Optical Fields

Overview: In the following, we introduce a 2-D mathematical model of a plane semiconductor waveguide with ridge-geometry. The geometry of the waveguide is used to perform several approximations (see sections 1,1, 1.2, and 1.3) in order to obtain the final form of the 2-D wave-equation (1.34).

For a passive device the material polarisation of the semiconductor can be neglected. To describe active devices like semiconductor lasers and amplifiers the material polarisation (induced by pumping the medium electrically or optically) can be used to describe light generation/absorption via recombination/generation of electrons-hole pairs.

1  Waveguiding and Optical Fields
    1.1  Slowly Varying Envelope Approximation
    1.2  Paraxial Approximation
    1.3  Effective Index Approximation

Our aim is to calculate the optical fields inside the cavity of an edge-emitting QDSOA. A simplified sketch of a waveguide geometry is presented in Fig. 1.1 showing the transverse direction 'x', the vertical direction 'y' and the propagation direction of the light fields 'z'.

Figure 1.1: Semiconductor waveguide with ridge geometry. In a semiconductor laser or amplifier the active area (cyan colored region) contains quantum wells or quantum dotsand is electrically pumped. The rigde of the structure determines the pumped region.

In order to obtain a wave-equation that can be easily dealt with numerically, we make a series of approximations. These include a slowly varying amplitude, paraxial and effective index approximation, respectively, and are introduced in sections 1.1− 1.3. The starting point for the derivation of the wave equation describing the propagation of the light fields in a semiconductor waveguide with rigde geometry are Maxwell's equations. The equations are expressed in SI units, and the dependence on space and time coordinates is not written explicitly [6].

∇·D = ρ,          ∇×E+ ∂ ∂t B= 0, (1.1) ∇·B = 0,          ∇×H− ∂ ∂t D= J. (1.2)

In equations (1.1) − (1.2), D represents the electric displacement, ρ the charge density, E the electric field, B the magnetic induction, H the magnetic field, and J the current density. We assume that constitutive relations for non-magnetic materials are valid, and that the source terms can be neglected:

B = μ0 H,          D= ϵE+ P, (1.3) ρ = 0,          J= 0, (1.4)

where Pdenotes the material polarization induced by charge carriers localized in QDs, while ϵ is the background electric permittivity and μ₀ is the magnetic permeability.
The rotation operator is applied to the second Maxwell equation in (1.1). Inserting the second equationin (1.2) and the first constitutive equation (1.4), we obtain:

∇×( ∇×E) = −μ0 ∂2 ∂t 2 D.

(1.5)

Using the second constitutive equation, we can substitute D in (1.5) to obtain the wave-equation:

⎡ ⎣ ∆− μ0ϵ ∂2 ∂t 2 ⎤ ⎦ ⎛ ⎝ E+ 1 ϵ P ⎞ ⎠ = − 1 ϵ ∇×(∇×P).

(1.6)

We then split the wave-equation into a transverse and a longitudinal part. To achieve this we explicitly use the transverse and longitudinal component of the nabla operator [7]:

∇ = ∇T + ez  ∂ ∂z .

(1.7)

In a next step, we assume that the main propagation direction is along the longitudinal axis of the cavity `z' and that the optical fields can be described by a slowly varying envelope field multiplied by a component that is oscillating at optical frequencies. In this context `slow' means that the envelope field varies little during the time interval related to the period of the optical field. In this case, the optical fields can be described in terms of the slowly varying envelope of the optical field.

In order to describe forward '+' and backward '-' propagating optical fields in the amplifier cavity, we choose the following ansatz [4]:

E ±= (ET ±+ez Ez ±) e±i βz − i ω0t       and       P ±= (PT ±+ez Pz±) e±i βz − i ω0t, (1.8)

where β is the propagation constant in longitudinal direction and ω ₀is the central frequency of the optical fields. With this ansatz, the total electric field and the material polarisation take the following form:

E= 1 2 ( E + e i βz − i ω0t+ E − e −i βz − i ω0t+ c. c. ), (1.9) P= 1 2 ( P + e i βz − i ω0t+ P − e −i βz − i ω0t+ c. c. ). (1.10)

Inserting (1.7) and (1.8) into (1.6), we obtain the transverse part of the wave-equation:

⎡ ⎣ ∆T − μ0ϵ ⎛ ⎝ ∂2 ∂t 2  − 2iω ∂ ∂t  − ω2 ⎞ ⎠ + ⎛ ⎝ ∂2 ∂z 2  ± 2iβ ∂ ∂z  − β2 ⎞ ⎠ ⎤ ⎦ ⎛ ⎝ E T ± +  1 ϵ  PT ± ⎞ ⎠ =  − 1 ϵ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎝ ∂ ∂z ±iβ ⎞ ⎠ ∇T Pz±− ⎛ ⎝ ∂2 ∂z2 ±2iβ ∂ ∂z − β2 ⎞ ⎠ PT±+ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂2 ∂yx Py±− ∂2 ∂y 2 Px± ∂2 ∂xy P x ±− ∂2 ∂x 2 P y ± 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , (1.11)

and the longitudinal part of the wave-equation:

⎡ ⎣ ∆T − μ0ϵ ⎛ ⎝ ∂2 ∂t2  − 2iω ∂ ∂t  − ω2 ⎞ ⎠ + ⎛ ⎝ ∂2 ∂z 2  ± 2iβ ∂ ∂z  − β2 ⎞ ⎠ ⎤ ⎦ ⎛ ⎝ Ez± +  1 ϵ Pz± ⎞ ⎠ =     − 1 ϵ ⎡ ⎣ ⎛ ⎝ ∂ ∂z ±iβ ⎞ ⎠ ∇T·PT ±− ∆T Pz± ⎤ ⎦ . (1.12)

Equations (1.11) and (1.12) describe the dynamics of the transverse and longitudinal part of the slowly varying field envelope. They have been obtained by assuming that the main propagation direction is along the optical cavity and that the optical field can be split into a fast component (oscillating at optical frequencies) and a slowly varying envelope. We can further simplify the wave-equations by assuming that the optical fields propagate parallel (or at least at small angles) with respect to the longitudinal axis of the cavity. We follow the approach of Lax et al. using the paraxial approximation presented in [7].

To perform the paraxial approximation, the wave-equations are written in a scaled form using the following scaling parameters [7]:

l  =  βw2           diffraction length, (1.13) τ  =  l  √   μ0ϵ       propagation time ( related to diffraction length ), (1.14) w                  width of beam with Gaussian profile. (1.15)

In the following, dimensionless variables are indexed with ρ and transform according to:

x= w xρ         y= w yρ         z= l zρ         t= τ tρ .

(1.16)

We define a dimensionless quantity f : = [w/l] which in our case is a small quantity and serves as expansion parameter. Introducing the new variables, the transverse wave-equation reads:

⎡ ⎢ ⎣ f ∆Tρ+ f 3 ⎛ ⎝ ∂2 ∂z2ρ  −  ∂2 ∂t2ρ ⎞ ⎠ + 2if ⎛ ⎝ ∂ ∂tρ  ±  ∂ ∂zρ ⎞ ⎠ + f w 2(ϵr k02− β2) ⎤ ⎥ ⎦ E Tρ±= − 1 ϵ ⎛ ⎝ f 2 ∂ ∂zρ ±i ⎞ ⎠ ∇Tρ Pzρ± − 1 ϵ ⎡ ⎢ ⎣ f ∆Tρ ⎛ ⎜ ⎝ f 3 ⎛ ⎝ ∂2 ∂t2ρ − 2if ⎛ ⎝ ∂ ∂tρ − 1 f ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ PTρ±− f ϵ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂2 ∂yρ ∂zρ P±yρ −  ∂2 ∂yρ2 P±xρ ∂2 ∂xρ ∂yρ P±xρ −  ∂2 ∂xρ2 P±yρ 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,         (1.17)

where k ₀is the propagation constant in vacuum and ϵ_r: = ϵ/ϵ ₀is the background dielectric constant. In scaled form, the longitudinal part of the wave-equation reads:

⎡ ⎢ ⎣ f ∆Tρ+ f 3 ⎛ ⎝ ∂2 ∂z2ρ  −  ∂2 ∂t2ρ ⎞ ⎠ + 2if ⎛ ⎝ ∂ ∂tρ  ±  ∂ ∂zρ ⎞ ⎠ + f w 2(ϵr k02− β2) ⎤ ⎥ ⎦ Ezρ±= − 1 ϵ ⎛ ⎝ f 2 ∂ ∂zρ ±i ⎞ ⎠ ∇Tρ · PTρ± − 1 ϵ ⎡ ⎢ ⎣ f ∆Tρ+ f 3 ⎛ ⎝ ∂2 ∂z2ρ  −  ∂2 ∂t2ρ ⎞ ⎠ + 2if ⎛ ⎝ ∂2 ∂tρ  ±  ∂2 ∂zρ ⎞ ⎠ + f w 2(ϵr k02− β2) ⎤ ⎥ ⎦ Pzρ±           (1.18)

In order to obtain a consistent system of coupled differential equations, the electric field and the nonlinear polarisation are written as power series of the form[7]:

ETρ± = ETρ±,(0)+ f 2 ETρ±,(2)+ …, Ezρ± = f Ezρ±,(1)+ f 3 Ezρ±,(3)+ …, PTρ± = f 2 PTρ±,(2)+ f 4 PTρ±,(4)+ …, Pzρ± = f 3 Pzρ±,(3)+ f 5 Pzρ±,(5)+ …, (1.19)

Inserting the expressions (1.19) into (1.17), the transverse wave-equation in lowest expansion order reads:

⎡ ⎢ ⎣ f∆Tρ + 2if ⎛ ⎝ ∂ ∂tρ  ±  ∂ ∂zρ ⎞ ⎠ + f w 2(ϵr k02− β2) ⎤ ⎥ ⎦ ETρ±,(0)= − f ϵ0 k02 β2 PTρ±,(2)

(1.20)

Inserting (1.19) into (1.18), one obtains the longitudinal wave-equation in lowest expansion order:

⎡ ⎢ ⎣ f∆Tρ + 2if ⎛ ⎝ ∂ ∂tρ  ±  ∂ ∂zρ ⎞ ⎠ + f w 2(ϵr k02− β2) ⎤ ⎥ ⎦ Ezρ±,(1)= − f ϵ (±i) ∇Tρ · PTρ±,(2)

(1.21)

Rewritten in dimensionalised units, the transverse part of thewave-equation reads:

⎡ ⎣ ∆T+ 2i ⎛ ⎝ μ0ϵω0 ∂ ∂t ±β ∂ ∂z ⎞ ⎠ + (ϵr k02− β2) ⎤ ⎦ ET±= − k02 ϵ0 PT±,

(1.22)

whereas the longitudinal part of the wave-equation reads:

ϵr k02 Ez±= ±iβ  ∇T · ET±

(1.23)

Equations (1.22) and (1.23) represent the transverse and longitudinal part of the wave-equation after applying the slowly varying envelopeand the paraxial approximation, respectively. Taking into account the vertical wave-guiding properties of the double hetero-structure, one can eliminate the dependence of (1.22) and (1.23) on the vertical coordinate y. In order to obtain a 2-D wave-equation, we use the effective index approximation.

In order to eliminate the vertical coordinate, in the paraxial wave equations the following substitution is made [1], [2]:

ET± (x,y,z,t) ⇒ ET±(x,z,t)  Φ(x,y),

(1.24)

PT±(x,y,z,t) ⇒ PT±(x,z,t)  Φ(x,y)  Ψ(y),

(1.25)

where Φ(x,y) is the steady state mode profile in vertical direction and Ψ(y) is of the form:

Ψ(y) = ⎧ ⎪ ⎨ ⎪ ⎩ 1     for    |y| ≤ d/2 0     for    |y| ≥ d/2,

(1.26)

restricting the material polarization in vertical direction to the active area of the device.Inserting this ansatz into (1.22) and neglecting terms including [(∂Φ(x,y))/(∂x)] and [(∂ ²Φ(x,y))/(∂x ²)] by assuming that Φ(x,y) is slowly varying with x one obtains:

Φ ∂2 ∂x 2 ET± + ET± ⎡ ⎣ ∂2 ∂y 2 Φ+ (ϵr k02− β2) Φ ⎤ ⎦ + 2i ⎛ ⎝ μ0ϵω0 ∂ ∂t ±β ∂ ∂z ⎞ ⎠ ET±Φ = − k 0 2 ϵ0 ΦΨPT±. (1.27)

The second term in equation (1.27) vanishes if ϵ_r does not depend on x. This follows from the defining equation of the steady state mode profile [8]:

∂2 ∂y 2 Φ+ (ϵr k02− β2) Φ = 0,

(1.28)

keeping in mind that Φ is the eigenfunction and the effective propagation constant β [3] is the eigenvalue. If the refractive index varies in transverse direction the solution of (1.28) is non-trivial. Here we consider a wave-guide with weak index guiding in transverse direction provided by a ridge along the optical cavity (see Fig. 1.1). We approximate the dielectric constant ϵ_r by:

ϵr(x,y) = ⎧ ⎪ ⎨ ⎪ ⎩ ϵa+ δϵa(x) |y| ≤ d/2 dielectric constant active area ϵc |y| ≥ d/2 dielectric constant cladding layer.

(1.29)

For small variations of ϵ_r in transverse direction Eq. (1.28) can be solved perturbatively [8]:

β2= β02+ (∆β(1))2= β02+ ⌠ ⌡ d/2 −d/2  dy |Φ|2 δϵa k02.

(1.30)

where β₀ is the effective propagation constant calculated using (1.28), neglecting any variation of ϵ_r in transverse direction. In a weakly index-guided structure ( δϵ_a≤ 10 ⁻⁴), the small transverse variation of the dielectric constant justifies neglecting higher order correction terms. Following [5], (1.27) is multiplied by Φ ^∗, the complex conjugate of the mode profile function. Integrating over the vertical coordinate and dividing by ∫ _−∞ ^∞ dy |Φ|² leads to:

⎡ ⎣ ∂2 ∂x 2 + δϵa k02 Γ+ i - ϵ   p  k02 ⎤ ⎦ ET±+2i ⎛ ⎝ μ0ϵ0ϵpω0 ∂ ∂t ±β ∂ ∂z ⎞ ⎠ ET±= − k02 ϵ0   Γ PT±,

(1.31)

where the following definitions have been used:

Γ = ⎛ ⎝ ⌠ ⌡ d/2 −d/2   dy|Φ|2 ⎞ ⎠ ⎛ ⎝ ⌠ ⌡ ∞ −∞  dy |Φ|2 ⎞ ⎠ −1   confinement factor (1.32) ϵp = β2/k02 effective dielectric constant (1.33)

In order to describe wave-guide losses, we add a small complex component to the effective dielectric constant ϵ_p such that: ϵ_p⇒ ϵ_p+i―ϵ_p and: |―ϵ_p| << |ϵ_p|.
In this case the effective attenuation constant can be defined as [6]: α =―ϵ_pβ/ϵ_p. Rewriting (1.31) in standard form, we obtain the 2-D wave-equation:

⎡ ⎣ ∂ ∂t ± β k 0 c ϵp ∂ ∂z ⎤ ⎦ ET±= ⎡ ⎣ i 2 ω0 ⎛ ⎝ c2 ϵp ⎞ ⎠ ∂2 ∂x2 + iω0 2 δϵa ϵp Γ− β k0 c ϵp α 2 ⎤ ⎦ ET±+ iω0 2 Γ ϵ0ϵp PT±.

(1.34)

The terms on the left side of (1.34) describe the forward ′+′ and backward ′−′ propagation of the light fields in the wave-guide cavity. The terms on the right side describe light diffraction, wave-guiding, light absorption and the coupling of the light fields and the material polarisation. To calculate the light fields in the optical cavity, (1.34) can be integrated numerically, provided that the material polarisation P_T ^± is known. A model describing the dynamics of the material polarisation in a semiconductor medium containing quantum dots is presented here.

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