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Broad-band Mach-Zehnder interferometry is analytically described and experimentally demonstrated as an analytical tool capable of high accuracy refractive index measurements over a wide spectral range. Suitable photonic engineering of the interferometer sensing and reference waveguides result in sinusoidal TE and TM spectra with substantially different eigen-frequencies. This allows for the instantaneous deconvolution of multiplexed polarizations and enables large spectral shifts and noise reduction through filtering in the Fourier Transform domain. Due to enhanced sensitivity, optical systems can be designed that employ portable spectrum analyzers with nm range resolution without compromising the sensor analytical capability. Practical detection limits in the 10−6-10−7 RIU range are achievable, including temperature effects. Finally, a proof of concept device is realized on a silicon microphotonic chip that monolithically integrates broad-band light sources and single mode silicon nitride waveguides. Refractive index detection limits rivaling that of ring resonators with externally coupled laser sources are demonstrated. Sensitivities of 20 μm/RIU and spectral shifts in the tens of a pm are obtained.
©2014 Optical Society of America
1. Introduction
Spectrally resolved devices usually involve ring resonators, microspheres or whispering galleries that rely on high Q factors to achieve wavelength resolutions of the order of a fraction of a pm. Their design, however, while guaranteeing sharp resonances limit the refractive index (RI) sensitivity in the range from 30 to 700 nm/RIU [1], the lower limit corresponding to extremely high Q factor microsphere resonators. As a result, their practical detection limit (DL) in liquid samples is about 10−6 RIU [2,3] if the effects of temperature fluctuations are included [2]. Another aspect of such resonators is the “focusing” of the information at the resonance wavelength (plus or minus few pm) while most of the spectral free range goes with no interrogation. In single optical path resonators the resonance wavelength λr and its shift δλ to a change δΝs in the sensing waveguide effective index Νs are related through [4,5]
where Νgs is the sensing waveguide group effective index. Equation (1) gives also the spectral sensitivity in nm/RIU if the right hand side numerator is replaced by the where nc is the cover medium RI. Ιn most cases Νgs ≈Ns and, consequently, the resonator sensitivity is limited to well under λr/Ns (nm/RIU).If on the other hand the Mach-Zehnder (MZ) configuration shown in Fig. 1 is selected as the interferometer, then due to the differential nature of the device Eq. (1) becomes [4,6]
where now the denominator term ΔNrs is the difference between the reference and sensing arm effective indices Nr and Ns, respectively, ΔNrs = Nr- Ns. In the right hand side in Eq. (3) the denominator term is two orders of magnitude less than the denominator in Eq. (1) and, as a consequence, the sensitivity is equally higher. Seeing the MZ as an optical filter, its transfer function is the ratio of the output Iout over the input Iin power:where φ(λ) is the phase difference between the two arms, L is the exposed sensing arm length, and ΔΚrs is the MZ interferometer propagation constant difference:In Eq. (4) we assume that the optical power waveguided in each of the two arms is the same. The transfer function T(λ) ιs governed by this propagation constant difference and the interaction length L through the cosine expression in Εq. (4), that is in terms of well known parameters, as opposed to the ring-resonator type of devices where only the resonant frequencies are known as functions of effective indices. As is later outlined, this sinusoidal dependence and the much higher sensitivities due to Eq. (3) allow accurate measurements of index changes even if nanometer resolution spectrometers are employed, provided T(λ) and its analyte induced changes are analyzed over a broad spectrum. The following section outlines how the wavelength and analyte derivative of the propagation constants define the output spectrum oscillatory behavior and the sensitivity. Sections 3 and 4 provide the noise and detection limit analysis, while in section 5 a broad-band monolithic silicon MZ device is presented and its performance as a RI sensor is demonstrated.
Fig. 1 A Mach-Zehnder interferometer with a broad-band input and an output modulated according to Eq. (4). The green lines are monomodal waveguides. The sensing arm exposed to the cover medium has a length L. Pink indicates the overcladding area.
2. Sinusoidal responses
We assume, and later prove, that through proper photonic engineering of the two arms the propagation constant difference can be made nearly linear with respect to λ in the spectral range of interest:
The terms a and b are the slope and the zero λ intercept of the ΔNrs(λ)/λ function. Considering the MZ interferometer as a resonator, the Free Spectral Range (FSR), FWHM and Q factor become 1/(La), 1/(2La), 2Laλ, respectively. To a first look, the gains in sensitivity in (2) seem to be lost due to the effect on Q of the slope factor, a, that decreases with the diminishing effective index difference, ΔNrs(λ). Here, however, we do not track a sharp resonance at a specific wavelength but rather a broad-band spectrum is recorded through spectrometers the spectral resolution of which needs only be much smaller than the free spectral range. The spectral shifts are derived from the entire recorded spectrum so that effective resolutions in the pm range are possible even though portable commercial spectrometers with nm range resolution are employed, as is pointed out in the next sections. The sinusoidal nature of the transfer function in Eq. (7) facilitates the spectral shift study. The use of Discrete Fourier Transform (DFT) in such an output spectrum leads to a dominant peak in the wavenumber domain and the phase is identified with the argument of the complex DFT value at the main peak while white noise is filtered by ignoring non resonant frequencies in the spectrum. The peak will appear at 2πaL and the phase at the peak is 2πbL. The oscillation frequency determining slope factor, a, is polarization dependent and two distinct peaks will appear at different positions in the DFT domain when both polarizations are multiplexed at the input of the interferometer. Peak splitting allows simultaneous demultiplexing of either polarization with obvious benefits in the effective cover medium characterization, as demonstrated in the experimental section.By considering the m order peak, parameters a and b in Eq. (7) are expressed in terms of L and the Free Spectral Range (FSR = λm-λm-1):
The expression for a in Eq. (9) indicates how to calculate the slope factor a through the oscillation period in the MZ spectral output. If the wavelength derivative of the propagation constant difference defines the frequency of the transfer function, it is the derivative of the same variable with respect to the analyte index that defines sensitivity. Here, the analyte is taken as the cover medium refractive index, nc. A differential change 2π(δNs/λ) in δΚs due, to a change of δnc, results in a spectral shift δλ of the transfer function T(λ) in Eq. (7):And the spectral shift becomesDepending on the waveguide geometry [6], the shift is a red one when ΔNrs(λ)/λ is an increasing function of λ (a>0, λm>λm-1) or a blue one in case of a decreasing function(a<0, λm<λm-1). If δΝs/λ is assumed constant in the spectral range of interest, then the sinusoidal output spectrum will undergo a solid shift δλ as given by Eq. (11). The spectral shifts then, can accurately be determined by monitoring the changes of the phase of the complex Fourier value at the peak wavenumber, 2πaL.As stated, the waveguides are assumed to be properly engineered so that the slope, a, remains almost constant over a wide spectral range. This is shown in Fig. 2(a)-2(b) where numerically calculated ΔNrs(λ)/λ values are shown for the two polarizations in a MZ made from silicon nitride with an exposed and reference arm thickness of 150 and 167 nm, respectively. The monomodal rib waveguides had a width of 1.25 μm and an etch depth of 4 nm. Cladding of silicon dioxide is assumed except for the sensing arm where the overcladding was water. The selection of the waveguide thicknesses was such as to guarantee more or less uniform slopes in the spectral range between 550 to 900 nm, Fig. 2(a)-2(b), and will be further elaborated in the experimental section. The sign of the slope and the linear region position depends mainly on the waveguide geometry and the reference and sensing waveguide thickness differential [6]. In the spectral region of interest and in addition to wavelength linearity, the difference between the black and red curves in Fig. 2(a)-2(b) is the cover medium induced δNs/λ differential caused by the nc change of 0.01 RIU. To a good approximation this differential is constant with λ, especially for the TE polarization. Additionally, the slope for TM is about 2.5 times higher, Fig. 2(b), than the slope for TE, Fig. 2(a), which explains why the output spectral oscillations for TE, Fig. 2(c), are 2.4 times lower than that of TM, Fig. 2(d). Therefore, if both polarizations are multiplexed at the MZ input, then demultiplexing is possible in the Fourier transform domain. Α shift δλ will translate in a phase change (δφp):
Fig. 2 Simulated propagation constant difference, cover medium effects and output spectra for the two polarizations for a MZ interferometer with L = 600μm and geometrical characteristics as described in the main text. In (a) and (b) the ratio ΔNrs/λ is shown for nc = 1.33 (black) and nc = 1.34 (red) for the TE (a) and TM (b) polarization. The fundamental mode effective indices Nr, Ns were obtained through simulations with FemSIM software package (SYNOPSYS). The chromatic dispersions of the nitride core and oxide claddings were accounted for through independent spectroscopic ellipsometry measurements in separate nitride and oxide thin films deposited on silicon. The vertical shifts with increasing nc in (a) and (b) define the δΝs/λ ratio in Eq. (11). The decreasing propagation constant difference with λ results in blue spectral shifts when the cover medium refractive index goes from 1.33 (black) to 1.34 (red) as shown in (c) and (d) for TE and TM. The TE shift is about a period while the TM shift is 2.4 periods.
The last equation can be derived from Eq. (11) by observing that a spectral shift equal to λm-λm-1 corresponds to a 2π phase shift. Therefore, the interferometer phase sensitivity or the δφp/δnc ratio is the product of L times the sensing arm propagation constant derivative with respect to nc. The phase shift sensitivity term depends only on the sensing arm. The reference arm effective index is not involved in the phase sensitivity but determines the oscillation frequency of the spectral fringes through ΔNrs and the slope factor a.
Equation (12) is the same as the expression for the monochromatic MZ interferometer. However, in the latter case the phase starting value is not known, unless a peak to peak transition is induced, something highly unlikely for small RI changes. On the contrary, here the phase is directly readable on the output spectra and no phase ambiguity and signal fading issues are present, anymore, as performance limiting intrinsic factors characteristic of monochromatic MZ interferometers [7,8].
The above equations indicate the differences of a spectrally resolved Mach-Zehnder interferometer compared to the standard ring or microsphere resonators. In our case the observable is not δλ but rather a phase difference and we have two parameters in our disposal: ΔNrs(λ) which determines the slope factor and L. Length L determines the sensitivity, while the slope factor determines the Free Spectral Range and the Q factor, in association with L. In a standard resonator the observable is δλ and is determined by sweeping the bias of the laser diode to change the emission wavelength while monitoring the resonator output to observe the sudden drop or peak. The wavelength range interrogated is a part of a free spectral range and usually few pm. The resonator length L is rather irrelevant as far as sensitivity is concerned. Here, the measured phase shift is proportional to L while the entire spectrum is recorded through commercially available spectrometers with spectral resolution of the order of 1 nm as a result of the wide spectral shifts and the broad-band nature of the light source. In addition, the thermal drift effects of the core and undercladding refractive index are cancelled out because of the differential nature the effective index involved, ΔNrs(λ).
3. Fundamental phase noise calculations
Here we calculate the phase noise due to the intrinsic shot noise in each of the spectrometer channels. Let s(λ) be the output signal of the MZ interferometer and n(λ) the spectral noise. For the Discrete Fourier Transform in the wavenumber domain, k, we assume a number of M spectrometer channels with a unit width of Δλ = W/M, where W is the spectral bandwidth of the input light source. The input photon flux is assumed constant over W, resulting in:
where i runs from 1 to M, λi = iΔλ + λ0, and A is the input photon count per channel per measurement (spectrum recording). The DFT for the signal and the noise is: Again the sums run from 1 to M. Subscripts R and J indicate the real and the imaginary parts of the transforms. The DFT of the recorded signal is S(k) + N(k). The phase at the peak k0 = 2πaL is given by:where the approximation is meant in the sense that the peak signal is well above the noise level. For the sake of simplicity, and without loss of generality, we will assume that b = 0 in Eq. (13) and therefore Sj(k0) = 0. Consequently, at the peak k0 the phase noise is:and the standard deviation of becomes:where σΝj is the standard deviation of the noise imaginary part. From Eq. (15):Taking the shot noise of the signal as the dominant noise source then:and from Eq. (19) and Eq. (13) with b = 0:where we assume M>>1. Similarly, from Eq. (13) and Eq. (14), Sr(k0) = (1/4)AM and through Eq. (21) and Eq. (18):Here, we note that AM/2 is the total number of photons detected in a single recording of the spectrum. If instead of a sharp peak at k0 = 2πaL the resonance is spread over a number of points δM in the DFT domain, then the expression in Eq. (22) becomes:where the phase is calculated as an average over the δM peak points.4. Limits of detection
The detection limit, DL, in a resonator is proportional to the resolution R, the smallest detectable wavelength shift, and inversely proportional to the sensitivity S = δλ/δnc [2]:
In our case the observable is the phase shift. Consequently, the relation above holds with R being the smallest detectable phase shift, 3σφn, where σφn is the rms phase noise value. The phase sensitivity S is given by (12):In addition to the fundamental phase noise, σφsn, due to the shot noise of the spectrometer signal itself, other factors like the thermal fluctuations and the spectrometer finite resolution contribute to σφn. From Eqs. (24) and (25):If the spectrometer channel shot noise induced σφsn is dominant, then the detection limit is set by the fundamental shot noise, σφn = σφsn, and DL converges to the intrinsic DLi value:The equation above shows that the intrinsic detection limit is inversely proportional to the sensing arm length (L), the waveguide sensitivity (δΝs/δnc) and the square root of the photon count in a single spectrum recording. Using a commercial portable spectrometer having a 16 bit vertical resolution with a thousand channels in the broad emission band of the light source, then the available counts in a single measurement are ~64000x103 and the phase intrinsic noise, σφsn as in Eq. (22) becomes (4000)−1 or 0.25 mrad. The phase noise is translated into a spectral noise through Eqs. (11)-(12) as:
From Fig. 2, a = 0.410−7 (nm)−2 (TE) and a = 1.410−7 (nm)−2 (TM) and Eq. (28) translates to pm values for the spectral noise, as shown in Table 1 where the phase and spectral noises are listed for two L values. Along with σφsn, the noise σφr related to the finite resolution of the spectrometer (0.43 nm wavelength resolution, 16 bit vertical resolution) is listed in Table 1. This noise is calculated as the uncertainty on the δφp/δnc ratio when the δnc values vary in the range from 10−7 to 10−2 RIU. Due to the Fourier transform domain processing of the spectral data, the phase is calculated from the entire spectrum so that the expected uncertainty due to the finite spectrometer resolution is suppressed to well under the shot noise related limit φsn. As far as the temperature fluctuations induced spectral noise, σλt, the MZI configuration is inherently less susceptible compared to the ring resonator case due to the compensating effect of the reference arm. The MZ interferometer temperature induced noise is due to the temperature dependence of the sensing arm cover medium, nc. Even in the case of the same effective index sensitivity to temperature fluctuations as in ring resonators and by adopting the σλt = 10 fm for the standard deviation value of the ring resonator case [2], the corresponding MZ value is σλt times the ratio of the MZ spectral sensitivity (nm/RIU, Eq. (3) over the ring resonator spectral sensitivity in Eq. (1). Then, from Eqs. (1), (3), (6) and (28) the temperature fluctuation related noise expressed in phase terms isThe detection limits listed in Table 1 are derived from the total phase noise σφn calculated as the geometrical sum of σφsn, σφr and σφt and by taking into account the waveguide sensitivities from Fig. 2(a)-2(b): The detection limits predicted by Eq. (25) are in the 10−6-10−7 RIU range depending on polarization and exposed arm length (L). The temperature related noise tends to dominate at longer L because σφt is linear with L, Eq. (29), while σφsn is independent from L, as Eq. (22) shows. If L is long enough then the limiting factor is the temperature noise λt, then from Eqs. (10) and (15)Ιn Eq. (32) the ratio in the right hand side is the inverse of the spectral sensitivity of the sensing arm as a resonant structure. The sensing arm derivative is 0.36x10−5 and 0.34x10−5 (nm)−2 for TE and TM, respectively, therefore the temperature imposed detection limits are 0.63x10−7 and 0.26x10−7 for TE and TM, respectively.
Table 1. Root mean square values of the various noise sources and detection limits for the two polarizations and two values of L. The spectrometer has a 16 bit vertical resolution and a thousand channels in the spectral region of interest.
From Table 1 one concludes that 2 mm of length are enough to approach the temperature fluctuation limit. If the temperature fluctuation induced noise is somehow suppressed, then the intrinsic detection limit DLi of 1.58x10−7 RIU can be obtained from Eq. (27) for TM and a 2 mm sensing arm. If this limit is to be brought to 10−8 then an L value of 25 mm would be required. The resulting detection limits are equal or even better than the ones obtained with high Q resonators [2] due to the high sensitivity and the processing of the entire spectrum that suppresses noise. Additionally, in MZI is a single pass device and the spectral sensitivity and quality factor do not degrade with the cover medium buffer, as the case is with the quality factor in very sharp resonators relying on high photon circulation [2].
5. Experimental results-monolithic silicon microphotonic MZ device
The theory presented above will be experimentally verified through an interferometric device has as the main building block the monolithic optocoupler shown in Fig. 3. Light emitting diodes (LEDs) and silicon nitride planar waveguides are monolithically integrated and optically coupled on a silicon die, Figs. 3(a)-3(b). The LEDs are silicon avalanchediodes that emit white light, Figs. 3(c)-3(d), when biased beyond their breakdown voltage [9].The integration of the active and passive optical components is achieved by employing mainstream planar technology and self-alignment techniques that allow efficient coupling between the LED and the waveguide [10–12]. The LED P++ emitter is implanted through the silicon nitride layer, the boron atoms compensate the preexisting base N+ implant so that the metallurgical junction of the LED is precisely placed under the up-going segment of the waveguide. The SiO2 spacer next to the LED provides for the smooth waveguide bending and minimizes bending losses, Fig. 3(a). The light is first coupled to a multimode silicon nitride waveguide, Figs. 3(b)-3(d), then goes through a mode filter followed by the MZ interferometer before it delivers the modulated signal to the spectroscopic detector through an external fiber, Figs. 3(a)-3(b). A fluidic compartment on top of the exposed sensing waveguide, Fig. 3(a), allows changing the cover medium. The realization of such an optocoupler addresses a long standing issue of optoelectronics, as far as light source integration and coupling to submicron waveguides is concerned [10–12]. Here, it is extended so that monomodal rib waveguides, Figs. 3(e)-3(f), are employed to enable interferometry.
Fig. 3 Monolithic integration and basic optical configurations of the optocoupler device. (a). Schematic showing the LED, 1, the MZ interferometer, 2, and the silicon nitride rib waveguide, 3. Also shown is the waveguide bending over the SiO2 spacer, 4. The top cladding layer is removed over the sensing arm to expose it to the analyte (b). Photonic pathway and mode filtering. The waveguide starts as a 2μm wide multimode strip waveguide and then connects to the mode filter that rejects higher order modes and keeps the fundamental within the following shallow etched rib waveguide. The single mode rib waveguide is 1.25 μm wide and etched up to 4 nm. The mode filter consists of two back to back tapers: from 2-to-8 μm (strip) and 8-to-1.25 (rib). (c). Recorded spectra of the waveguided TE and TM modes at the emitting edge. (d) The LED and the emitting junction (white arrow) coupled to the multimode strip waveguide heading down. (e). The rib waveguide schematic. (f). The emitting edge of the single mode waveguide. Through a polarizer, the TE modes are shown along with the core width, 1.25μm, and the undercladding and overcladding thicknesses, S1 = 3 μm and S2 = 2μm. The slab between the overcladding and the undercladding is the nitride core.
The phase tracking measurement scheme and the monolithic LED eliminates issues such as fluctuations in external light insertion losses due to vibrations and mechanical misalignments in the coupling optics.
The MZ interferometer of Fig. 3 with an exposed and reference arm thickness at 150 nm, and 167 nm, respectively, and a length L = 600 μm has a spectral response shown in Fig. 4. The exposed sensing arm was thinned down from the as deposited 167 nm to 150 nm by inserting the completed chip into a reactive ion etching reactor and eching away 17 nm of the silicon nitride core. The etching depth was selected so that either polarization had a clearly defined sinusoidal spectral response. The overcladding SiO2 layer would protect the reference arm from the etching process. The spectral output obtained without polarizers, Fig. 4(a), clearly exhibits two characteristic frequencies, a lower one for the TE and a higher for the TM since both polarizations are excited at the input. After DFT analysis two well separated regions are obtained in the wavenumber domain, Fig. 4(b). Each region contains one peak, the lower corresponds to TE and the higher corresponds to TM polarization. Through signal splitting in the Fourier transform domain the two polarizations are deconvoluted from the composite signal and found in good agreement with the signals obtained by employing polarizers and separate recordings, Figs. 4(c)-4(d). In either the TE or the TM case the positions of the spectrum peaks and valleys are the same for the polarizer involving measurements as well as for the signals deconvoluted from the composed response. The amplitude differences observed are due to the fact that in the emission spectral range the polarizer transmittance varies between 80% and 65% and increases with the wavelength. Also, the insertion of the polarizer requires refocusing between the chip emitting edge and the collecting lenses, which affects the fiber coupling losses and envelope function of the signal. In any case, such differences have no effect on the signal minima and maxima positions that bear the phase information.
Fig. 4 Total signal monitoring and polarization separation through signal splitting in the DFT domain for a MZ with L = 600μm. The spectrometer employed was Maya Pro 2000, Ocean Optics. The LEDs are driven by a current source at 10 mA. (a). Total spectral response (TE + TM). Two frequencies are apparent. (b). DFT analysis of (a) showing the two distinct regions, TE and TM. The wavenumber is multiplied by the spectrometer spectral bandwidth (200-1100 nm) to express the peaks in DFT integers. (c). TE spectra as obtained with a polarizer and as obtained from signal splitting in the DFT domain. (d). TM spectra as obtained with a polarizer and as obtained from signal splitting in the DFT domain.
The spectral shifts due to a cover medium change are obtained from the change in the argument of the complex DFT value at the point of the peak intensity for either polarization. A fluidic compartment was mounted on top of the chip and through pumping, the cover medium changed from water to a 16.66% isopropanol solution in water (1 volume of isopropanol to 5 volumes of water) and then back to water. The supply rate was set at 20 μl/min. The RI step is 1.2x10−2 RIU, as independently measured by a commercialrefractometer. The cover medium transition is shown in Fig. 5, where the shifts of the deconvoluted signals are shown, Fig. 5(a), and compared with the ones obtained through polarizers and separate recordings, Fig. 5(b), for the same cover medium change. The agreement is excellent and demonstrates the potential of this interferometry as a dual polarization analytical tool. The sensitivity formula in Eq. (25), with a L = 600 μm and SnTE and SnTM as in Eqs. (30)-(31), predicts phase shifts 7.46 and 17.18 rads for TE and TM, respectively, in very good agreement with Fig. 5.
Fig. 5 Phase responses for TE and TM for a cover medium transition of 1.2x10−2 RIU. The transition starts at 100s and ends in 450s. The results are obtained by processing the real time spectral measurements. The plot in (a) presents the TE and simultaneous TM responses obtained by polarization deconvolution through signal splitting in the DFT domain and peak argument tracking. The plot in (b) presents the TE and separate TM responses as phase shifts monitored separately by employing a polarizer and two successive measurements.
To increase sensitivity and demonstrate the refractrometric potential of the device, another MZ with L = 2 mm was chosen while, now, the exposed arm thickness was 155 nm to ensure that the TM oscillation period is well above the spectrometer resolution (0.43 nm). The spectral response and shifts due to cover medium change is demonstrated in Fig. 6. Spectral shifts of 34 nm for RI change of 1.8x10−3 RIU are obtained for TE, corresponding to a sensitivity of 19 μm/RIU, which is one to two orders of magnitude higher than ring resonator sensitivities [1]. The sensitivity in terms of spectral shifts for TM is smaller due to the much higher oscillation frequency in the TM spectrum which counterbalances the higher phase sensitivity. The sensitivity plot as a phase shift against cover media RI was obtained by pumping over the chip water solutions with varying isopropanol content and refractive index. Again the transition from water to water- isopropanol solution was chosen with isopropanol volume dilutions starting from 1/6 and going down to 1/20.000. The results are shown in Fig. 7. The highest dilution corresponds to a RI change of 3.6x10−6 RIU. Both the TE and TM simultaneously deconvoluted shifts are displayed as a linear function of the isopropanol dilution. The maximum phase change in Fig. 7 for is 23.5 and 56.53 rad, for TE and TM respectively, and for a 1/6 dilution with a RI change of 1.2x10−2 RIU. These phase shift values divided by the similar shifts in Fig. 5, with the same RI change but with L = 600 μm, give ratios of 3.24 (TE) and 3.36 (TM), very close to the length ratio (2000/600 = 3.33). This is a confirmation of Eq. (25) in the sense that phase sensitivity does scale with L.
Fig. 6 Spectral responses for the TE (a) and TM (b) polarization for water (black line) and water-2.5% isopropanol solution (red line). Here, δnc = 1.810−3 RIU and L = 2mm. The blue shifts in TE is nearly 34 nm (about 60% of a period) while in TM is 14 nm (about one and a half period). In (a) the shot signal noise is evident in the spectrum peaks as opposed to the valleys.
Fig. 7 Calibration curve as phase shift per dilution ratio for L = 2mm. The highest dilution ratio (1/20000) corresponds to RI change of 3.610−6 RIU.
At the maximum dilution (1/20000) the average value of TM phase shift obtained from Fig. 7 is 1.8x10−2 rads which, through Eq. (28) and a spectral period λm-λm-1 of 7.9 nm, Fig. 6(b), corresponds to a spectral shift of 22.6 pm, way below the spectrometer resolution of 0.43 nm. Τhe maximum dilution, δnc = 3.6x10−6RIU, is still visible especially in the TE polarization, as shown in Fig. 8. The rms value of the phase noise, σφn, is measured at 1.8 mrad and 7.2 mrad for TE and TM respectively. We now compare these numbers to the theoretical ones derived in section 3. By applying σφsn = 2/(AM)1/2 to TE with A = 3000 (average counts/channel) and a bandwidth of 200 nm, as obtained from Fig. 6(a), with a channel number M = 200 nm/0.43 nm, a calculated σφsn value of 1.7 mrad is obtained. For TM, Eq. (23) is employed for the calculation since the resonance peak is spread over 8 points, as shown in Fig. 4(b). Now A = 1800 (average counts/channel), M = 200nm/0.43nm, Fig. 6(b), and σφsn = 6.5 mrad. The calculated values are in very good agreement with the measured ones, confirming the noise analysis earlier presented.
Fig. 8 Phase response in the case of isopropanol dilution 1/20.000. The in arrow indicates the transition water to isopropanol solution and the out arrow the reverse transition.
The previous noise figures also point out that for signal levels an order of magnitude lower than the one assumed in Table 1 it is the shot noise of the spectrometer that determines the detection limit. With the experimentally determined phase sensitivities at S = 1958 rad/RIU (TE) and 4710 rad/RIU (TM), Fig. 7, and with the measured rms noise levels as above, the resulting detection limits, DL = 3σφn/S, are 2.75x10−6 RIU (TE) and 4.58x10−6 RIU (TM). The later numbers indicate the sensitivity limits of the monolithic devices presented here for the specific L value of 2 mm. Higher sensitivities would require either higher signal levels of longer sensing arms. We note that the above phase sensitivities, if expressed per unit of exposed arm lengths, are both higher than the corresponding sensitivities from MZ interferometers based on silicon nitride slot waveguides [13].
6. Conclusions
A broad-band Mach-Zehnder device is proposed and demonstrated as an analytical tool capable of high accuracy RI measurements over a wide spectral range and for either polarization. The broad-band nature of the device allows deconvolution of multiplexed TE and TM modes through Discrete Fourier Transform of the spectral response, while its differential topology results in high sensitivities. The sensing and reference arms are engineered so that the propagation constant differences are linear functions of the wavelength resulting in sinusoidal TE and TM spectra with substantially different eigen-frequencies. In the wavenumber domain the two polarizations exhibit sharp resonances. The phase at these peaks is a sensitive function of exposed arm cover medium while noise is filtered out by ignoring nonresonance wavenumbers. The high spectral sensitivity makes possible the use of portable spectrum analyzers with nm range resolution as detectors while the DFT domain processing of the output results in pm range spectral resolutions. The intrinsic phase noise scales as the inverse of the square root of the photons recorded in a single measurement while the phase sensitivity scales with the exposed arm length. Practical limits of detection in the 10−6-10−7 RIU range can be achieved by increasing the sensing arm length until the temperature fluctuation noise limit is reached. No deep submicron lithography is required to couple the waveguide to the resonant structure. Finally, such a device was realized on a silicon microphotonic chip with monolithically integrated broad-band light sources linked to single mode silicon nitride waveguides. Sensitivities of nearly 20 μm/RIU are obtained by phase tracking of the peaks in the Fourier Transform domain. Refractive index detection limits in the vicinity of 10−6 RIU are demonstrated.
Acknowledgments
This work was supported by the EU-funded Projects “PYTHIA” (FP7-ICT-224030) www.pythia-project.eu and “FOODSNIFFER” (FP7-ICT-318319) www.foodsniffer.eu.
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