Accepted: July 08, 2022 | Published Online: July 11, 2022

Citation: Kidd GE (2022) A DFB Diode Laser Model for Direct Digital Absorption Spectroscopy. Int J Exp Spectroscopic Tech 7:029.

Copyright: © 2022 Kidd GE. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Abstract

A mathematical model of the operational behavior of tunable DFB diode lasers is presented for use in absorption spectroscopy instruments based on computer analysis and control. The laser active region and its behaviour are defined by an electronic model and a thermal model from which the laser intensity and wavenumber are developed as non-linear functions of the power-in, power-radiated-out difference by the modulation current through changes of the index of refraction with temperature and carrier density. The model equations define four essential control variables which permit the accurate determination of the laser intensity function and which permit a linear wavenumber function for the defined non-linear current function. The model also contains many physical and operational coefficients which require only approximate calibration or estimation through the action of the control variables. The accuracy of the model is tested on a DFB diode laser at a wavenumber of 7168.4 (cm^-1).

Index Terms

Tunable laser, IR spectroscopy, IR laser.

Introduction

Instruments using laser absorption spectroscopy can accurately and rapidly measure the concentrations of a number of low molecular weight molecules i.e., H₂O, CO₂, CH₄, N₂O, etc. and biomarker gases methane, acetone, ammonia, etc. Modern DFB diode lasers and ICLs can produce wavelengths in the 1 to 6 um range which covers numerous high line strength values for the named gases. To realize very accurate results in instruments using a synthetic reference, an accurate laser model is required to define the intensity function and the wavenumber function. Direct absorption spectroscopy is based on the equation:

$$I_{DET}\left(i\left(x\right)\right)\text{ = }{I}_{LSR}\left(i\left(x\right)\right)e^{-S\left(v\left(x\right)\right)}\text{ (1)}$$

Where, i(x) is typically a current ramp to modulate the laser radiation wavenumber across the absorption feature (line) of an absorbing gas, I_DET(i(x)) is the detected power function, I_LSR(i(x)) is the laser modulation power output function, $e^{-S\left(v\left(x\right)\right)}$ is the power transmission function and S(v(x)) is the sample absorbance function of the laser wavenumber, v(x), which is modulated by current and temperature. The sample absorbance function [1], which is obtained from the logarithm of EQ.1, is digitally integrated to obtain the mole fraction (MF) of the absorbing gas.

A mole fraction gain error exists for a non-linear v(x) and an offset error is produced by the ln(I_LSR(i(x))) operation. The model developed for direct digital absorption spectroscopy makes v(x) a linear function on x (∆v = constant) and I_LSR(i(x)) is modeled very accurately and divided out of EQ.1 to eliminate the offset error. For the equations, the independent variable, x, is an integer with a range of -x_M ≤ x ≤ x_M (x_M typically 50) and the "(" and ")" brackets are replaced with "[" and "]" brackets in the computer application.

The Basis Equations for the Laser Active Region (LAR)

The LAR basis equations are developed from the electronic model of Figure 1 and the thermal model of Figure 2. For the electronic model, the current and voltage relationships are modeled as a diode. For the thermal model, the LAR is modeled as a thermal resistance and capacitance.

Laser Current: $i\left(x\right)\text{ = }{i}_0+\frac{\Delta i}{2}g\left(x\right)\left(ma\right);$

$$g\left(x\right)\text{ = }\frac{x}{x_M}+g_2{\left(\frac{x}{x_M}\right)}^2+g_3{\left(\frac{x}{x_M}\right)}^3+g_4{\left(\frac{x}{x_M}\right)}^4$$

i₀ is the median laser current; ∆i is the modulation current range; coefficients g₂, g₃ and g₄ are to be determined.

Laser Terminal Voltage (volts):

$$V_D\left(x\right)\text{ = }\frac{T_C}{T_{CO}}{V}_{D0}+\frac{T_C}{T_{CO}}\frac{nk{T}_{CO}}{e}1\text{n}\left(1.0+ag\left(x\right)\right)$$

$$a\text{ = }\frac{\Delta i}{2{\text{i}}_a}\text{ (3)}$$

V_D0 is the median LAR voltage at i(x = 0) = i₀ and T_C = T_C0;

T_C (K) is the controlled laser case temperature [4].

LAR power "IN" minus the median:

$$P_{IN}\left(x\right)\text{ = }i\left(x\right)V_D\left(x\right)-P_{INO}\text{ (mW)}$$

$P_{INO}\text{ = }\frac{T_C}{T_{Ca}}{i}_0{V}_{D0}$ is the median power input at x = 0. (5)

LAR power "OUT" minus the median:

$$P_{OUT}\left(x\right)\text{ = }{I}_L\left(i\left(x\right)\right)-i_{TH}\left.\left(x\right)\right)-I_{L0}\left(\text{mW}\right)\text{ (6)}$$

I_L0 is the median radiation power out at x = 0

LAR threshold current (ma):

$$i_{TH}\left(x\right)\text{ = }\left(i\left(x\right)-i_0\right)\frac{\partial i_{TH}}{\partial i}+i_{TH0}\left(ma\right)\text{ (7)}$$

$$i_{TH0}\text{ = }{i}_{T{H}_{-}CONTROL}+i_{TH0}{}_{{}_{-}CAL}+\frac{\partial i_{TH}}{\partial T}\left(T_C-T_{C0}\right)$$

The median threshold current, $i_{TH0}$, is a control variable as described later.

LAR temperature (K):

$$T_{LAR}\left(x\right)\text{ = }{T}_{LAR0}+\frac{\partial T}{\partial P}\left(P_{IN}\left(x\right)-P_{OUT}\left(x\right)\right);\text{ (8)}$$

T_LAR0 = LAR median temperature;

$\frac{\partial T}{\partial P}$ is the LAR thermal resistance (K/mW).

LAR index of refraction:

$$n_L\left(x\right)\text{ = }{n}_{L0}+\frac{\partial n}{\partial i}\left(i\left(x\right)-i_0\right)+\frac{\partial n}{\partial T}\left(T_{LAR}\left(x\right)-T_{LAR0}\right)\text{ (9)}$$

n_L0 is the LAR median index of refraction; $\frac{\partial n}{\partial i}$ is index of refraction change with current; $\frac{\partial n}{\partial T}$ is index of refraction change with temperature.

Laser external wavenumber:

$$v\left(x\right)\text{ = }\frac{v_L\left(x\right)}{n_L\left(x\right)}\text{ = }\frac{L_0}{L\left(x\right)}\frac{v_{L0}}{n_L\left(x\right)}\left(c{m}^{-1}\right);\text{ (10)}$$

v_L0 is the median internal wavenumber.

LAR length:

$$L\left(x\right)\text{ = }{L}_0\left(1.0+k_T\left(T_{LAR}\left(x\right)-T_{LAR0}\right)\right);\text{ (11)}$$

L₀ is the median LAR length; k_T is the thermal coefficient of expansion of the LAR material.

Laser Power "IN" Operational Equation Development

For the product term in EQ.4, the ln() function is expanded as a power series:

$$i\left(x\right)v\left(x\right)\text{ = }{i}_0\left(1+ag\left(x\right)\right)\frac{T_C}{T_{C0}}{V}_{D0}f\left(x\right)\text{ (12)}$$

$$f\left(x\right)\text{ = 1+}bag\left(x\right)-\frac{b{\left(ag\left(x\right)\right)}^2}{2}+\frac{b{\left(ag\left(x\right)\right)}^3}{3}-\frac{b{\left(ag\left(x\right)\right)}^4}{4}$$

$$b\text{ = }\frac{nk{T}_{C0}}{e{V}_{D0}}$$

On combining EQ.4, EQ.5 and EQ.12, the operational form of the power "IN" equation is:

$${\text{P}}_{\text{IN}}\left(\text{x}\right)\text{ = }\frac{{\text{T}}_{\text{C}}}{{\text{T}}_{\text{C0}}}\frac{\Delta \text{i}}{2}{\text{V}}_{\text{D0}}\left({\text{A}}_{\text{P}}\text{g}\left(\text{x}\right){\text{+B}}_{\text{P}}\text{g}{\left(\text{x}\right)}^{\text{2}}{\text{+C}}_{\text{P}}\text{g}{\left(\text{x}\right)}^{\text{3}}{\text{+D}}_{\text{P}}\text{g}{\left(\text{x}\right)}^{\text{4}}\right)\left(\text{mW}\right)\text{ (13)}$$

$$where{A}_P=(1+b),B_P=\frac{ab}{2};C_P=-\frac{a^{2}b}{6};D_P=\frac{a^{3}b}{12}.$$

Laser Power "OUT" Operational Equation Development

The radiated power or laser intensity (mW) is expressed as a general power series:

$$I_L\left(i\left(x\right)\right)\text{ = }{k}_1\left(i\left(x\right)-i_{TH}\left(x\right)\right)+k_2{\left(i\left(x\right)-i_{TH}\left(x\right)\right)}^2+k_3{\left(i\left(x\right)-i_{TH}\left(x\right)\right)}^3+k_4{\left(i\left(x\right)-i_{TH}\left(x\right)\right)}^4+\mathrm{...}+\text{ (14)}$$

EQ.14 has many unknown coefficients which are extremely difficult to calibrate beyond k2. With no loss of definition, EQ.14 can be greatly simplified by assuming:

$$k_3\text{ = }\frac{k_2{}^2}{k_1}\text{ and }{k}_4\text{ = }\frac{k_2{}^3}{k_1{}^2}\text{ , etc}\text{. (15)}$$

After substitution of EQ.15 into EQ.14 and simplifying, the laser power "OUT" or intensity function is expressed as:

$$I_L\left(i\left(x\right)\right)\text{ = }\frac{k_1\left(i_O-i_{THO}\right)\left(1.0+hg\left(x\right)\right)}{1.0-\frac{k_2}{k_1}\left(i_O-i_{THO}\right)\left(1.0+hg\left(x\right)\right)}\left(ma\right);$$

$$h\text{ = }\frac{\frac{\Delta i}{2}\left(1.0-\frac{\partial i_{TH}}{\partial i}\right)}{\left(i_O-i_{THO}\right)}\text{ (16)}$$

The coefficients k₁, k₂ and $\frac{\partial i_{TH}}{\partial i}$ require calibration but not accurately. EQ.16 is divided into EQ.1, to eliminate the offset error in the mole fraction estimates. The operational form of the power "OUT" equation is given as:

$$P_{OUT}\left(x\right)\text{ = }\frac{k_1\left(i_O-i_{THO}\right)\left(1.0+hg\left(x\right)\right)}{1.0-\frac{k_2}{k_1}\left(i_O-i_{THO}\right)\left(1.0+hg\left(x\right)\right)}-\frac{k_1\left(i_O-i_{THO}\right)}{1.0-\frac{k_2}{k_1}\left(i_O-i_{THO}\right)}\text{ (17)}$$

The power "OUT" equation, EQ.17, must be expanded into its power series form to be combined with the power "IN" equation, EQ.13 and substituted into EQ.6. This form is expressed as:

$$P_{OUT}\left(x\right)\text{ = }\frac{{\text{T}}_{\text{C}}}{{\text{T}}_{\text{C0}}}\frac{\Delta \text{i}}{2}{\text{V}}_{\text{D}0}\left({\text{A}}_{\text{I}}g\left(\text{x}\right)+{\text{B}}_{\text{I}}g{\left(\text{x}\right)}^2+{\text{C}}_{\text{I}}g{\left(\text{x}\right)}^3+{\text{D}}_{\text{I}}g{\left(\text{x}\right)}^4\right)\text{ (18)}$$

Where

$$A_1\text{ = }\frac{k_1}{\frac{T_C}{T_{CO}}{V}_{DO}}c\left(1+2\frac{k_2}{k_1}d+3{\left(\frac{k_2}{k_1}\right)}^2{d}^2+4{\left(\frac{k_2}{k_1}\right)}^3{d}^3\right)$$

$$B_1\text{ = }\frac{k_1}{\frac{T_C}{T_{CO}}{V}_{DO}}{c}^2\left(\frac{k_2}{k_1}+3{\left(\frac{k_2}{k_1}\right)}^{2}d+6{\left(\frac{k_2}{k_1}\right)}^3{d}^2\right)\frac{\Delta i}{2}$$

$$C_1\text{ = }\frac{k_1}{\frac{T_C}{T_{CO}}{V}_{DO}}{c}^3\left({\left(\frac{k_2}{k_1}\right)}^2+4{\left(\frac{k_2}{k_1}\right)}^{3}d\right){\left(\frac{\Delta i}{2}\right)}^2$$

$$D_1\text{ = }\frac{k_1}{\frac{T_C}{T_{CO}}{V}_{DO}}{c}^4{\left(\frac{k_2}{k_1}\right)}^3{\left(\frac{\Delta i}{2}\right)}^3$$

$$c\text{ = }\left(1.0-\frac{\partial i_{TH}}{\partial i}\right);d\text{ = }{i}_0-i_{TH0}$$

LAR Operational Temperature Equation Development

The LAR temperature equation is now given as:

$$T_{LAR}\left(x\right)-T_{LAR0}\text{ = }\frac{\partial T}{\partial P}\frac{{\text{T}}_{\text{C}}}{{\text{T}}_{\text{C0}}}\frac{\Delta i}{2}{\text{V}}_{\text{D0}}\left(\text{Ag}\left(\text{x}\right)+\text{Bg}{\left(\text{x}\right)}^2+\text{Cg}{\left(\text{x}\right)}^3+\text{Dg}{\left(\text{x}\right)}^4\right)\left(\text{K}\right)\text{ (19)}$$

Where A = A_P-A_I, B = B_P-B_I, C = C_P-C_I and D = D_P-D_I.

Laser Wavenumber Operational Equation Development

After combining the power "IN" and power "OUT" equations and the LAR temperature equation, EQ.8, with the LAR index of refraction equation, EQ.9, the operational form of the laser external wavenumber equation is expressed as:

$$\begin{array}{l}v\left(x\right)\text{ = }\frac{L_0}{L\left(x\right)}\frac{v_{L0}}{\left(n_{L0}+\frac{\partial n}{\partial T}\left({\text{T}}_{\text{LAR}}\left(\text{x}\right)-{\text{T}}_{\text{LAR0}}\right)+\frac{\partial n}{\partial i_{RI}}\left(i\left(x\right)-i_0\right)\right)}\text{ (20)}\\ \text{ = }\frac{L_0}{L\left(x\right)}\frac{v_0}{\left(1+\frac{1}{n_{L0}}\frac{\partial n}{\partial T}\left({\text{T}}_{\text{LAR}}\left(\text{x}\right)-{\text{T}}_{\text{LAR0}}\right)+\frac{1}{n_{L0}}\frac{\partial n}{\partial i_{RI}}\frac{\Delta i}{2}g\left(x\right)\right)}\\ \text{ = }\frac{v_0}{\left(1+k_T\left({\text{T}}_{\text{LAR}}\left(x\right)-{\text{T}}_{\text{LAR0}}\right)\right)f\left(x\right)}\end{array}$$

$$f\left(x\right)\text{ = 1+}\frac{1}{n_{L0}}\frac{\partial n}{\partial T}\left({\text{T}}_{\text{LAR}}\left(\text{x}\right)-{\text{T}}_{\text{LAR0}}\right)+\frac{1}{n_{L0}}\frac{\partial n}{\partial i_{RI}}\frac{\Delta i}{2}g\left(x\right)$$

For a buried heterostructure laser, the refractive index is proportional to the net electron density or current, which is the drive current less the leakage current. The model described here considers the leakage current to be negligible. The linear wavenumber method (LWF) [2] requires a function of the form $A^{\ast }g\left(x\right)+B^{\ast }g{\left(x\right)}^2+C^{\ast }g{\left(x\right)}^3+D^{\ast }g{\left(x\right)}^4$. Therefore, the denominator terms must be converted to their inverted power series equivalents and multiplied together. The LWF equation for is given as:

$$v\left(x\right)\text{ = }{v}_0-\left(v_0{k}_T+\frac{v_0}{n_{L0}}\frac{\partial n}{\partial T}\right)\left({\text{T}}_{\text{LAR}}\left(\text{x}\right)-{\text{T}}_{\text{LAR0}}\right)-\frac{v_0}{n_{L0}}\frac{\partial n}{\partial i_{RI}}\frac{\Delta i}{2}g\left(x\right)\text{ (21)}$$

Or

$$\text{v}\left(\text{x}\right)\text{ = }{v}_0+{\text{A}}^{\text{*}}\text{g}\left(\text{x}\right){\text{+B}}^{\text{*}}\text{g}{\left(\text{x}\right)}^{\text{2}}{\text{+C}}^{\text{*}}\text{g}{\left(\text{x}\right)}^{\text{3}}{\text{+D}}^{\text{*}}\text{g}{\left(\text{x}\right)}^{\text{4}}$$

Where

$$A^{\ast }\text{ = -}\left({\text{k}}_T{c}_1+\frac{1}{n_{L0}}\frac{\partial n}{\partial T}{c}_1+\frac{1}{n_{L0}}\frac{\partial n}{\partial i_{RI}}{c}_2\right)\frac{\Delta i}{2}A{v}_0$$

$$B^{\ast }\text{ = -}\left({\text{k}}_{\text{T}}{c}_1+\frac{1}{n_{L0}}\frac{\partial n}{\partial T}{c}_1\right)\frac{\Delta i}{2}B{v}_0$$

$$C^{\ast }\text{ = -}\left({\text{k}}_{\text{T}}{c}_1+\frac{1}{n_{L0}}\frac{\partial n}{\partial T}{c}_1\right)\frac{\Delta i}{2}C{v}_0$$

$$D^{\ast }\text{ = -}\left({\text{k}}_{\text{T}}{c}_1+\frac{1}{n_{L0}}\frac{\partial n}{\partial T}{c}_1\right)\frac{\Delta i}{2}D{v}_0$$

$$c_1\text{ = }\frac{\partial T}{\partial P}\frac{T_C}{T_{C0}}{V}_{D0,}{c}_2\text{ = }\frac{1.0}{A}$$

In EQ.21, the coefficients $k_T,\frac{1}{n_{L0}}\frac{\partial n}{\partial T}$ and $\frac{1}{n_{L0}}\frac{\partial n}{\partial i_{RI}}$ are sufficiently small to exclude product and higher order terms. Clearly, if g(x) is linear, v(x) is highly non-linear.

Linear wavenumber method (LWF) solution for g₂, g₃ and g₄

The LWF method is duplicated from [2]. Expanding A^*g(x), B^*g(x)², C^*g(x)³ and D^*g(x)⁴ and using only terms to the fourth power, the equations are expressed as

$$A^{\ast }g\left(x\right)\text{ = }{A}^{\ast }\left(\frac{x}{x_M}+g_2{\left(\frac{x}{x_M}\right)}^2+g_3{\left(\frac{x}{x_M}\right)}^3+g_4{\left(\frac{x}{x_M}\right)}^4\right)$$

$$B^{\ast }g{\left(x\right)}^2\text{ = }{B}^{\ast }\left({\left(\frac{x}{x_M}\right)}^2+2g_2{\left(\frac{x}{x_M}\right)}^3+\left(2g_3+g_2^2\right){\left(\frac{x}{x_M}\right)}^4\right)$$

$$C^{\ast }g{\left(x\right)}^3\text{ = }{C}^{\ast }\left({\left(\frac{x}{x_M}\right)}^3+3g_2{\left(\frac{x}{x_M}\right)}^4\right)$$

$$D^{\ast }g{\left(x\right)}^4\text{ = }{D}^{\ast }{\left(\frac{x}{x_M}\right)}^4$$

Next, the following three equations are derived to eliminate the (x/x_M)², (x/x_M)³ and (x/x_M)⁴ terms:

$$A^{\ast }{g}_2+B^{*}\text{ = 0; }{A}^{\ast }{g}_3+2B^{*}{g}_2+C^{\ast }\text{ = 0}$$

$$A^{\ast }{g}_4+B^{*}\left(2g_3+g_2^2\right)+3C^{\ast }{g}_2+D^{*}\text{ = 0}$$

Solving for the g₂, g₃ and g₄ coefficients, their values are defined as:

$$g_2\text{ = -}\frac{B^{*}}{A^{\ast }}$$

$$g_3\text{ = 2}{\left(\frac{B^{*}}{A^{\ast }}\right)}^2-\frac{C^{\ast }}{A^{*}}\text{ (22)}$$

$$g_4\text{ = -5}{\left(\frac{B^{*}}{A^{\ast }}\right)}^3+5\frac{B^{*}}{A^{\ast }}\frac{C^{\ast }}{A^{*}}-\frac{D^{\ast }}{A^{*}}$$

With the above solutions for the coefficients g₂, g₃ and g₄, the laser wavenumber expression becomes linear and is expressed

$$\text{v}\left(\text{x}\right){\text{ = v}}_0+{\text{A}}^{*}\frac{\text{x}}{{\text{x}}_{\text{M}}}\left({\text{cm}}^{-1}\right)$$

Laser Calibration Coefficients and Control Variables

The laser coefficients used in the model are listed in Table 1. The laser coefficients require a non-accurate calibration due to the action of the control variables through digital control loops. Coefficients 1-7 are proprietary to the laser manufacturer but can be estimated from the properties of similar materials. Coefficients 8-18 can be calibrated using simple methods. The control variables are listed in Table 2. The control loops for i₀ (software control) and T_C (electronic control) keep the laser median wavenumber aligned to the centre wavenumber of the absorption line. The control loop (software control) for ∆i [5] maintains the laser wavenumber modulation width, 2A*, equal to the reference wavenumber width, ∆v_R. The control loop for i_TH0 maintains the correct shape of I_L(i(x)) and minimizes the mole fraction offset error.

Model Testing

The model was tested on a Laser Components DFB diode laser for the HITRAN [3] H₂O line at ν₀ = 7168.4 (cm^-1). The calibration values of the control variables and coefficients for operation at room temperature and pressure are listed in Table 3.

Table 3: Control variable and coefficient values for the test DFB diode laser

To test the accuracy of the linear wavenumber part of the model, the function $\frac{-1\text{n}\left(I_D\left(i\left(x\right)\right)/\left(I_L\left(i\left(x\right)\right)\right)\right)}{R\left(x\right)}$ is used where the reference absorbance function, which is a Lorentz function [1] at atmospheric pressure, is defined as:

$$\text{R}\left(\text{x}\right)\text{ = }\frac{\text{SNL}\ast {\text{MF}}_{\text{R}}}{{\text{πγ}}_{\text{R}}}\frac{1}{1+{\left(\frac{\Delta {\text{v}}_{\text{R}}\text{x}}{2{\text{γ}}_{\text{R}}{\text{x}}_{\text{M}}}\right)}^2}\text{ (23)}$$

Wherein, Δv_R ia a software control variable from which Δi is calculated (see Appendix 'B'). The numerator of the test function is essentially S(x) + constant where S(x) is the sample absorbance function which should have the same form as R(x) but with the actual values of HITRAN and laser parameters. The constant comes from the ln(detector gain) and ln(amplifier gain). Figure 1 presents a plot of S(x) (S(x) = (S(x) + S(-x))/2-constant to align S(0) = R(0)) and R(x) and the test function after calibration. Clearly, S(x) equals R(x) in shape and magnitude. Therefore, the laser wavenumber function is highly linear. The R(x) function is synthesized using the HITRAN 2008 [3] data base 7168.4 (cm^-1) line parameters. To test the intensity compensation function, I_L(x), the laser temperature is set to a value of 22.9 to shift the median wavenumber by about -0.85 cm^-1 and obtain a detector signal with low absorption.

The threshold current value was adjusted by (0.4*(22.9-21.18) = 0.688 ma). A screen dump of the LWF graphic is illustrated in Figure 2. With no absorption, the left side of EQ.1 divided by I_L(i(x)) becomes almost a straight line and therefore the intensity is highly accurate. The end anomalies in Figure 1 and Figure 2 are artifacts of the temperature transient after current zeroing before the current ramp.

For Figure 1, the open path sampling head was placed in a closed volume which contained a beaker of H₂O. The sample temperature was used to calculate the saturated vapour pressure which was used in the R(x) equation. The equation coefficients are calculated using the laser parameter values from Table 3 and listed in Table 4.

Control Algorithms

After taking the logarithm of EQ.1 divided by EQ.16, the sample absorbance function (SAF) is obtained through two steps defined as:

$$SAF\left(x\right)\text{ = }S\left(v\left(x\right)\right)-\frac{1}{2}\left(ln\left(\frac{I_{LSR}\left(i\left(x\right)\right)}{I_{REF\left(i\left(x\right)\right)}}\right)+ln\left(\frac{I_{LSR}\left(i\left(-x\right)\right)}{I_{REF}\left(i\left(-x\right)\right)}\right)\right)\text{ (24)}$$

$$SAF\left(x\right)\text{ = }SAF\left(x\right)-SAF\left(x_M\right)$$

This formulation uses the forward/reverse average to make the ln() term symmetrical about x = 0 and the function is end zeroed, which also removes constant values from the ln() operation. The third term of EQ.24 is in general non-linear and of class parabolic. S(v(x)) is already symmetrical about x = 0 as the laser wavenumber is controlled to be linear. The reference absorbance function is calculated as:

$$\text{RAF}\left(\text{x}\right)\text{ = R}\left(\text{v}\left(\text{x}\right)\right)\text{-R}\left(\text{v}\left({\text{x}}_{\text{M}}\right)\right)\text{ (25)}$$

In general, the discrete (Fourier) cosine transform (DCT) [6] is expressed as;

$$\text{DCT}\left(\text{y}\left(\text{x}\right)\right)\text{ = Y}\left[\text{k}\right]$$

$$\text{Y}\left[\text{k}\right]\text{ = }\frac{\text{1}}{{\text{2x}}_{\text{M}}}{\displaystyle \sum _{{\text{-x}}_{\text{M}}}^{{\text{x}}_{\text{M}}}\text{y}\left(\text{x}\right)}\text{cos}\left(\text{π}\left(\text{k-}\frac{\text{1}}{\text{2}}\right)\frac{\text{x}}{{\text{x}}_{\text{M}}}\right)$$

$$\text{k = 1,2,3,,, (26)}$$

The Fourier set defined by $cos\left(\pi \left(k-\frac{1}{2}\right)\frac{x}{x_M}\right)$ are orthogonal over -x_M ≤ x ≤ x_M and symmetrical about x = 0. Also, since SAF(x) and RAF(x) have a zero value at x = ±x_M and they are of class Lorentz, their DCTs are exponential functions. They are expressed as:

$$SAF\left[k\right]\text{ = }{A}_S{e}^{\text{-2π}\left(k-1/2\right)\frac{{\gamma }_S}{\Delta v_S}}+\text{IERR}\left[\text{k}\right]$$

$$\text{k = 1,2,3,,,}$$

$$SAF\left[k\right]\text{ = }{A}_S{e}^{\text{-2π}\left(k-1/2\right)\frac{{\gamma }_S}{\Delta v_S}}+\text{IERR}\left[\text{k}\right]$$

$$\text{RAF}\left[\text{k}\right]{\text{ = A}}_{\text{R}}{\text{e}}^{\text{-2π}\left(\text{k-}\text{1}/\text{2}\right)\frac{{\text{γ}}_{\text{R}}}{{\text{Δv}}_{\text{R}}}}\text{ (28)}$$

Where IERR[k] is the DCT of the third term in EQ.24. Typically, the laser modulation range, Δv_S, is chosen so that S(Δv_S/2) is close to a zero value. Next the ratio of SAF[k] to RAF[k] is given as

$$:\frac{SAF\left[k\right]}{RAF\left[\text{k}\right]}\text{ = }\frac{A_S}{A_R}{e}^{\text{-2}\pi \left(k\text{-}\text{1}/\text{2}\right)\left(\frac{{\gamma }_S}{\Delta v_S}-\frac{{\gamma }_R}{\Delta v_R}\right)}+\frac{\text{IERR}\left[\text{k}\right]}{\text{RAF}\left[\text{k}\right]}$$

$$\text{k = 1,2,3,,, (29)}$$

The intensity error function, IERR[k]/RAF[k], although not formally defined shows a distinct pattern when I_LSR(i(x))≠I_L(i(x)). This error pattern, as illustrated in Figure 3, shows an error which alternates in sign with k and with amplitude which is proportional to the magnitude of the third term of EQ.24 and which decreases in magnitude with k.

The distinct error pattern for the second term of EQ.24 is a linear function of k with a slope proportional to $\left(\frac{{\gamma }_S}{\Delta v_S}-\frac{{\gamma }_R}{\Delta v_R}\right)$ as illustrated in Figure 4.

EQ.29 establishes the sum of both error functions but the intensity errors are typically the smallest in magnitude. The calibration control loop manages the shape of the reference absorbance function and uses Δv_R as the control variable through the equation given as:

$${\text{Δv}}_{\text{R}}{\text{ = Δv}}_{\text{R0}}+\text{GAIN}\left({\displaystyle \sum _{\text{k = 1}}^{\text{K}}\frac{\text{SAF}\left[\text{k}\right]}{\text{RAF}\left[\text{k}\right]}-{\displaystyle \sum _{\text{k = 1}}^{\text{K}}\frac{\text{SAF}\left[\text{k+K}\right]}{\text{RAF}\left[\text{k+K}\right]}}}\right)\text{ (30)}$$

EQ.30 continuously adjusts the value of Δv_R to keep $\left(\frac{{\gamma }_S}{\Delta v_S}\text{ = }\frac{{\gamma }_R}{\Delta v_R}\right)$. This form of EQ.30 produces negligible error when are non-zero. The reference intensity control loop manages the shape of the reference intensity function and uses i_TH0 as the control variable and its control equation is given as:

$$i_{TH0}\text{ = }{i}_{TH0\_CAL}+GAIN\left({\displaystyle \sum _{\text{k = 1}}^{\text{K}}\frac{\text{SAF}\left[\text{2k-1}\right]}{\text{RAF}\left[\text{2k-1}\right]}-{\displaystyle \sum _{\text{k = 1}}^{\text{K}}\frac{\text{SAF}\left[\text{2k}\right]}{\text{RAF}\left[\text{2k}\right]}}}\right)-E1\text{ (31)}$$

$$E1\text{ = }\frac{1}{K}GAIN\left({\displaystyle \sum _{\text{k = 1}}^{\text{K}}\frac{\text{SAF}\left[\text{k}\right]}{\text{RAF}\left[\text{k}\right]}-{\displaystyle \sum _{\text{k = 1}}^{\text{K}}\frac{\text{SAF}\left[\text{k+K}\right]}{\text{RAF}\left[\text{k+K}\right]}}}\right)\text{ (32)}$$

EQ.31 continuously adjusts i_TH0 to manage the shape of the reference intensity function. Typically a value of K = 4 gives very good results. The parameter, GAIN, is set to give stable loop operation. E1 corrects for slope errors when $\frac{{\gamma }_S}{\Delta v_S}\ne \frac{{\gamma }_R}{\Delta v_R}$. The control variables for both loops can be low pass filtered to increase loop stability and reduce effects from laser noise.

Conclusions

A model of a DFB laser has been presented which accurately describes the laser intensity function and produces a linear wavenumber function for a non-linear current function to minimize gain and offset errors in the mole fraction estimates in tunable diode laser absorption instruments based on digital computing.

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