Week 5: Semiconductor Laser Design Principles

Question 1

What does LASER stand for?

Answer 1

Laser stands for light amplification of simulated emission of radiation.

Question 2

What two basic things do you need to build a laser?

Answer 2

One: You need a resonator, a place where the light will bounce around in a cavity and come back with the same amplitude and phase if it is on resonance.

Two: You need a gain material, a medium that provides the amplification. In this class, that’s a semiconductor.

Question 3

Describe at a high level what makes a laser.

Answer 3

A laser is made up of two parallel mirrors, with a gain medium between them that provides amplification.

Question 4

How would you describe a laser oscillation?

Answer 4

A laser oscillation is a plane wave that has complex propagation coefficients and goes through successive reflections within the cavity between the parallel mirrors.

Question 5

What do you need from the electric field to produce laser resonance or oscillation?

Answer 5

You need the field to reproduce itself in amplitude and phase exactly after one round trip. This can be expressed in the oscillation condition.

Oscillation condition:
r_1 * r_2 * exp(-2* Gamma * L) = 1

r_1: reflection coefficient of first mirror
r_2: reflection coefficient of second mirror
Gamma: complex propagation coefficient
L: length separating the mirrors

Question 6

What does Gamma represent in the oscillation equations? How do you calculate it?

Answer 6

Gamma is the complex propagation coefficient for the cavity between the mirrors. It takes into account the phase that’s accumulated from propagating through the cavity as well as any gain or loss.

Gamma = gamma + ink

Gamma != gamma
(They’re the upper and lower case versions.)

gamma: the gain or loss term
i: the imaginary number
n: the index of refraction
k: free space wave number

Question 7

What is the expression for the transmitted electric field in the laser?

Answer 7

The expression is:

E_t = [ (t_1t_2E_iexp(-GammaL) / (1 - r_1r_2exp(-2Gamma*L) ]

t_1: transmission coefficient of first mirror
t_2: transmission coefficient of the second mirror
r_1: reflection coefficient of first mirror
r_2: reflection coefficient of second mirror
Gamma: complex propagation coefficient
L: length separating the mirrors
E_i: incident electric field

Question 8

What is k, the free space wave number, equal to?

Answer 8

Free space wave number, k:

k = (2*pi)/lambda

Question 9

How do you calculate gamma, the term for the gain or loss?

Answer 9

gamma = 1/2 * (alpha + g)

alpha = loss/meter
g = gain/meter

Question 10

How would rewrite the oscillation condition without directly using Gamma?

Answer 10

You can rewrite it by substituting in for Gamma. This makes the requirement for laser oscillation becomes:

r_1 * r_2 * exp[(g - alpha)L] * exp[-2i * (2pin/lambda)L] = 1

Question 11

How would you get the amplitude condition from the oscillation condition? How does this relate to power reflectance?

Answer 11

The amplitude condition is one part of the oscillation condition (the other is the phase condition).

The amplitude condition is: r_1r_2exp(alpha - gamma)*L = 1

Power reflectance, R, is the ratio of reflected intensities, so the ratio of reflected intensity to the incident intensity.

R = abs| E_r |^2 / abs| E_i | ^2 = r^2
E_i: incident electric field
E_r: reflected electric field

We can rewrite the amplitude condition in terms of power relfectances, R_1 and R_2:

g = alpha - (ln(R_1R_2) / 2L)

Question 12

How do you get the phase condition part of the oscillation condition? What does it tell you about the number of half waves needed to get oscillation?

Answer 12

The phase condition can be obtained from the following relation:

mlambda = 2n*L

where m is an integer number. This tells you that in order to get oscillation, you have to have an integer number of half waves fit in the cavity.

Question 13

How would you calculate the expected wavelength spread of the laser?

Answer 13

∂lambda = lambda^2 / (2 * n * L(1 - (lambda/n)(∂n/∂lambda)))

Question 14

Explain the principles behind a wave guide.

Answer 14

When we want to make a wave guide, we want there to be total internal reflection so that no light escapes the material in the middle of the sandwich. To do this, we need to have the middle material have an index of refraction greater than that of the cladding material.

Question 15

What is the confinement factor?

Answer 15

The confinement factor is the fraction of the optical mode that overlaps the active region relative to the total power in the mode.

So if you have an area with index of refraction, n_2, that is the gain area or active region, then this is the fraction of power in the optical mode that’s in this active region of n_2 versus the whole guide.

Question 16

What are the two boundary conditions for the effective index method?

Answer 16

The two boundary conditions for the effective index method are:

1) The perpendicular components of the magnetic flux and the electric displacement are continuous.
2) The tangential components of the electric field and the magnetic field are also continuous.

Question 17

Give a high level description of what the electric field looks like mathematically inside the guide and outside the guide in the cladding region.

Answer 17

Inside the guide the field is a sinusoidally oscillating field which we know from the cosine term in the typical y solution equation. Outside the guide, within the cladding, the field is going to decay exponentially which we recognize from the exponential term in the typical y solution equation.

Question 18

What are two of the most important design parameters for semiconductor lasers?

Answer 18

Two of the most important design parameters for semiconductor lasers are the single mode condition and the transverse and longitudinal mode confinement factor.

Question 19

What is desirable about a single mode? How do you determine the layer thickness needed to maintain a single mode? What is a normal value for the thickness?

Answer 19

The case were the wave guide only supports a single transverse mode is a desirable one because it means you don’t have a lot of mode competition.

For a single mode, we have:

D < pi and K_0 = 2*pi / lambda

we know that:
D = k_0 * sqrt(n_2^2 - n_1^2) * d where d is the thickness of the wave guide.

So we can find the maximum thickness of the wave guide by:

d < (lambda/2)*(1 /sqrt(n_2^2 - n_1^2))

Normally, d is less than 200 nanometers and this almost always satisfies the single mode condition.

Question 20

What fraction of a wavelength can fit inside the active region to achieve the single mode condition?

Answer 20

In order to achieve the single mode condition, we need to restrict the thickness of the slab guide so that only a half wavelength can fit inside the active region.

Question 21

What are the maximum thicknesses of the active regions for indium gallium arsenide phosphide, gallium arsenide, and aluminum arsenide?

Answer 21

Active region thickness for:

indium gallium arsenide phosphide:
< 480 nanometers

gallium arsenide and alluminum arsenide:
< 870 nanometers

Question 22

What controls the Fermi level in the case of doped and undoped semiconductors?

Answer 22

In the case of an undoped (intrinsic) semiconductor, the Fermi level is controlled and moved away from the mid gap by the ratio of the effective gasses. In the case of the doped semiconductors, the Fermi level position (assuming that the dopant is much greater than the intrinsic concentration) is controlled by the doping level.

Question 23

What is the threshold current in a semiconductor laser and what is it determined by?

Answer 23

The threshold current in a semiconductor laser is the current required to initiate laser action. It’s determined by the energy dependent gain coefficient, the light confinement factor, the width of the laser, and the spontaneous emission and losses.

Question 24

What is the relationship between light and carrier injection before and after the threshold is reached?

Answer 24

The light increases linearly with carrier injection until the threshold is reached. This is due to spontaneous emission. Once threshold is reached, stimulated emission takes over and laser action begins.

Question 25

Describe the relationship between carrier density and current.

Answer 25

When we look at carrier density as a function of current, we can see that the carrier density is clamped at the threshold value. Meaning that the current density will increase as a function of current until the threshold current is reached and then once that point is reached, the current density just plateaus.

Question 26

Write down expression for J_th​, threshold current density in terms of:

    N_th, threshold carrier density
    q, the charge on an electron
    d, the thickness of the active region
    τ_e, minority carrier lifetime
    η_i, the injection efficiency

Answer 26

Threshold current density, J_th:

J_th ​= ​q×d×N_th​​ / τ_e​×η_i

where:

    N_th, threshold carrier density
    q, the charge on an electron
    d, the thickness of the active region
    τ_e, minority carrier lifetime
    η_i, the injection efficiency

Question 27

Write an expression for the carrier density at threshold in terms of:

    N_o, the carrier density at transparency,
    α_m, losses of mirror,
    α_i, internal wave guide losses
    a, differential gain
    Γ, confinement factor.

Answer 27

The carrier density at threshold can be written as:

N_th​ = N_o​+ ((α_m​+α_i​​) / aΓ)