Optical micro-knot resonator hydrophone

J M De Freitas; T A Birks; M Rollings

doi:10.1364/OE.23.005850

1. Introduction

There has been a lot of interest recently in optical micro-resonators for sensing applications [1–4] that has attracted the attention of the defence industry, particularly for naval applications such as flank and towed arrays. Optical micro-knots, micro-loops and micro-rings are a class of miniature resonators that allow light to achieve resonance in closed circular optical paths. It is not unusual for a distinction to be made within this broad class of devices into two large subgroups covering micro-rings and micro-knots. Micro-rings are fabricated using CMOS facilities which allow photonic circuits to be built on Si-based platforms using well-tried electronics fabrication techniques [4]. On the other hand, micro-knots and micro-loops are generally fabricated from optical micro-fibres to form small loops or simple knots [1–3,5]. Caspar and Bachus [6] were some of the earliest workers in 1989 to realize a resonant loop of diameter 2 mm with fibre diameter of 8.5 μm operating at 1550 nm. They demonstrated a free spectral range of 30 GHz with a finesse of about 3. These devices have been targeted for their potential applications, for example, as channel dropping filters [7], add-drop multiplexers [8], delay lines [9], refractive index sensors [10,11], bio-medical sensors [12], accelerometers [13], ultrasound detectors [14–16], and in-air acoustic sensors [17,18].

This paper presents preliminary results of what are believed to be the first demonstration of an underwater acoustic sensor using an optical micro-knot resonator. In Section 2, we will discuss the theoretical basis of the acoustic detection mechanism, whereas in Sections 3 and 4 we will present the experimental setup, results and a discussion. The discussion will focus on two aspects, namely, the inferences we can draw from the nonlinear least squares parameter fits to the micro-knot transmission profile, and the sensitivity of the embedded micro-knot as an underwater acoustic sensor.

2. Optical micro-knot resonator

Figure 1(a) shows an optical micro-knot resonator. These sensors are formed by tapering (heating and stretching) a single mode telecoms optical fibre down to very small diameters (of the order of 1 μm) then forming a knot with a diameter of the order of a few hundred μm. Light entering the micro-knot at A (Fig. 1(a)) is split in the coupling region between the ring and the output at B. Moreover, additional light entering A will combine with light leaving the ring and travel towards B, while feeding the ring with more light. Light in the ring will go into resonance if the length of the fibre along the circular path is such that the round-trip phase is an integer multiple of 2π. At resonance, light at the resonant wavelength emerging from the ring is completely out of phase with that entering the ring at A, so that the collective interference of the different fractions of light results in a characteristic dip in the transmissionspectrum of the output signal at B (Fig. 1(b)). The minimum of the dip depends on the balance between coupling and absorption losses. When these match, the output intensity can fall to zero, sometimes referred to as ‘critical resonance’.

Fig. 1 (a) Optical micro-knot resonator. (b) Transmission intensity profile.

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The sensitivity of the micro-knot depends on the instantaneous path length change, or circumferential strain, in the loop when exposed to an acoustic wave-field. The circumferential strain is influenced by both the Poisson ratio ν and the Young’s modulus E of silica and its encapsulant; in fact, for thin circular ring geometries, the circumferential (or, hoop) strain is inversely proportional to E and directly proportional to (2 − ν) [19]. Moreover, it is known that the sensitivity of the sensor can be enhanced by embedding it in a material whose Young’s modulus and Poisson ratio are both lower than that of silica [20].

A secondary elasto-optic effect is also possible where the refractive indices of the core and cladding of the waveguide undergo change [20,21]. Together they cause a corresponding shift in the resonance wavelength which can be observed either on a spectrometer, or via the phase change measured in an interferometer. To date, it is not known which effect is dominant or whether they are of the same order of magnitude for micro-fibres, although there are indications that path length changes may be dominant; e.g. for the larger 125µm diameter fibre, the elasto-optic contribution is about 22% and in the opposite sense of that due to the physical path length change effects [22].

A model could be built up around these ideas as follows: The input electric field at A is E_ie^jωt, where E_i is the wavefield amplitude, ω is the angular frequency of the input wavetrain, t is time and j is the fundamental complex unit. For any input wavetrain the fraction of the wavefield towards B in the first instance is $\sqrt{ρ} E_{i} e^{j ω t}$ , ρ is the power fraction of the light split towards B. The fraction of the wavefield entering the loop is $j \sqrt{τ} E_{i} e^{j ω t}$ , where τ is the power fraction of the light entering the loop. After the first round trip, the electric field exiting the loop towards B is ${(j \sqrt{τ})}^{2} E_{i} e^{j (ω t + φ)} e^{- α L}$ , where α is the loss per unit length in the loop, L = 2πR is its circumference, φ = 2n_effπL/λ is the phase acquired by the wave-train around the loop, n_eff is the effective refractive index of the fundamental mode propagating in micro-fibre and λ is the wavelength of the wave-train. If we assume that following the second trip the light bundle contains a portion of light that has undergone a first round trip, then the light exiting the loop is

{(j \sqrt{τ})}^{2} \sqrt{ρ} E_{i} e^{j (ω t + 2 φ)} e^{- 2 α L} + {(j \sqrt{τ})}^{2} E_{i} e^{j (ω t + φ)} e^{- α L} .

The third round consists of the two previous wavetrains in addition to light entering the loop for the first time. This leads to

{(j \sqrt{τ})}^{2} {(\sqrt{ρ})}^{2} E_{i} e^{j (ω t + 3 φ)} e^{- 3 α L} + {(j \sqrt{τ})}^{2} \sqrt{ρ} E_{i} e^{j (ω t + 2 φ)} e^{- 2 α L} + {(j \sqrt{τ})}^{2} E_{i} e^{j (ω t + φ)} e^{- α L} .

where the first term corresponds to the wavetrain that has circulated three times around the loop, whereas the second and third terms correspond two cycles and one cycle around the loop respectively.

It is not difficult to see that for continuously circulating wavetrains, the output wavefield E_Le^jωt exiting the loop is

E_{L} e^{j ω t} = - τ E_{i} e^{j (ω t + φ)} e^{- α L} \sum_{n = 0}^{\infty} {(\sqrt{ρ} e^{j φ - α L})}^{n}

Since the output from the loop will always mix with the transmitted component

\sqrt{ρ} E_{i} e^{j ω t}

, then using Eq. (1) the total wavefield amplitude at B becomes

E_{o} = \sqrt{ρ} E_{i} - \frac{τ E_{i} e^{j φ} e^{- α L}}{1 - \sqrt{ρ} e^{j φ} e^{- α L}}

The normalized transmission intensity I = |E_o/E_i|² at B is given by

I = \frac{ρ + {(τ + ρ)}^{2} e^{- 2 α L} - 2 \sqrt{ρ} (τ + ρ) e^{- α L} \cos φ}{[1 + ρ e^{- 2 α L} - 2 \sqrt{ρ} e^{- α L} \cos φ]}

We assume in general that τ + ρ + γ = 1, where γ is the loss exclusively in the coupling region. Resonance occurs when φ = 2mπ (m a whole number). Figure 1(b) shows a family of transmission profiles for various split ratios when γ = 0. The Full Width Half Depth (FWHD) Δλ of the transmission spectrum is given by

Δ λ = \frac{\tan^{- 1} (\frac{1 - ρ e^{- 2 α L}}{2 \sqrt{ρ} e^{- α L}}) λ^{2}}{n_{e f f} π L}

The Q-factor is a measure of the quality of the resonance and is defined as (λ/Δλ) which can readily be deduced from Eq. (4). Furthermore, this factor represents the ratio of the energy stored per cycle to the energy gained or lost from the resonator [7]. Thus the smaller the transmitted power fractionτ (larger ρ) into or out of the ring and the smaller the loop absorption factor α, the higher the Q-factor. Note that the FWHD in terms of angular measure, Δφ = (Δλ/λ)φ.

The maxima of Eq. (3) occur when $φ = (2 n + 1) π$ ( $n = 0, 1, ...$ ), so that

I_{\max} = \frac{{[\sqrt{ρ} + (τ + ρ) e^{- α L}]}^{2}}{{(1 + \sqrt{ρ} e^{- α L})}^{2}}

whereas the minima occur at

φ = 2 n π

(

n = 0, 1, ...

),

I_{\min} = \frac{{[\sqrt{ρ} - (τ + ρ) e^{- α L}]}^{2}}{{(1 - \sqrt{ρ} e^{- α L})}^{2}}

For the purposes of this paper we define a re-normalized intensity I* = I/I_max, and furthermore

I * = \frac{A + B \cos φ}{C + D \cos φ}

where

A = \frac{{[1 + \sqrt{ρ} e^{- α L}]}^{2} [ρ + {(τ + ρ)}^{2} e^{- 2 α L}]}{{[\sqrt{ρ} + (τ + ρ) e^{- α L}]}^{2}}

B = - \frac{{[1 + \sqrt{ρ} e^{- α L}]}^{2} [2 \sqrt{ρ} (τ + ρ) e^{- α L}]}{{[\sqrt{ρ} + (τ + ρ) e^{- α L}]}^{2}}

C = 1 + ρ e^{- 2 α L}

D = - 2 \sqrt{ρ} e^{- α L}

3. Investigation of embedded micro-knot resonators

The fabrication technique [23] involves making a large knot with the starting (125 μm single mode) fibre, tapering a length of the fibre down in size, then carefully drawing the knot tighter around the narrowed section. While the smaller knot is being drawn the optical characteristics of the micro-knot were monitored and fine tuned on an optical spectrometer. The micro-knot samples fabricated in this way consisted typically of fibre diameters of 4 μm with loop diameters of 800 μm.

The micro-knots were each placed in small polytetrafluorethylene (PTFE) trays and embedded in silicone rubber. The silicone rubber has a lower refractive index than silica and exhibits good transmission properties thereby facilitating wave-guiding in the micro-fibre. Moreover our experience suggests it possesses relatively good opto-mechanical qualities that are favorable for acoustic pressure transmission to the micro-knot. Figure 2(a) shows a packaged micro-knot sample. The length of the sensor package was 10 cm and was covered with a glass plate for additional protection. The extinction ratios of the in-air sensors ranged between 3 and 5 dB with a free spectral range of about 1 nm.

Fig. 2 (a) Optical micro-knot resonator embedded in silicone rubber. (b) Bending the package as a means of straining the micro-knot resonator.

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The resonance characteristics of the micro-knots can be measured in straightforward experiments aimed at determining the line-width, quality factor, and other coefficients. A simple technique involves stretching the micro-knot by slightly bending the PTFE tray while interrogating the sensor with a relatively narrow line-width laser. This is done by gently pushing the centre upwards while pulling downwards on the ends of the package (see Fig. 2(b)), then releasing the forces; other stretch combinations are obvious. The plasticity of the PTFE tray/silicone rubber combination facilitates a relatively slow temporal relaxation of the micro-knot back to equilibrium conditions after stretching. The result is that the micro-knotresonances are scanned across the fixed laser line as it returns to equilibrium. This is similar to the techniques utilized by [16]. The dependence on polarization state of the input light was not explicitly investigated. We assume this is small due to the cylindrical geometry of the micro-fibre and the nature of physical contact between the fibres in the coupling region. Our assumption is confirmed by the deep resonances (Fig. 4) which show no evidence of polarization splitting and would be much shallower if present (e.g [24].).

Figure 3 shows a schematic of the experimental set-up. Light from a Profile LDS1550 source with 1549.59nm wavelength and 0.092nm line-width was launched into the micro-knot sensors and the output was detected in an InGaAs detector (Thorlabs DF400FC). Each transmission spectrum was collected by a Tektronix digital oscilloscope where it represents a time series of the changing spectrum.

Fig. 3 Experimental set-up to investigate the optical properties of a micro-knot resonator embedded in silicone rubber.

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The data collected in this manner was normalized by subtracting the detector offset then dividing the time series by the maximum within the selected portion of the time series. This corresponds to Eq. (7) above. Figure 4 shows a typical normalized time series of the micro-knot transmission spectrum undergoing relaxation after gently bending the package. This suggests that path length changes are effective in creating resonance shifts, and possibly the dominant effect in acoustic mediated micro-knot sensors.

Fig. 4 Normalized time series of micro-knot spectrum relaxing back to equilibrium after straining the resonator by bending the package. Inset: Spectral dip fitted to a quadratic.

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For the purposes of analysis, we sought to pick three adjacent spectral dips that were of similar extinction ratios, symmetric about a vertical line through the dip, and equidistant in time. Although the spectrum is a function of time, we can readily convert it to one of phase by assuming a 2π phase difference between the spectral dips. To maintain the high resolution of the data, the lower portions of the dips were fitted to a quadratic function (see inset Fig. 4) and the minimum point selected as the start/end points of the phase change across 6π radians in total. The main assumption here is that there is a linear change between consecutive dips which would be true if the dips are equidistant. Based on the criteria above we selected three consecutive spectral dips around the −0.3s region in Fig. 4 for analysis. A non-linear least squares fit was then performed on each spectral dip independently using the Levenberg-Marquardt procedure against the model introduced in Eq. (7).

Figure 5 shows the non-linear least squares fit to the experimental data for one of the spectral dips. Table 1 shows the values obtained for parameters A, B, C and D. The second column in the table shows the parameters obtained from Fig. 5 using the backward difference operator; the other parameters were obtained using the forward difference operator. The micro-knot sensor line-width (FWHD) was found to be 46.4pm ± 0.8pm with a Q-factor of 33450. Since the line-width of the laser is similar in magnitude to that of the micro-knot sensor, it is reasonable to expect that the convolution of the two line-widths would result in an increase in the measured micro-knot line-width. When the laser is modeled as a Lorentzian line, the deconvolved micro-knot line-width turns out to be 37.7pm ± 0.8pm with the Q-factor increasing to 41100. This is believed to be the highest Q-factor obtained to date for an encapsulated micro-knot resonator.

Fig. 5 Nonlinear least squares fit to one of the spectral dips.

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Table 1. Nonlinear least squares fit parameters

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The parameters A to D can give us some insight into the dynamics of the coupling region, and the transmission and reflection properties of the resonator. We can estimate the ρe^−2αL term by averaging the values obtained from different combinations of the fit parameters and over several spectral dips. Table 2 shows the relations used for the estimation of the ρe^−2αL term, where a mean value of 0.60 ± 0.01 has been found. We do not assume in our model that the coupling region is lossless; in fact, we suspect this may be substantial i.e. γ is non-zero.

Table 2. Estimation of the value of the termρe^−2αL

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Using a similar approach to the estimation of ρe^−2αL, we estimated the term (1−γ)e^−2αL = (τ + ρ)e^−2αL to be 0.6096 and τe^−2αL to be 0.009 (Note that $(1 - γ) e^{- 2 α L} = \sqrt{ρ} e^{- α L} (1 - K) / (1 + K)$ where $K = \sqrt{I_{\min}^{*}} (1 - \sqrt{ρ} e^{- α L}) / (1 + \sqrt{ρ} e^{- α L})$ and $I_{\min}^{*}$ is the minimum of Eq. (7), i.e. when φ = 0.). However, in each case we were unable to separate the two terms because of the complexity of the model and our approach to the problem. One way to determine the linear absorption coefficient α, would be to prepare an identical length of straight section tapered fibre (comparable to that used to make the micro-knot), embed it in silicone rubber and make careful fibre loss measurements. Model validation is useful in taking the technology forward to a real world application where repeatability in both fabrication and performance must be achieved.

4. Embedded micro-knot resonator-based hydrophones

A hydrophone is a device that detects underwater acoustic wavefields, i.e. an underwater microphone. An optical micro-knot resonator-based hydrophone could be realized by embedding the resonator in a material that is conducive to both optical waveguiding as well as the efficient transmission of acoustic pressure variations to the resonator. For a free floating micro-knot resonator in the embed material, changes in the external pressure due to the acoustic wavefield will cause changes in optical path length in the loop, in addition to changes in refractive index in the core and cladding of the fibre (elasto-optic effects, e.g [20,21].).

The approach to underwater acoustic detection involves the use of a relatively fast spectrometer that grabs the transmission spectrum as it changes in time as described by [16]. The sensor here is considered to be the combination of the micro-knot and its packaging in the silicone rubber encapsulation. Figure 6 shows a schematic of the experimental layout. A BaySpec spectrometer (FGBA-F-1525-1565-FA) with a maximum of 5kHz sampling rate, working pixel resolution 0.09−0.14nm, and spectral range between 1525nm to 1565nm was used for this work. We used a commercially available 1550nm ASE broadband IR source (ILXLightwave MPS-8033).

Fig. 6 Experimental set-up to investigate optical micro-knot-based hydrophones.

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In these experiments the micro-knot was immersed in a vibrating column water tank that was driven vertically at several acoustic frequencies to determine the shift in the spectral dips with respect to input acoustic pressure variations. This arrangement creates pressure variations along the water column and is useful for low frequency calibration (less than 1kHz) where the acoustic wavelengths are of the order of metres. The setup is capable of absolute pressure calibrations [25]. A reference hydrophone (Reson TC4013-4) was used to record the acoustic pressure close to the micro-knot sample. Any spectrometer pixel monitoring the half-depth position on a given spectral dip, where the slope of the profile is a maximum, will undergo the largest variation in output voltage. Figure 7 shows such a response of the micro-knot to acoustic frequencies at 25Hz and 300Hz. This is believed to be the first demonstration of a micro-knot based hydrophone.

Fig. 7 Spectrometer output from a micro-knot-based hydrophone corresponding to pixel readout at the full-width half-depth on the spectral dip for 25Hz and 300Hz.

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The intrinsic sensitivity of the micro-knot has been determined from measurements made with a reference hydrophone at 40 Hz. The RMS pressure δp delivered to the micro-knot sensor is 1.15x10¹⁰ μPa at 40 Hz. Moreover, the slope on the flank of the dip was 2660 counts/nm, and at 40 Hz the peak amplitude was 253 counts so that the shift δλ in wavelength is 0.095 nm. This corresponds to an RMS value of 0.067 nm. The intrinsic sensitivity of the micro-knot packaged in this way turns out to be 67pm/1.15x10⁴ Pa = 5.83x10⁻³ pm/Pa with a noise equivalent pressure (NEP) of 3.22 Pa/√Hz over 750Hz. The noise floor is limited by the dynamic range of the spectrometer and as an intensity-based method, is not usually consideredthe best means for low-noise/high dynamic range sensor performance. Moreover, it is readily seen from Eq. (7) that direct modulation of the micro-knot produces signals proportional to cos(φ_osinωt) which are highly nonlinear, contains the main carrier signal at ω with several Bessel sidebands around the carrier that depend on the modulation amplitude φ_o; this is often undesirable. In these circumstances an interferometer with an interrogation system capable of demodulation (e.g [19].) can be used.

The normalized sensitivity (δλ/δp)/λ is 3.76x10⁻¹⁵ (μPa)⁻¹, or −288 dB re (μPa)⁻¹. Typical normalized sensitivities for mandrel based fibre-optic hydrophones are −300 dB re (μPa)⁻¹ [19]. The use of polymer micro-ring resonators for the detection of ultrasound as discussed in [14–16] showed a sensitivity of 200pm/MPa [16] – a much lower sensitivity than that obtained in this work (5830pm/MPa). However, this difference is likely to be due to the requirement for the resonator to respond to acoustic pressure changes in a different part of the acoustic spectrum. For polymer micro-ring sensors, the resonators are somewhat constrained physically by being attached to the stiffer silicon platform, but this is consistent with the need to measure higher acoustic pressure changes in the tens of MHz region, whereas in this work the resonators are less constrained leading to higher sensitivity. The directionality of the micro-knot resonator was not measured during this work, however its directional response is expected to be similar to those of polymer micro-ring resonators [16].

A shift in wavelength also corresponds to a change in phase that can be measured in an interferometer. An interferometer is usually considered more sensitive than a spectrometer for these purposes and can have phase resolutions reaching a few μrad/√Hz. As such a calibration exercise was carried out to determine the corresponding micro-knot phase sensitivity.

For a 10m interferometer arm mismatch in a Michelson configuration, the sensitivity becomes 0.44 rad/Pa, or −127.1 dB re rad/μPa. This may be compared to typical values between −120 dB rad/μPa to −150 dB rad/μPa for conventional fibre-optic mandrel-based hydrophones over a range of sensor lengths [19]. As a specific case for the 10 m interferometer, the phase noise floor in a Michelson interferometer could be around 10 μrad/√Hz, so that for a sensor with sensitivity of 4.4x10⁻⁷ rad/μPa (i.e. −127.1 dB re rad/μPa), the acoustic noise floor turns out to be 22.5 μPa/√Hz. This is a realistic estimate and is almost 5 times (14 dB) lower than the designated deep sea state zero (DSS0) value of 100 μPa/√Hz at 1 kHz, the latter being a benchmark for underwater sensing applications [19]. However, one of the key challenges of the interferometric approach is the assumption that light used in the readout interferometer must originate from light circulating in the loop of the knot. In other words, the light used in the interferometer should be the resonant wavelength, since changes in wavelength correspond to changes in phase. Therefore, there needs to be a means to extract the light circulating in the micro-knot that is unmixed with the input light.

Another key challenge is to achieve a line width that is small enough to maintain a reasonably low noise floor. We assumed earlier a noise floor of 10 μrad/√Hz which is usually achieved with fibre lasers whose line widths are ~10kHz [19]; the line width of the light circulating in our micro-knot was ~300MHz after deconvolution (37.7pm; Q–factor ~41100). From our model of the circulating light in the loop (second term in Eq. (2)), to achieve a narrower line width, $\sqrt{ρ} e^{- α L} ≅ 1$ . This means that the linear absorption coefficient α must be very small (i.e. almost no losses in the loop) making ρ ~1. A further implication of this is that τ must be very small, i.e. the amount of light in the loop will be low, possibly requiring amplification on exit and hints at the power budget issues to be tackled.

4. Conclusions

This paper reported on what is believed to be the first observation of a micro-knot acoustic sensor for underwater applications. The normalized sensitivity of the sensor was found to be −288 dB re (μPa)⁻¹ and is comparable to conventional mandrel-based fibre-optic hydrophones, or alternatively a sensitivity of 5.83 fm/Pa. Furthermore, the Q-factor reported here is believed to be the highest to date for any encapsulated micro-knot. Estimates for combinations of optical parameters such as ρe^−2αL, (1−γ)e^−2αL and τe^−2αL could be made but not explicitly separated out due to our re-normalization process; suggestions were made on how to separate out these terms. Model validation is useful if the technology is to be taken forward as a viable alternative to conventional electrical systems where repeatability in fabrication and performance are very important. Some challenges were identified, such as those relating to sensor resonance line-width and the use of interferometers were discussed. Future work is likely to focus on developing a more robust micro-optical resonator that has a relatively low noise floor and can be multiplexed into a large-scale system.

Acknowledgments

This work was sponsored by the Centre for Defence Enterprise and the Defence Science and Technology Laboratory, UK. The authors would also like to thank L Xiao for developing the fabrication technique for fibre micro-knot resonators.

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Parameter	Spectral Dip (λ_source = 1549.59nm; Δλ_source = 92pm)
Parameter	1st dip^a	1st dip	2nd dip	Mean		SD
A	1.5757	1.5837	1.5837	1.5810	± 0.0046
B	−1.5670	−1.5753	−1.5744	−1.5722	± 0.0046
C	1.5909	1.5958	1.5968	1.5945	± 0.0032
D	−1.5529	−1.5586	−1.5571	−1.5562	± 0.0030
L-M Corr. Coef. ^b	0.9976	0.9968	0.9944	−	−
Line Width, Δφ (deg) ^c	25.1	24.8	25.6	25.2	± 0.4
Line Width, Δλ (pm)	46.2	45.7	47.2	46.4	± 0.8
Decov. Line Width (pm)	37.4	37.0	38.6	37.7	± 0.8
Q-Factor ^d	33614	33908	32830	33451	± 557
Extinction Ratio (dB)	12.8	12.9	12.6	12.8	± 0.2

	ρe^−2αL
Relation	1st dip^a	2nd dip	3rd dip	1st dip
C−1	0.5909	0.5968	0.6002	0.5958
D²/4	0.6029	0.6061	0.6037	0.6073
${[\sqrt{A - B} - 1]}^{2}$	0.5972	0.6052	0.6031	0.6043
Mean	0.5970	0.6027	0.6023	0.6025
SD	± 0.006	± 0.005	± 0.002	± 0.006
<ρe^−2αL>	0.60 ± 0.01

Parameter	Spectral Dip (λ_source = 1549.59nm; Δλ_source = 92pm)
Parameter	1st dip^a	1st dip	2nd dip	Mean		SD
A	1.5757	1.5837	1.5837	1.5810	± 0.0046
B	−1.5670	−1.5753	−1.5744	−1.5722	± 0.0046
C	1.5909	1.5958	1.5968	1.5945	± 0.0032
D	−1.5529	−1.5586	−1.5571	−1.5562	± 0.0030
L-M Corr. Coef. ^b	0.9976	0.9968	0.9944	−	−
Line Width, Δφ (deg) ^c	25.1	24.8	25.6	25.2	± 0.4
Line Width, Δλ (pm)	46.2	45.7	47.2	46.4	± 0.8
Decov. Line Width (pm)	37.4	37.0	38.6	37.7	± 0.8
Q-Factor ^d	33614	33908	32830	33451	± 557
Extinction Ratio (dB)	12.8	12.9	12.6	12.8	± 0.2

	ρe^−2αL
Relation	1st dip^a	2nd dip	3rd dip	1st dip
C−1	0.5909	0.5968	0.6002	0.5958
D²/4	0.6029	0.6061	0.6037	0.6073
${[\sqrt{A - B} - 1]}^{2}$	0.5972	0.6052	0.6031	0.6043
Mean	0.5970	0.6027	0.6023	0.6025
SD	± 0.006	± 0.005	± 0.002	± 0.006
<ρe^−2αL>	0.60 ± 0.01

Optical micro-knot resonator hydrophone

Abstract

1. Introduction

2. Optical micro-knot resonator

3. Investigation of embedded micro-knot resonators

4. Embedded micro-knot resonator-based hydrophones

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Tables (2)

Equations (13)

Optics Express