1. Introduction

Noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE-OHMS) is a technique that combines frequency modulation spectroscopy (FMS) for reduction of noise, in particular 1/f type of noise, with cavity enhanced absorption spectroscopy (CEAS) to enhance the interaction length between the light and a (gaseous) sample [1–4]. By locking the modulation frequency to the free spectral range (FSR) of the cavity, the technique will acquire an immunity to the frequency-to-amplitude noise that normally is the limiting factor in CEAS [5]. To obtain the NICE-OHMS signal, the beat signal between various pairs of spectral components of the laser beam is demodulated at the modulation frequency. For the case when the detected light field comprises a balanced set of components, the beat signals between various pairs of spectral components will cancel, giving rise to no net NICE-OHMS signal. This implies that the technique is inherently background free. These two features, noise immunity and being background free, give the technique an extraordinary potential for ultra-sensitive detection of molecular constituents.

Although the technique demonstrated, in one of its first realization, based on a fix-frequency laser and addressing a Doppler-free response, an impressive and so far unrivaled detection sensitivity of 1 × 10⁻¹⁴ cm⁻¹ [2], subsequent realizations of the technique, based upon tunable lasers, which mostly have utilized Doppler-broadened detection, have not been able to demonstrate equally impressive extraordinary number of merits [6–11]. The main reason for this is that in the absence of a sample there is in practice often a certain non-zero signal present, mainly originating from an unbalancing of the set of spectral components of the laser beam before the cavity, referred to as a background signal. There are several possible origins for such a background signal, of which the most common ones are multiple reflections between various surfaces (so called etalons) [12] or processes in the electro optic modulator (EOM) [13]. In any case, the noise and drifts of these processes set, in practice, the limit for the performance of the instrumentation. We will here define noise as the short time variation of the signal that can be reduced by increased averaging while drift will denote the long term changes of the background signal that increase with time. It is therefore of highest importance to reduce or eliminate all possible types of background signals, in particular their drifts. Several means of how to realize this have been developed over the years [7,14–16]. However, despite this, they still often remain at some level, limiting the performance of the system.

It is customary to detect the NICE-OHMS signal carried in the transmitted light (henceforth, referred to as in transmission). However, since the molecular NICE-OHMS signal (i.e. the signal originating from the molecules addressed in the cavity) originates from light that is “reflected back and forth” in the cavity, thus propagating in both directions, there will, in fact, be a molecular NICE-OHMS signal also in the reflected light. An alternative to the conventional mode of operation of NICE-OHMS, which so far has not yet been extensively investigated, is therefore to detect the signal in the reflected light (hereafter, in reflection). This was suggested already in 1998 by Jun Ye et al. in one of their seminal NICE-OHMS papers, in which they stated that “…it seems to be advantageous to detect the molecular signal in cavity reflection, provided that the optical impedance match of the cavity is adequate.” [2]. Except for a pair of demonstrations by Ye [17] and Silva [18], the NICE-OHMS signal has not, according to the authors’ knowledge, been detected in reflection.

The main reason for the preference of detecting the NICE-OHMS signal in transmission is that the S/N ratio in reflection is not as high as in transmission. This originates from the fact that some part of the reflected light is not mode matched to any longitudinal mode of the cavity, and thus unconditionally reflected from the cavity. As this light does not carry any information about the molecules in the gas sample, it only contributes noise.

However, as is shown below, the molecular and the background NICE-OHMS signals detected in reflection differs from those in transmission in one important aspect; if they have the same sign in transmission, they have opposite in reflection (and vice versa). Based on this, we propose and demonstrate in this paper that the simultaneous detection of NICE-OHMS signals in transmission and reflection, followed by subtraction of the latter from the former (properly weighted), henceforth referred to as differential NICE-OHMS, the influence of background signals originating before the cavity can be reduced (and in certain cases even eliminated), while the two molecular signals will constructively add. This can thus be used to reduce the influence of background signals originating from the optical system before the cavity and thereby significantly improve on the detectability of the system, in particular reduced their drifts.

The paper provides, first of all, expressions for the molecular and the background NICE-OHMS signals, detected both in transmission and reflection and verifies, by use of these, the aforementioned signal polarities. For simplicity, the expressions were derived for the conditions with a cavity constructed by two identical mirrors, for a low modulation index, i.e. when the light field comprises a triplet, and when the conventional low-absorption description of NICE-OHMS is valid [19]. It then derives the conditions for maximum reduction of the influence of background signals on differential NICE-OHMS. It thereafter presents simulations of some of the properties of the system, in particular the magnitudes of both molecular and background NICE-OHMS signals, detected in both reflection and transmission, for some typical sets of conditions. It then presents measurements of molecular NICE-OHMS signals, detected both in transmission and reflection, performed by a fiber laser working at around 1.53 µm, addressing a transition in C₂H₂, before it demonstrates, by construction of a differential NICE-OHMS signal, the principle and efficiency of the proposed methodology for background signal reduction. In the end, it demonstrates, by the use of an Allan-Werle plot, that this novel detection methodology will lead to both an improved resilience to drifts (reduced by almost one order of magnitude) and a new unprecedented detection sensitivity for Doppler broadened NICE-OHMS performed by a tunable laser (4.7 × 10⁻¹⁴ cm⁻¹ for an integration time of 170 s).

2. Theory

2.1. Mode matched versus non mode matched light

In NICE-OHMS the spectral components of the frequency modulated (FM) laser are tightly locked to longitudinal modes of the cavity using the Pound-Drever-Hall (PDH) [20] and DeVoe-Brewer (DVB) techniques [21]. As is shown in Fig. 1 in [19], this implies that the laser light consists of a number of spectral components of which the ones that are used for FMS are in resonance with various (often a set of consecutive) cavity mode while the ones that are produced for the PDH locking are not so. However, not all of the light that impinges upon the cavity is properly mode matched (MM) to longitudinal modes of the cavity. The non-mode matched (NMM) light can consist of both spatially and frequency NMM light. The latter, in turn, consists of the PDH sidebands and the parts of the spectral components of the laser light that are not locked to the cavity modes due to a limited locking bandwidth. As is shown below, the MM part of the light that impinges upon the cavity will be transmitted, reflected, or lost (scattered or absorbed by cavity mirrors), depending on the properties of the light and the cavity, while the NMM light will unconditionally be reflected. To treat this situation adequately, we will, as schematically is shown in Fig. 1, decompose the power of the light that impinges upon the cavity, $P_{\mathrm{in}}$, into two parts, viz. one that is MM to a longitudinal mode of the cavity, denoted $P_{in}^{MM}$, and one that is NMM, referred to as $P_{in}^{NMM}$.

 

Fig. 1 Schematic illustration of the nomenclature used for the differential NICE-OHMS realization presented in this work. The various entities are defined in the text. DM: Demodulation; Diff: Subtraction of the reflection signal (weighted by the factor $a$) from the transmitted signal.

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2.2. Amplitude transmission and reflection functions of a cavity filled with gas

The fate of the MM part of the light, which thus can produce a NICE-OHMS signal from molecules in the cavity, can be assessed by the use of the complex transmission and reflection functions for the electric field of narrow linewidth light addressing a longitudinal mode of a cavity.

Let us for simplicity consider the case when the carrier of the laser light addresses the $q$-th longitudinal mode of a FP cavity made of two identical mirrors. Since noise-immune conditions in general can be obtained when the frequency separation of consecutive spectral components of the laser light is equal to a multiple of the FSR of the cavity, ${\nu }_{\text{FSR}}$ [22,23], we will here consider the general case when the NICE-OHMS modulation frequency, ${\nu }_{\mathrm{mod}}$, is locked to a multiple of the FSR, ${\nu }_{\text{FSR}}$. This implies that the $j$-th spectral component of the frequency modulated MM light addresses the $[q+({\nu }_{mod}/{\nu }_{\text{FSR}})j]$-th longitudinal mode of the cavity. The complex cavity transmission and reflection functions for the electric field of the $j$-th spectral component of the light, in the presence of molecules in the cavity, ${\chi }_{\text{q,j}}^{\text{c,tran,mol}}$ and ${\chi }_{\text{q,j}}^{\text{c,refl,mol}}$, can be written as [19]

2.3. On-resonance empty cavity reflection and transmission

A cavity consisting of two identical mirrors with non-zero losses does not provide impendence matched conditions [3]. This implies that although a given spectral component is tightly locked on-resonance (i.e. to the center of its cavity mode), a part of the light will still be reflected. The relative on-resonance transmitted and reflected powers of a single MM component of the laser light, denoted $T_{\text{c}}$ and $R_{\text{c}}$ and defined as $P_{\text{tran}}^{\text{MM}}/P_{\text{in}}^{\text{MM}}$ and $P_{\text{refl}}^{\text{MM}}/P_{\text{in}}^{\text{MM}}$, where $P_{\text{tran}}^{\text{MM}}$ and $P_{\text{refl}}^{\text{MM}}$ are the transmitted and reflected powers of the MM part of the component, respectively, can be assessed as ${\left|{\chi }_{\text{q,j}}^{\text{c,tran,mol}}\right|}^2$ and ${\left|{\chi }_{\text{q,j}}^{\text{c,refl,mol}}\right|}^2$ evaluated on resonance for an empty cavity. For the case when the laser light consists of more than one component, and each of these simultaneously addresses its own cavity mode on resonance, $T_{\text{c}}$ and $R_{\text{c}}$ represent the relative transmission and reflection powers of the entire MM light field.

An evaluation of ${\left|{\chi }_{\text{q,j}}^{\text{c,tran,mol}}\right|}^2$ and ${\left|{\chi }_{\text{q,j}}^{\text{c,refl,mol}}\right|}^2$ on resonance for an empty cavity, by the use of the Eqs. (1) and (2), shows that $T_{\text{c}}$ and $R_{\text{c}}$ can alternatively be written

2.4. Total power of light reaching the transmission and reflection detectors

Under the condition that all transmitted and reflected MM light impinges upon their corresponding detectors, it is possible to conclude that $P_{\text{tran}}^{\text{MM}}$ and $P_{\text{refl}}^{\text{MM}}$ can be written as $T_{\text{c}}{P}_{\text{in}}^{\text{MM}}$ and $R_{\text{c}}{P}_{\text{in}}^{\text{MM}}$, respectively, while the power of the NMM light detected by the reflection detector, termed $P_{\text{refl}}^{\text{NMM}}$, which is unconditionally reflected, is equal to $P_{\text{in}}^{\text{NMM}}$. This implies that the total power that impinges upon the transmission and reflection detectors, denoted $P_{\text{tran}}^{}$ and $P_{\text{refl}}^{}$, respectively, can be written as

It should be noticed that while the relative amount of MM light that is reflected on-resonance, $R_c$, which is caused by the impedance mismatch, constitutes an intrinsic property of the cavity, i.e. it is a fixed quantity for a given cavity, the relative amount of NMM light, $\epsilon$, can, to a certain extent, be tailored with minor changes of the setup, e.g. by adjusting the spatial mode matching or the modulation index of the modulation for the PDH locking.

2.5. Molecular transmission and reflection NICE-OHMS signals

For simplicity we will consider the case of a small FM modulation index, for which there is only one pair of sidebands [24], and a small single-pass absorption, defined as ${\alpha }_{0}L<<\pi /F$, where ${\alpha }_0$ is the on-resonance absorption coefficient and $L$ is the cavity length. Under these conditions, the NICE-OHMS signal detected in transmission can be represented by an FMS signal enhanced by a cavity elongation factor, given by $2F/\pi$ [2,19]. Under the additional condition that $R_{\text{c}}\ne 0$, i.e. that there is always some of the MM light being reflected, it is possible to derive an expression for the NICE-OHMS signal detected in reflection by following the same procedure as for the transmission signal, as for example is done in [19]. Under these conditions [25], the molecular NICE-OHMS signals detected in transmission and reflection, denoted $S_{\mathrm{mol}}^{\text{tran}}$ and $S_{\text{mol}}^{\text{refl}}$ respectively, can be expressed as

The ${(-1)}^{{\nu }_{\mathrm{mod}}/{\nu }_{\text{FSR}}}$ factor in Eq. (8) originates from the fact that the transmission function for the electric field, ${\chi }_{\text{q,j}}^{\text{c,tran,mol}}$, takes an alternating sign for consecutive cavity modes, given by the first part of Eq. (3), which makes up an $\exp \{-[q+({\nu }_{\mathrm{mod}}/{\nu }_{\text{FSR}})j]\pi \text{i}\}$ factor. In the literature, this term is often absorbed in the demodulation phase, whereby it is seldom explicitly expressed. However, to properly establish the relation between various types of signal we will keep this sign-factor here. The minus sign in Eq. (9) originates from the fact that a small absorption event in the cavity gives rise to an increased electric field in reflection, as can be deduced from Eq. (2), opposite to what it does in transmission, Eq. (1).

A comparison of the expressions above [the Eqs. (8) and (9)] shows that the molecular reflection and transmission NICE-OHMS signals can be related to each other according to

2.6. Background transmission and reflection NICE-OHMS signals

In the following we assume that the NICE-OHMS background signals originate from an imbalance between the various spectral components of the modulated light field prior to the cavity. Although many of the optical components of the instrumentation were placed at etalon-immune distances (EID) [16], the NICE-OHMS background signal usually originates from either remaining etalons in the optical system, which can be considerable in fiber coupled components, or residual amplitude modulation created in the EOM [26]. Since these effects are difficult to eliminate they are often the dominating contribution to NICE-OHMS background signals.

Under the conditions that we can neglect the fact that the frequency NMM light might be slightly shifted from the carrier, which, in general, is a valid approximation (caused by the fact that for Doppler broadened NICE-OHMS the dominating background signal has a weak frequency dependence, with a small variation over the scale of the PDH-modulation frequency [27]), it is possible to consider the NICE-OHMS background signals detected in transmission and reflection, $S_{\text{BG}}^{\text{tran}}$ and $S_{\text{BG}}^{\text{refl}}$, to be proportional to the total power of light that impinges upon each detector, i.e. $P_{\text{tran}}^{}$ and $P_{\text{refl}}$, respectively. Hence, they can be written as

This implies that the reflection and transmission NICE-OHMS background signals can be related to each other according to

2.7. Total and differential NICE-OHMS signals – background signal canceling

The total transmission and reflection NICE-OHMS signals, $S_{\text{tot}}^{\text{tran}}$ and $S_{\text{tot}}^{\text{refl}}$, are given by

By subtracting the total NICE-OHMS signal detected in reflection, $S_{\text{tot}}^{\text{refl}}$, from that detected in transmission, $S_{\text{tot}}^{\text{tran}}$, with the former weighted by $a$, it is possible to obtain a differential NICE-OHMS signal, $S_{\text{diff}}$, that has no influence of the background signals, viz. as

This shows that it is possible to eliminate the influence of the background signals created in the optical system prior to the cavity by simultaneously measuring the NICE-OHMS signals in reflection and transmission and thereafter constructing a weighted difference between the two.

2.8. Mode matching and cavity parameters as function of powers

It could be concluded from above that the relevant powers and signals, including their ratios, can be written in terms of $T_{\text{c}}$, $R_{\text{c}}$, and $\epsilon$, where that the former two, for the case with two identical mirrors, can be expressed in terms of $F{l}^2/\pi$ [28]. It is therefore suitable to investigate the dependence of the powers and signals of interest in terms of $F{l}^2/\pi$ and $\epsilon$.

By the use of the definition of $P_{\text{in}}$, the fact that $P_{\text{refl}}^{\text{NMM}}$ = $P_{\text{in}}^{\text{NMM}}$, and the Eqs. (4) - (7), in a manner similar to that of Hood et al. [29], it is possible to show that $F{l}^2/\pi$ can be assessed from a combination of three measurable quantities, viz. as

Since $F{l}^2/\pi$ can be written as $r{l}^2/(l^2+t^2)$, it is also possible to conclude that $F{l}^2/\pi$ can, for any cavity made of high reflectance mirrors ($r\approx 1$), be interpreted as the fraction of the total mirror losses, $l^2+t^2$, that is scattered or absorbed, $l^2$. Hence, it can, in practice, range from zero, which it takes for low or medium finesses cavities (for which $t^2$ is $>>l^2$), up to a maximum of unity, which it can take for cavities made by high reflectivity mirrors with small transmission coefficients ($t^2<<l^2$). It can be concluded, however, that it is, in general, the values $F{l}^2/\pi \le$0.5 that are of practical importance.

The $\epsilon$ entity can be alternatively be assessed as

2.9. Shot noise limited conditions

The shot noise equivalent absorption coefficient, at which the NICE-OHMS signal is equal to the standard deviation of the shot noise, in transmission, i.e. ${({\alpha }_0)}_{\text{SN}}^{\text{tran}}$, can be estimated from Eq. (6).35) in [4] and can be written as

3. Simulations

To utilize the response of a resonant Fabry-Perot cavity for differential NICE-OHMS in an optimal manner, some properties of the system were simulated for a few sets of typical conditions. For simplicity, in all simulations, the instrumentation factors have been assumed to be identical, i.e. ${\eta }_{\text{V}}^{\text{refl}}={\eta }_{\text{V}}^{\text{tran}}$.

Figure 2 illustrates, in panel (a), by the black solid and the set of red dotted curves, the cavity transmitted and reflected powers, $P_{\text{tran}}$ and $P_{\mathrm{refl}}$, normalized by $P_{\text{in}}^{\text{MM}}$, which, according to the Eqs. (6) and (7), are given by $T_{\text{c}}$ and $R_{\text{c}}+\epsilon$ respectively (the left axis), as functions of $F{l}^2/\pi$. As can be deduced from the Eqs. (13) and (14), the same curves represent the magnitude of the two background signals, $S_{\text{BG}}^{\text{tran}}$ and $S_{\text{BG}}^{\mathrm{refl}}$, normalized to the transmission NICE-OHMS background signal for a cavity with lossless mirrors,${S_{BG}^{tran}|}_{F{l}^2/\pi =0}$ (right axis). Panel (b) displays, by the black solid and the red dotted curves, normalized molecular transmission and reflection NICE-OHMS signals, respectively, as functions of the same parameter. The normalization is in both cases done with respect to the molecular transmission NICE-OHMS signal for the case with lossless mirrors, i.e. as $S_{mol}^{tran}/{S_{\mathrm{mol}}^{\text{tran}}|}_{F{l}^2/\pi =0}$ and $S_{\mathrm{mol}}^{refl}/{S_{\text{mol}}^{\text{tran}}|}_{F{l}^2/\pi =0}$. In panel (a), the various dotted curves represent different relative amounts of NMM light, viz. $\epsilon$ values of 0, 0.1, 0.2, and 0.3, respectively, as indicated in the figure. The solid curve in panel (a) and both curves in panel (b) are valid for all values of $\epsilon$. The three vertical dotted lines in the figure mark $F{l}^2/\pi$ values of 0.02, 0.23, and 0.5, corresponding to typical cavities made of mirrors with a given loss (12.6 ppm, representing the experimental conditions in this work) but dissimilar reflectivities, representing finesse values of 5 000, 55 000, and 125 000, respectively, where the middle one represents the experimental conditions in this work while the latter one corresponds to the case when $t^2=l^2$.

 

Fig. 2 Simulations of some of the properties of a FP cavity constructed by two identical mirrors. Panel (a): left y-axis: normalized transmitted and reflected power, $P_{\text{tran}}/P_{\text{in}}^{\text{MM}}$ and $P_{\mathrm{refl}}/P_{\text{in}}^{\text{MM}}$, respectively; right y-axis: normalized NICE-OHMS background signals in transmission and reflection, i.e. $S_{\text{BG}}^{\text{tran}}/{S_{BG}^{tran}|}_{F{l}^2/\pi =0}$ and $S_{\text{BG}}^{\text{refl}}/{S_{BG}^{tran}|}_{F{l}^2/\pi =0}$, respectively; Panel (b): normalized molecular NICE-OHMS signals in transmission and reflection, i.e. $S_{\text{mol}}^{\text{tran}}/{S_{mol}^{tran}|}_{F{l}^2/\pi =0}$ and $S_{mol}^{refl}/{S_{mol}^{tran}|}_{F{l}^2/\pi =0}$. In both panels, the black solid and the red dashed curves represent transmission and reflection, respectively. The three vertical dotted lines represent $F{l}^2/\pi$ values of 0.02, 0.23, and 0.5.

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Panel (a) shows, first of all, that the transmitted power (the black solid curve) decreases monotonically with $F{l}^2/\pi$ while the reflected counterparts (the dotted curves) increase. It also indicates that, for the somewhat fictitious case with no NMM light (i.e. $\epsilon$ = 0, the lowermost dotted curve), the reflected power is 0.04% and 9% of the transmitted for $F{l}^2/\pi$-values of 0.02 and 0.23, respectively, while they are equally large for the case with $F{l}^2/\pi$ being 0.5. For the case with finite amounts of NMM light (i.e. $\epsilon$ > 0), the other dotted curves illustrate that the reflected power is a given value ($\epsilon$) larger than when the light solely is composed of MM light. This implies that for the case with $\epsilon$ being 0.1, the reflected power is higher than 10% of the transmitted power irrespective of the $F{l}^2/\pi$ value. Since the transmission and reflection NICE-OHMS background signals, i.e. $S_{\text{BG}}^{\text{tran}}$ and $S_{\text{BG}}^{\text{refl}}$, are proportional to the transmitted and reflected powers, the same quantitative conclusions can be drawn for the transmission and reflection NICE-OHMS background signals.

Panel (b) illustrates that the molecular transmission NICE-OHMS signal (the black solid curve) decreases monotonically with $F{l}^2/\pi$. The molecular reflection NICE-OHMS signal (the red dotted curve), on the other hand, increases gradually, up to the point when $F{l}^2/\pi$ is 0.5, after which it decreases. The reflection signal becomes 2% and 30% of the transmission signal for $F{l}^2/\pi$ values of 0.02 and 0.23, respectively, while the two are again equally large for the case with $F{l}^2/\pi$ being 0.5. Note that these results are valid regardless of the amount of NMM light.

From this figure, it can be concluded that although low finesse cavities can create only small molecular NICE-OHMS signals in reflection [the red dotted curve in panel (b)] there can be considerable NICE-OHMS background signals [the red dotted curves in panel (a)], created by the NMM light. As is concluded below, this is the basis for using the reflection detector as a reference detector for differential NICE-OHMS to reduce or eliminate the influence of the background signal on an assessment of the response of the molecules under study.

Another important conclusion that can be drawn from the figure is that, opposed to the case with transmission, the reflected molecular signal is not proportional to the reflected power, not even in the absence of NMM light.

Figure 3 displays the behavior of the factor $a$, given by Eq. (15), by which the reflection NICE-OHMS signal needs to be multiplied to yield a differential NICE-OHMS with a minimum of drift. The values of $a$ are presented as a function of $F{l}^2/\pi$ for the four different relative amounts of NMM light previously considered, i.e. for $\epsilon$values of 0, 0.1, 0.2, and 0.3. The three vertical dotted lines represent the same $F{l}^2/\pi$ values as in Fig. 2.

 

Fig. 3 Simulations of the factor a by which the reflection NICE-OHMS signal needs to be multiplied before being subtracted from the transmission signal to yield a differential NICE-OHMS signal that is not affected by any drifts. The four solid curves represent different relative amounts of NMM light, expressed in term of $\epsilon$ values, viz. 0, 0.1, 0.2, and 0.3 (as indicated). The vertical lines have the same meaning as in Fig. 2.

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Since the reflected NICE-OHMS signal usually has a poorer signal-to-noise ratio (S/N) than the transmitted [2,18], a large value of $a$ can lead to a low S/N ratio for the differential NICE-OHMS signal. Excessive values of $a$ are therefore in general not beneficial. Figure 3 shows that $a$ decreases monotonically with $F{l}^2/\pi$ for all presented values of $\epsilon$. It also shows that for $\epsilon$ being zero, $a$ is getting huge for small $F{l}^2/\pi$ values (in fact, infinitely large when $F{l}^2/\pi$ is zero). This behavior originates from the fact that, as is shown by Fig. 2 (a), in the absence of NMM light, the reflection NICE-OHMS background signal, which is used to compensate for the transmission NICE-OHMS background signal, goes to zero as $F{l}^2/\pi$ decreases. For more realistic values of $\epsilon$, however, it takes lower values and it stays finite for all $F{l}^2/\pi$ (e.g. for $\epsilon$ > 0.2, $a$ stays below 5). This suggests that a given amount of mode mismatch, in particular when small $F{l}^2/\pi$ values are used, is preferred for the differential NICE-OHMS technique.

4. Experimental

The experimental setup, which is schematically depicted in Fig. 4, is, to a large degree, identical to that in Ref [27]. One exception is though that the NICE-OHMS signal was detected also in reflection. In short, the light emitted from an erbium-doped fiber laser (EDFL) was sent to a fiber-coupled acousto-optic modulator (f-AOM) that frequency shifted the light around 110 MHz. The light then passed through a fiber-coupled polarizer (f-POL), a fiber-coupled electro-optic modulator (f-EOM) with a proton exchanged waveguide (to avoid residual amplitude modulation [26],), and a collimator (f-C) into free space. The light then propagated through a half-wave plate (λ/2), a mode-matching lens, a polarizing beam splitter cube (PBS), and a quarter-wave plate (λ/4) before it impinged onto the cavity. NICE-OHMS signals were obtained from both the transmitted and reflected light by use of the two photodiodes (PD₁ and PD₂) using the usual demodulation schemes for both arms.

 

Fig. 4 Schematic illustration of the NICE-OHMS instrumentation. EDFL, Er-doped fiber laser; f-AOM, fiber-coupled acousto-optic modulator; f-POL, fiber-coupled polarizer; f-EOM, fiber-coupled electro-optic modulator; f-C, fiber-coupled collimator; λ/2, half-wave plate; PBS, polarizing beam splitter; λ/4, quarter-wave plate; PD_1-2, photodiodes; PS, power splitter; PDH servo, servo for the Pound-Drever-Hall locking; DVB servo, servo for the DeVoe-Brewer locking.

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The cavity consisted of two high reflectivity mirrors, producing a cavity finesse of 55 000 (assessed by both cavity ring down and cavity mode width measurements), that were mounted on cylindrical piezo-electric transducers (PZT) and separated by a Zerodur spacer with a length of 39.4 cm, which gave rise to a cavity with a FSR of 380 MHz. The pressure inside the cavity was controlled by a turbo pump system and monitored by a vacuum gauge. Two photodiodes (PD₁ and PD₂) with a bandwidth of 1 GHz were used to detect transmitted and reflected light. To minimize the amount of etalons in the system, the positions of the optical components were carefully aligned to EIDs [30].

The EOM was fed with two radio frequencies at 20 and 380 MHz to generate sidebands for the Pound-Drever-Hall (PDH) locking [20] and the frequency modulation (FM). The error signal for the PDH locking, which was acquired by demodulating the reflected light with a 20 MHz reference signal by use of a mixer followed by a low pass filter, was sent into a proportional–integral–derivative (PID) servo that provided one low frequency and one high frequency feedback signal for control of the laser light, fed to the PZT-transducer in the EDFL and the f-AOM, respectively, providing a locking bandwidth of 100 kHz. The modulation frequency of the FM was locked to the FSR of the cavity by use of the DVB technique [21]. The error signal for this, which was obtained by demodulating the reflected light at 360 MHz, was fed back to the 380 MHz source via a DVB servo. To maximize the NICE-OHMS signal, the modulation index of the FM modulation was set to 1.0 [24]. The modulation index of the modulation for the PDH locking was taken as 0.1 since this gave a sufficient S/N ratio of the PDH error signal.

The calibration of the system was made in a few steps. First, based on tabulated data for the transition used, the absorption coefficient on resonance, ${\alpha }_0$, was calculated for the experimental conditions (temperature and pressure) used for the calibration. A NICE-OHMS signal was then measured in transmission under these conditions. This provides a value for the instrumentation factor, ${\eta }_{\text{V}}^{\text{tran}}$. The ratio of the two instrumentation factors, i.e. ${\eta }_{\text{V}}^{\text{refl}}/{\eta }_{\text{V}}^{\text{tran}}$, was then estimated from the ratio of the two analytical NICE-OHMS signals (transmission and reflection) by the use of Eq. (12) and the assessed value of $F{l}^2/\pi$ (see below). It was found that this ratio was 0.58, which was mainly attributed to losses in the RF power splitter in the reflection path.

5. Result

Before any experimental assessments could be performed, the system had to be characterized with respect to the cavity and mode matching parameters.

5.1. Assessment of cavity and mode matching parameters

To assess values of $F{l}^2/\pi$ and $\epsilon$, according to the Eqs. (19) and (20), all powers in the system had to be measured. Since it is not possible to measure $P_{\text{refl}}$ with a power meter while the laser is locked, it was assessed using the DC response of the reflection detector. For this, the DC response was calibrated using a power meter under unlocked conditions. A characterization then revealed that for an input power, $P_{\text{in}}$, of 2.05 mW, the reflected and transmitted powers, $P_{refl}$ and $P_{tran}$, were 0.42 and 1.02 mW, respectively. This implied that $F{l}^2/\pi$ could be assessed to 0.23 and $\epsilon$ to 0.19 [31].

5.2. NICE-OHMS measurements

Figure 5 displays, by the solid black curves in the two panels (a) and (b), a pair of typical NICE-OHMS signals from 10 ppm of C₂H₂ in N₂ at 100 mTorr, addressing the P_e(11) line at 1531.5877 nm, which corresponds to an absorption coefficient on resonance ${\alpha }_0$ of 1.6 × 10⁻⁸ cm⁻¹, at dispersion phase, detected in transmission and reflection, respectively. The corresponding dotted lines represent fits of the Doppler broadened response according the Eqs. (8) - (10). The lower windows illustrate the residuals. As can be seen from these, which basically only comprise the sub-Doppler response, the theory predicts the Doppler broadened response of the system well, both in transmission and reflection. This verifies the description of the reflection NICE-OHMS signal given above. The ratio of the NICE-OHMS signals detected in transmission and reflection was found to be 5.8.

 

Fig. 5 The panels (a), (b), and (c) show, by the black solid curves, the transmission, reflection, and differential NICE-OHMS signals (the latter using a value for $a$ of − 4.2) from 10 ppm of C₂H₂ in N₂ at 100 mTorr detected at 1531.5877 nm, which corresponds to an absorption coefficient on resonance,${\alpha }_0$, of 1.6 × 10⁻⁸ cm⁻¹. The panels (d), (e), and (f) illustrate the drift of the background over 2 hours, represented by the difference between two empty cavity NICE-OHMS signals measured at two instances, separated by 2 hours, for the transmission, reflection and differential signals respectively. The signal drifts in panel (d) - (f) are given relative to the peak to peak of the corresponding signals in the panels (a) - (c). In all cases, the dotted red curves show the corresponding fits. The assessed values of ${\alpha }_0$ retrieved by the fits in the panels (d) - (f), which are measures of the drift of the background signal, are given in lower parts of the panels. Note that this drift is one order of magnitude smaller for the differential NICE-OHMS than for the other two signals.

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To elucidate the drifts of the background signals, the difference between two empty cavity NICE-OHMS signals with a time delay of 2 hours, referred to as the relative background drift, was considered. The panels (d) - (f) show the result for the transmission, reflection and differential NICE-OHMS modes of detection, respectively. The signals are given relative to the peak to peak values of the corresponding molecular signals displayed in the panels (a) - (c). The dotted curves again represent fits of the Doppler broadened response, this time made with the same center frequency and line shape as for the analytical NICE-OHMS signal in the panels (a) - (c).

Based on the ratio of the instrumentation factors and the cavity and mode matching parameters given above, the value of $a$ was assessed by the use of Eq. (15) to be −4.2. Panel (c) in the same figure therefore displays the differential NICE-OHMS signal obtained from the signals shown in the panels (a) and (b) using this value of $a$. These panels show that the magnitude of the differential NICE-OHMS signal supersedes both of the other two signals (i.e. those detected in transmission and reflection).

The panels (d) and (e) illustrate first of all that the background drift consists of a structured background with a rather short periodicity (around 125 MHz, presumably originating from remaining etalons between the second cavity mirror and the reflection detector). The fits in the same panels also indicate that the background drift also contains a broader structure that resembles the analytical NICE-OHMS signal. Panel (f) illustrates that although the differential NICE-OHMS signal still comprises a certain amount of the high frequency structure, it has virtually no remaining broad structure that can be picked up by the fit. The values for ${\alpha }_0$ given in the lower parts of the panels (d) - (f) are the result from the fits and show that fitted absorption coefficient, which is a measure of the drift of the background signal in the system, is more than one order of magnitude lower for the differential than for the transmission or reflection mode of detection.

It is of interest to note that the structure of the background drift with short periodicity contribute virtually not at all to the fit. This is in agreement with the findings of Silander et al. which shows (by Fig. 4 in that work) that a fit of a Doppler broadened NICE-OHMS response picks up very little of etalons with a periodicity <200 MHz [27]. It is also of interests to note that these curves verify the predicted signs of the various signals; while the analytical and background signals have the same sign in transmission, they have the opposite in reflection. This illustrates well the basis of the differential mode of detection, viz. that by subtracting the two signals, the influence of the background can be reduced while the analytical signal will be enhanced.

To assess the performance of the differential mode of detection of NICE-OHMS for various measurement times, Fig. 6 shows the Allan deviation of the absorption coefficient retrieved from fits to NICE-OHMS spectra measured with an empty cavity at dispersion phase over 12 hours, here referred to as the noise equivalent absorption per unit length (NEAL), in terms of Allan-Werle plots [32]. The blue, red, and green curves show the NEAL of the transmission, reflection, and the differential NICE-OHMS signals, the latter constructed as $S_{\text{tot}}^{\text{tran}}-a{S}_{\text{tot}}^{\text{refl}}$ with the value of $a$ of −4.2. The dashed lines illustrate the white-noise (t^−1∕2) dependence for short integration times.

 

Fig. 6 The Allan-Werle plot of the transmission, the reflection, and two differential NICE-OHMS signals (blue, red, and green and black curves, respectively), where the latter two were constructed as $S_{\text{tot}}^{\text{tran}}-a{S}_{\text{tot}}^{\text{refl}}$, with a taken as −4.2 and −5.5, respectively.

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The data show first of all that although the reflection NICE-OHMS signal (the red curve) has a larger amount of white noise than the transmission NICE-OHMS signal (the blue curve), 9.7 × 10⁻¹³ cm⁻¹ Hz^−1/2 as compared to 4.1 × 10⁻¹³ cm⁻¹ Hz^−1/2, respectively, the long-term drifts of the two modes of detection (transmission and reflection) are similar, caused by the fact that their molecular to background signal ratios are similar.

It can then be noticed that differential NICE-OHMS based on the estimated optimum value of $a$, −4.2, gives rise to a signal that has a similar white noise response but significantly less long-term drifts than the transmission signal (a factor of 4 smaller for integration times above 500 s). The latter finding confirms that the presented novel differential methodology indeed is capable of providing NICE-OHMS signals with reduced amounts of drifts.

The white noise response of the differential and transmission NICE-OHMS signals are both a few times above the shot-noise limit, which can be estimated to be around 1 × 10⁻¹³ cm⁻¹ Hz^−1/2. This indicates that a large part of the white noises in the transmission and reflection signals are uncorrelated; if they would be fully uncorrelated, the white-noise of the differential NICE-OHMS signal would be larger (4.9 × 10⁻¹³ cm⁻¹ Hz^−1/2) than that actually measured.

However, as is illustrated by the black curve in the same figure, it was also found that differential NICE-OHMS signals with even less long-term drift could be obtained by using a larger value of $a$. The black curve in the figure therefore shows the differential NICE-OHMS signal with $a$ being −5.5. In this case the drifts were reduced with respect to the transmission NICE-OHMS signal even more, approximately by a factor of 7 (by a factor of 6 after 1000 s and a factor of 8 after 4500 s), however to the expense of a slightly larger white noise response (4.6 × 10⁻¹³ cm⁻¹ Hz^−1/2). This verifies the prediction that the differential mode of detection can be used to significantly reduce the influence of long-term drifts in NICE-OHMS.

The difference between the estimated $a$ factor and the one that experimentally was found to yield the smallest amount of drift of the differential NICE-OHMS signal can be attributed to a coupling between etalon effects in the transmission and the reflection arms (it was observed that artificially induced etalons in transmission can appear also in the reflection signal). This was neither covered by the model above, nor investigated further in this work.

This substantial reduction of long-term drifts implies that, for integration times in the drift regime (in this case > 100 s), the differential NICE-OHMS signal has almost one order of magnitude less drift and is stable almost one order of magnitude longer than the conventional NICE-OHMS signal (detected in transmission). This implies that a minimum NEAL, defined as the minimum of the Allan-Werle plot, of 4.7 × 10⁻¹⁴ cm⁻¹, could be obtained for an integration time of 170 s. This is a factor of 1.6 better than when the signal is detected solely in transmission, which was of 7.4 × 10⁻¹⁴ cm⁻¹ (detected over 50 s) and 1.4 times better than the previously best NEAL for NICE-OHMS obtained by a tunable laser, 6.6 × 10⁻¹⁴ cm⁻¹ detected over 150 s using a whispering-gallery-mode laser [11]. The minimum NEAL reported here is thus the lowest NEAL value so far demonstrated for a NICE-OHMS system based on a tunable laser.

6. Summary and conclusions

In this paper, we have predicted, simulated, realized, and demonstrated a novel realization of NICE-OHMS, termed differential NICE-OHMS, in which the influence of (long-term) drifts of the background signals that are created before the cavity is eliminated while the molecular NICE-OHMS signal is enhanced. It is based on simultaneous detection of NICE-OHMS signals in transmission and reflection (where the latter signal originates from an impedance mismatch of the cavity). The differential NICE-OHMS signal is constructed by a subtraction of the reflection NICE-OHMS signal (properly weighted) from the transmission signal.

A theoretical analysis based on expressions for the transmitted and reflected powers as well as both molecular and background NICE-OHMS signals in both transmission and reflection from a cavity constructed by two identical mirrors was performed and indicated that the influence of the drifts of the background signal can be eliminated (or strongly reduced) from an NICE-OHMS assessment if the differential NICE-OHMS signal is constructed by subtracting the reflection NICE-OHMS signals weighted by a given factor from the transmission signal. The analysis also provided an explicit expression for the weighting factor required to obtain a differential NICE-OHMS signal to which there are no contributions from the drifts of background signals created before the cavity.

A differential NICE-OHMS system, based on an erbium-doped fiber laser and a cavity with a finesse of 55 000, was used to verify the predictions made by simulations based on the expressions derived. The instrumentation utilized incorporated a number of the previously found measures to reduce the influences of background signals in NICE-OHMS, e.g. the use of an EOM with a proton exchanged waveguide that cannot support the propagation of light along the ordinary axis [14]; an AOM for improved locking of the frequency of the laser to a cavity mode [15], and EIDs [16]. It was demonstrated that the differential realization of NICE-OHMS could reduce long-term drifts of the background signal with respect to the ordinary realization of NICE-OHMS (i.e. with detection either in transmission or reflection) even more, and do it significantly (in our case up to a factor of 7). The remaining drifts are assumed to originate from background signals that are created in paths of the light that are not common to the transmission and reflection detection arms.

As a consequence of this, it was found that the differential NICE-OHMS system could demonstrate a noise equivalent absorption per unit length (NEAL), defined as the minimum of the Allan-Werle plot of the absorption coefficient retrieved from fits of a conventional expression for NICE-OHMS to measured NICE-OHMS signals, of 4.7 × 10⁻¹⁴ cm⁻¹, obtained for an integration time of 170 s. This is a factor of 1.4 times better than the previously best NEAL for NICE-OHMS achieved by a tunable laser, which was 6.6 × 10⁻¹⁴ cm⁻¹ detected over 150 s using a whispering-gallery-mode laser [11]. The value reported here therefore represents the lowest NEAL value so far demonstrated for a NICE-OHMS system based on a tunable laser.

It is assumed that differential NICE-OHMS, with its improved NEAL, can enhance the use of the NICE-OHMS for various types of applications, in particular for trace gas analysis when long measurement procedures are needed (e.g. when gases need to be monitored over appreciable times) or when the systems are suffering from large amounts of background signal (which, for example, can take place under field applications). This opens up for assessments of rare gas molecules, e.g. isotopologues with doubly or triply substituted isotopes or those that are radioactive, or transitions with exceptionally small dipole moments.