Interferometric optical gyroscope based on an integrated silica waveguide coil with low loss

Danni Liu; Danni Liu; Hui Li; Hui Li; Xiao Wang; Xiao Wang; Huilan Liu; Huilan Liu; Peiren Ni; Peiren Ni; Ning Liu; Ning Liu; Lishuang Feng; Lishuang Feng

doi:10.1364/OE.392510

1. Introduction

Smaller gyros enable smaller guidance and navigation systems in smart, self-guiding applications such as drones that were previously considered unrealizable [1,2]. Integrated optical gyros are emerging as a promising alternative for miniaturized inertial sensors [3]. Their lower vibrational sensitivity and rate random walk of integrated optical gyros make them superior to microelectromechanical (MEMS) gyroscopes [4]. The properties of integrated optical gyros have led to numerous studies of the integration of interferometric [4–12] and resonant [13–24] optical gyroscopes. At present, monolithic integration cannot be realized due to the limitation of the micro-nano fabrication technology. Most researches focuses on optical gyros based on integrated devices.

Integrated optical gyros accuracy is a crucial factor for navigation. Researchers have thus devoted most of their effort to improving their accuracy. The resonant integrated optical gyroscope (RIOG) system improves accuracy by reducing waveguide propagation loss and suppressing noise [16,23]. Most RIOG systems have adopted SiO₂ waveguides to build the resonator, as they have the lowest propagation loss in current industrial applications. The highest reported RIOG accuracy in the open literature is based on a SiO₂ resonator [16]. However, Fresnel backreflection, backscattering, and polarization within the source coherence length have negative influences on RIOG performance [25,26]. As a result, RIOGs are not sufficiently developed for commercial application.

The use of a broadband source in an interferometric optical gyroscope (IOG) can minimize the effects of the above-mentioned noise [27]. Interferometric fiber optical gyroscopes (IFOG) have been widely used in navigation, tactical, and industrial applications [1,27,28]. Therefore, many research institutions prefer to use an interferometric system to realize integrated optical gyroscopes [5,6,10,12].

The sensing coil insertion loss fundamentally limits the performance of the IOG. It includes the waveguide propagation loss induced by material absorption and imperfect fabrication process, the bend loss induced by curved waveguides, cross talk of adjacent waveguides loops, and coupling loss induced by end-fire coupling. The first three terms are referred to as the sensing coil propagation loss. There are many promising integrated platforms for the IOG sensing coils such as indium phosphorus (InP) [21], silicon nitride (Si₃N₄) [4,7–10,29,30], silicon-on-insulator (SOI) [5,12], and SiO₂ [23,25,31]. InP platforms enable the fabrication of active and passive building blocks, and are thus the most promising to realize monolithic integration. The resonator reported by di Bari exhibits a Q factor of approximately $6 \times {10^5}$ and the propagation loss of the InP waveguide is 0.3 dB/cm [12]. The Si₃N₄ waveguides have attracted recent attention because they provide passive waveguide coils with low propagation loss (0.1 dB/m) [4]. However, the fundamental mode sizes of the waveguides do not match the butt-coupling fiber mode, increasing the insertion loss. Single-mode SOI waveguides have a more compact footprint that leads to larger loss, which is also detrimental to IOG performance [12]. The SiO₂ waveguide has the lowest unit propagation loss among them [25]. Therefore, SiO₂ is a more competitive material to build low insertion loss sensing coils for realizing high accuracy IOGs.

Some waveguides have been used to construct sensing coils and to verify the performance of the gyroscopes. The University of California, Santa Barbara reported an integrated 3-m-long large-area Si₃N₄ sensing coil. The angle random walk (ARW) and bias drift of the IOG set up by this integrated coil were measured to be $8.52{\kern 1pt} \;{^\circ \mathord{\left/ {\vphantom {^\circ {\sqrt h }}} \right. } {\sqrt h }}$ and 58.7 °/h, resulting in the first demonstration of a commercial grade IOG [4,8]. Huazhong Science and Technology University demonstrated an IOG based on a SOI platform with a footprint of 600 × 700 µm, with bias drift of 51.3 °/s [12]. California Institute of Technology developed a nanophotonic optical gyroscope on a Si platform with ARW of $650{\kern 1pt} \;{^\circ \mathord{\left/ {\vphantom {^\circ {\sqrt h }}} \right. } {\sqrt h }}$ and bias drift of 1 r.p.m. by using a reciprocal sensitivity enhancement technique [6]. A comparison of the sensing coil properties and gyroscope performances of the reported IOGs is shown in Table 1. However, the techniques necessary to build a low insertion loss sensing coil using a SiO₂ waveguide and verify the high accuracy gyroscope effect have not been reported to date.

Table 1. Comparison of the reported performances of IOGs.

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In this paper, we present a detailed report of the design and characterization of an IOG based on an integrated SiO₂ waveguide sensing coil. We begin with the design of the SiO₂ waveguide. The minimum bend radius of the SiO₂ waveguides and the space between adjacent loops of the waveguide are designed to minimize the sensing coil insertion loss. Section 3 discusses the theoretical sensitivity of the IOG, so as to decide the optimal length of the sensing coil according to the tradeoff between the sagnac loop length and the detection limit. Section 4 describes the fabrication and characterization of the sensing coil. Sinusoidal wave biasing modulation improves the system detection sensitivity. Section 5 proposes a new method to find the eigen frequency of the sensing coil then decides the proper modulation parameter of the gyroscope. The gyroscope output measurement and accuracy characterization are discussed followed by a summary and conclusion of the work.

2. SiO₂ waveguide design

The structure of the SiO₂ waveguide from top to bottom comprises a cladding layer with thickness of 20 µm, the core which is 6.5 µm thick and 6.5 µm wide, a buffer layer with thickness of 15 µm, and a silicon layer with a thickness of 0.625 mm, as shown in Fig. 1. The refractive index of the Si is 3.45. The refractive index of the cladding layer and the buffer layer is 1.4448. The refractive index of the core is 1.4513. The wavelength (${\lambda _0}$) in free space is 1550 nm for all the simulations.

Fig. 1. The structure of the SiO₂ waveguide.

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To investigate the mode distribution of the waveguide, a full-vectorial finite-difference method was performed using the commercial FIMMWAVE software package produced by PhotonDesign. Both quasi-TE and quasi-TM modes are supported in the SiO₂ waveguide. Figure 2 shows simulated electric field intensity profiles for the quasi-TE and quasi-TM modes. The upper and lower boundary of the simulation area is the cladding and buffer edge. The width of the simulation area is 30 µm. The sizes of the waveguide satisfy single-mode conditions. The effective refractive index of the optical mode is 1.4471, and the simulated mode size is $8.17{\kern 1pt} \; \times 8.17{\kern 1pt} \;\mathrm{\mu} \textrm{m}$. It should be note that dimensions are not drawn to real scale to see the optical mode distribution clearly.

Fig. 2. Simulated electric field intensity distributions of the fundamental (a) quasi-TE mode and (b) quasi-TM mode.

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To find the optimal bend radius of the waveguide sensing coil, the relationship between the bend radius and mode loss was investigated. As shown in Fig. 3(a), the propagation loss of the SiO₂ waveguide initially increases as the bend radius decreases. In order to compromise between the wafer size and sensing coil area, the bend radius of the sensing coil and the bend loss are chosen to be 15 mm and 0.5 dB/m, respectively. To investigate the space and crosstalk of adjacent loops of the waveguides, an eigenmode expansion algorithm was performed using the FIMMPROP software package produced by PhotonDesign. The relationship between the spacing of adjacent loops of the waveguides and the mode coupling ratio was simulated. As shown in Fig. 3(b), when the space between adjacent waveguide loops is larger than 25 µm, crosstalk between the adjacent waveguides is weak. Thus, this is the minimum spacing required to provide immunity from cross-term interference.

Fig. 3. Relationship between (a) the bend radius and mode loss, (b) the spacing of adjacent loops of the waveguide and mode coupling ratio.

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The above simulations determine the minimum bend radius and space between adjacent loops of the waveguide when the sensing coil is formed of the SiO₂ waveguide. These are necessary parameters of sensing coil design and are described in the next section.

3. Integrated sensing coil design

The integrated sensing coil structure must be designed after the waveguide parameters have been determined. Then, the optimal length of the sensing coil is determined according to the tradeoff between the sagnac loop length and the detection limit for a given optical power and insertion loss.

3.1 Structure design

The Archimedean spiral structure is used to realize reciprocal rotation sensing part. It is an ideal geometry to achieve maximum enclosed area and length for a given chip footprint. The Archimedian spiral can be defined as:

(1)$$\rho \textrm{ = }a + {{b\theta } \mathord{\left/ {\vphantom {{b\theta } {2\pi }}} \right.} {2\pi }}$$

where a is the minimum radius and b is the waveguide spacing.

The length of single Archimedean spiral loop is:

(2)$${s_n} = \int_0^{2\pi } {\sqrt {{a^2} + {{\left( {\frac{{{b_n}}}{{2\pi }}} \right)}^2} + {{\left( {\frac{{{b_n}}}{{2\pi }}} \right)}^2} \cdot {\theta ^2}} \;} d\theta$$

where n refer to the nth circle, so ${b_n} = n \cdot b$.

The length of whole Archimedean spiral waveguide is:

(3)$$s = \sum\limits_1^N {{s_n}}$$

The schematic diagram of the proposed IOG with an integrated SiO₂ waveguide sensing coil realized by an Archimedean spiral structure is shown in Fig. 4. A beam of light is injected into the integrated coil after being polarized, split and phase modulated. Fiber pigtails are used for the highly efficient butt coupling between the phase modulator based on a lithium niobate (LiNbO₃) Y-branch and the SiO₂ waveguide sensing coil. The counterclockwise (CCW) and clockwise (CW) light interfere with each other after propagating in the integrated coil. The modulation signal applied to the Y-branch phase modulator is a sine wave, which does not require a phase modulator with a flat frequency response. The modulation frequency is tens of MHz because the lengths of waveguide sensing coils are usually several meters. The use of a sine wave, compared to a square wave with the same frequency, can reduce the circuit bandwidth requirements. The gyroscope output can be obtained by synchronous demodulation.

Fig. 4. The schematic diagram of the IOG with an integrated SiO₂ waveguide sensing coil.

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3.2 Optimal length design

The integrated sensing coil length is decided based on constraints due to the optimal detection limit, which is determined by the noise level. The equivalent angular rotation rate of noise intensity is related to the modulation-demodulation method.

3.2.1 Reciprocal biasing modulation-demodulation

For a sinusoidal phase modulation applied on both counter propagation waves, the electric field of light after being split and phase modulated can be expressed as:

(4)$${E_{\textrm{CW}}} = \frac{{\sqrt 2 }}{2}{e^{{{ - \alpha L} \mathord{\left/ {\vphantom {{ - \alpha L} 2}} \right.} 2}}}{E_0}{e^{i[\omega t + {\varphi _m}sin(2\pi {f_m}t) + {\phi _\tau } - {\varphi _m}sin(2\pi {f_m}(t - \tau )) + {{{\phi _R}} \mathord{\left/ {\vphantom {{{\phi_R}} 2}} \right.} 2}]}}$$

(5)$${E_{C\textrm{C}W}} = \frac{{\sqrt 2 }}{2}{e^{{{ - \alpha L} \mathord{\left/ {\vphantom {{ - \alpha L} 2}} \right.} 2}}}{E_0}{e^{i[\omega t - {\varphi _m}sin(2\pi {f_m}t) + {\phi _\tau } + {\varphi _m}sin(2\pi {f_m}(t - \tau )) - {{{\phi _R}} \mathord{\left/ {\vphantom {{{\phi_R}} 2}} \right.} 2}]}}$$

with

(6)$${\varphi _m} = \frac{V}{{{V_\pi }}}\pi $$

where V is the amplitude of sine wave.${f_m}$ is corresponding frequency. We assume that the output power of ASE (amplified spontaneous emission) source is ${P_0} = E_0^2$ and CW, CCW incident light after splitting by the Y branch have the same amplitudes of electric field. The insertion loss of splitter and modulator is ignored to simplify the discussion.$\alpha $ is the waveguide loss, and L is the length of the waveguide coil. $\omega $ is the frequency of light. ${\varphi _m}$ refers to the modulation depth. $\tau $ is transit delay period, and ${\phi _\tau }$ is the phase difference induced by sensing coil. ${\phi _R}$ is the phase difference induced by rotation angular rate. The half-wave voltage of Y-branch phase modulator is ${V_\pi }$.

Then the interference signal becomes:

(7)$$\begin{array}{l} P({\phi _R}) = \frac{1}{2}{e^{ - \alpha L}}{P_0}\{{1 + \cos [{{\phi_R} + {\phi_m}} ]} \}\\ = \frac{1}{2}{e^{ - \alpha L}}{P_0}\{{1 + \cos [{{\phi_R} + 2{\varphi_m}\sin (2\pi {f_m}t) - 2{\varphi_m}\sin (2\pi {f_m}(t\textrm{ - }\tau ))} ]} \}\\ = \frac{1}{2}{e^{ - \alpha L}}{P_0}\left\{ {1 + \cos \left[ {{\phi_R} + 4{\varphi_m}\sin (\pi {f_m}\tau )\cos (2\pi {f_m}(t\textrm{ - }\frac{\tau }{2}))} \right]} \right\} \end{array}$$

Using the ${J_n}$ Bessel function expansion, this becomes:

(8)$$\begin{array}{l} P({\phi _R}) = \frac{1}{2}{e^{ - \alpha L}}{P_0} + \frac{1}{2}{e^{ - \alpha L}}{P_0}\cos ({\phi _R})\left\{ \begin{array}{l} {J_0}[4{\varphi_m}\sin (\pi {f_m}\tau )]\\ + 2{J_2}[4{\varphi_m}\sin (\pi {f_m}\tau )]cos(4\pi {f_m}(t\textrm{ - }\frac{\tau }{2})) + \cdots \end{array} \right\}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{1}{2}{e^{ - \alpha L}}{P_0}\sin ({\phi _R})\left\{ \begin{array}{l} 2{J_1}[4{\varphi_m}\sin (\pi {f_m}\tau )]sin(2\pi {f_m}(t\textrm{ - }\frac{\tau }{2}))\\ + 2{J_3}[4{\varphi_m}\sin (\pi {f_m}\tau )]sin(6\pi {f_m}(t\textrm{ - }\frac{\tau }{2})) \cdots \end{array} \right\} \end{array}$$

With a synchronous demodulation, this yields a biased signal:

(9)$${P_d}({\phi _R}) = {e^{ - \alpha L}}{P_0}{J_1}[4{\varphi _m}\sin (\pi {f_m}\tau )]sin({\phi _R})$$

The curves of the unmodulated interference response and the demodulated biased signal after modulation are shown in Fig. 5. The signal slope of the unmodulated interference output is zero at maximum. Reciprocal biasing modulation aids obtaining a nonzero response to get a higher sensitivity on the rotation induced phase difference. On a stable bias, the signal slope with respect to the phase difference is:

(10)$$\frac{{\partial {P_d}}}{{\partial {\phi _R}}}({\phi _R} = 0) = {e^{ - \alpha L}}{P_0}{J_1}(4{\varphi _m}\sin (\pi {f_m}\tau )) \approx 0.53{e^{ - \alpha L}}{P_0}$$

Fig. 5. Unmodulated interference response and demodulated biased signal at the different rotation induced phase difference.

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The maximum sensitivity is obtained for $4{\varphi _m}\sin (\pi {f_m}\tau ) \approx 1.8\;\textrm{rad}$.

3.2.2 Theoretical sensitivity of the IOG

The theoretical sensitivity of the IOG need to be recalculated because of the use of a waveguide sensing coil. It is necessary to take into account the thermo-refractive noise in addition to the thermal noise of the preamplifier, shot noise, and relative intensity noise of the source which contribute to the theoretical sensitivity of IFOGs [9]. Moreover, the different equivalent phase difference introduced by sinusoidal phase modulation should be considered. Firstly, we recalculated the shot noise. The expression for shot noise can be written as [28]:

(11)$${{{\sigma _P}} \mathord{\left/ {\vphantom {{{\sigma_P}} {\sqrt {{B_s}} }}} \right. } {\sqrt {{B_s}} }} = \sqrt {2 \cdot ({{hc} \mathord{\left/ {\vphantom {{hc} \lambda }} \right.} \lambda }) \cdot {P_{}}}$$

where h is Planck constant and ${B_s}$ is the counting bandwidth. For $\lambda = 1550nm$,

(12)$${{{\sigma _P}} \mathord{\left/ {\vphantom {{{\sigma_P}} {\sqrt {{B_s}} }}} \right. } {\sqrt {{B_s}} }} = \textrm{ 5}\textrm{.06} \times \textrm{1}{\textrm{0}^{ - 10}} \cdot \sqrt {{P_{}}}$$

Practical semiconductor PIN diodes yield an increase of shot noise by a factor 1.1 when the flow of photons is converted into a primary current of electron [28]. Such a shot noise yields a measured phase difference as below:

(13)$${\phi _{shot}} = \frac{{{\sigma _p}}}{{\sqrt {{B_s}} \frac{{\partial {P_d}}}{{\partial {\phi _R}}}}} = \textrm{ }\frac{{1.05 \times \textrm{1}{\textrm{0}^{ - 9}}}}{{\sqrt {{e^{ - \alpha L}}{P_0}} }}\left( {{{rad} \mathord{\left/ {\vphantom {{rad} {\sqrt {Hz} }}} \right. } {\sqrt {Hz} }}} \right)$$

The broadband optical source suffers from relative intensity noise (RIN) because of the random beating between all its frequency components [32]. The optical source centered at 1550 nm we used in the experiment has a full width at half maximum (FWHM) of 15 nm. It has a 2 THz frequency width. The expression for the measured phase difference induced by RIN can be written as [9,32]:

(14)$${\phi _{\textrm{RIN}}}\textrm{ = }\frac{k}{{\sqrt {\Delta {f_{source}}} }}\left( {rad/\sqrt {Hz} } \right)$$

where $\Delta {f_{source}}$ is the frequency width of optical source. k is related to modulation-demodulation method. In this case, $k = 1.89$.

The expressions for measured phase difference induced by the thermal noise of the preamplifier and thermo-refractive noise can be written as [9,32,33]:

(15)$${\phi _{\textrm{thermal}}}\textrm{ = }\frac{{{{10}^{ - 9}}}}{{{e^{ - \alpha L}}{P_0}\sqrt R }}\left( {rad/\sqrt {Hz} } \right)$$

(16)$${\phi _{\textrm{thermal - refractive}}}\textrm{ = }4.3 \times {10^{ - 7}}\sqrt {\frac{L}{{40}}} \left( {rad/\sqrt {Hz} } \right)$$

The relationship between angular rotation rate and non-reciprocal phase shift is often expressed as:

(17)$$\;\Delta \theta = \frac{{2\pi LD}}{{c{\lambda _{}}}}\Omega $$

where $L$ and D are the length and the diameter of the Sagnac loop, respectively. $\Omega $ is the angular rotation rate, and $\Delta \theta $ is the non-reciprocal phase shift.

From the following relation, the smallest detectable angular rotation rate is:

(18)$${\Omega _{minimum}} = \frac{{{\phi _{shot}} + {\phi _{\textrm{RIN}}} + {\phi _{\textrm{thermal}}} + {\phi _{\textrm{thermal - refractive}}}}}{{{\raise0.7ex\hbox{${\partial ({\Delta \theta } )}$} \!\mathord{\left/ {\vphantom {{\partial ({\Delta \theta } )} {\partial \Omega }}} \right.}\!\lower0.7ex\hbox{${\partial \Omega }$}}}}$$

Figure 6 shows the contribution from each noise term to the rotation noise spectral density for a given source power and waveguide insertion loss. In this simulation, we assume that the output power of the ASE is 20 mW and waveguide loss is 4 dB/m. The insertion loss of optical components and junctions between them is taken into account. The tradeoff between sagnac loop length and detection limit shows that there is an optimum choice of sensing coil length for the given waveguide loss and source power to achieve the optimum sensitivity. The minimum detection limit is $1.03{{^ \circ } \mathord{\left/ {\vphantom {{^ \circ } {\sqrt {\textrm{h} }}}} \right. } {\sqrt{ \textrm{h} }}}$ when sensing coil length is 2.81 m under these simulation conditions.

Fig. 6. Contributions from various noise sources to the rotation noise spectral density.

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4. Integrated sensing coil fabrication and characterization

The scheme of the integrated sensing coil based on an Archimedean spiral structure is shown in Fig. 7(a). The coil consists of a 1.89-m-length spiral, starting at 30-mm radius and ending at 31.5-mm radius, with a waveguide space of 150 µm, turning 10 and a half times. Input and output are carried out through straight waveguides. One of the straight waveguides passed through a semicircle for a smooth transition. The bend radius of the semicircle is 15 mm. The other straight waveguide is directly tangent to the Archimedean spiral coil. The length of the straight waveguide is 55 mm. Thus, the entire sensing coil consists of 2.14-m-long waveguide. The coil has 11 crossings in and then 11 more crossings out, 22 in total.

Fig. 7. (a) Illustration of the sensing coil based on Archimedean spiral structure; (b) Top view of fabricated waveguide coil.

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The integrated waveguide coil is fabricated on silicon substrate. A 20-µm-thick bottom layer is grown by wet thermal oxidation in 1150 °C for a month. A 6.5-µm-thick layer of SiO₂ doped germanium is then deposited using plasma enhanced chemical vapor deposition (PECVD), as shown in Fig. 8(a). The waveguide cores are defined with UV lithography and a dry etch process. A 20-µm-thick SiO₂ doped boron and phosphorus as the top cladding layer is then deposited using PECVD. By controlling the flow rate of reaction gas, the concentration of doped ions in cladding and core can be adjusted to get required refractive index. The wafer is annealed at 800 °C. The cross section of the fabricated SiO₂ waveguide is shown in Fig. 8(b). The top view of the fabricated waveguide coil illuminated using a red laser is shown in Fig. 7(b).

Fig. 8. The cross section of the (a) deposited core layer, (b) fabricated SiO₂ waveguide.

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The integrated sensing coil based on the SiO₂ waveguide was connected to other optical devices in the IOG system by the fiber tail coupling. Thus, we need to estimate the insertion loss of the interference loop. The measurement system scheme is shown in Fig. 9. The light source pigtail is connected to the waveguide pigtail through an adapter. The optical power output from the interference ring is 0.918 mW. The measured optical power in the breakpoint behind the splice point was 6.3 mW. The insertion loss of the interference loop was 8.37 dB. In the actual gyroscope experimental system, the optical path between phase modulator and sensing coil is connected by a pigtail fusion method, which increases the loss by 2.38 dB.

Fig. 9. The schematic diagram of insertion loss measurement system.

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5. Gyroscope characterization and performance measurements

5.1 Modulation parameters

The reciprocal biasing technique is implemented with a sine modulation, which does not require a phase modulator with a flat frequency response. According to Ref [28]., the frequency of modulation signal should be the inverse of twice the transit delay period:

(19)$${f_p} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {2\tau }}} \right.}\!\lower0.7ex\hbox{${2\tau }$}}$$

This frequency is known as the eigenfrequency of the sensing coil to achieve perfect rejection of all the even harmonics.

After determining the frequency of the modulation signal, the amplitude is chosen to realize maximum detection sensitivity. According to Eq. (10), the amplitude is obtained for

(20)$$V = \frac{{1.8{V_\pi }}}{{4\pi }}$$

where ${V_\pi } = 5.76V$. The amplitude of sinusoidal signal driven to the modulator is 0.83 V.

To find the proper modulation frequency, a linear frequency sweep of the sine signal is applied to the phase modulator, as shown in Fig. 10. The output of the detector without angular rotation rate can be written as:

(21)$$P({f_m}) = \frac{1}{2}{e^{ - \alpha L}}{P_0}\{{1 + \cos [{2{\varphi_m}\sin (2\pi {f_m}t) - 2{\varphi_m}\sin (2\pi {f_m}(t\textrm{ - }\tau ))} ]} \}$$

The simulation results of the interference signal received by the photon detector (PD) with modulation frequency sweeping are shown in Fig. 11. When the modulation signal is a direct current (DC) signal, that is to say, ${f_m} = 0$, the output of the PD is a DC signal. When the modulation signal frequency is increased from zero, the PD output changed to an alternating current (AC) signal. When the modulation frequency is equal to twice the eigenfrequency, the PD output is also DC. The simulation results show that when the DC signal repeats for the first time, the modulation frequency is equal to twice the eigenfrequency.

Fig. 10. Setup for characterization of modulation frequency.

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Fig. 11. The simulation results of interference signal with modulation frequency sweeping.

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We increased the frequency of the modulation signal from zero in the setup of Fig. 10. When the PD output is the same as the applied DC signal, the frequency of the signal is twice the eigenfrequency. It is known from experiments that the proper frequency was 15.6 MHz. The measured value agrees well with the expected value of proper frequency based on actual length of the sensing coil (2.14 m waveguide coil + fiber pig tails ∼4.46 m).

5.2 Rotation signal measurement and noise characterization

A prototype was constructed based on Fig. 4. A broadband ASE with 2 THz linewidth was employed. Its central wavelength and power were 1550 nm and 20 mW, respectively. A $1 \times 2$ splitter was placed between the modulator and the ASE to realize detection of the interference light. After passing through the splitter, the light was then injected into the modulator, which was based on a proton-exchanged LiNbO₃ Y-branch waveguide and lumped push-pull electrode. The phase modulator had an optical bandwidth of 300 MHz and ${V_\pi }$ at each single arm of 5.76 V. The frequency and amplitude of the sinusoidal signal driving the modulator were 15.6 MHz and 0.83 V, respectively. The detector was based on InGaAs. The noise level was 3.0${{\textrm{pW}} \mathord{\left/ {\vphantom {{\textrm{pW}} {\sqrt {\textrm{Hz}} }}} \right. } {\sqrt {\textrm{Hz}} }}$ at peak responsivity. The gyroscope output can be obtained by demodulating the detector signal at the same modulation frequency.

To correlate the measured output value of the gyroscope to the input angular rotation rate, we placed the setup on a calibrated rotation stage and measured the scale factor. The scale factor accuracy test was conducted at various angular rotation rates of ±1 °/s, ±2 °/s, ±3°/s,±4 °/s, and ±5 °/s. The gyroscope output from the scale factor experiment is shown in Fig. 12(a). The corresponding scale factor errors are shown in Fig. 12. The root mean square (RMS) can be taken to evaluate the scale factor accuracy:

(22)$${E_{sf}} = \sqrt {\frac{{\sum {{{({{{({{{\tilde{\omega }}_i} - {\omega_i}} )} \mathord{\left/ {\vphantom {{({{{\tilde{\omega }}_i} - {\omega_i}} )} {{\omega_i}}}} \right.} {{\omega_i}}}} )}^2}} }}{n}} $$

where ${\tilde{\omega }_i}$ is the measured value of angular rotation rate, ${\omega _i}$ is the input angular rotation rate, and n represents the number of measurement. It is indicated that the scale factor accuracy is better than 0.8469%.

Fig. 12. (a) Gyroscope output of scale factor experiment; (b) Scale factor error distribution.

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Finally, the IOG output was captured at a rate of 10 Hz for 5 h, as shown in Fig. 13(a). The standard deviation of this result (10 s smoothing) is 34.4 °/h. The plot of Allan variance for 5 h of data is shown in Fig. 13(b). A −0.5 slope line was fit to the data to extract the ARW of the system to be $1.26{{^ \circ } \mathord{\left/ {\vphantom {{^ \circ } {\sqrt {\textrm{h}} }}} \right. } {\sqrt {\textrm{h}} }}$. The bias drift of the gyroscope evaluated from the flat portion (slope = 0) of the Allan deviation plot is 7.32°/h.

Fig. 13. (a) Static test results of IOG; (b) Allan deviation measurement of IOG.

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6. Conclusion

The sensing coil insertion loss determines the IOG accuracy. The relationship between the integrated sensing coil structure parameters and the loss can be established by separating the insertion loss. A smaller insertion loss is achieved after optimizing the structural parameters using a numerical method. Then the sensing coil is fabricated and characterized. A sinusoidal phase modulation method is applied to achieve perfect rejection of all the even harmonics and realize maximum detection sensitivity. This method lowered the hardware circuit requirements for detection of the angular rotation rate of the IOG with high eigen frequency. In addition, a convenient method to determine the proper modulation parameters based on the setup of the IOG was developed. The bias drift of the IOG using a 2.14-m-long integrated SiO₂ waveguide sensing coil is 7.32°/h and the ARW is $1.26{{^ \circ } \mathord{\left/ {\vphantom {{^ \circ } {\sqrt {\textrm{h}} }}} \right. } {\sqrt {\textrm{h}} }}$, which agree well with the theoretical sensitivity of the IOG. Experimental results show that the reduction in integrated sensing coil insertion loss can further improve the performance of the IOG. The integrated sensing coil insertion loss could be lowered to get obtain a higher- precision IOG.

Funding

National Natural Science Foundation of China (61973019).

Disclosures

The authors declare no conflicts of interest.

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Material	Loss			Length	Footprint	ARW	Bias drift
Material	Propagation loss	Coupling loss	Insertion loss	Length	Footprint	ARW	Bias drift
InP [21]	0.3 dB/cm	\	\	\	\	\	\
Si₃N₄ [4]	0.015 dB/cm	11.7dB	16.2dB	3 m	∼12.57 cm²	$8.52^{\circ} / \sqrt{h}$	58.7 °/h
SOI [12]	∼2.10 dB/cm	\	\	∼29 mm	0.42 mm²	\	51.3 °/s
SOI [6]	\	\	\	∼7 mm	2 mm²	$650^{\circ} / \sqrt{h}$	1 r.p.m.

Interferometric optical gyroscope based on an integrated silica waveguide coil with low loss

Abstract

1. Introduction

2. SiO₂ waveguide design