1. Introduction

So far two general approaches have been applied in traditional distance measurement. The first one is using incoherent detection of light pulses to convert time-of-flight information to large distance measurement [1], but limited in the resolution of the available electronic instruments. The second is utilizing classical wavelength-based optical interferometers based on the superposition principle of two coherent waves to achieve excellent measurement of small displacements [2]. In order to improve the accuracy of precision displacement measurement, various ways have been proposed to improve measurement resolutions, such as, by using the femtosecond ultrashort pulse lasers [3], the combined use of interferometric and time-of-flight measurements [4], optical comb by overlapping two interfering pulses with adjustment of the pulse repetition [5], and the phase demodulation of sinusoidal phase modulating interferometer [6, 7]. In the two-beam path configurations of traditional interferometers, the additional energy losses and phase delays introduced in the splitting and combining processes cannot be avoided with adverse effects for the accuracy and compactness of displacement measurement system.

Zero-index metamaterial (ZIM) is a kind of artificial composite material whose refractive index effectively equals zero. The electromagnetic waves (EMW) experience uniform phase and infinite wavelength inside these ZIMs, and such materials have been applied to realize electromagnetic wave wavefront shaping [8], cloaking [9], beam self-collimation [10], tunneling waveguides [11] and extremely convergent lens [10]. Different from the conventional metamaterials [12] comprising metallic composites with strong resonance loss at higher frequencies, photonic metamaterials [13–15], such as photonic crystals (PhCs) [14, 15], are made entirely of dielectric or semiconductor materials with the evident benefits of low loss and immunity to the optical and electronic noises, whose dielectric constant is periodically modulated on a length scale comparable to the desired wavelength of operation. By proper design, the accidental degeneracy of two dipolar modes and a single monopole mode generates at the Brillouin zone center of photonic metamaterial [16, 17]with ε_eff = μ_eff = 0, which can be regarded as ZIM. Since permittivity ε_eff and permeability μ_eff approach zero simultaneously and linearly, the effective impedance of such PhC is a finite constant with the non-zero group velocity at the Dirac-like point (DLP), light enters a state of quasi-infinite phase velocity and infinite wavelength as if it were not there.

In this paper, we proposed a novel measurement mechanism utilizing a single-beam interferometry to realize precise displacement measurement. Different from the single-beam polarization interferometry by the phase-shift difference between the s and the p waves of a laser beam reflecting from the poled polymer film [18], the displacement is measured in a normal reflection configuration without optical bias and the change of polarization state, all-dielectric photonic metamaterial with effective zero-index is used to break through the limit of microscopic size to visually reflect the perturbance of stationary wave formed by the superposition of incident and reflected beams in the transmission line. The numerical simulations are compared with the results of the displacement measurement, and excellently agree with theoretical analyses and experiment.

2. Principle and design

As an Ohmic loss-free alternative to metallic structures, dielectric photonic metamaterials with effectively zero permittivity and permeability can retain the phase and amplitude states of incident wave with the uniform distributions. In this work, a square-lattice PhC composed of high-dielectric cylinders embedded in air background was adopt as the photonic metamaterial, with the structure parameters of lattice constant a = 1, cylinder radius r = 0.213a, permittivity ε_r = 10 and permeability μ = 1. By applying plane-wave expansion method [19] for two-dimension, the band structure for TM-polarized mode is shown in Fig. 1(a) with the frequency of normalized quantity a/λ (λ is wavelength in free space). The second and fourth bands (i.e. TM2 and TM4) cross each other linearly to form a Dirac cone intersected by the additional flat TM3 band at the degenerate point, i.e. Dirac-like point, which is located at the center of the Brillouin zone with k = 0 at the DLP frequency of ω_D = 0.568 a/λ. According to the effective medium theory [20], the PhC with a Dirac cone at the Γ point is an effective ZIM with ε_eff = μ_eff = 0 (i.e. n_eff = 0) at the DLP frequency. As long as the size is large enough, the PhC array can behave as a ZIM [21], thus, the incident EMW can pass through the ZIM as if it were not there at the DLP frequency.

 

Fig. 1 Band diagram of the square-lattice PhC for TM mode with the parameters of a = 1, r = 0 0.213a, ε_r = 10 and μ = 1 with the insets of Brillouin zone and lattice structure.

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When a monochromatic linear-polarization wave normally incidents on a mirror with high reflectivity, the superposition of incident wave E_i and reflected wave E_r in opposite direction brings about a stationary wave along the transmission line, which can be expressed as

Figure 2(a) gives the schematic diagram of the displacement measuring system with the aforementioned PhC and a high reflector placed in parallel to each other along the optical axis. The PhC array is fixed at the position of x = 0. The reflector is mounted upon a measured object and displaces along the transmission line. At the DLP frequency, the stationary wave formed by the interference between the incident and reflected waves penetrates the PhC array and excites a linearly proportional uniform electro-field oscillation in it. At the surface of the reflector, the phase difference between the incident and reflected waves is π (radian) because of half-wave loss in external reflection. Hence, a constant state of node with the weakest field-intensity is achieved at the reflective interface. Supposing the distance between the PhC and reflector is D, at the position of x = 0, the phase difference between the incident and reflected waves can be expressed as:

 

Fig. 2 (a) Schematic diagram of the single optical path displacement measuring system, which is composed of PhC array with effective zero-index and mirror with high reflectivity; (b) In PhC array, the strongest field-intensity is achieved at the anti-node locations with the distances of odd multiples of λ/4 and the weakest field-intensity is achieved at the node locations with the distances of even multiples of λ/4.

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3. Numerical simulation and analysis

To verify the theory above, the proof-of-concept simulation (Fig. 3) was implemented by FDTD Solutions [23]. Since PhCs are structures in which the dielectric constant is periodically modulated on a length scale comparable to the desired wavelength of operation, the fulfillment frequency can shift higher or lower by scaling down or up the PhC dimensions. Therefore, the working frequency band of demonstrated PhC can be easily extended from microwave to THz or optical frequencies as long as the lattice structure of PhC can be designed on the scale of working wavelength. Herein, the simulation and experiment were carried out in the microwave regime, as these were readily available to us. Via meticulous design, the dielectric rods with ε_r = 10 in radius r = 2.5mm were used to construct a square-lattice PhC array with the size of 15a × 10a (a = 11.76mm). A TM-polarized wave normally incidents on the input interface of the PhC array at the DLP frequency of 14.49 GHz (i.e. the wavelength of λ = 20.7mm in air). The reflector was set to be a perfect electric conductor (PEC) with an ideal reflectivity R≈1.

 

Fig. 3 (a) Calculated electric-field intensity and phase with the increase of distance D between PhC and reflector. The corresponding electric-field distributions at the locations of anti-node and node with the strongest and weakest intensities are shown in (b) and (c), respectively.

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With the reflector shifting away from the fixed PhC array, the distance dependence of the calculated electric-field intensity (black solid line) and phase (red dash line) in the PhC array are shown in Fig. 3(a). The measuring result of electric-field intensity with near six elementary cycles is shown in 60mm spatial displacement, revealing the periodic perturbance of stationary wave oscillation along the transmission line. Regardless of initial phase, the phase keeps linearly proportional to the distance D with a cycle about 10.35mm, which is consistent with the effective half-wavelength of PhC array at the DLP frequency of 14.49GHz, due to the double phase accumulations of EMW in the round trip. Further investigations were performed at different probing positions in the PhC array, the similar relationships of field-intensity, phase and distance D were obtained to demonstrate the PhC characteristics of uniform phase distributions at the DLP frequency. The corresponding electric-field distributions for the PhC array located at different locations of anti-node and node are shown in Fig. 3(b) and 3(c), respectively. In any case, the electric-field distributions in the PhC array almost kept uniform without phase delay at the DLP frequency, attaining the strongest at anti-nodes and the weakest near zero at node, which further confirm our theoretical predictions in Fig. 2(b). Therefore, the PhC array with effective zero-index at the frequency of 14.49 GHz can be used to break through the limit of microscopic size to visually reflect the characteristic of stationary wave oscillation at any x-point.

4. Experiment and results

As a proof-of-concept demonstration, this experiment was implemented to verify the above proposed theory. The square-lattice PhC array composed of highly pure alumina ceramic rods with the same dimensions in above simulations was adopted to facilitate the measurements. From Drude’s treatment [24], no matter in optical or the lower frequencies, the metals can be regarded as perfect conductors with high reflectivity of R≈1. Herein, the aluminum metal is adopted as the reflector in the microwave domain. As shown in Fig. 4(a), a microwave planar waveguide formed by two upper and lower aluminum plates was used to ensure the invariable TM-polarized mode along the z axis. The rectangular waveguide with the cross-sectional dimension of 15.8mm × 7.9mm were utilized as the waveguide adapters to excite and probe plane waves with electric-field E parallel to the z-axis. In the Ku-band region (12∼18 GHz), only the dominant mode of TE10 can travel through the planar waveguide. The PhC array was fixed in the microwave planar waveguide with the input surface perpendicular to the incident beam. A polished aluminum bar with the height of h = 7.8mm was parallelly placed at the end of the PhC array as the movable high reflector along the transmission line.

 

Fig. 4 (a) Experimental layout of the single-beam interferometry measurement system; (b) The periodic variation of transmittance measured in the PhC array with the increase of distance D; (c) Measured phase spectra for PhC located at the antinode with the strongest field-intensity and the node with the weakest field-intensity.

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To further confirm the effects of distance D on the electric-field intensity and phase in PhC, the transmittance and phase were measured in the PhC array as the aluminum bar moving away from the PhC array. It can be seen clearly from Fig. 4(b) that the experimental results of square symbols are nearly accordance with the simulation result denoted by solid line, although with slight offsets due to the size mismatch of different waveguides and the fabrication imperfection of PhC array. Five cycles of field-intensity rise and falling to fluctuate appear in the process of increasing the distance D from 0 to 52 mm, which further validates our measurement mechanism and indicates that continuous multi-wavelength displacement can be easily measured in real-time. Figure 4(c) gives the phase spectra for the PhC array located at the sequential node (solid line) and anti-node (dash line) with a red circle marking the values at the DLP frequency. Consistent with the effective medium theory [20], two phase spectra approximate linear around the DLP frequency and the distinct phase difference of π between them at the DLP frequency is in good agreement with the calculated result in Fig. 3(a), which verifies the sub-wavelength displacement measurement can also be readily realized by this mechanism. The follow-up experimental results show that the similar phenomena occur alternately at the subsequent locations of node and anti-node with the increasing of distance between the PhC and reflector. Therefore, utilizing the single-beam interferometry by employing photonic metamaterial with effective-zero-index, we successfully experimentally achieved satisfying performance of precise displacement measurement.

5. Discussion

The proposed displacement measurement mechanism dictates all-dielectric photonic metamaterials rather than metallic metamaterials can be used to realize the required effective zero-refractive-index. The different practicable frequencies of the measurement system can be easily adjusted by scaling the size of all-dielectric PhC with Dirac-like point. For example, when the lattice constant of all-dielectric PhC is scaled down to 600nm, the DLP frequency proportionally blue shifts to the optical frequency of 211 THz (λ₀ = 1422nm) [17]. For this reason, the frequency of the demonstrated displacement measurement system can be easily extended from microwave to THz regime as long as the effective zero-index of photonic metamaterials can be obtained at the operating frequency.

Different from the non-ambiguity range of λ/2 between the strongest and weakest field-intensities of interference fringes in the traditional optical interferometry [25], the higher sensitivity and resolution have been realized by this displacement measurement mechanism due to the subtle distance interval of λ/4 between the strongest and weakest field-intensities of stationary wave in the compact uniaxial reflection configuration. In addition, the nanometer accuracy could even be achieved for precision displacement measurement supposing using the phase generated carrier homodyne detection scheme [26] in our design strategy.

It should be noted that the working frequency in this mechanism is not necessarily limited to the strict Dirac-like frequency. Using the effective medium theory [20] for all-dielectric PhC, the effective refractive index near the DLP is generally far less than 1 [16], with the effective wavelength long enough to break through the limit of microscopic size in air to be distinguished readily. At the fixed detecting position of PhC array, the phase delay induced by the PhC array is constant and independent of the distance D for the frequency slight deviating from DLP, therefore, the phase shift Δδ is still linearly proportional to the displacement ΔD, which is in agreement with the relationship of Eq. (2). The obvious evidence has been shown in Fig. 4(c), where the almost constant phase difference between the linear phase spectra around the DLP frequency from 14.4∼14.6GHz illustrates moderate deviation of working frequency will not impact the accuracy of the predesignated measuring device based on the proposed mechanism.

6. Conclusion

In summary, we experimentally and numerically demonstrated the single-beam interferometry mechanism by employing photonic metamaterial with effective zero-index for precise displacement measurement. All-dielectric PhC with Dirac-like point was applied to visually response the perturbance of stationary wave oscillation formed by the interference between the incident and reflected beams in the transmission line. Displacement over multi-wavelength or sub-wavelength can be measured in real-time by the proposed mechanism with the compact uniaxial reflection configuration. The intrinsic non-ambiguity range of λ/4 of stationary wave oscillation promotes the accuracy of displacement measurement. The operational frequency is adjustable by scaling the overall dimension of the PhC. Other dielectric-based photonic metamaterials with effective zero-index and homodyne detection scheme could also be applied in the design strategy for the nanometer accuracy. This work provides an easy way to design and construct precise displacement measuring system and gives guidelines to its applications in both theory and practice.