Methodology for materials analysis using swept-frequency feedback interferometry with terahertz frequency quantum cascade lasers Thomas Taimre,1 Karl Bertling,2 Yah Leng Lim,2 Paul Dean,3 Dragan Indjin,3 and Aleksandar D. Raki´c2,∗ 1 School
of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia 2 School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia 3 School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK ∗ [email protected]
Abstract: Recently, we demonstrated an interferometric materials analysis scheme at terahertz frequencies based on the self-mixing effect in terahertz quantum cascade lasers. Here, we examine the impact of variations in laser operating parameters, target characteristics, laser–target system properties, and the quality calibration standards on our scheme. We show that our coherent scheme is intrinsically most sensitive to fluctuations in interferometric phase, arising primarily from variations in external cavity length. Moreover we demonstrate that the smallest experimental uncertainties in the determination of extinction coefficients are expected for lossy materials. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (140.3430) Laser theory; (140.5965) Semiconductor lasers, quantum cascade; (120.4530) Optical constants.
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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.
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18635
Fig. 1. Three mirror model for a laser under feedback. Light recirculates in the laser cavity between mirrors M0 and Ms — solid arrows in laser cavity of length L. Light exits through Ms , is reflected from the external target Mext , and is reinjected into the laser cavity — solid arrows in external cavity of length Lext .
2. 2.1.
Background Optical properties of the target
We model the optical properties of the target material at a particular spatial point as a (nonmagnetic) homogeneous half-space with √ complex refractive index (using the positive time convention e+jωt ) nb = n − j k (with j ≡ −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 θR . The dependence of (R, θR ) on (n, k) is given through the pair of relations [27] 2n0 k (n0 − n)2 + k2 R= , θ = − arctan , (1) R (n0 + n)2 + k2 n20 − n2 − k2 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, θR ) is given through the pair of relations [27] √ 1−R 2 R sin (θR ) √ √ n= , k= . (2) 1 + R − 2 R cos (θR ) 1 + R − 2 R cos (θR ) 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. 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 #213350 - $15.00 USD (C) 2014 OSA
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18636
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 − θR , (3) ϕ(t) = ϕ0 + T where ϕ0 is the round-trip transmission phase delay in the external cavity at the start of the frequency sweep, Φ∆ is the interferometric phase deviation caused by the current (frequency) sweep, and θR 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’) ϕS − ϕFB = C sin(ϕFB + arctan α) ,
(4)
where ϕFB represents the total external round-trip phase at the perturbed laser frequency, ϕS 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 ϕS and the resulting feedback phase ϕFB 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 ϕS as the phase stimulus varying linearly in time according to (3) and of ϕFB 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 = V0 + β cos (ϕFB ) ,
(5)
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 ϕFB (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 θR 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] p τext C ≡ C(α) = κext 1 + α 2 , (6) τL where τext is the round-trip propagation time in the external cavity, τL is the round-trip delay of the solitary laser, and κext is the coupling coefficient that depends on the reflectivity of the
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emitting mirror of the laser Rs and the reflectivity of the target R (associated with mirrors Ms and Mext in Fig. 1, respectively), as r R κext = ε (1 − Rs ) . (7) Rs 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 (RM and θRM ) which are related to the physical properties of the target (R and θR ) in a known way. We will term them “proxies” for the actual reflectivity RA ≡ R (sometimes termed reflectance) and phase-shift on reflection θRA ≡ θR , 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 RA as RM , (theoretically) given by √ √ C (1 − Rs ) √ Lext RA , RM = √ =ε (8) nin Lin Rs 1 + α2 √ √ so that RM is a multiple of RA in the ideal case. For the reflection phase proxy θRM we conveniently choose the actual reflection phase with the reference plane translated to coincide with the plane of the laser exit mirror. Therefore, θRM contains not only the actual reflection phase θRA , 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: 4π θRM = θRA − ν0 Lext , (9) c so that, ideally, θRM is θRA together with an√additive term corresponding √ to the total phase-shift on transmission. Notice that, ideally, both RM and θRM are linear in RA and θRA , 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 RM and θRM in Sections 3 and 4. 2.4.
System calibration
√ √ Due to the linear dependence of RM on RA 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 √ √ RM = aR + bR RA , (10) where RA is the actual reflectivity of the material under test, aR and bR are unknown parameters to be determined, and RM = C2 /(1 + α 2 ), as defined earlier [see (8)], is representative of the material’s measured, but uncalibrated reflectivity. Along similar lines, to account for systematic
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18638
phase uncertainties, we express θR [see (9)] as θRM = aθ + bθ θRA ,
(11)
where θRA is the actual phase shift on reflection, aθ and bθ are unknown parameters to be determined, and θRM (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, aR , bR , aθ , and bθ , which can be determined by √ model fitting (see Appendix C) from two measurements on materials with known ( RA , θRA ) 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 A M A M A M A (RM 1 , R1 ) (θR,1 , θR,1 ), and (R2 , R2 ) (θR,2 , θR,2 ) respectively, the set of four linear equations are q q q q A M A M A = a +b , RA1 = aR + bR RM R RM R R 1 2 2 , θR,1 = aθ + bθ θR,1 , θR,2 = aθ + bθ θR,2 . (12) The solution of this system of equations is straightforward and provides values for aR , bR , aθ , and bθ . In particular, q q q q q q RA1 RM RM RA2 RA1 − RA2 2 − 1 q q q aR = , bR = q , (13) M M− M − RM R R R 1 2 1 2 aθ =
A θM −θM θA θR,1 R,2 R,1 R,2 M −θM θR,1 R,2
,
bθ =
A −θA θR,1 R,2 M −θM θR,1 R,2
.
(14)
Once these values have been obtained we can readily calculate actual values for θRA and RA for the material under test using (10) and (11). √ It is clear that an error in calibration standard ∆ RA and ∆θRA will propagate as both an additive error (through the constant calibration terms) and as a multiplicative error (through √ the linear terms) to each measurement ( RM or θRM ). 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.
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Changes in external cavity length (∆L), lasing frequency (∆ν),√phase-shift on reflection (∆θRA ), and amplitude reflection coefficient of the external target (∆ RA ) — all of which can take positive or negative values — enter the excess phase equation as changes to (3) through ϕ + ∆ϕ =
4π (ν0 + ∆ν) (Lext + ∆L) − θRA + ∆θRA , c
(15)
as well as interacting nonlinearly with changes in the feedback parameter (6) through the solution of (4) via C + ∆C = ε
√ p √ (1 − Rs ) √ (Lext + ∆L) 1 + α2 RA + ∆ RA . nin Lin Rs
(16)
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 θRM = θRA + ∆θRA −
4π (ν0 Lext + ν0 ∆L + ∆νLext + ∆ν∆L) . c
(17)
From (17), we see that changes in external cavity length and changes in lasing frequency affect the proxy for the phase-shift on reflection θRM in the same way. Directly from (8) and (16), these changes also impact the “measured” reflectivity through √ √ √ (1 − Rs ) √ (Lext + ∆L) RM = ε RA + ∆ RA . nin Lin Rs
(18)
From (18), we see that changes in external cavity length and changes in reflectivity affect the √ proxy for the reflectivity RM 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 (∆L = γλ0 , so that the fractional change in external cavity length relative to λ0 is γ = ∆L/λ0 ), the external cavity length in terms of the unperturbed laser wavelength (Lext = Γλ0 ), and the change in lasing frequency in terms of the unperturbed lasing frequency ν0 (∆ν = ην0 , so that the fractional change in frequency is η = ∆ν/ν0 ), (17) can be cast as θRM = θRA + ∆θRA − 4π {Γ + γ + η (Γ + γ)} .
(19)
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 6 θRM + 4πΓ − θRA 6 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 ∆θRA 1 ∆θ A 1 − 6 γ + η (Γ + γ) 6 R + . 4π 4 4π 4
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(20)
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18640
Fig. 2. Systematic sensitivity of the system when √ η ≈ 0 and ηΓ ≈ 0, in terms of equivalent change in amplitude reflection coefficient ∆ RA and phase-shift on reflection ∆θRA , for a range of laser wavelengths (frequencies). Blue contours are lines of constant λ0 γ. Red contours are lines of constant γ.
Similarly, (18) can be cast as √ √ √ (1 − Rs ) √ (Γ + γ) λ0 RM = ε RA + ∆ RA . (21) nin Lin Rs √ When γ Γ (that is ∆L Lext ), RM does not change significantly with small changes in the surface profile. Hence, θRM 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 ∆ RA and phase-shift on reflection ∆θRA . 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 ∆ RA = 10−4 . If one wishes to determine the corresponding uncertainty in θRA , one examines the horizontal red γ = 0.1 line to find uncertainty equivalent to ∆θRA = 4π · 10−1 rad. As such we can confirm that θRA is more sensitive than RA to changes in cavity length. 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, tem-
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18641
perature 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 through to (n, k) parameter space. We assume a bivariate Gaussian error ( RM , θRM ) propagates √ M distribution in ( R , θRM ) space, denoted here by √ RM ∼ N (µ , Σ) , (22) θRM where
√ E RM µ= , EθRM
√ √ M) R Cov( RM , θRM ) Var( √ and Σ = . Cov( RM , θRM ) Var(θRM )
(23)
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 Lext ), reflectivity RA , and phase-shift on reflection θRA , but all other quantities are fixed. Under the √ mild assumption of mutual independence of the random variations in (θRA , ν, Lext ) and in ( RA , Lext ), meaning in particular that all covariances between the random variations in these quantities are zero, we have expected “measurements” [see also (8) and (9)] √ (1 − Rs ) √ A √ E R ELext , E RM = ε nin Lin Rs
EθRM = EθRA −
4π EνELext . c
(24)
The covariance and variances of the “measurements” are √ √ √ √ (1 − Rs ) 2 √ Var( RM ) = ε Var(Lext )Var( RA ) + (ELext )2 Var( RA ) + (E RA )2 Var(Lext ) , nin Lin Rs (25) 2 4π A Var(θRM ) = Var(θM )+ Var(Lext )Var(ν) + (ELext )2 Var(ν) + (Eν)2 Var(Lext ) , (26) c √ (1 − Rs ) 4π √ A √ Cov( RM , θRM ) = −ε E R EνVar(Lext ) . (27) nin Lin Rs c We note particularly that variations in Lext contribute most to spurious variations in θRM . This can be seen by examining the second and third terms in the parenthesis of Var(θRM ), and noting that the coefficient in front of the variance in (ELext )2 Var(ν) is much smaller in magnitude than that in front of (Eν)2 Var(Lext ) for most cases, reinforcing the earlier point that target √ smoothness is usually much more important than a pure spectrum. Moreover, as Lext√and RA are typically of the similar order, variations in Lext are as important as variations in RA , once again reinforcing our earlier analysis. Assuming a calibrated linear laser–target system relationship between a “measured” pair
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18642
√ √ ( RM , θRM ) and the actual pair ( RA , θRA ) (see Section 2.4), we may write √ √ RA RM =A + b ∼ N Aµ + b , AΣAT , θRM θRA
(28)
where A contains the calibration terms bR and bθ on the diagonal, and b contains the calibration terms aR and aθ . Then, via the delta method (see e.g. [42]), we may write the (local) distribution in (n, k) space as √ n RM , θRM ∼approx N g(Aµ + b) , Jg AΣAT JgT , (29) =g k √ where Jg is the Jacobian of the transformation from g ≡ (n, k) 7→ ( RA , θRA ) evaluated at Aµ + b (see Appendix B). In practice, an average of measurements is used rather than a single data point, so that √ nb M, θ M , R = g R b k
√ 1 N √ RM = ∑ RM i , N i=1
θRM =
1 N M ∑ θR,i , N i=1
(30)
where, by assuming measurement errors are uncorrelated and therefore independent due to the Gaussian model here, √ ! RM ∼ N (µ , Σ/N) . (31) θRM Therefore, the error-reduction factor of 1/N will propagate through the linear √ transformation (and Jacobian) to reduce the corresponding (local) covariance matrices in ( RA , θRA ) 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 (RA , θRA ) 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 θRA 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 RA translates to similar levels of uncertainty in n. 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 Lext , RA , and θRA . The waveforms with random error show what one might expect to see in practice using a digital oscilloscope with simulated trace persistence. 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 #213350 - $15.00 USD (C) 2014 OSA
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18643
√ A Fig. 3. Sensitivity of the system to random variations in ( RA , θq R ) for a range of plas√ tics. 95% confidence intervals are drawn for each plastic, with Var( RA ) ≈ 0.00316 q √ (Var( RA ) = 10−5 ) and Var(θRA ) ≈ 0.00316 rad (Var(θRA ) = 10−5 rad2 ), at ν = 2.59 THz p with Var(ν) = 10 kHz (Var(ν) = 108 Hz2 ), and an external cavity of Lext = 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).
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 RM and θRM 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.
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18644
Fig. 4. Sensitivity of swept-frequency self-mixing signals to systematic and random changes. (a) Systematic change in external cavity length (∆L) of 0, 250, and 500 wavelengths on Lext = 500 mm, for fixed RA = 0.01 and θRA = 0.05 rad. (b) Systematic change in RA from 0.001, 0.01, and 0.1, for fixed Lext = 500 mm and θRA = 0.05 rad. (c) Systematic change in θRA from 0.01 rad, 0.05 rad, and 0.1 rad, for fixed Lext = 500 mm and RA = 0.01. (d)–(f) As (a)–(c), with √ additive zero-mean Gaussian variation in Lext with standard dep viation 10−4 λ0 m; RA with standard deviation 10−2 ; and θRA with standard deviation 10−2 rad.
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18645
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 √ RM i = β0 + βx (xi − x0 ) + βy (yi − y0 ) , (32) for each observation, where (xi , yi ) 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 , βx , βy ) 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 1 x1 − x0 y1 − y0 β0 1 x2 − x0 y2 − y0 , β = βx , (33) A = . . . .. .. .. βy 1 xN − x0 yN − y0 where A is the design matrix for the linear model, and β is the vector of unknown parameters. The least-squares solution βb can be found by solving the linear system of equations (the normal equations) (AT A)βb = AT v . (34) The tilt-removed values (keeping the constant term) would then be vei = vi − βbx (xi − x0 ) − βby (yi − y0 ) ,
i = 1, . . . , N .
(35)
√ Appendix B: Jacobian of the transformation from (n, k) to ( R, θR ) √ The Jacobian of the transformation in (2) from g ≡ (n, k) 7→ ( R, θR ) is given by ! ∂n ∂n Jg =
√ ∂ R ∂√k ∂ R
∂ θR ∂k ∂ θR
,
where each element of Jg can be calculated as √ √ −2 R + (1 + R) cos (θR ) 2 R(−1 + R) sin (θR ) ∂n ∂n √ =2 √ √ 2 , ∂ θ = 2 , ∂ R R 1 + R − 2 R cos (θR ) 1 + R − 2 R cos (θR ) √ √ −2 R + (1 + R) cos (θR ) ∂k 2(−1 + R) sin (θR ) ∂k √ = = −2 R √ √ 2 , 2 . ∂ θR ∂ R 1 + R − 2 R cos (θR ) 1 + R − 2 R cos (θR )
(36)
(37)
(38)
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, θR , Φ∆ ,V0 , and β . The information about the complex refractive index to be extracted is encoded mainly in C, α, and θR . 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 , . . . ,Vn ) at times (t1 ,t2 , . . . ,tn ), the goal is to obtain a parameter vector θ ∗ that explains the systematic varia#213350 - $15.00 USD (C) 2014 OSA
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18646
tion 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 ∗ ∈ argmin S(θ ) , θ ∈Θ
n
with S(θ ) =
∑ wk (Vk −V (tk ; θ ))2 ,
(39)
k=1
where Θ is the permissible parameter space, and {wk } 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 , . . . ,tn }. 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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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18647
of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia 2 School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia 3 School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK ∗ [email protected]
Abstract: Recently, we demonstrated an interferometric materials analysis scheme at terahertz frequencies based on the self-mixing effect in terahertz quantum cascade lasers. Here, we examine the impact of variations in laser operating parameters, target characteristics, laser–target system properties, and the quality calibration standards on our scheme. We show that our coherent scheme is intrinsically most sensitive to fluctuations in interferometric phase, arising primarily from variations in external cavity length. Moreover we demonstrate that the smallest experimental uncertainties in the determination of extinction coefficients are expected for lossy materials. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (140.3430) Laser theory; (140.5965) Semiconductor lasers, quantum cascade; (120.4530) Optical constants.
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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.
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18635
Fig. 1. Three mirror model for a laser under feedback. Light recirculates in the laser cavity between mirrors M0 and Ms — solid arrows in laser cavity of length L. Light exits through Ms , is reflected from the external target Mext , and is reinjected into the laser cavity — solid arrows in external cavity of length Lext .
2. 2.1.
Background Optical properties of the target
We model the optical properties of the target material at a particular spatial point as a (nonmagnetic) homogeneous half-space with √ complex refractive index (using the positive time convention e+jωt ) nb = n − j k (with j ≡ −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 θR . The dependence of (R, θR ) on (n, k) is given through the pair of relations [27] 2n0 k (n0 − n)2 + k2 R= , θ = − arctan , (1) R (n0 + n)2 + k2 n20 − n2 − k2 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, θR ) is given through the pair of relations [27] √ 1−R 2 R sin (θR ) √ √ n= , k= . (2) 1 + R − 2 R cos (θR ) 1 + R − 2 R cos (θR ) 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. 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 #213350 - $15.00 USD (C) 2014 OSA
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18636
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 − θR , (3) ϕ(t) = ϕ0 + T where ϕ0 is the round-trip transmission phase delay in the external cavity at the start of the frequency sweep, Φ∆ is the interferometric phase deviation caused by the current (frequency) sweep, and θR 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’) ϕS − ϕFB = C sin(ϕFB + arctan α) ,
(4)
where ϕFB represents the total external round-trip phase at the perturbed laser frequency, ϕS 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 ϕS and the resulting feedback phase ϕFB 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 ϕS as the phase stimulus varying linearly in time according to (3) and of ϕFB 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 = V0 + β cos (ϕFB ) ,
(5)
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 ϕFB (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 θR 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] p τext C ≡ C(α) = κext 1 + α 2 , (6) τL where τext is the round-trip propagation time in the external cavity, τL is the round-trip delay of the solitary laser, and κext is the coupling coefficient that depends on the reflectivity of the
#213350 - $15.00 USD (C) 2014 OSA
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18637
emitting mirror of the laser Rs and the reflectivity of the target R (associated with mirrors Ms and Mext in Fig. 1, respectively), as r R κext = ε (1 − Rs ) . (7) Rs 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 (RM and θRM ) which are related to the physical properties of the target (R and θR ) in a known way. We will term them “proxies” for the actual reflectivity RA ≡ R (sometimes termed reflectance) and phase-shift on reflection θRA ≡ θR , 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 RA as RM , (theoretically) given by √ √ C (1 − Rs ) √ Lext RA , RM = √ =ε (8) nin Lin Rs 1 + α2 √ √ so that RM is a multiple of RA in the ideal case. For the reflection phase proxy θRM we conveniently choose the actual reflection phase with the reference plane translated to coincide with the plane of the laser exit mirror. Therefore, θRM contains not only the actual reflection phase θRA , 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: 4π θRM = θRA − ν0 Lext , (9) c so that, ideally, θRM is θRA together with an√additive term corresponding √ to the total phase-shift on transmission. Notice that, ideally, both RM and θRM are linear in RA and θRA , 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 RM and θRM in Sections 3 and 4. 2.4.
System calibration
√ √ Due to the linear dependence of RM on RA 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 √ √ RM = aR + bR RA , (10) where RA is the actual reflectivity of the material under test, aR and bR are unknown parameters to be determined, and RM = C2 /(1 + α 2 ), as defined earlier [see (8)], is representative of the material’s measured, but uncalibrated reflectivity. Along similar lines, to account for systematic
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18638
phase uncertainties, we express θR [see (9)] as θRM = aθ + bθ θRA ,
(11)
where θRA is the actual phase shift on reflection, aθ and bθ are unknown parameters to be determined, and θRM (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, aR , bR , aθ , and bθ , which can be determined by √ model fitting (see Appendix C) from two measurements on materials with known ( RA , θRA ) 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 A M A M A M A (RM 1 , R1 ) (θR,1 , θR,1 ), and (R2 , R2 ) (θR,2 , θR,2 ) respectively, the set of four linear equations are q q q q A M A M A = a +b , RA1 = aR + bR RM R RM R R 1 2 2 , θR,1 = aθ + bθ θR,1 , θR,2 = aθ + bθ θR,2 . (12) The solution of this system of equations is straightforward and provides values for aR , bR , aθ , and bθ . In particular, q q q q q q RA1 RM RM RA2 RA1 − RA2 2 − 1 q q q aR = , bR = q , (13) M M− M − RM R R R 1 2 1 2 aθ =
A θM −θM θA θR,1 R,2 R,1 R,2 M −θM θR,1 R,2
,
bθ =
A −θA θR,1 R,2 M −θM θR,1 R,2
.
(14)
Once these values have been obtained we can readily calculate actual values for θRA and RA for the material under test using (10) and (11). √ It is clear that an error in calibration standard ∆ RA and ∆θRA will propagate as both an additive error (through the constant calibration terms) and as a multiplicative error (through √ the linear terms) to each measurement ( RM or θRM ). 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.
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18639
Changes in external cavity length (∆L), lasing frequency (∆ν),√phase-shift on reflection (∆θRA ), and amplitude reflection coefficient of the external target (∆ RA ) — all of which can take positive or negative values — enter the excess phase equation as changes to (3) through ϕ + ∆ϕ =
4π (ν0 + ∆ν) (Lext + ∆L) − θRA + ∆θRA , c
(15)
as well as interacting nonlinearly with changes in the feedback parameter (6) through the solution of (4) via C + ∆C = ε
√ p √ (1 − Rs ) √ (Lext + ∆L) 1 + α2 RA + ∆ RA . nin Lin Rs
(16)
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 θRM = θRA + ∆θRA −
4π (ν0 Lext + ν0 ∆L + ∆νLext + ∆ν∆L) . c
(17)
From (17), we see that changes in external cavity length and changes in lasing frequency affect the proxy for the phase-shift on reflection θRM in the same way. Directly from (8) and (16), these changes also impact the “measured” reflectivity through √ √ √ (1 − Rs ) √ (Lext + ∆L) RM = ε RA + ∆ RA . nin Lin Rs
(18)
From (18), we see that changes in external cavity length and changes in reflectivity affect the √ proxy for the reflectivity RM 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 (∆L = γλ0 , so that the fractional change in external cavity length relative to λ0 is γ = ∆L/λ0 ), the external cavity length in terms of the unperturbed laser wavelength (Lext = Γλ0 ), and the change in lasing frequency in terms of the unperturbed lasing frequency ν0 (∆ν = ην0 , so that the fractional change in frequency is η = ∆ν/ν0 ), (17) can be cast as θRM = θRA + ∆θRA − 4π {Γ + γ + η (Γ + γ)} .
(19)
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 6 θRM + 4πΓ − θRA 6 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 ∆θRA 1 ∆θ A 1 − 6 γ + η (Γ + γ) 6 R + . 4π 4 4π 4
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(20)
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18640
Fig. 2. Systematic sensitivity of the system when √ η ≈ 0 and ηΓ ≈ 0, in terms of equivalent change in amplitude reflection coefficient ∆ RA and phase-shift on reflection ∆θRA , for a range of laser wavelengths (frequencies). Blue contours are lines of constant λ0 γ. Red contours are lines of constant γ.
Similarly, (18) can be cast as √ √ √ (1 − Rs ) √ (Γ + γ) λ0 RM = ε RA + ∆ RA . (21) nin Lin Rs √ When γ Γ (that is ∆L Lext ), RM does not change significantly with small changes in the surface profile. Hence, θRM 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 ∆ RA and phase-shift on reflection ∆θRA . 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 ∆ RA = 10−4 . If one wishes to determine the corresponding uncertainty in θRA , one examines the horizontal red γ = 0.1 line to find uncertainty equivalent to ∆θRA = 4π · 10−1 rad. As such we can confirm that θRA is more sensitive than RA to changes in cavity length. 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, tem-
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18641
perature 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 through to (n, k) parameter space. We assume a bivariate Gaussian error ( RM , θRM ) propagates √ M distribution in ( R , θRM ) space, denoted here by √ RM ∼ N (µ , Σ) , (22) θRM where
√ E RM µ= , EθRM
√ √ M) R Cov( RM , θRM ) Var( √ and Σ = . Cov( RM , θRM ) Var(θRM )
(23)
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 Lext ), reflectivity RA , and phase-shift on reflection θRA , but all other quantities are fixed. Under the √ mild assumption of mutual independence of the random variations in (θRA , ν, Lext ) and in ( RA , Lext ), meaning in particular that all covariances between the random variations in these quantities are zero, we have expected “measurements” [see also (8) and (9)] √ (1 − Rs ) √ A √ E R ELext , E RM = ε nin Lin Rs
EθRM = EθRA −
4π EνELext . c
(24)
The covariance and variances of the “measurements” are √ √ √ √ (1 − Rs ) 2 √ Var( RM ) = ε Var(Lext )Var( RA ) + (ELext )2 Var( RA ) + (E RA )2 Var(Lext ) , nin Lin Rs (25) 2 4π A Var(θRM ) = Var(θM )+ Var(Lext )Var(ν) + (ELext )2 Var(ν) + (Eν)2 Var(Lext ) , (26) c √ (1 − Rs ) 4π √ A √ Cov( RM , θRM ) = −ε E R EνVar(Lext ) . (27) nin Lin Rs c We note particularly that variations in Lext contribute most to spurious variations in θRM . This can be seen by examining the second and third terms in the parenthesis of Var(θRM ), and noting that the coefficient in front of the variance in (ELext )2 Var(ν) is much smaller in magnitude than that in front of (Eν)2 Var(Lext ) for most cases, reinforcing the earlier point that target √ smoothness is usually much more important than a pure spectrum. Moreover, as Lext√and RA are typically of the similar order, variations in Lext are as important as variations in RA , once again reinforcing our earlier analysis. Assuming a calibrated linear laser–target system relationship between a “measured” pair
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18642
√ √ ( RM , θRM ) and the actual pair ( RA , θRA ) (see Section 2.4), we may write √ √ RA RM =A + b ∼ N Aµ + b , AΣAT , θRM θRA
(28)
where A contains the calibration terms bR and bθ on the diagonal, and b contains the calibration terms aR and aθ . Then, via the delta method (see e.g. [42]), we may write the (local) distribution in (n, k) space as √ n RM , θRM ∼approx N g(Aµ + b) , Jg AΣAT JgT , (29) =g k √ where Jg is the Jacobian of the transformation from g ≡ (n, k) 7→ ( RA , θRA ) evaluated at Aµ + b (see Appendix B). In practice, an average of measurements is used rather than a single data point, so that √ nb M, θ M , R = g R b k
√ 1 N √ RM = ∑ RM i , N i=1
θRM =
1 N M ∑ θR,i , N i=1
(30)
where, by assuming measurement errors are uncorrelated and therefore independent due to the Gaussian model here, √ ! RM ∼ N (µ , Σ/N) . (31) θRM Therefore, the error-reduction factor of 1/N will propagate through the linear √ transformation (and Jacobian) to reduce the corresponding (local) covariance matrices in ( RA , θRA ) 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 (RA , θRA ) 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 θRA 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 RA translates to similar levels of uncertainty in n. 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 Lext , RA , and θRA . The waveforms with random error show what one might expect to see in practice using a digital oscilloscope with simulated trace persistence. 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 #213350 - $15.00 USD (C) 2014 OSA
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18643
√ A Fig. 3. Sensitivity of the system to random variations in ( RA , θq R ) for a range of plas√ tics. 95% confidence intervals are drawn for each plastic, with Var( RA ) ≈ 0.00316 q √ (Var( RA ) = 10−5 ) and Var(θRA ) ≈ 0.00316 rad (Var(θRA ) = 10−5 rad2 ), at ν = 2.59 THz p with Var(ν) = 10 kHz (Var(ν) = 108 Hz2 ), and an external cavity of Lext = 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).
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 RM and θRM 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.
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18644
Fig. 4. Sensitivity of swept-frequency self-mixing signals to systematic and random changes. (a) Systematic change in external cavity length (∆L) of 0, 250, and 500 wavelengths on Lext = 500 mm, for fixed RA = 0.01 and θRA = 0.05 rad. (b) Systematic change in RA from 0.001, 0.01, and 0.1, for fixed Lext = 500 mm and θRA = 0.05 rad. (c) Systematic change in θRA from 0.01 rad, 0.05 rad, and 0.1 rad, for fixed Lext = 500 mm and RA = 0.01. (d)–(f) As (a)–(c), with √ additive zero-mean Gaussian variation in Lext with standard dep viation 10−4 λ0 m; RA with standard deviation 10−2 ; and θRA with standard deviation 10−2 rad.
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Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18645
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 √ RM i = β0 + βx (xi − x0 ) + βy (yi − y0 ) , (32) for each observation, where (xi , yi ) 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 , βx , βy ) 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 1 x1 − x0 y1 − y0 β0 1 x2 − x0 y2 − y0 , β = βx , (33) A = . . . .. .. .. βy 1 xN − x0 yN − y0 where A is the design matrix for the linear model, and β is the vector of unknown parameters. The least-squares solution βb can be found by solving the linear system of equations (the normal equations) (AT A)βb = AT v . (34) The tilt-removed values (keeping the constant term) would then be vei = vi − βbx (xi − x0 ) − βby (yi − y0 ) ,
i = 1, . . . , N .
(35)
√ Appendix B: Jacobian of the transformation from (n, k) to ( R, θR ) √ The Jacobian of the transformation in (2) from g ≡ (n, k) 7→ ( R, θR ) is given by ! ∂n ∂n Jg =
√ ∂ R ∂√k ∂ R
∂ θR ∂k ∂ θR
,
where each element of Jg can be calculated as √ √ −2 R + (1 + R) cos (θR ) 2 R(−1 + R) sin (θR ) ∂n ∂n √ =2 √ √ 2 , ∂ θ = 2 , ∂ R R 1 + R − 2 R cos (θR ) 1 + R − 2 R cos (θR ) √ √ −2 R + (1 + R) cos (θR ) ∂k 2(−1 + R) sin (θR ) ∂k √ = = −2 R √ √ 2 , 2 . ∂ θR ∂ R 1 + R − 2 R cos (θR ) 1 + R − 2 R cos (θR )
(36)
(37)
(38)
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, θR , Φ∆ ,V0 , and β . The information about the complex refractive index to be extracted is encoded mainly in C, α, and θR . 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 , . . . ,Vn ) at times (t1 ,t2 , . . . ,tn ), the goal is to obtain a parameter vector θ ∗ that explains the systematic varia#213350 - $15.00 USD (C) 2014 OSA
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18646
tion 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 ∗ ∈ argmin S(θ ) , θ ∈Θ
n
with S(θ ) =
∑ wk (Vk −V (tk ; θ ))2 ,
(39)
k=1
where Θ is the permissible parameter space, and {wk } 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 , . . . ,tn }. 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).
#213350 - $15.00 USD (C) 2014 OSA
Received 3 Jun 2014; revised 17 Jul 2014; accepted 17 Jul 2014; published 24 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018633 | OPTICS EXPRESS 18647
