrical Characteristics

The fit to the electrical characteristics of the NLL is shown in Figures 5 and 6. It is clear that the heater IV-curve fit the data very well, while the gain section IV-curve lack in accuracy. On the other hand the characteristics of the gain section IV-curve is close to that of the experimental data, seen in the approximately linear relation for currents greater than Ithand with a small discontinuity in slope at Ith.

0

Heater current 0

Heatervoltage

Experimental Model

Figure 5: IV-curve of the heater pads for model and experimental data.

0 I_th

Gain current

Gainvoltage

Experimental Model

Figure 6: IV-curve of the gain section for model and experimental data.

5.1.2 Lasing Power

The lasing power is studied with the PI-curve (lasing power over gain current) shown in Figure 7. The threshold characteristics are clearly modeled. What is missing is the slight non-linearity of the region with currents greater than Ith.

0 I_th

Lasing power Gaincurrent 0

Experimental Model

Figure 7: Laser output power over gain current for model and experimental data.

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5.1.3 Lasing Frequency

Analysing the lasing frequency can be done in many different ways. One common representation is a mode map. The mode map (shown in Figure 8) is a map of lasing frequency over both rear and lead reflector tuning temperature. To analyse the accuracy of the model we plot the one dimensional sweep along the main diagonal and one cross diagonal of the mode map. These sweeps are shown in Figure 9.

It is clearly illustrated that the model has the same characteristic behaviour as the experimental data for the lasing frequency over tuning temperature.

The result also shows that the slope and average cavity mode distance of the one dimensional sweeps are not exact, yet have the same characteristics.

5.2 Code Performance

One other result of interest is code performance in the sense of speed and calculation heaviness. To quantify this we measure the time taken to perform a certain task. Averaging many such measurements gives us the average time taken of a calculation. Ta- ble 1 presents the sampling frequency of three different measurements for the model and a real device.

It clearly shows that evaluation of lasing frequency is the slowest evaluation in the model. Sampling frequency of real devices are governed by measurement instrument and are in general around 2 kHz. Thus the table also shows that the performance matches or outperforms the real measurements, which is the desired behavior.

The data of Table 1 was generated on a Macbook Air 1,4 GHz Dual-Core Intel Core i5, using Python and the vector library NumPy. The resolution of the frequency array mentioned in chapter 4 was 1 GHz per index.

Table 1: Average sampling frequency in kilohertz (kHz) for the model and a real device.

Measurement Sample freq. [kHz]

Heater voltage 1900

Gain voltage 46

Lasing frequency 2.6

6 Discussion

In general the model fits the experimental data well and the characteristic behaviours of most of the

NLL’s measured proprieties are mimicked by the model. For the electrical characteristics the model operates almost instantly and the calculations at a speed faster than measuring instruments can record and measuring lasing frequency match real sampling frequencies.

6.1 Applicability

These results are definitely satisfactory to the ex- tent of accuracy. They model all the characteristic behaviours of the NLL and would pass most tests and extractions. This makes the model useful in a test environment.

The strength of the model in this environment is that it will always give a predictable result, which can not be guaranteed with real devices. This is useful when debugging data extraction code or designing algorithms, but can also be argued to be a disadvantage. Since the code written for testing should handle real device data it must be able to cope with irregularities. The advantage of our very modular design is that these types of irregularities can be incorporated in the model if of interest.

6.2 Improvements and Future Work

Although useful as is, the model has many areas of improvement. As a result of the already modular design of the model most of these improvements can be done by simply exchanging mathematical models or implementation principles in single components, which is just what is wanted.

First of all one major improvement would be to base all the mathematical models on design parameters of the NLL, such as lengths or materials and not on experimentally extracted parameters. This would make the model easier to adapt to new design even before the first tests can be run. Of special interest would be to more accurately model the gain section after this principle since that might model the non linearities more accurately as explained by Coldren [2].

Further the model has possible implementation improvements. The model is slow on lasing frequency calculation and the main computational time is spent on the intensive array multiplications and ex- tracting iterative processes. Although the implementation is very good from the modular stand- point, the number of unnecessary computations is

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10 20 30 40 50 60 70 Rear reflector tuning [^◦C]

10 20 30 40 50 60 70

Leadreflectortuning[◦ C]

191 192 193 194 195 196

Frequency[THz]

Figure 8: Mode map generated by the model. Dashed red lines mark sweeps shown in Figure 9.

0 20 40 60

Rear reflector tuning temperature [^◦C]

Frequency

Experimental Model

(a) Diagonal sweep. Left: entire diagonal. Right: magnification of left image’s rectangle.

0 10 20 30 40 50

Rear reflector tuning temperature [^◦C]

Frequency

Experimental Model

(b) Cross-diagonal sweep.

Figure 9: Sweeps along dashed red lines shown in Figure 8.

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high. There are possible solutions to this, most ef- fecting the generality of the model. This paper does not test any other solutions to the implementation, but suggested in Appendix A is an implementation that might make the extraction of lasing frequency more efficient.

7 Conclusions

When summarizing it is clear that the model ful- filled the aims set out in the beginning of the project. It does accurately model the lasing power and frequency of the laser over all relevant tuning domains. Also, in a satisfactory way, it models the electrical characteristics of the device. The implementation is modular in the sense that each component can easily be altered or adjusted to represent a new design or parametric changes. Although the implementation is computationally heavy it is in all aspects faster that real life measurements. Thanks to this the model will be able to replace real devices in a test environment. To be fully useful the model would need to be based only on design parameter and this is where future work should focus.

This model is one further step towards a more efficient workflow in the design of new lasers. With an even more accurate model this can become an essential tool for data extraction design and in the future also a tool for design of components and entire devices.

References

[1] John Carroll. Distributed Feedback Semiconduc- tor Lasers., volume v. PM52 of Circuits, De- vices and Systems. Institution of Engineering and Technology, 1997.

[2] L. A. Coldren. Diode lasers and photonic integrated circuits. Wiley series in microwave and optical engineering. Wiley, New York, 1995.

[3] Urban Eriksson, Jan-Olof Wesstrom, Yitong Liu, Stefan Hammerfeldt, Martin Hassler, Bjorn Stoltz, Niclas Carlsson, Salehe Siraj, Edgard Goobar, and Yasuhiro Matsui. High performance narrow linewidth thermally tuned semiconductor laser. In 2015 European Conference on Optical Communication (ECOC), pages 1–3.

Viajes el Corte Ingles, VECISA, 2015.

[4] John N Fox. Temperature coefficient of resistance. Physics Education, 25(3):167–169, 1990.

[5] Ilian H¨aggmark. Fiber bragg gratings in temperature and strain sensors, 2014.

[6] Daniele Melati, Abi Waqas, Andrea Alippi, and Andrea Melloni. Wavelength and composition dependence of the thermo-optic coefficient for ingaasp-based integrated waveguides. Journal of Applied Physics, 120(21), 2016.

[7] R¨udiger Paschotta. Laser Applications in the Encyclopedia of Laser Physics and Technol- ogy. https://www.rp-photonics.com/laser_

applications.html. Accessed: 2020-04-25.

[8] Joseph A Thie. Nuclear Medicine Imaging:

An Encyclopedic Dictionary. Springer, 1st ed.

2012.. edition, 2012.

[9] Alexey A. Vasiliev, Anton V. Nisan, Gleb N.

Potapov, Nikolay N. Samotaev, Konstantin Yu.

Oblov, and Anastasia V. Ivanova. Mems sensors based on very thin ltcc. Proceedings, 1(4):324, 2017.

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A Suggested Lasing Frequency Algorithm

Presented here is a suggested implementation for a faster and more computationally light weight calculation of the lasing frequency.

A.1 Method

Implement each component of the model with a method evaluating the transmission/reflection for a given frequency.

1. Given the two grating reflectivity peaks, find at an instant which two pairs of lead and rear peaks that are closest to each other in frequency. Denote these pairs (f_1l, f_1r) and (f_2l, f_2r).

2. Evaluate the round trips of each peak using the methods mentioned above. Denote these complex numbers as X(f_1l), X(f_1r), X(f_2l) and X(f_2r).

3. Assume a linear relation of the round trip phase between X(f_l) and X(f_r) (where a non-indexed frequency denotes any of the peak pairs). This assumption holds for the models above thanks to the linear nature of all transmissions and the liniarization around reflector peak frequencies.

4. Find the frequency fc centered between the peaks in each pair as fc= ^f^r^+f₂ ^r.

5. Find the frequencies just left and right (indexed₋ and+) of the center frequency with argument equal to 0. We have that these frequencies can be found by

f₊= f_c+2π − arg_2π(X(f_c))

k , (30)

f₋ = fc+−arg2π(X(fc))

k , (31)

with

k = arg_ϕ(X(f_l)) − arg_ϕ(X(f_r)) fl− fr

, (32)

where arg2π is function returning the phase modulo 2π and argϕis a function returning the phase as is.

6. Let the lasing frequency be f_l such that |X(f_l)| = sup{|X(f₁₋)|, |X(f₁₊)|, |X(f₂₋)|, |X(f₂₊)|}.

A.2 Comments

This method assumes some lineareties and other properties that the models fulfill today, but might not in the future. This is to be seen as a possible method for optimizing the code in a situation where code performance is key.

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Modeling a Tunable Narrow Linewidth Laser

Modeling a Tunable Narrow Linewidth Laser

CHRISTOFFER EJEMYR

Abstract

Abstrakt

Contents

1 Introduction

2 The Narrow Linewidth Laser

3 Models

4 Implementation

5 Results

6 Discussion

7 Conclusions

References

A Suggested Lasing Frequency Algorithm