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Abstract
We present a novel design for an angle sensor based on photon coupling to internal optical modes of a two dimensional photonic crystal. We show in simulation that an implementation of this design could achieve sensitivities as high as 1.61 × 106 V/rad, which in principle allows for angle measurements with a noise floor of 2.98 × 10−14 rad$/\sqrt{\textrm{Hz}}$ at the photodiode noise equivalent power. We discuss the limitations of this design and predict the impact these limitations have on the sensitivity as well as the possible ways to further increase the devices sensitivity. As a proof of concept, we demonstrate experimentally a photonic crystal with an angle sensitive mode.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High precision angle sensing has become a focus in recent years due to the growing recognition that much useful seismic information could be obtained if the angular degrees of freedom were measured. Advanced seismic isolation systems of gravitational wave detectors LIGO and Virgo require accurate measures of ground rotation to eliminate the cross coupling between ground rotation and horizontal motion found in most seismometers. This would allow the isolation systems to be improved. Commonly rotation measurement is performed using the optical lever scheme. More recent iterations of this scheme include the auto-collimator and walk-off sensor. Devices based on these schemes have been show to be sensitive to angle changes as small as nano radians [1,2]. In this work we present an alternate scheme which utilises two dimensional photonic crystals.
A photonic crystal is a metamaterial where a periodic dielectric structure defines the optical properties of the device [3]. Photonic crystals are appealing because they allow design of optical properties that may not be possible with unstructured materials. Fabrication of photonic crystals is possible using current semiconductor technology.
Two dimensional photonic crystals are created by a periodic structure over a thin membrane. This is easily achieved by a pattern of holes across the membrane. The optical properties of two dimensional photonic crystals come from the internal electromagnetic resonance structure of the device [4].
Broadly two dimensional photonic crystal can be divided into two classes: uniform, where the periodic lattice is unbroken; and defective, where the periodic lattice has some defect which usually takes the form of a missing or varied cell in the periodic structure. Some applications of defective two dimensional photonic crystals include wave guides [5–7], micro cavities [8–10], photonic logic circuits [11–13] and single photon sources [14,15]. Uniform photonic crystals have application in sensing across various fields (bio-sensing, chemical sensing, gas sensing, refractive index sensing etc.) [16–19], in lasers [20–22] and as reflectors [23,24] which are used in opto-mechanics [25,26] as well as being a candidate light sail in the Starshot initiative [27,28].
There are various designs for photonic crystals to be used as angle sensors. Zhang et al. [29] designed an in fiber photonic crystal tilt sensor that works by having an in fiber photonic crystal partially submerged in a solution. As the tilt is changed the amount of the photonic crystal submerged in the fluid changes which shifts the wavelength of the resonant mode inside the fiber cable. By measuring the wavelength of the resonance dip in transmission it is possible to measure the tilt of the cable relative to the fluid. This achieves a sensitivity of −1.5461 nm/degree to −30.1244 nm/degree when measuring from −35.1$^{\circ }$ to 37.05$^{\circ }$. Chen et al. designed an in fiber torsion sensor using an in fiber photonic crystal that measures the torsion angle by the shift in wavelengths of valleys in the transmission spectrum [30]. The resulting sensor has a sensitivity as high as 0.9354 nm/degree. Angle tunable narrow band filters made from dielectric multilayers [31] may be repurposed as one dimensional photonic crystal angle sensors, however these generally have poor sensitivities to changes in angle. A summary of angle sensing technology is presented in Table 1.
Table 1. Summary of angle sensing technology
Previous photonic crystal based angle sensing technology has in common that the readout scheme is a wavelength measurement of some feature in the optical spectrum of the device. This requires the use of broadband sources and optical spectrum analysers which complicates the readout schemes. For our work we propose the use of a single frequency source and a direct intensity measurement as the readout. It is possible to implement this scheme on the previous photonic crystal based angle sensors presented above, however we propose a simpler scheme in which a single two dimensional photonic crystal is used as the sensor by coupling strength to its internal optical modes.
Coupling strength to internal resonant optical modes of a photonic crystal depends on the incident angle of the light. This can be used as a measure of the relative angle between the photonic crystal and an incident light beam by measuring the transmitted light intensity. By utilising high Q modes photonic crystal modes the sensitivity to angle can be maximised. A simple conceptual technique for angle measurement based on the coupling strength to internal modes is to illuminate the photonic crystal with a laser source and measures the transmitted intensity through the photonic crystal. A key advantage of this design over current angle sensor technologies is that this sensor can be made relatively small as well as being very simple. Furthermore compared to most photonic crystal sensors it uses a direct intensity measurement rather than a measurement of optical resonant wavelength which is far more difficult.
In this work we will cover the following. In section 2. we demonstrate through simulation that angle sensing based on this design is capable of achieving sensitivities as high as 1.61 $\times$ $10^{6}$ V/rad which in principle allows for angle measurements with a noise floor of 2.98 $\times$ $10^{-14}$ radians. In section 3. we examine the sensitivity limitations of this design. In section 4. we discuss an enhanced angle sensing arrangement which allows for increased sensitivities by integrating an angle sensitive photonic crystal with existing angle enhancement techniques. Finally in section 5. we demonstrate an angle sensing photonic crystal as a proof of concept, that corresponds to a maximum sensitivity of 46 V/rad and a noise floor of 1.04 $\times$ $10^{-9}$ radians.
2. Simulation
For design of photonic crystals the software package $S^{4}$ is used to simulate directly the transmission of various photonic crystal structures. $S^{4}$ computes the transmission by solving linear "Maxwells equations in a periodic layered structure using rigorous coupled wave analysis" [32]. Simulations are performed to search for modes with the highest transmission gradient with the angle between the incident light and the photonic crystal. The parameters hole spacing and radius are varied along with the angle of incidence of the light to find a combination with a high sensitivity to changes in the angle of incidence. The search was done in two stages, an initial broad sweep to find an appropriate geometry and a focused sweep to get an accurate measure of the geometries potential sensitivity to angle. The membrane design parameters are presented in Table 2. Figure 1(a) shows the maximum transmission gradient for different simulated photonic crystal geometries, the different high transmission gradient lines are the different modes which could be used for angle sensing.
Fig. 1. (a) Simulated maximum transmission gradient with photonic crystal geometry for S. (b) Simulated 1064 nm light transmission with angle of incidence with photonic crystal radius of 286 nm and spacing of 1076 nm which shows a maximum transmission gradient of 3.5 $\times$ 10$^{5}$/rad.
Table 2. Parameter spaces in broad search for high angle sensitivity photonic crystal (where $\lambda$ is the optical wavelength in $\mu$m)
The transmission vs incident angle for the simulated geometry with the highest transmission gradient is as shown in Fig. 1(b). The maximum transmission gradient in this figure is 3.5 $\times$ 10$^{5}$/rad. Table 3 shows a selection of simulated crystal designs that have high angular sensitivity.
Table 3. Selection of simulated photonic crystal designs with high angle sensitivity.
To assess the feasibility of the simulated photonic crystals an example implementation is proposed. A 6.6 mW laser illuminates the photonic crystal and a Thorlabs PDB450C photo-diode [34] measures the transmitted intensity, parameters for these devices can be found in Table 4. The choice of devices is somewhat arbitrary, these devices were chosen for their suitability. In this scheme the angle measurement is the relative change in angle between the input laser and the photonic crystal. A measurement sensitivity for this scheme can be estimated using Eq. (1).
where $\kappa _\textrm{photonic}$ is the photonic crystal sensitivity, $\Delta$V is the voltage change, $\alpha$ is the angle change, $\beta$ is the transmission gradient, $P$ is the input laser power, $R$ is the photo-diode responsivity and $G$ is the photo-diode gain. Equation (1) can be populated with the values presented in Table 4. The highest sensitivity achieved in the searched parameter space is therefore 1.61 $\times 10^{6}$ V/rad.Table 4. Parameters for sensitivity estimations. Values for the photo-diode assume a Thorlabs PDB450C [34].
3. Sensitivity limitations
In this section potential issues that could limit the sensitivity of these devices are examined. There are broadly four sources of the limitations in these devices: (1) noise from the photonic crystal; (2) noise from the laser source; (3) noise from the photodetector; and (4) dynamic range.
3.1 Photonic crystal
3.1.1 Mechanical thermal noise
Mechanical thermal noise of the photonic crystals surface profile will couple to the angle measurements. Using the small angle approximation and integrating changes in the surface profile over the photonic crystals surface it can be shown that the angle changes are equivalent to the out of plane surface displacement of the photonic crystal. The out of plane motion was estimated using Eq. (2) which models a simple mass-spring harmonic oscillator system [35].
Table 5. Parameters for the mechanical thermal noise estimation [36].
3.1.2 Fabrication
Inconsistency in the fabrication process may limit angular sensitivity. For the sample presented in section 5. focused ion beam lithography was used for fabrication, this process was carefully tuned to get accurate hole size and hole spacing. The maximum accuracy for the focused ion beam is 4 nm. Hole spacing and hole size accuracy must be sub nanometer for a high angle sensitivity design for a specific laser wavelength. However if the laser wavelength can be tuned to match the fabricated sensors wavelength of maximum sensitivity, the requirement on hole size and spacing accuracy can be significantly relaxed.
3.2 Laser
3.2.1 Shot noise
The laser shot noise can be estimated using Eq. (3) [37].
where $\Delta P$ is the laser shot noise, $\omega _{0}$ is the laser angular frequency and $P$ is the laser power. Assuming the values in Table 4 this limits the minimum detectable angle to 4.69 $\times$ $10^{-15}$ rad$/\sqrt{\textrm{Hz}}$.3.2.2 Intensity noise
Laser intensity noise is significant as it couples directly to the angle measurement since the readout is an intensity measurement. This requires laser intensity noise to be suppressed in order to facilitate the precision measurements. The intensity noise of commercial lasers [38] will limit measurements to a noise floor of 8.57 $\times$ $10^{-10}$ radians at 10 Hz using the same assumptions as in section 2. However this condition could be alleviated by intensity stabilisation of the laser which has been shown to achieve a relative intensity noise of $10^{-8}/\sqrt{\textrm{Hz}}$ at 10 Hz [39]. This level of intensity noise will limit measurements to a noise floor of 2.86 $\times$ $10^{-14}$ rad$/\sqrt{\textrm{Hz}}$.
3.2.3 Frequency noise
Frequency noise of the laser will also couple to the angle measurement and degrade the sensitivity as the coupling strength to the photonic crystal resonance depends on the laser frequency. In order for the device to maintain the angle sensitivity of 3.5 $\times$ 10$^{5}$ /rad simulations show the frequency noise needs to be limited to $<$ 50 MHz. This is easily achieved with commercial lasers [38].
3.2.4 Beam divergence
In the simulations it was assumed that the photonic crystal interacts with incident plane waves. In practice however, the beam will not be perfect plane waves, but have some curvature. This means that there will be some distribution of angles of incidence. The range of angles will be determined by the position of the photonic crystal relative to the beam waist and the waist size of the beam. At the waist the plane wave approximation is valid, therefore by putting the photonic crystal at the beams waist this effect can be minimised. However, there is finite accuracy to which this position can be set. So the effect of wave-front curvature on device sensitivity is investigated. The transmission curve with wavefront mismatch can be calculated by the convolution of the distribution of angles of incidence and the ideal plane wave transmission curve as shown in Eq. (4). Figure 2 shows the effects of beam divergence, parameterised by waist size and distance from waist, on the maximum angle sensitivity.
where $T$ is the ideal plane wave transmission curve, $T^{\dagger }$ is the transmission curve with wavefront mismatch, $D$ is the distribution of angles of incidence, $\theta$ is the angle of incidence, $\omega$ is the beam waist and $s$ is the distance between the beam waist and the photonic crystal.Fig. 2. Simulated sensitivity [using the parameters from Fig. 1(b)] with changes in the input beam waist size and photonic crystal position relative to the waist
In our case the size of photonic crystals is limited to the order of 200 $\mu$m diameter by the time it takes to fabricate samples of this size. If the beam exceeds the photonic crystal then excess light is transmitted through the membrane resulting in a reduction in the sensitivity. This restricts the beam waist to 30 $\mu$m in order to confine the beam to the photonic crystal. For this waist size the position of the photonic crystal needs to be confined to within 100 nm of the beam waist in order to maintain sensitivity with 10$\%$ of maximum achievable sensitivity. This is easily achievable with commercially available microscope stages [40].
3.2.5 Polarisation
The polarisation of the laser beam will impact the sensitivity of the readout system. The simulations assume a pure polarisation state of the input beam. For the high sensitivity geometry, light with a polarisation state orthogonal to the high sensitivity state will be transmitted. This results in the sensitivity as a function of polarisation angle following a cosine shape. A change in the polarisation angle of >25$^{\circ }$ would reduce the sensitivity by 10$\%$. Commercially available polarises easily achieve extinction rates of $10^{5}$, which limits the distribution of polarisation such that it is negligible and it is trivial to adjusted the angle to within 25$^{\circ }$. Therefore polarisation will not be a significant issue for these photonic crystal angle sensors.
3.3 Photodetector
The readout of the device will be an intensity measurement from a photodetector. Therefore noise in the photodetector will couple to the angle measurement. Assuming the values from Table 4 the Thorlabs PDB450C [34] photodetector allows for measurements of angle change as low as 2.98 $\times$ $10^{-14}$ rad$/\sqrt{\textrm{Hz}}$ at the level of the noise equivalent power of the photo-diode.
3.4 Dynamic range
The dynamic range of this scheme is inversely proportional to the maximum sensitivity. Figure 3 shows this relation in the range of simulated sensitivities. The sensitivity and dynamic range can be selected by choice of photonic crystal mode. The simulated modes from Table 3 are represented in dark blue in Fig. 3. The sensitivity/dynamic range trade-off can be tuned for a particular crystal geometry by changing the angle operating point and the laser wavelength. In Fig. 3 this is demonstrated for the geometry defined in the 3rd line of Table 3. The dark blue line in the figure represents that range of sensitivity and hence dynamic range that can be achieved by tuning the operating angle and the laser wavelength between 3.5 $\times$ $10^{-4}$ and 7 $\times$ $10^{-3}$ radians and 1064.1 nm and 1063.2 nm respectively.
Fig. 3. Working range with sensitivity for a photonic crystal angle sensor. Dark blue points represent the geometries defined in Table 3 rows 1, 2, and 4. The dark blue line represents the geometry defined in row 3 over a range of angle and laser wavelength tuning conditions.
The limited dynamic range for high sensitivity angle sensing can be mitigated with active control of the photonic crystals angle. This allows the angle sensitivity to be maintained and the dynamic range to be increased to the angle actuator dynamic range. In this scheme the angle measurement is present in the control signal.
4. Enhanced angle sensing arrangement
It is possible to create a compound angle sensor of increased sensitivity by integrating an angle sensitive photonic crystal with optical beam angle enhancing techniques. This could be as simple as passing a laser beam deflected at position $d_i$ through a lens of power $f$ at position 0 that focuses the deflection point to position $d_o$ to give an angle magnification of $d_i/d_o$. By combining this with a walk-off sensor arrangement [2] the angle amplification can be further enhanced. The walk-off sensor behaves similar to an optical lever. An optical lever measures the relative angle change between a mirror and a laser by reflecting the laser off the mirror and measuring changes in the output beam angle. A walk off sensor includes an additional mirror that reflects the beam back onto the target mirror $N$ times. This enhances the the relative angle between the two mirrors by the number of times the beam is reflected back and forth between the mirrors. In this scheme the angle measured is the relative angle between the two mirrors of the walk-off sensor arrangement. By using a walk-off sensor followed by a lens that projects the deflection point onto the photonic crystal it is possible to increase the sensitivity of the compound angle sensor device by the factor provided in Eq. (5). The basic geometry of this scheme is shown in Fig. 4.
where $\kappa _\textrm{compound}$ is the compound device sensitivity, $\kappa _\textrm{photonic}$ is the photonic crystal sensitivity, $N$ is the number of bounces, $h$ is the mirror spacing of the walk-off sensor, $L_o$ is the distance from the walk off sensor to the focusing lens and $f$ is the focusing lens power. With similar parameters to that of McCann et al. [2] and a 5 cm focusing lens it is possible to achieve a sensitivity enhancement of $\sim$145 times for a total sensitivity of $\sim$ 2.3 $\times$ $10^{8}$ V/rad.Fig. 4. Geometry of the combined walk off sensor/lens scheme for angle enhancement onto an angle sensitive photonic crystal.
5. Experimental proof of concept
As a proof of the concept that it is possible to measure the relative angle between a single frequency laser source and an angle sensitive photonic crystal with a direct intensity measurement in this section we test the angular sensitivity of an available photonic crystal. This sample was designed for high reflectivity and minimum sensitivity to angle, so measured angle sensitivity is expected to be very low. However, the agreement with simulation gives confidence that a high sensitivity device could be fabricated. This sample was fabricated by focused ion beam lithography from a silicon nitride membrane from Norcada [41]. This sample was characterised for its transmission using the setup in Fig. 5. A broadband superluminescent light emitting diode (SLED) source (Exalos EXS210010-01, wavelength 1 $\mu$m - 1.125 $\mu$m) exposes the sample with a focused beam slightly offset from the beam waist. A pinhole is used to select the angle of incidence of light. The inset of Fig. 5 shows how shifting the pinhole across the beam selects different angles of incidence. The transmitted light is collected by a fibre optical coupler and is recorded on an Agilent 86142B optical spectrum analyser. Readings from the samples are compared to reference readings from free space transmission which allows the spectrum for transmission to be measured.
Fig. 5. Optical setup used for measuring broadband transmission spectra of photonic crystals. The light source is a fibre coupled SLED which emits light in a range of 1-1.125 $\mu$m. The light is directed through a fibre-free space coupler (FC1). A half wave plate (1/2 WP) is used to control the polarisation of the beam. Lenses L1 and L2 are used to control the beam spot size on the sample. A pinhole (PH) mounted on a translation stage assembly (TSA) is used to select the light incidence angle to be measured. A second fibre-free space coupler (FC2) couples the light into a fibre so that it can be measured by an Agilent 86142B optical spectrum analyser (OSA). The inset shows how a specific angle of incidence can be selected.
Figure 6 shows an experimental measurement of the transmission with angle of incidence of the sample photonic crystal with simulations overlayed. The simulations show good agreement with the experimental results. The optical wavelength of the measurement is 1070 nm chosen since it has the maximum angular sensitivity in the measurement band.
Fig. 6. Experimental measurement of the transmission with angle of incidence for the sample photonic crystal at an optical wavelength of 1070 nm
This shows that it is possible to measure the angle change between a photonic crystal and a laser source using a direct intensity measurement of the transmitted light. The angular sensitivity of the sample photonic crystal used in this experiment is far less than the maximum simulated sensitivity achieved above. Figure 6 shows a maximum transmission gradient of 10/rad. This is expected as this sample was not designed for maximum angular sensitivity but rather for minimum angular sensitivity for use in other experiments, and was chosen for this work due to its availability.
Measurements in Fig. 6 show higher values for the uncertainty in the transmitted intensity than would be expected from the results in section 3.. This is because the experimental setup used here is not that proposed earlier in this work for a high sensitivity angle sensor but rather a broadband scheme meant to identify the wavelength of features in the optical transmission spectrum of the sample, as such the uncertainties on these measurements are higher than would be permissible in an angle sensor scheme. Here the measurement sensitivity is limited by the optical power of the light source. The optical power is significantly reduced when measuring over a small wavelength range and is further reduced by the process of selecting the angle of incidence of the light using a pinhole. Furthermore the method of selecting both a wavelength range and an angle range limits the resolution in both these parameters, so there is some trade-off between the uncertainty in the transmitted intensity and the resolution in the incident angle and the wavelength. This problem can be alleviated with increased optical power.
Points two and three in Fig. 6 show significant deviation from the simulation. The deviation is thought to be due to uncertainties that were not accounted for. This could be related to the experimental having additional sources of error that were not considered. This may be related to the fabrication process for the sample. For this sample focused ion beam lithography with a gallium ion beam was used to directly pattern a silicon nitride membrane. The use of a gallium ion beam for patterning leads to gallium contamination of the sample which changes the optical properties of the material. Contamination of the sample was accounted for in the simulation by adding a uniform contamination layer gallium nitride [42–44] with additional optical loss [45]. This model is relatively simple and may not account for the contamination exactly which could explain the residual deviation.
6. Conclusion
We have presented a novel photonic crystal angle sensor design based on coupling strength to photonic crystal internal modes. Previous photonic crystal based angle sensors measure angle by the change in the wavelength of a feature in the optical spectrum of the device. Our design allows for direct intensity measurements to be used as a measure of the relative angle between a laser source and a phonic crystal. Furthermore we use only a single two dimensional photonic crystal which makes the sensors far simpler. Simulations show that this sensor concept is capable of achieving sensitivities as high as 1.61 $\times$ $10^{6}$ V/rad and in principle can measure angles as small as 2.98 $\times$ $10^{-14}$ rad$/\sqrt{\textrm{Hz}}$ making it competitive with state of the art angle sensing technology. The limitations of such a device were discussed and it is was shown that the sensor will likely be limited by technical noise sources of the photodetector. This technique can also be combined with current angle enhancing techniques, such as the walk-off sensor, in order to create a compound angle sensor of increased sensitivity. Experimentally tilt sensitivity to angle of incidence in a photonic crystal was demonstrated as a proof of concept.
In future works a high transmission gradient photonic crystal will be tested in combination with the walk-off sensor and lens angle enhancement scheme to experimentally verify the maximum transmission gradient simulated in section 2. and the limiting noise sources outlined in section 3..
Funding
Australian Research Council (CE170100004, DP170104424).
Acknowledgments
The authors acknowledge the facilities, and the scientific and technical assistance of the Australian National Fabrication Facility at the Centre for Microscopy, Characterisation & Analysis, The University of Western Australia, a facility funded by the University, State and Commonwealth Governments. Benjamin Neil was supported by an Australian Government Research Training Program (RTP) Scholarship and RTP Fee-Offset Scholarship through the University of Western Australia.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper is not publicly available at this time but may be obtained from the authors upon reasonable request.
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