Methodology for materials analysis using swept-frequency feedback interferometry with terahertz frequency quantum cascade lasers

Thomas Taimre; Karl Bertling; Yah Leng Lim; Paul Dean; Dragan Indjin; Aleksandar D. Rakić

doi:10.1364/OE.22.018633

1. Introduction

In [1], we exploited the self-mixing effect in terahertz (THz) quantum cascade lasers (QCLs) to create an ultimately compact and easy to implement materials analysis and coherent imaging scheme for remote targets. Our scheme enables the complex refractive index of targets to be measured by combining a local oscillator, mixer, and the detector all in a single THz QCL through forming a self-mixing interferometer. This approach is non-destructive, and has been demonstrated in the reflection mode, permitting its use on thick samples. Potential applications areas in the THz frequency range include biomedical, pharmaceutical, security, industrial, and cultural heritage [2–16]. Moreover, the self-mixing interferometer at the heart of our scheme has been demonstrated over a wide range of devices and wavelengths, including THz QCLs [1,17–20], mid-infrared QCLs [21–23] and interband cascade lasers [24], and near infrared [25] and visible diode lasers [26]. In this work, we quantify the impact of variations in system parameters on our scheme, evaluating its usefulness for different laser–target conditions. Throughout, we point to the THz QCL as an exemplar device for which we have demonstrated the effectiveness of the scheme. However, the reader should bear in mind that our scheme is not fundamentally linked to the particular emission frequency of the device nor to the particular device type.

Our method for materials analysis involves interrogating an external target by applying a small linear frequency sweep to the laser and measuring the resulting interferometric signal. This signal is imprinted with information about the complex refractive index of the target, which can be retrieved through model fitting and system calibration. There are a number of potential sources of systematic and random error in the scheme, including variations in the laser operating parameters (fluctuations in frequency, relative intensity noise), variations in the target (surface profile and roughness, material homogeneity and granularity), variations in the laser–target system (tilt due to misalignment, vibrations of the target), and variations in system calibration (imperfect calibration standards). In this paper, we first detail how we describe the optical properties of the target and how the interferometric signal resulting from our scheme is modelled. In the process, it becomes clear which characteristics of the interferometric system impact the signal, and the nature by which the optical constants of the target are imprinted on the signal. In addition, we detail the system calibration scheme, and remark on the propagation of calibration errors. Second, we examine how systematic changes to elements of the system manifest as changes in the signal, and identify which of these are dominant. Third, we examine how random fluctuations in the system components affect the interferometric signal, and identify when consideration of these random fluctuations becomes important. Finally, we conclude with a discussion of the implications and applicability of the scheme, drawing on the analysis conducted herein.

2. Background

2.1. Optical properties of the target

We model the optical properties of the target material at a particular spatial point as a (non-magnetic) homogeneous half-space with complex refractive index (using the positive time convention e^+jωt) n̂ = n − jk (with $j \equiv \sqrt{- 1}$ , where n is referred to as the refractive index of the target material, and k is known as its extinction coefficient. Equivalent information is contained in the reflectivity of the material R and the phase-shift incurred by THz radiation on reflection θ_R. The dependence of (R, θ_R) on (n, k) is given through the pair of relations [27]

R = \frac{{(n_{0} - n)}^{2} + k^{2}}{{(n_{0} + n)}^{2} + k^{2}}, θ_{R} = - arctan (\frac{2 n_{0} k}{n_{0}^{2} - n^{2} - k^{2}}),

where n₀ is the refractive index of the incident material (typically air), which we take to be n₀ = 1 from this point onward. Conversely, the dependence of (n, k) on (R, θ_R) is given through the pair of relations [27]

n = \frac{1 - R}{1 + R - 2 \sqrt{R} cos (θ_{R})}, k = \frac{1 \sqrt{R} sin (θ_{R})}{1 + R - 2 \sqrt{R} cos (θ_{R})} .

2.2. Modelling the self-mixing signal

We use a three mirror model to describe the laser system under feedback [28] (see Fig. 1), which is equivalent to the steady-state solution to the well-established model for a laser experiencing optical feedback proposed by Lang and Kobayashi [29]. In this model, only one round-trip in the external cavity is considered (solid arrows in Fig. 1). When the external target is displaced longitudinally, the laser system is swept through a set of compound cavity resonances [19]. The equivalent effect may be obtained by changing the laser frequency, which is accomplished in [1] by applying a linear modulation of the laser driving current. The primary effect of this current sweep is a modulation of both the laser power and the voltage across the laser terminals. The secondary effect, which is of most importance here, is a chirp of the lasing frequency.

Fig. 1 Three mirror model for a laser under feedback. Light recirculates in the laser cavity between mirrors M₀ and M_s — solid arrows in laser cavity of length L. Light exits through M_s, is reflected from the external target M_ext, and is reinjected into the laser cavity — solid arrows in external cavity of length L_ext.

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The total phase delay in the external cavity can be decomposed into the transmission phase delay arising from the round-trip through the cavity and the phase change on reflection from the target, which is material dependent. The second order effect of the linear current sweep is a linear chirp of the lasing frequency (600 MHz on 2.59 THz for a change in driving current of 50 mA in [1]), leading to a linear dependence of transmission phase with time. As a consequence of the slow current (and hence frequency) modulation of the laser, this scheme operates the laser as a swept-frequency continuous-wave homodyning transceiver, similar to self-mixing range finders [30–32]. Due to the slow rate of the sweep, the laser under feedback transitions from one steady-state to another, effectively stepping continuously through a series of operating frequencies, which results in linear dependence of phase in time. Therefore the time-dependent external phase delay (interferometric phase) φ over one frequency modulation period T as a function of time is of the form

φ (t) = φ_{0} + \frac{Φ_{Δ}}{T} t - θ_{R},

where φ₀ is the round-trip transmission phase delay in the external cavity at the start of the frequency sweep, Φ_Δ is the interferometric phase deviation caused by the current (frequency) sweep, and θ_R is the phase change on reflection from the material under test (which is assumed not to change significantly with the small frequency sweep). Clearly, φ is a function of the instantaneous laser frequency, which depends on the level of feedback in the laser system.

According to the Lang and Kobayashi model for a semiconductor laser under optical feedback in a steady state [29], the laser frequency satisfies the phase condition (sometimes called the ‘excess phase equation’)

φ_{S} - φ_{FB} = C sin (φ_{FB} + arctan α),

where φ_FB represents the total external round-trip phase at the perturbed laser frequency, φ_S represents the total external round-trip phase at the solitary laser frequency, C is the feedback parameter that depends on the amount of light coupled back into the laser cavity, and α is the linewidth enhancement factor [33–37]. Note that both the solitary phase φ_S and the resulting feedback phase φ_FB will be of the form in (3) and that we have suppressed the explicit time dependence in (4) for notational simplicity. However, for concreteness, it is useful to think of φ_S as the phase stimulus varying linearly in time according to (3) and of φ_FB as solved from (4) being the resultant feedback phase. Solutions to (4) are not possible in closed form and therefore require numerical solution [38, 39]. The interferometric phase change is directly observable through the change in emitted optical power, or equivalently through the change in voltage across the laser terminals [1, 40]. In our scheme, the linear current sweep leads to a quasi-linear voltage change near threshold. As such, the self-mixing signal embedded in the modulated voltage signal is related to the phase change through

V = V_{0} + β cos (φ_{FB}),

where V is the voltage waveform obtained after the removal of a common reference slope to remove the primary effect of the linear current sweep, V₀ is a dc component of this signal (corresponding to a material-dependent voltage offset from the reference slope), and β is the modulation index [1]. For this modulation scheme, note that V is a function of time through its dependence on the interferometric phase φ_FB (whose explicit time dependence has been suppressed for notational simplicity).

The refractive index n and the extinction coefficient k directly affect the self-mixing voltage in our model through the phase-shift on reflection θ_R in (3), which can be obtained from (1). Moreover, the reflectivity of the target R is directly linked to the model parameters C and α through the definition of the feedback parameter C, given by [36]

C \equiv C (α) = \frac{τ_{ext}}{τ_{L}} κ_{ext} \sqrt{1 + α^{2}},

where τ_ext is the round-trip propagation time in the external cavity, τ_L is the round-trip delay of the solitary laser, and κ_ext is the coupling coefficient that depends on the reflectivity of the emitting mirror of the laser R_s and the reflectivity of the target R (associated with mirrors M_s and M_ext in Fig. 1, respectively), as

κ_{ext} = ε \sqrt{\frac{R}{R_{s}}} (1 - R_{s}) .

Here ε represents the fraction of emitted power that is re-injected into the laser cavity, accounting for power loss on transmission in the external cavity and partial re-injection into the laser cavity. The reflectivity of the external target is usually not known — however we shall later make use of the form of (7) to recover unknown reflectivities using a pair of laser–target system calibration standards.

2.3. Measured reflectivity and phase-shift on reflection proxies

We introduce two observable variables (R^M and $θ_{R}^{M}$ ) which are related to the physical properties of the target (R and θ_R) in a known way. We will term them “proxies” for the actual reflectivity R^A ≡ R (sometimes termed reflectance) and phase-shift on reflection $θ_{R}^{A} \equiv θ_{R}$ , where we have introduced the superscript A to make the distinction of actual target properties clear from measured target properties (which have been designated with the superscript M). From (6) and (7), we take our proxy for $\sqrt{R^{A}}$ as $\sqrt{R^{M}}$ , (theoretically) given by

\sqrt{R^{M}} = \frac{C}{\sqrt{1 + α^{2}}} = ε \frac{(1 - R_{s})}{n_{in} L_{in} \sqrt{R_{s}}} L_{ext} \sqrt{R^{A}},

so that

\sqrt{R^{M}}

is a multiple of

\sqrt{R^{A}}

in the ideal case. For the reflection phase proxy

θ_{R}^{M}

we conveniently choose the actual reflection phase with the reference plane translated to coincide with the plane of the laser exit mirror. Therefore,

θ_{R}^{M}

contains not only the actual reflection phase

θ_{R}^{A}

, but also the phase delay due to the transmission to the target and back. This is in accordance with (3), evaluated at a particular (unperturbed) frequency ν₀ during the frequency sweep:

θ_{R}^{M} = θ_{R}^{A} - \frac{4 π}{c} ν_{0} L_{ext},

so that, ideally,

θ_{R}^{M}

is

θ_{R}^{A}

together with an additive term corresponding to the total phase-shift on transmission. Notice that, ideally, both

\sqrt{R^{M}}

and

θ_{R}^{M}

are linear in

\sqrt{R^{A}}

and

θ_{R}^{A}

, respectively. Of course in practice the ideal case is not achieved, and so we must first calibrate the linear dependence of the reflectivity and phase proxies on their actual counterparts. Our approach to calibration is detailed next in Section 2.4. Following calibration, we will examine precisely how systematic and random changes in laser–target system parameters impact

\sqrt{R^{M}}

and

θ_{R}^{M}

in Sections 3 and 4.

2.4. System calibration

Due to the linear dependence of $\sqrt{R^{M}}$ on $\sqrt{R^{A}}$ in practice, which may result from external reflections other than that from the target (including reflections from the cryostat shield and the window), we write

\sqrt{R^{M}} = a_{R} + b_{R} \sqrt{R^{A}}

where R^A is the actual reflectivity of the material under test, a_R and b_R are unknown parameters to be determined, and R^M = C²/(1 + α²), as defined earlier [see (8)], is representative of the material’s measured, but uncalibrated reflectivity. Along similar lines, to account for systematic phase uncertainties, we express θ_R [see (9)] as

θ_{R}^{M} = a_{θ} + b_{θ} θ_{R}^{A},

where

θ_{R}^{A}

is the actual phase shift on reflection, a_θ and b_θ are unknown parameters to be determined, and

θ_{R}^{M}

(which we call the phase proxy) is representative of the uncalibrated phase shift on reflection with the reference plane translated to the laser exit mirror. Equations (10) and (11) contain four unknown parameters, a_R, b_R, a_θ, and b_θ, which can be determined by model fitting (see Appendix C) from two measurements on materials with known (

\sqrt{R^{A}}

,

θ_{R}^{A}

) values, which can be viewed as a set of four linear equations with four unknowns. Denoting the calibration pairs of measured and actual reflectivities and phase-shifts for our two standards as (

R_{1}^{M}

,

R_{1}^{A}

) (

θ_{R, 1}^{M}

,

θ_{R, 1}^{A}

), and (

R_{2}^{M}

,

R_{2}^{A}

) (

θ_{R, 2}^{M}

,

θ_{R, 2}^{A}

) respectively, the set of four linear equations are

\sqrt{R_{1}^{A}} = a_{R} + b_{R} \sqrt{R_{1}^{M}}, \sqrt{R_{2}^{A}} = a_{R} + b_{R} \sqrt{R_{2}^{M}}, θ_{R, 1}^{A} = a_{θ} + b_{θ} θ_{R, 1}^{M}, θ_{R, 2}^{A} = a_{θ} + b_{θ} θ_{R, 2}^{M} .

The solution of this system of equations is straightforward and provides values for a_R, b_R, a_θ, and b_θ. In particular,

a_{R} = \frac{\sqrt{R_{1}^{A}} \sqrt{R_{2}^{M}} - \sqrt{R_{1}^{M}} \sqrt{R_{2}^{A}}}{\sqrt{R_{1}^{M}} - \sqrt{R_{2}^{M}}}, b_{R} = \frac{\sqrt{R_{1}^{A}} - \sqrt{R_{2}^{M}}}{\sqrt{R_{1}^{M}} - \sqrt{R_{2}^{M}}},

a_{θ} = \frac{θ_{R, 1}^{A} θ_{R, 2}^{M} - θ_{R, 1}^{M} θ_{R, 2}^{A}}{θ_{R, 1}^{M} - θ_{R, 2}^{M}}, b_{θ} = \frac{θ_{R, 1}^{A} - θ_{R, 2}^{A}}{θ_{R, 1}^{M} - θ_{R, 2}^{M}} .

Once these values have been obtained we can readily calculate actual values for

θ_{R}^{A}

and R^A for the material under test using (10) and (11).

It is clear that an error in calibration standard $Δ \sqrt{R^{A}}$ and $Δ θ_{R}^{A}$ will propagate as both an additive error (through the constant calibration terms) and as a multiplicative error (through the linear terms) to each measurement ( $\sqrt{R^{M}}$ or $θ_{R}^{M}$ ). Therefore, for this scheme, it is crucial that the optical constants of calibration standards be known accurately in order to minimise the introduction of calibration error, which would compound other errors present in the system.

3. Impact of systematic variations

In practice, we combine several measurements from the same (homogeneous) target material, either from repeated measurements at the same location on the target, from multiple spatial locations of the same target material, or both. In an ideal situation, all of these measurements would be identical. However, in practice there will be some error leading to differences in recorded results which will need to be accounted for.

There are a number of potential sources of systematic errors which contribute to changes in the self-mixing signal. One of the potential sources in the laser operating parameters is a systematic change in lasing frequency ν (arising for example from phase-instability or large linewidth) from the unperturbed frequency ν₀ (other than the intended change caused by the frequency sweep). Potential sources in target characteristics are systematic local variation in external cavity length due to the target surface profile, and systematic spatial variation in optical constants of the target material (unintentional material inhomogeneity). Another potential source of error in laser–target system characteristics is the tilt of the target, leading to systematic change in external cavity length for different lateral positions of the probing beam on the target.

Changes in external cavity length (ΔL), lasing frequency (Δν), phase-shift on reflection ( $Δ θ_{R}^{A}$ ), and amplitude reflection coefficient of the external target ( $Δ \sqrt{R^{A}}$ ) — all of which can take positive or negative values — enter the excess phase equation as changes to (3) through

φ + Δ φ = \frac{4 π}{c} (ν_{0} + Δ ν) (L_{ext} + Δ L) - (θ_{R}^{A} + θ_{R}^{A}),

as well as interacting nonlinearly with changes in the feedback parameter (6) through the solution of (4) via

C + Δ C = ε \sqrt{1 + α^{2}} \frac{(1 - R_{s})}{n_{in} L_{in} \sqrt{R_{s}}} (L_{ext} + Δ L) (\sqrt{R^{A}} + Δ \sqrt{R^{A}}) .

Note that the approach taken in this analysis is similar to the concept of noise-equivalent phase [25, 41].

From (9) and (15), changes in external cavity length and changes in lasing frequency impact the “measured” phase-shift on reflection (9) through

θ_{R}^{M} = θ_{R}^{A} + Δ θ_{R}^{A} - \frac{4 π}{c} (ν_{0} L_{ext} + ν_{0} Δ L + Δ ν L_{ext} + Δ ν Δ L) .

From (17), we see that changes in external cavity length and changes in lasing frequency affect the proxy for the phase-shift on reflection

θ_{R}^{M}

in the same way. Directly from (8) and (16), these changes also impact the “measured” reflectivity through

\sqrt{R^{M}} = ε \frac{(1 - R_{s})}{n_{in} L_{in} \sqrt{R_{s}}} (L_{ext} + Δ L) (\sqrt{R^{A}} + Δ \sqrt{R^{A}}) .

From (18), we see that changes in external cavity length and changes in reflectivity affect the proxy for the reflectivity

\sqrt{R^{M}}

in the same way.

In practice, several measurements of the target material are acquired across its spatial extent. Clearly, if systematic variations in lasing frequency (the known frequency sweep) and external cavity length (such as tilt of the target surface) can be accounted for (see Appendix A), then the remaining variations in the self-mixing signal can be viewed as a consequence of variations in the reflectivity and phase-shift on reflection of the material (provided the shape of the target surface is known, e.g. it is flat).

Equations (17) and (18) can be rewritten in terms relative to wavelength and frequency. Writing the change in external cavity length in terms of the unperturbed laser wavelength λ₀ (ΔL = γλ₀, so that the fractional change in external cavity length relative to λ₀ is γ = ΔL/λ₀), the external cavity length in terms of the unperturbed laser wavelength (L_ext = Γλ₀), and the change in lasing frequency in terms of the unperturbed lasing frequency ν₀ (Δν = η ν₀, so that the fractional change in frequency is η = Δ ν/ν₀), (17) can be cast as

θ_{R}^{M} = θ_{R}^{A} + Δ θ_{R}^{A} - 4 π {Γ + γ + η (Γ + γ)} .

Therefore, with reference to a particular external cavity length and unperturbed laser wavelength (resulting in a particular phase-shift on transmission 4πΓ), and based on the condition

- π ⩽ θ_{R}^{M} + 4 π Γ - θ_{R}^{A} ⩽ π

, an absolute limit on the ability to uniquely recover the phase-shift on reflection (with respect to a particular phase-shift on transmission 4πΓ) from (19) is

\frac{Δ θ_{R}^{A}}{4 π} = \frac{1}{4} ⩽ γ + η (Γ + γ) ⩽ \frac{Δ θ_{R}^{A}}{4 π} + \frac{1}{4} .

Similarly, (18) can be cast as

\sqrt{R^{M}} = ε \frac{(1 - R_{s})}{n_{in} L_{in} \sqrt{R_{s}}} (Γ + γ) λ_{0} (\sqrt{R^{A}} + Δ \sqrt{R^{A}}) .

When γ ≪ Γ (that is ΔL ≪ L_ext),

\sqrt{R^{M}}

does not change significantly with small changes in the surface profile. Hence,

θ_{R}^{M}

will usually be comparatively more sensitive to such changes, and thereby contribute most to changes in n and k.

Figure 2 shows quantitatively how the systematic changes discussed here are manifested as equivalent changes in amplitude reflection coefficient $Δ \sqrt{R^{A}}$ and phase-shift on reflection $Δ θ_{R}^{A}$ . For example, consider an optically-flat target with uncertainty in external cavity length on the order of λ₀/10 (γ = 0.1). For all semiconductor laser wavelengths, it is possible to polish a target sample to λ₀/10; for terahertz frequencies, this is particularly easy to accomplish. From the diagonal blue γ = 0.1 line and with λ₀ = 100 μm, one obtains uncertainty equivalent to $Δ \sqrt{R^{A}} = 10^{- 4}$ . If one wishes to determine the corresponding uncertainty in $θ_{R}^{A}$ , one examines the horizontal red γ = 0.1 line to find uncertainty equivalent to $Δ θ_{R}^{A} = 4 π \cdot 10^{- 1}$ rad. As such we can confirm that $θ_{R}^{A}$ is more sensitive than R^A to changes in cavity length.

Fig. 2 Systematic sensitivity of the system when η ≈ 0 and ηΓ ≈ 0, in terms of equivalent change in amplitude reflection coefficient $Δ \sqrt{R^{A}}$ and phase-shift on reflection $Δ θ_{R}^{A}$ , for a range of laser wavelengths (frequencies). Blue contours are lines of constant λ₀γ. Red contours are lines of constant γ.

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4. Impact of random variations

There are a number of noise sources that contribute to noise in the self-mixing signal. Potential sources related to operation of the laser are shot-noise associated with the current source, temperature fluctuations which lead to instantaneous frequency deviations, laser intensity and phase noise, and laser sensitivity to fluctuations in the optical feedback as the frequency is swept. Potential noise sources in target characteristics are random local variation in the external cavity length arising from surface profile variation, and random spatial variation in optical constants. Potential noise sources in laser–target system characteristics are fluctuations in external cavity length due to vibrations, and spatial variations in the material properties of the external cavity.

Here, we examine how the distribution of random fluctuations in the measurement variables ( $\sqrt{R^{M}}$ , $θ_{R}^{M}$ ) propagates through to (n, k) parameter space. We assume a bivariate Gaussian error distribution in ( $\sqrt{R^{M}}$ , $θ_{R}^{M}$ ) space, denoted here by

(\begin{matrix} \sqrt{R^{M}} \\ θ_{R}^{M} \end{matrix}) ~ N (μ, Σ),

where

μ = (\begin{matrix} 𝔼 \sqrt{R^{M}} \\ 𝔼 θ_{R}^{M} \end{matrix}), and Σ = (\begin{matrix} Var (\sqrt{R^{M}}) & Cov (\sqrt{R^{M}}, θ_{R}^{M}) \\ Cov (\sqrt{R^{M}}, θ_{R}^{M}) & Var (θ_{R}^{M}) \end{matrix}) .

Note that μ and Σ can be estimated directly from measured data — once the systematic error has been removed. Moreover, we can quantify how random temporal or spatial fluctuations in lasing frequency (such as relative intensity noise), external cavity length (for example due to surface roughness), and material properties (for example due to sample inhomogeneity) impact the covariance of the measurement errors Σ.

We assume that there are fluctuations in lasing frequency ν, surface deviation from known systematic tilt (manifested through L_ext), reflectivity $\sqrt{R^{A}}$ , and phase-shift on reflection $θ_{R}^{A}$ , but all other quantities are fixed. Under the mild assumption of mutual independence of the random variations in ( $θ_{R}^{A}$ ,ν, L_ext) and in ( $\sqrt{R^{A}}$ ,L_ext), meaning in particular that all covariances between the random variations in these quantities are zero, we have expected “measurements” [see also (8) and (9)]

𝔼 \sqrt{R^{M}} = ε \frac{(1 - R_{s})}{n_{in} L_{in} \sqrt{R_{s}}} 𝔼 \sqrt{R^{A}} 𝔼 L_{ext}, 𝔼 θ_{R}^{M} = 𝔼 θ_{R}^{A} - \frac{4 π}{c} 𝔼 ν 𝔼 L_{ext} .

The covariance and variances of the “measurements” are

Var (\sqrt{R^{M}}) = {(ε \frac{(1 - R_{s})}{n_{in} L_{in} \sqrt{R_{s}}})}^{2} (Var(L_{ext}) Var (\sqrt{R^{A}}) + {(𝔼 L_{ext})}^{2} Var (\sqrt{R^{A}}) + {(𝔼 \sqrt{R^{A}})}^{2} Var (L_{ext})),

Var (θ_{R}^{M}) = Var (θ_{M}^{A}) + {(\frac{4 π}{c})}^{2} (Var (L_{ext}) Var (ν) + {(𝔼 L_{ext})}^{2} Var (ν) + {(𝔼 ν)}^{2} Var (L_{ext})),

Cov (\sqrt{R^{M}}, θ_{R}^{M}) = - ε \frac{(1 - R_{s})}{n_{i n} L_{i n} \sqrt{R_{s}}} \frac{4 π}{c} 𝔼 \sqrt{R^{A}} 𝔼 ν Var (L_{e x t}) .

We note particularly that variations in L_ext contribute most to spurious variations in

θ_{R}^{M}

. This can be seen by examining the second and third terms in the parenthesis of Var(

θ_{R}^{M}

), and noting that the coefficient in front of the variance in (𝔼L_ext)²Var(ν) is much smaller in magnitude than that in front of (𝔼ν)²Var(L_ext) for most cases, reinforcing the earlier point that target smoothness is usually much more important than a pure spectrum. Moreover, as L_ext and

\sqrt{R^{A}}

are typically of the similar order, variations in L_ext are as important as variations in

\sqrt{R^{A}}

, once again reinforcing our earlier analysis.

Assuming a calibrated linear laser–target system relationship between a “measured” pair ( $\sqrt{R^{M}}$ , $θ_{R}^{M}$ ) and the actual pair ( $\sqrt{R^{A}}$ , $θ_{R}^{A}$ ) (see Section 2.4), we may write

(\begin{matrix} \sqrt{R^{A}} \\ θ_{R}^{A} \end{matrix}) = A (\begin{matrix} \sqrt{R^{M}} \\ θ_{R}^{M} \end{matrix}) + b ~ N (A μ + b, A Σ A^{T})

where A contains the calibration terms b_R and b_θ on the diagonal, and b contains the calibration terms a_R and a_θ. Then, via the delta method (see e.g. [42]), we may write the (local) distribution in (n, k) space as

(\begin{matrix} n \\ k \end{matrix}) = g (\sqrt{R^{M}}, θ_{R}^{M}) ~_{approx} N (g (A μ + b), J_{g} A Σ A^{T} J_{g}^{T}),

where J_g is the Jacobian of the transformation from

g \equiv (n, k) \mapsto (\sqrt{R^{A}}, θ_{R}^{A})

evaluated at Aμ +b (see Appendix B).

In practice, an average of measurements is used rather than a single data point, so that

(\begin{matrix} \hat{n} \\ \hat{k} \end{matrix}) = g (\bar{\sqrt{R^{M}}}, \bar{θ_{R}^{M}}), \bar{\sqrt{R^{M}}} = \frac{1}{N} \sum_{i = 1}^{N} {\sqrt{R^{M}}}_{i}, \bar{θ_{R}^{M}} = \frac{1}{N} \sum_{i = 1}^{N} θ_{R, i}^{M},

where, by assuming measurement errors are uncorrelated and therefore independent due to the Gaussian model here,

(\begin{matrix} \bar{\sqrt{R^{M}}} \\ \bar{θ_{R}^{M}} \end{matrix}) ~ N (μ, Σ / N) .

Therefore, the error-reduction factor of 1/N will propagate through the linear transformation (and Jacobian) to reduce the corresponding (local) covariance matrices in (

\sqrt{R^{A}}

,

θ_{R}^{A}

) and (n, k) space by a factor of 1/N.

The impact of random variations in optical constants is illustrated in Fig. 3, where lines of constant n (black) and k (red) are plotted for a range of (R^A, $θ_{R}^{A}$ ) pairs. Superimposed on Fig. 3 are the optical constants of several plastics at 2.59 THz, including those used in our study [1]. Particularly evident from Fig. 3 is that, as k decreases, the same variation in $θ_{R}^{A}$ translates to greater uncertainty in k. This means that the technique performs better for lossy materials. On the other hand, over the range of plastics considered here, the same variation in $\sqrt{R^{A}}$ translates to similar levels of uncertainty in n.

Fig. 3 Sensitivity of the system to random variations in ( $\sqrt{R^{A}}$ , $θ_{R}^{A}$ ) for a range of plastics. 95% confidence intervals are drawn for each plastic, with $\sqrt{Var (\sqrt{R^{A}})} \approx 0.00316$ (Var( $\sqrt{R^{A}}$ ) = 10⁻⁵) and $\sqrt{Var (θ_{R}^{A})} \approx 0.00316$ rad (Var( $θ_{R}^{A}$ ) = 10⁻⁵ rad²), at ν = 2.59 THz with $\sqrt{Var (ν)} = 10$ kHz (Var(ν) = 10⁸ Hz²), and an external cavity of L_ext = 500 mm. Black contours are lines of constant n. Red contours are lines of constant k. Plastics at 2.59 THz are POM (dot), PA6 (circle), PVC (×), HDPE (cross), PTFE (star), PMMA (square), HDPE Black (diamond), and PC (triangle).

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Figure 4 shows the sensitivity of swept-frequency self-mixing signals as the external cavity or material properties change, synthesised for a 2.59 THz laser with a periodic sawtooth frequency chirp of 600 MHz and an external cavity length of 500 mm. Panels on the left are error-free self-mixing signals, and panels on the right are instances of the same signals with random error at every time-instant in L_ext, $\sqrt{R^{A}}$ , and $θ_{R}^{A}$ . The waveforms with random error show what one might expect to see in practice using a digital oscilloscope with simulated trace persistence.

Fig. 4 Sensitivity of swept-frequency self-mixing signals to systematic and random changes. (a) Systematic change in external cavity length (ΔL) of 0, 250, and 500 wavelengths on L_ext = 500 mm, for fixed R^A = 0.01 and $θ_{R}^{A}$ = 0.05 rad. (b) Systematic change in R^A from 0.001, 0.01, and 0.1, for fixed L_ext = 500 mm and $θ_{R}^{A}$ = 0.05 rad. (c) Systematic change in $θ_{R}^{A}$ from 0.01 rad, 0.05 rad, and 0.1 rad, for fixed L_ext = 500 mm and R^A = 0.01. (d)–(f) As (a)–(c), with additive zero-mean Gaussian variation in L_ext with standard deviation 10⁻⁴ $\sqrt{λ_{0}}$ m; $\sqrt{R^{A}}$ with standard deviation 10⁻²; and $θ_{R}^{A}$ with standard deviation 10⁻² rad.

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5. Discussion

In [1], we demonstrated an imaging and material analysis scheme based on self-mixing with a THz QCL. There are many possible sources of systematic and random variation which may influence the scheme, arising from the laser operating parameters, the target characteristics, and the laser–target system characteristics. Here, we analysed the impact of such systematic and random errors on our material analysis scheme. As this scheme is coherent, it is not surprising that it is intrinsically most sensitive to fluctuations in interferometric phase. The main contributor to phase fluctuations was seen to be both systematic and random variations in the external cavity length, for example due to variation in surface profile of the remote target, tilt of the target, or mechanical vibration. Nevertheless, effects arising from sample tilt can be corrected for as described in Appendix A. Analysis of random noise sources also indicates that lossy materials lead to smaller experimental uncertainty in k. Moreover, the quality of calibration standards was seen to be crucial for the scheme. Characteristics of lasers ill-suited to our scheme include phase-instability under feedback, high phase and intensity noise, susceptibility to polarisation switching, and shorter wavelengths. As such, the THz QCL is particularly well suited to our scheme due to its narrow linewidth, long wavelength, and remarkable stability under feedback [22].

Appendix A: accounting for systematic variation in the external cavity

If systematic variation in the external cavity (due to alignment or surface profile) is known, then it can clearly be accounted for directly in (17) and (18). On the other hand, if systematic variation in the external cavity is not known, but can be assumed to be of a particular form, then it can be removed from $\sqrt{R^{M}}$ and $θ_{R}^{M}$ by fitting the known (spatial) error description to data in the least-squares sense and accounting for it as above. This approach will be particularly effective if the target (or parts of the target) are of the same (homogeneous) material.

For example, if there is known to be systematic planar tilt over a particular (homogeneous) material patch on the target, then we may fit a plane to that region and remove, by writing

{\sqrt{R^{M}}}_{i} = β_{0} + β_{x} (x_{i} - x_{0}) + β_{y} (y_{i} - y_{0}),

for each observation, where (x_i, y_i) represents the spatial coordinates on the target, (x₀, y₀) represents a particular relative coordinate (such as the center or a corner of the target), and (β₀, β_x, β_y) represents the (unknown) parameters of the plane.

If we arrange N (one-dimensional) measured values into a column vector v — such as “measured” reflectivity —, then we may write the planar model as a (general) linear model v = Aβ +ε, where ε is a zero-mean vector of random errors, and

A = (\begin{array}{l} 1 & x_{1} - x_{0} & y_{1} - y_{0} \\ 1 & x_{2} - x_{0} & y_{2} - y_{0} \\ ⋮ & ⋮ & ⋮ \\ 1 & x_{N} - x_{0} & y_{N} - y_{0} \end{array}), β = (\begin{matrix} β_{0} \\ β_{x} \\ β_{y} \end{matrix}),

where A is the design matrix for the linear model, and β is the vector of unknown parameters. The least-squares solution β̂ can be found by solving the linear system of equations (the normal equations)

(A^{T} A) \hat{β} = A^{T} v .

The tilt-removed values (keeping the constant term) would then be

\tilde{v_{i}} = v_{i} - \hat{β_{x}} (x_{i} - x_{0}) - \hat{β_{y}} (y_{i} - y_{0}), i = 1, \dots, N .

Appendix B: Jacobian of the transformation from (n, k) to ( $\sqrt{R}$ _,θ_R)

The Jacobian of the transformation in (2) from $g \equiv (n, k) \mapsto (\sqrt{R}, θ_{R})$ is given by

J_{g} = (\begin{array}{l} \frac{\partial n}{\partial \sqrt{R}} & \frac{\partial n}{\partial θ_{R}} \\ \frac{\partial k}{\partial \sqrt{R}} & \frac{\partial k}{\partial θ_{R}} \end{array},)

where each element of J_g can be calculated as

\frac{\partial n}{\partial \sqrt{R}} = 2 \frac{(- 2 \sqrt{R} + (1 + R) cos (θ_{R}))}{{(1 + R - 2 \sqrt{R} cos (θ_{R}))}^{2}}, \frac{\partial n}{\partial θ_{R}} = \frac{2 \sqrt{R} (- 1 + R) sin (θ_{R})}{{(1 + R - 2 \sqrt{R} cos (θ_{R}))}^{2}},

\frac{\partial k}{\partial \sqrt{R}} = \frac{2 (- 1 + R) sin (θ_{R})}{{(1 + R - 2 \sqrt{R} cos (θ_{R}))}^{2}}, \frac{\partial k}{\partial θ_{R}} = - 2 \sqrt{R} \frac{(- 2 \sqrt{R} + (1 + R) cos (θ_{R}))}{{(1 + R - 2 \sqrt{R} cos (θ_{R}))}^{2}} .

Appendix C: materials analysis model fitting

Our scheme contains a parametric model, based directly on the steady state solution to the Lang and Kobayashi model, that describes well the set of experimentally acquired time domain traces. Equations (3)–(5) form a model with six key parameters, namely C, α, θ_R, Φ_Δ, V₀, and β. The information about the complex refractive index to be extracted is encoded mainly in C, α, and θ_R.

To extract self-mixing model parameters, we fit the model to data in the least-squares sense, for each spatial pixel of the target. Given an experimental signal (V₁,V₂,...,V_n) at times (t₁,t₂,...,t_n), the goal is to obtain a parameter vector θ^* that explains the systematic variation inherent in the observed self-mixing signal. This is achieved by finding a set of model parameters θ that minimises the (weighted) sum of squared differences between the (discrete) self-mixing signal and the corresponding discretely evaluated model. In other words,

θ^{*} \in \underset{θ \in Θ}{argmin} S (θ), with S (θ) = \sum_{k = 1}^{n} w_{k} {(V_{k} - V (t_{k}; θ))}^{2},

where Θ is the permissible parameter space, and {w_k} are (optional) weights. To proceed, for each candidate parameter vector θ ∈ Θ, we need to solve the excess-phase equation (4), and compute our modelled waveform V in (5) at the collection of time-points {t₁,t₂,...,t_n}. Note that solving (4) requires care [38, 39]. This (weighted) least-squares approach is particularly useful when the acquired self-mixing signal is extremely noisy, as is typically the case with low-reflectivity targets.

Acknowledgments

This research was supported under Australian Research Council’s Discovery Projects funding scheme (DP 120 103703) and the European Cooperation in Science and Technology (COST) Action BM1205. Y.L.L. acknowledges support under the Queensland Government’s Smart Futures Fellowships programme. P.D. acknowledges support from the EPSRC (UK).

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Methodology for materials analysis using swept-frequency feedback interferometry with terahertz frequency quantum cascade lasers

Abstract

1. Introduction

2. Background

2.1. Optical properties of the target

2.2. Modelling the self-mixing signal

2.3. Measured reflectivity and phase-shift on reflection proxies

2.4. System calibration

3. Impact of systematic variations

4. Impact of random variations

5. Discussion

Appendix A: accounting for systematic variation in the external cavity

Appendix B: Jacobian of the transformation from (n, k) to ( $\sqrt{R}$ _,θ_R)

Appendix C: materials analysis model fitting

Acknowledgments

References and links

Cited By

Figures (4)

Equations (39)

Optics Express

Abstract

1. Introduction

2. Background

2.1. Optical properties of the target

2.2. Modelling the self-mixing signal

2.3. Measured reflectivity and phase-shift on reflection proxies

2.4. System calibration

3. Impact of systematic variations

4. Impact of random variations

5. Discussion

Appendix A: accounting for systematic variation in the external cavity

Appendix B: Jacobian of the transformation from (n, k) to ( R,θR)

Appendix C: materials analysis model fitting

Acknowledgments

References and links

Cited By

Figures (4)

Equations (39)

Optics Express

Appendix B: Jacobian of the transformation from (n, k) to ( $\sqrt{R}$ _,θ_R)