We present a stochastic procedure to investigate the correlation spectra of quantum dot superluminescent diodes. The classical electric field of a diode is formed by a polychromatic superposition of many independent stochastic oscillators. Assuming fields with individual carrier frequencies, Lorentzian linewidths and amplitudes we can form any relevant experimental spectrum using a least square fit. This is illustrated for Gaussian and Lorentzian spectra, Voigt profiles and box shapes. Eventually, the procedure is applied to an experimental spectrum of a quantum dot superluminescent diode which determines the first- and second-order temporal correlation functions of the emission. We find good agreement with the experimental data and a quantized treatment. Thus, a superposition of independent stochastic oscillators represents the first- and second-order correlation properties of broadband light emitted by quantum dot superluminescent diodes.
Modern-day optical applications like optical coherence tomography [
Adding a new perspective to the investigation of QDSLDs, we discuss a stochastic model for the emission in this article. Stochastic approaches have long proven to have a wide-ranging field of applications in biology [
The article is organized as follows: the stochastic model of emission spectra is developed in Section 2. It consists of individual classical fields, which are described by a distinct stochastic differential equation. After investigating properties of these fields, relevant spectra are modelled as a superposition. This is applied to a specific experimental spectrum produced by a QDSLD [
The classical electric field of a diode results from a superposition of stochastic fields. Hence, the electric field outside of the diode reads
ε d ( t ) = ∑ j = 1 N ε j ( t ) , (1)
with the number of fields N and ε j ( t ) the j-th complex field amplitude.
An individual classical field ε ( t ) ∈ ℂ is modelled as a complex Ornstein-Uhlenbeck process [
d ε ( t ) = ( i ν 0 − γ ) ε ( t ) d t + D d W ( t ) , (2)
with the carrier frequency ν 0 , the linewidth γ , the diffusion constant D = γ I , the mean intensity of the electric field I = lim t → ∞ 〈 | ε ( t ) | 2 〉 and the complex Wiener noise increment d W ( t ) ∈ ℂ , whose properties are given by 〈 d W ( t ) 〉 = 0 and 〈 | d W ( t ) | 2 〉 = d t .
The stationary first-order temporal correlation function reads [
G s ( 1 ) ( τ ) = l i m t → ∞ 〈 ε * ( t ) ε ( t + τ ) 〉 = I e − γ | τ | − i ν 0 τ . (3)
The spectral power density is given by the Fourier transform
S ( ν ) = 1 2 π ∫ − ∞ ∞ d τ G s ( 1 ) ( τ ) e i ν τ (4)
of (3) in accordance to the Wiener-Khintchine theorem [
S ( ν ) = I 2 π γ ( ν − ν 0 ) 2 + γ 2 , 1 2 π ∫ − ∞ ∞ d ν S ( ν ) = I . (5)
Furthermore, the stationary normalized second-order temporal correlation function is given by the Siegert relation [
g s ( 2 ) ( τ ) = l i m t → ∞ 〈 ε * ( t ) ε * ( t + τ ) ε ( t + τ ) ε ( t ) 〉 〈 ε * ( t ) ε ( t ) 〉 〈 ε * ( t + τ ) ε ( t + τ ) 〉 = 1 + e − 2 γ | τ | . (6)
In addition to analytical results, we perform numerical simulations of (2). In order to obtain an efficient simulation procedure, we separate the rapid oscillating carrier frequency by the transformation ε ( t ) = η ( t ) e − i ν 0 t yielding
d η ( t ) = − γ η ( t ) d t + D d W ( t ) . (7)
As the diffusion constant D is independent of the electric field amplitude η ( t ) , the Euler scheme [
η ( t i + 1 ) = η ( t i ) − γ η ( t i ) Δ t + D Δ W , (8)
with the discrete time step Δ t = t i + 1 − t i and Δ W a complex Gaussian random process with mean 〈 Δ W 〉 = 0 and variance 〈 | Δ W | 2 〉 = Δ t .
The first-order temporal correlation function of (3) is calculated from a sample average over M realizations
G ( 1 ) ( τ ) = 1 M ∑ m = 1 M ( ε ( m ) ( t s ) ) * ε ( m ) ( t s + τ ) (9)
long after the transient regime t s ≫ 1 / γ j . ε ( m ) ( t ) is the m-th realization of the electric field. This result can be used to calculate the spectral power density of the emission using a Fourier transformation.
The determination of the normalized second-order temporal correlation function (6) can be split into two separate calculations. The first-order temporal correlation functions in the denominator can be simulated according to (9), while the second-order correlation function in the numerator can be calculated as
G ( 2 ) ( τ ) = 1 M ∑ m = 1 M | ε ( m ) ( t s + τ ) ε ( m ) ( t s ) | 2 . (10)
Simulation results as well as analytical calculations of the first- and second-order temporal correlation properties of an individual field can be seen in
The simulations show good agreement with the analytical results for the given parameters ( γ = 0.5 THz , I = 1 , ν 0 = 10 THz , Δ t = 0.01 ps , M = 10 4 ).
The emission of a diode (1) is described as the superposition of N independent classical fields with individual linewidths γ j , mean intensities I j and central frequencies ν j . The stationary first-order temporal correlation function reads
G d ( 1 ) ( τ ) = l i m t → ∞ 〈 ε d * ( t ) ε d ( t + τ ) 〉 = l i m t → ∞ ∑ j = 1 N 〈 ε j * ( t ) ε j ( t + τ ) 〉 . (11)
Thus, the spectral power density is the incoherent sum of the individual spectra
S d ( ν ) = ∑ j = 1 N S j ( ν ) . (12)
This model can be used to approximate a wide range of shapes through the adjustment of the 3 N free parameters γ j , I j and ν j in (12) by means of a least square fit, minimizing the error functional
e = ∑ i ( S t ( ν i ) − S d ( ν i ) ) 2 (13)
for a test spectrum S t ( ν ) at discrete frequencies ν i . Examples of interest are given by Gaussian spectra [
S g ( ν ) = 1 σ 2 e − ( ν − ν 0 ) 2 2 σ 2 , (14)
Lorentzian spectra [
S l ( ν ) = 2 π γ ( ν − ν 0 ) 2 + γ 2 , (15)
Voigt profiles [
S v ( ν ) = 1 σ 2 Re { e − z 2 erfc ( − i z ) } , z = ν + i γ σ 2 , (16)
with the complementary error function erfc ( z ) , and bandwidth limited box shapes
S b ( ν ) = { 2 π / γ , for | ν | ≤ γ / 2 , 0, else . (17)
This is illustrated in
1 2 π ∫ − ∞ ∞ d ν S ( ν ) = 1. (18)
Superluminescent diodes are semiconductor-based light sources, which are characterized by spatially directed emission and spectral widths in the THz regime. The experiments with QDSLDs [
Therefore the developed formalism is used to describe the emission of the diode, which is modelled by N = 30 individual oscillators. Using a least square fit (see (13)) to an experimental spectrum S e ( ν ) [
The optical power spectrum emitted by the QDSLD is simulated numerically. For this, the central frequencies ν j , linewidths γ j and mean intensities I j describing the QDSLD emission determined in Sec. 2.4 are used to calculate the individual electric fields ε j ( t ) according to (8). The electric field emitted by the diode ε d ( t ) results as a superposition of the individual field ε j ( t ) according to (1). Subsequently, the stationary first-order temporal correlation function G d ( 1 ) ( τ ) is calculated according to (9) using M = 10 4 realizations of the diode field ε d ( t ) . The spectral power density of the emission S d ( ν ) is determined using a Fourier transformation.
The result of the simulation (see
b = 1 ∫ − ∞ ∞ d ν S 2 ( ν ) . (19)
This yields b d = 4.51 THz , implying a coherence time of τ c,d = 1 / b d = 221.9 fs , which matches the experimental results of b e = 4.29 THz and τ c,e = 233 fs very well.
The method of modelling emission spectra as a superposition of individual oscillators is therefore suitable to describe the first-order temporal correlation properties of QDSLDs.
By extracting appropriate simulation parameters from an experimental spectrum, the electric field emitted by the diode can be simulated numerically and can be used to calculate the stationary first-order temporal correlation function and optical power spectrum of the emission.
In addition to the investigation of the optical power spectrum, the developed formalism can be used to investigate the classical photon statistics of the QDSLD emission. For this, the electric field ε d ( t ) emitted by the diode already calculated in Sec. 3 can be reused. Instead of calculating first-order temporal correlation properties of the field, M realizations of ε d ( t ) are used to calculate the stationary normalized second-order temporal correlation function g d ( 2 ) ( τ ) of the emission according to (6, 9, 10).
The result for the central frequencies ν j , linewidths γ j and mean intensities I j determined in Section 2.4 is illustrated in
In this article, we study a stochastic model to describe experimental emission spectra. These are considered to result as a superposition of individual complex Ornstein-Uhlenbeck processes. The first- and second-order temporal correlation properties of these oscillators are investigated analytically and numerically. We can approximate Gaussian-, Lorentzian-, Voigt- and bandwidth limited spectra by determination of Lorentzian linewidths, carrier frequencies and amplitudes of the individual oscillators using least square fits.
The developed procedure is applied to the emission properties of quantum dot superluminescent diodes. Simulation parameters are extracted from a least square fit to an experimental spectrum [
The stochastic description of QDSLD emission offers a straightforward perspective on the process of light generation inside QDSLDs, describing it as a superposition of individual classical oscillators. More data on the emission characteristics of the constituents of QDSLDs can lead to a better understanding and contribute to the design of new diodes. Furthermore, this approach can be utilized in the investigation of other properties of QDSLDs. As it explains the statistical properties of the electric field emitted by the diode, it can be used in a classical explanation of temperature dependent intensity fluctuation suppression observed by Blazek et al.
We thank Sébastien Blumenstein for the provision of experimental data and Prof. Wolfgang Elsäßer for stimulating discussions.
The authors declare no conflicts of interest regarding the publication of this paper.
Hansmann, K.N. and Walser, R. (2021) Stochastic Simulation of Emission Spectra and Classical Photon Statistics of Quantum Dot Superluminescent Diodes. Journal of Modern Physics, 12, 22-34. https://doi.org/10.4236/jmp.2021.121003
Consider the Ito stochastic differential equation [
d x ( t ) = a ( x ( t ) ) d t + b ( x ( t ) ) d W ( t ) , (20)
with the drift term a ( x ) and the diffusion term b ( x ) . Identifying x ( t 0 ) = x 0 , the formal solution of this equation is given by integration:
x ( t ) = x 0 + ∫ t 0 t d t ′ a ( x ( t ′ ) ) + ∫ t 0 t d W ( t ′ ) b ( x ( t ′ ) ) (21)
The goal of time discrete maps x ( t i + 1 ) = F ( x i ) of stochastic differential equations is the approximation of a solution x ( t ) up to a order of convergence γ . Such a scheme is said to converge strongly with order γ > 0 , if for the final time instant T and N = T / Δ there is a finite ε and Δ 0 > 0 such that [
〈 | x ( T ) − x ( t N ) | 〉 ≤ ε Δ γ (22)
for any time discretization 0 < Δ < Δ 0 . A strong Taylor scheme of order γ can be constructed by considering the Ito-Taylor expansion, which is obtained by continuously applying the integral form of Ito’s formula [
f ( x ( t ) ) = f ( x 0 ) + ∫ t 0 t d t ′ L 0 f ( x ( t ′ ) ) + ∫ t 0 t d W ( t ′ ) L 1 f ( x ( t ′ ) ) , (23)
with L 0 = a ( x ( t ′ ) ) ∂ x + ( 1 / 2 ) b 2 ( x ( t ′ ) ) ∂ x 2 and L 1 = b ( x ( t ′ ) ) ∂ x , to nonconstant terms inside the integrals of the formal solution (21). A criterium [
A strong convergence scheme of order 1/2 is the Euler scheme
x ( t ) = x 0 + a ( x 0 ) ∫ t 0 t d t ′ + b ( x 0 ) ∫ t 0 t d W ( t ′ ) + R . (24)
Discretizing the time steps, discrete map can be developed which yields
x ( t i + 1 ) = x ( t i ) + a ( x ( t i ) ) Δ t + b ( x ( t i ) ) Δ W , (25)
where Δ W is a Gaussian random process with 〈 Δ W 〉 = 0 and 〈 Δ W 2 〉 = Δ t . To expand the Euler scheme to order 1.0 of strong convergence, the double stochastic integral appearing in the remainder R in (24) has to be included, yielding
x ( t ) = x 0 + a ( x 0 ) ∫ t 0 t d t ′ + b ( x 0 ) ∫ t 0 t d W ( t ′ ) + L 1 b ( x 0 ) ∫ t 0 t ∫ t 0 t ′ d W ( t ′ ) d W ( t ″ ) + R . (26)
This is called the Milstein scheme [
∫ t 0 t ∫ t 0 t ′ d W ( t ′ ) d W ( t ″ ) = 1 2 [ ( W ( t ) − W ( t 0 ) ) 2 − ( t − t 0 ) ] , (27)
which leads to the iteration rule for the Milstein method
x ( t i + 1 ) = x ( t i ) + [ a ( x ( t i ) ) − 1 2 b ( x ( t i ) ) ∂ x b ( x ( t i ) ) ] Δ t + b ( x ( t i ) ) Δ W + 1 2 b ( x ( t i ) ) ∂ x b ( x ( t i ) ) Δ W 2 . (28)
With increasing order of convergence γ , the simulation schemes become more complex and include an increasing number of stochastic increments.