1. Introduction

Spectrometers, the optical devices to measure the intensity varying of light with frequencies, are indispensable across a range of industrial processes and fundamental scientific research [1]. Traditional spectrometer design with ultra-high resolution has the drawback of the large scale [2], which is from both the large grating components and the long wave-propagation path [3–6]. The photonic crystals (PhCs) [7,8] based on-chip technology offers a potential compact design to minimize the ultra-high-resolution spectrometers. At first, the sharp turning of the equal frequency contours (EFCs) of PhC has attracted much attention, where both the $K$-vector super-prism (KVSP)effect, which is defined by a small change of incident angle resulting in large refraction angle change [9], and the $S$-vector super-prism (SVSP) effect, which is defined by a small change of frequency resulting large refraction angle change [10], is very strong [11–15]. The disadvantage of such a design is the beam diverges in the PhC greatly, which needs great efforts to obtain detectable signals [16,17]. Then, the super-collimation (SC) effect [18] which is from the zero-curvature section of EFCs and causes the propagation of finite-width beams in PhC without diverging or converging, is introduced in the design of PhC spectrometers. The finding of the local super-collimation (LSC) effect [19], which is from the turning point of EFCs, makes the design more feasible. The fundamental theory of PhC spectrometers has been established [20,21] in which the high-resolution region with large $r = q/p$ in $\vec {k}$-space is derived, where, $q = \frac {\partial \theta _t}{\partial \omega }$ is the parameter for the SVSP effect and $\theta _t$ is the propagation angle of the refractive beam, $p =\frac {\partial \theta _t}{\partial \theta _i}$ is the parameter for the LSC effect [18] and $\theta _i$ is the incident angle of the beam into the PhC. This theory is simple and straightforward to evaluate the beam-splitting capability of PhCs, since the large $q$ guarantees that two beams propagate in different directions with a small frequency difference, and the almost-zero $p$ guarantees almost no expansion for beams after long propagation in PhCs. Besides the existence of a large-$r$ region, other critical conditions still need to be considered for the design of compact high-resolution PhC spectrometers. Li et al. [22] find a wide-angle range with LSC around the frequency-sensitive super-collimation frequency [23] in rectangular lattice PhCs and show that the range could be used in the design of beam-splitting devices. This discovery has provided a solid foundation for designing PhC spectrometers, which are based on both the SVSP effect and the LSC effect of PhC. Gao et al. [24] derive the size requirement of PhC for the ultrahigh-resolution spectrometer under the condition of the very-wide-beam limit. Furthermore, Liu et al. [25] recently studied the propagation characteristics of the finite-wide beam around LSC and developed a theory to predict the beam behaviors which agrees with numerical simulations very well. However, a comprehensive study of ultra-high-resolution PhC spectrometers has not been done, especially lack of methods to optimize the working frequency range and the PhC size when the incident beam is not very wide. Even more, so far, there is no experimental demonstration of the PhC spectrometer design based on both SVSP and LSC effects to the best of our knowledge.

In this work, the theory for the design of an ultra-high-resolution PhC spectrometer is developed in which the methods are given to optimize the working bandwidth and the PhC size for the incident beam with finite width, and finally, the experiments are done for the first time to show the effectiveness of the PhC spectrometer design based on both SVPS and LSC effects. Specifically, first, the working bandwidth is expanded for a high-resolution PhC spectrometer by tilting the PhCs at a certain angle so that the rectangular lattice is not parallel to the interface between the air and PhC anymore. This oblique angle allows the PhC’s large-$r$ region to coincide with the equi-incident-angel path (EIAP) [21] in the bandwidth as much wider as possible. Then, we utilize the finite-wide beam propagation theory [25] to predict the splitting of beams with different frequencies after long propagation. To meet the requirement of the spectrometer resolution, we can self-consistently obtain the optimized incident beam width $and$ the corresponding PhC size. Finally, based on these analysis results, we design the spectrometer and conduct experiments in microwave frequencies to verify its spectroscopic capability. Our work provides both the fundamental principles and experimental validation for designing a compact high-resolution PhC spectrometer with a wider bandwidth. The potential applications of our work can extend to the research of ultra-high resolution, ultra-small infrared (or visible-light) PhC spectrometers. The theoretical size of the ultra-high-resolution PhC spectrometer from our design could be two orders smaller than the traditional design, e.g., the scale is about $11.2$ mm for our design with the resolution of 10000 at the $2.6$ $\mu$m wavelength, comparing the size of meter scale for traditional design.

2. Theory and design

In this work, the PhC model is a three-dimensional (3D) PhC slab structure with a perfect electric conductor (PEC) on both top and bottom boundaries, which are shown in Fig. 1(a). It consists of dielectric rods arranged in rectangular lattices with lattice constants $a$ and $b$ in $x$ and $y$ directions, respectively. The dielectric constant, height, and radius of the rods are represented by $\epsilon _r$, $h$, and $r_{rod}$, respectively. The aspect ratio of the rectangular lattice is $\beta = b/a$. The height, $h$, of the PhC rods needs to be optimized. First, we hope to prevent the appearance of high-order modes in the $z$-direction, so the optical length in the $z$-direction should be less than half the wavelength of the highest frequency in our working range. Second, very small $h$ is not a good choice too, since the coupling efficiency of light from waveguide could be very low. Hence, we generally choose the value of $h$ with the condition that the optical length in $z$-direction is between one-fourth and half of the wavelength. Our study focuses on the TM ($E_z$ polarization) modes. The 3D PhC slab described has the same photonic band gap as that of the 2D PhC [26]. In Fig. 1(b), the EFCs of the second band of PhC with $\beta = 1.8$, $r_{rod}=0.19a$ and $h=1.1a$ are shown in Fig. 1(b), in which the large-$r$ regions are signified by red color. Our goal is to design a compact high-resolution PhC spectrometer with enough wide working frequency range based on super-prism and LSC effects. The red part signified by the dashed-black lines of Fig. 1(b) presents a typical high-resolution region of PhC in $\vec {k}$-space with very large $r$, where $q$ is moderately large with the PhC refraction angle change of about $20$ degrees for a 10% frequency difference for incident light and $p \simeq 0$ since it’s near the turning points of EFCs. Theoretically, if the incident beam with different frequencies can excite the modes of PhC with very large $r$, the beams of different frequencies can be split after enough long propagation distances since the moderately large $q$ means the beams of different frequencies propagate in different directions and $p \simeq 0$ means that the beams almost don’t expand [22]. However, practically the parameters $r$, $p$, and $q$ alone are insufficient for designing a compact high-resolution PhC spectrometer with enough wide working frequency. Several other critical parameters must be carefully investigated, e.g., the incident angle, the oblique angle of PhC, the incident beam width and the PhC size, etc. Next, we will see that the wider working frequency range can be optimized by choosing the incident angle of the waveguide and the oblique angle of PhC, while the incident beam width and the PhC size can be optimized self-consistently to achieve certain resolution requirements of a spectrometer. After all these parameters are determined, the ideal design of the PhC spectrometer can be realized.

 

Fig. 1. (a) Schematic diagram of the 3D PhC structure, composed of periodic dielectric rods (grey) which is sandwiched by two metal slabs (yellow). (b )EFCs (black lines) of the second TM ($E_z$-polarization) band for the 3D PhC structure and the corresponding $\lg \lvert r\rvert$ factor shown by the color map in the $\vec {k}$- space. The deep red color represents the large-$r$ regions. The purple curve donates EIAP of the incident wave in $\vec {k}$- space, which crosses the large-$r$ region at one point. The arrows, perpendicular to the local EFCs, are to signify the group velocity of Bloch modes.

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2.1 Oblique angle $\phi$ and incident angle $\theta _i$

In this subsection, first, we will find the method to optimize the working frequency range of the PhC spectrometer. The enough large working-frequency range is very important for the design of a high-resolution spectrometer. Originally, the adjusting of incident angle is used to obtain the working frequency range [22]. However, as illustrated in Fig. 2(a) where the large-$r$ region is highlighted by red dots, if we only optimize the incident angle, the large-$r$ region does not typically align with the EIAPs, as illustrated by the blue lines representing specific incident frequency ranges at three different incident angles (8.3 degrees, 9.3 degrees, 10.3 degrees), incrementally increasing from bottom to top. This implies that for different incident angles the ideal working frequency range with large-$r$ is quite narrow, which is limited to the near vicinity of the cross-point(s).

 

Fig. 2. (a) A part of the Brillouin zone with the black lines as the EFCs of the second band of the PhC structure, the blue points as the large-$r$ regions, and the red line as the EIAP. The black points, $A$($k_{xA},k_{yA}$) and $B$($k_{xB},k_{yB}$), represent the expected frequency range for spectrometer design. (b) The schematic demonstration of the PhC tilting. We suppose that the $y$-axis always is the interface between air and PhC so that the interface is not parallel to the long lattice of PhC anymore after the tilting. (c) EFCs, the large-$r$ regions, and the EIAP of the tilted PhC. After tilting, an EIAP almost overlaps with the large-$r$ region totally, and passes through both points $A$($k'_{xA},k'_{yA}$) and $B$($k'_{xB},k'_{yB}$). Therefore, the working bandwidth can cover the expected working frequency range between $A$ point and $B$ point. (d) The position in the $\vec {k}$- space of points $A$ and $B$ before and after the tilting of PhC. Two red vectors represent $\vec {k}$ vectors of $A$ and $B$ frequencies with the same incident angle, whose $k_y$-components are the same as $k'_{yA}$ and $k'_{yB}$ which guarantees the EIAP passing both $A$ and $B$ points after tilting.

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To increase the working-frequency range, we improve the design by choosing a proper oblique angle of PhC, so that the EIAPs of different incident angles could be almost parallel to the large-$r$ region. Then, we can choose a proper incident angle to make sure that a part of the EIAP of that incident angle could overlap with the large-$r$ region. The oblique angle of PhC is defined in the following way: (i) we fix the $y$-axis as the interface between air and PhC; (ii) we suppose that the long lattice (or short lattice) of PhC is parallel to the $y$-axis (or $x$-axis) at first, then tilt the PhC a little, such that the long (or short) lattice of the PhC is no longer parallel to the $y$-axis (or $x$-axis); (iii)the angle $\phi$ between the interface or $y$-axis and the long lattice is called as the "PhC oblique angle." The definition of the oblique angle is schematically shown in Fig. 2(b). In Fig. 2(c), the impact of the oblique angle $\phi$ is illustrated, with the EIAPs at three different incident angles (4.4 degrees, 5.4 degrees, 6.4 degrees, incrementally increasing from bottom to top) represented by blue lines, which are almost parallel to the large-$r$ region indicated by red dots. In real design, the method to optimize the working bandwidth can be done in the following steps. First, we choose a proper working frequency range whose the large-$r$ region is relatively flat, which is signified by point A($k_{xA}, k_{yA}$) as the lowest frequency $\omega _A$ of the bandwidth and point B($k_{xB}, k_{yB}$) at the highest frequency $\omega _B$ of the bandwidth as shown in Fig. 2(a). Second, we can obtain the shift of point A and point B at $\vec {k}$-space to A($k_{xA}', k_{yA}'$) and point B($k_{xB}', k_{yB}'$) by rotating because of the introducing oblige angle $\phi$ as shown in Fig. 2(d). Third, we can obtain two corresponding points on the air dispersion (signified by the black lines in Fig. 2(d) by boundary conditions [27] of the same frequency and $k_y$ and make sure that two points are on the same direct line from the origin(equal incident angle). Then, we make sure that the EIAP is almost parallel and coincides with the large-$r$ region in $\vec {k}$-space. The ideal positions of point A($k_{xA}', k_{yA}'$) and point B($k_{xB}', k_{yB}'$) satisfy:

2.2 Incident beam width $W_0$ and the PhC size $L$

In this subsection, we will find the method to simultaneously optimize the width $W_0$ of the incident beam and the size of PhC which are essential for the design of a compact spectrometer. In Fig. 3, we have shown the beam-splitting mechanism for a high-resolution PhC spectrometer [28]. The incident beam with different frequencies is from a waveguide with the same incident angle. The splitting mechanism of the PhC spectrometer is based on the different refraction angles of different frequencies (the effect of the enough large $q$) and the small beam-width expanding angle (the effect of the very small $p$) so that the beams of different frequencies can separate from each other gradually after enough long propagation. The effects of beam width, which is determined by the waveguide width $W_0$, need to be carefully investigated. We will find that there is a dilemma in choosing $W_0$ by two extreme cases. On the one hand, we hope the beam width $W_0$ is as large as possible. According to the uncertainty theory of waves [29], the finite width of a beam is inversely proportional to the uncertainty of wavevector $\Delta \vec {k}$. Hence, if we hope to reduce the uncertainty of wavevector $\Delta \vec {k}$ by choosing a very large $W_0$, we can exactly select a point (a Bloch mode) with large $r$ in $\vec {k}$-space. On the other hand, we hope the beam width $W_0$ to be as small as possible. According to Fig. 3, if $W_0$ is very large, the beams of different frequencies have to propagate a very long path to be separated, which is against our goal of the compact spectrometer design. Hence, there exists an optimized width $W_0^{opt}$ for our compact spectrometer design. $W_0^{opt}$ should be large enough so that the uncertainty can be overcome and the large-enough $r$ region can be chosen, it also should not be very large so that the beams with small frequency difference can split from each other after not-very-long propagation path. Based on the above analysis, it found that the optimization of the $W_0$ and the PhC size $L$ and the resolution requirement of the spectrometer are correlated. We need a tricky and achievable process to obtain them self-consistently. In our previous work [24], with an assumption of very large $W_0\gg a$, we have approximately derived the needed propagation length of beams which could be expressed as: $L>W_0/{(tan(\Delta \omega \times q/2)-{c\times p}/{n_b\times \omega _0\times \pi \times W_0})}$ , where $\Delta \omega =\lvert \omega _1-\omega _2\rvert$ is the minimum detectable frequency difference, $\omega _0$ is the center working frequency, $n_b$ is the index of the background material and $c$ is the the velocity of light in vacuum. From the last equation, we can obtain the optimized width $W_0^{opt}$ approximately in the large-$W_0$ limit.

 

Fig. 3. Schematic diagram to interpret the beam-splitting mechanism of PhC spectrometer. With an incident beam with two different frequencies $\omega _1$ and $\omega _2$, since the difference deflection angles denoted by $\Delta \theta _t$ at these two frequencies (the super-prism effect) and the small divergence of two beams (the LSC effect), two deflected beams separate each other after enough long propagation distance.

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However, in this work, the assumption of very-large-$W_0$ is not generally satisfied. Therefore, we need to obtain the optimized width $W_0^{opt}$ and the size $L$ of PhC in a more strict and complex way. First, let’s call back the work [25], a method Eq. (3) is developed to precisely predict the field distribution profile of an LSC beam after propagating over a specific distance $L$ by integrating the complex amplitude of Bloch components. In the work, it is shown that the theoretical predictions agree with that of numerical experiments very well.

Then, the beam behavior at a certain propagation distance $L$, such as the details of the gradual broadening of beams at the LSC region, could be precisely obtained. Based on this method, we can predict the behaviors of beams with different frequencies, e.g., to determine if two beams are separated. Next, step by step, we will show how to optimize $W_0$ and the PhC size $L_{opt}$ simultaneously based on the resolution requirement. First, we suppose two beams of frequencies $\omega _1=\omega _c - \Delta \omega /2$ and $\omega _2=\omega _c + \Delta \omega /2$ are incident from the same waveguide, where $\omega _c$ is the central frequency and $\Delta \omega$ is the minimum frequency resolution of the designed spectrometer. From Eq. (3), we can calculate the field distribution $H_1(y, L)$, $H_2(y, L)$ of two incident beams at any propagation distance $L$ as shown in Fig. 4(a). Also, from the $q=\partial \theta _t / \partial \omega$, we can obtain the refraction angle difference $\Delta \theta _t = q \times \Delta \omega$ of two beam centers, from which the distance between the beam centers can be calculated by $d_b (L) = \Delta \theta _t \times L$. With the obtained field distribution $H_1(y, L)$, $H_2(y, L)$ and the distance $d_b(L)$ between the beam centers after propagation distance $L$, we can judge if two beams are separated or not by certain criterion. In this work, we use the overlap integral $\eta = \int _{-\infty }^{+\infty } I_1(y) I_2(y) dy \leq 0.4$ as the criterion to judge the separation of two beams, where $I_1(y) = H_1(y)^2 / \sqrt {A_1}$ and $I_2(y) = H_2(y)^2 / \sqrt {A_2}$, $A_1$ and $A_2$ are the normalization factors to guarantee $\int _{-\infty }^{+\infty } I_1(y)dy = \int _{-\infty }^{+\infty } I_2(y)dy = 1$. After scanning different values of $W_0$, we can find that there is a best choice of $W_0^{opt}$, for which two beams can be separated according to the criterion defined above with the smallest $L_{min}$. Hence, we can obtain the $W_0^{opt}$ and the scale of PhC $L_{min}$ simultaneously, which are shown in Fig. 4(b).

 

Fig. 4. (a)The diagram to show the criterion of two-beam separation after propagation distance $L$ by the overlapping-integral value of the field of two beams with different frequencies. (b)The overlapping-integral $\eta$ versus the incident beam width $W_0$ for a certain propagation length. Existing the smallest value for $\eta$ means that there is an optimized beam width $W_0^{opt}$.

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In the above discussion, we have not considered the errors in the real fabrication of PhCs. In the real design, we had better keep the redundancy for the PhC size $L$ for the tolerance of the fabrication errors. In our design of the PhCs, we choose real size $L = 1.3 \times L_{min}$ to increase the tolerance.

2.3 Spectrometer design

In this subsection, based on the design methods discussed above, we propose a compact high-resolution PhC spectrometer with a certain working bandwidth, which is shown in Fig. 5 schematically. The spectrometer comprises three main parts, an input waveguide at the left side of PhC, a PhC slab structure with PEC on both top and bottom boundaries, and the output waveguides at different angles on the right side of PhC. A beam with components of different frequencies is incident by the input waveguide from the left side, then the incident beam is separated into several beams with different frequencies by both super-prism and LSC effects of PhC. At last, these separated beams are sent to sensors(e.g., CCD [30] or CMOS [31]) by the output waveguides.

 

Fig. 5. Schematic diagram of the design of a compact high-resolution PhC spectrometer with optimized incident angle, the oblique angle, the incident beam width, the size of PhC, and the expected resolution.

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Because the scalability of electromagnetic waves is a direct conclusion from Maxwell’s equations if a theory or experiment is demonstrated at one frequency domain, it will work at all other frequency domains with the condition that the scale change could be strictly realized, i.e. the dielectric constants of materials and the relative geometry ratio keeping unchanged. Although in our design the dielectric constants of materials and the geometry parameters of the structure are quite moderate, the challenges in the fabrication of higher frequency ranges, e.g. the optical frequencies, still need to be considered carefully. According to experiments [19] about seventeen years ago, if we can control the processing error below $5{\%}$ the phenomena of SC and LSC are quite robust against the local defects since the LSC is from the averaged dispersion of PhC. With the technological improvement in the last decade, the fabrication error in micro- and nano-meter scales could be well controlled. Hence, our design could potentially be adapted across different frequency ranges, including Tera-hertz, infrared, visible light, ultraviolet, etc.

As discussed in the previous section, the size of PhC is determined by the expected resolution $R = \omega _c / \Delta \omega$ where $\omega _c$ is the central frequency of the working frequency range, and $\Delta \omega$ is the smallest frequency difference which could be resolved. To demonstrate the required PhC size of different frequency domains, we choose three typical wavelengths, 560 nm for visible light, $2.6$ $\mathrm{\mu}$m for the infrared, and 8.6 mm for the microwave. Assuming an expected resolution $R=1000$ for these three typical wavelengths, the corresponding sizes of the PhCs are $240$ $\mathrm{\mu}$m, $1.1$ mm, and $3.7$ m, respectively. For an ultra-high resolution of $R=10000$, the sizes for these three frequencies become $2.4$ mm, $11.2$ mm, and $37.1$ m. Comparing the meter scale of the traditional ultra-high resolution($R \geq 10000$) spectrometer for the infrared based on the traditional design by gratings, our PhC ultra-high resolution spectrometers are about two orders smaller [32]. Even more, our spectrometer could be realized on a silicon-on-insulator (SOI) chip as a thin-slab structure so that we can try the multi-layered design in which different layers of SOI are for different frequency ranges. We can increase the relative working bandwidth of the ultra-high-resolution spectrometer to $5{\%}$ by the multi-layer design, although the relative working bandwidth of a single layer is about $0.2{\%}$.

3. Experiment

In this section, we will show that the effectiveness of our spectrometer design is verified by the experiments at a microwave frequency range between $27.5$ GHz and $40$ GHz. With microwave horns as the source at the left side of PhC, and a wave-guide probe as the point receiver to detect the field on the right side of PhC, we can use a vector network analyzer (Keysight PNA-X N5245B) to measure the received signals. Two kinds of experiments have been done. The first is that, with a very wide horn as a similar plane-wave incidence, we can obtain the directional band gaps of PhC at $x$ and $y$ directions by detecting the transmitted field. The second is that, with an incident beam width of about $10.6 \times a = 39.6$ mm, we can detect the field distribution of the refracted beam on the right side of PhC of a certain frequency and then we can check the splitting of beams for different frequencies. The PhC is constructed from ceramic rods with a dielectric constant $\epsilon _r$ in the possible range of $[7.5$, $8.5]$, a height $h$ =$4$ mm and a diameter $r_{rod}=0.71$ mm ($r=0.19a$). These rods are arranged periodically between two metallic plates, which serve as the top and bottom PEC boundaries, as shown in Fig. 6. The lattice constants of the PhC in $x$ and $y$ directions are $a=3.741$ mm and $b=6.73$ mm, respectively. The upper and down metallic plates could be thought of as the perfect electric conductor(PEC) for microwave frequencies.

In the first step, we test the directional band gap for both $x-$ and $y-$directions to determine the dielectric constant $\epsilon _r$ of the rods and the quality of our fabrication. A Gaussian beam from a very wide horn, whose radiation can be approximately treated as the plane wave, is incident to the rectangular finite PhC slab with the length $L_x=371$ mm and the width $L_y=331$ mm. The test processing is shown schematically in Fig. 6(a). The transmitted field from PhC is detected so that the transmittance can be obtained. The transmittance results are shown in Fig. 6(b) and (c) for both $x-$ and $y-$directions. Theoretically, we also calculate the band gaps for different $\epsilon _r$ of the rods by plane-wave calculation method [33]. Comparing the experimental results of directional transmission with the calculated band gaps, we find that the best choice of $\epsilon _r$ is $7.8$. The band gap structure of PhC with $\epsilon _r ^{opt} = 7.8$ is shown in Fig. 6(d). The gap regions from our theoretical calculation are also signified by the shades in Fig. 6(b) and (c). We can see that the experimentally detected results agree very well with the directional gaps from the theoretical calculation. Now, we can be confident about the dielectric constant of the rods and the quality of our fabrication.

 

Fig. 6. The upper-left picture is the experimental demonstration of a PhC with two aluminum plates on both top and bottom. In the right picture, the top aluminum plate is removed and the PhC with periodic ceramic rods is shown. The down-left picture is to show details of PhC in the zoomed area of the right picture.

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In the second step, we focus on the splitting effect for spectrometers. Electromagnetic beams of different frequencies, each with a certain finite width, are incident from the left. The field distribution of the refracted beams is measured at various probing angles to observe the beam-splitting capability at different frequencies. This is achieved by scanning the circular edges of the PhC in $0.5$ mm steps which are much smaller than the wavelengths to ensure that the curve of the detected results looks continuous. The top view and the side view of the experimental setup are shown in Fig. 7(a) and 7(b). There is a red line signified in Fig. 7(a), which is treated as the interface between PhC and air, as the position of the $y$-axis. The rods of PhC at the left side of the red line, are removed to make the effect of tilting PhC with a certain oblique angle, which is also shown schematically in Fig. 5. To enhance coupling efficiency, a gradient (at $z$-direction) wave-guide is incorporated [34]. According to the theory in the upper section, the parameters are set as follows, the incident angle $\theta _i$ as $4$ degrees, the oblique angle $\phi$ of PhC as $5$ degrees, the incident beam width is $W_0=39.6$ mm and the radium of PhC plate as $335$ mm for the expected resolution as $40$.

 

Fig. 7. (a)Schematic diagram to show the testing method of the directional gap of PhC. The right dark-grey section symbolizes the horn antenna operating between $27.5$ and $40 GHz$, while the left dark-grey section represents a probe used to measure the transmitted field. (b)The measured transmittance $S12$ , without(red line) and with time-gating(black line), for the $x$ direction for the PhC. The shaded areas are the gap ranges obtained by theoretical calculation with $\epsilon _r =7.8$. (c)Similar to (b), the measured transmittance, without(red line) and with time-gating(black line), for the $y$ direction. The shaded areas are the gap ranges obtained by theoretical calculation with $\ epsilon_r =7.8$. (d)The band gap structure of TM($E_z$-polarization) modes of the PhC slab with $\beta = b/a = 1.8$, $r = 0.19a$, and $\epsilon _r =7.8$.

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Three frequencies, $34$ GHz, $35.2$ GHz, and $36$ GHz, are chosen as the incident frequencies. The field distribution of the refracted beam is probed, and the results are shown in Fig. 7(c), from which the beam-splitting effect of the PhC spectrometer can be seen. The theoretical predictions of the beam profiles from Eq. (3) are shown in Fig. 7(d). From the comparison between Fig. 7(c) and 7(d), we can see that the beam center position and the beam-splitting effect agree very well with the theoretical expectations. As a result, an incident beam with different frequencies is refracted at the interface between air and PhC because of the super-prism effect and split into beams at different directions according to their frequencies, and then these beams to different directions will be separated after propagating enough long path.

However, the beam widths of the experimental results are obviously larger than the theoretical prediction. There are several probable reasons for the larger beam widths in the experiment, e.g. the fabrication quality of PhC, the reflecting of the incident microwave beam(the incident beam is more complex than a Gaussian beam), etc.

After all, for the first time, we experimentally demonstrate that the PhC spectrometer works, which is designed based on both super-prism and LSC effects of PhC. The laboratory results match the goal of our design. Although our experiments are carried out in the microwave frequency range, with the evolving sophistication of micro-nano processing technology, we’re optimistic about extending our design to optical frequencies, such as infrared, visible light, and ultraviolet. According to our theoretical prediction, we could obtain the compact on-chip ultra-high-resolution PhC spectrometers, which are about two orders smaller than the traditional design by grating. The spectrometer in optical frequencies can be implemented on silicon-on-insulator(SOI) chips as a thin-slab structure so that the operating bandwidth can be expanded further through a multi-layer design.

4. Conclusion

In summary, we have proposed a design for a compact high-resolution PhC spectrometer with a wide working bandwidth. The methods, to optimize the incident angle and the tilting angle of PhC for a wider working bandwidth, and to optimize the incident beam width and the size of PhC for a required resolution, are developed. For the first time, we experimentally verified the effects of such a PhC spectrometer based on both super-prim and LSC effects at microwave frequencies. As micro-nano processing technology advances, we anticipate that the design could be extended to various optical frequency ranges, including infrared, visible-light, and ultraviolet ranges. This approach has significant potential for the development of compact and wide-working bandwidth spectrometers with ultra-high-resolution in optical frequency ranges. Theoretically, the size of the ultra-high-resolution PhC spectrometer in optical frequency ranges based on our design could be two orders smaller and much lighter than the traditional spectrometer with a grating design. The PhC spectrometers in optical frequencies can be implemented on silicon-on-insulator(SOI) chips as a thin-slab structure so that the operating bandwidth can be expanded further through a multi-layer design. Such PhC spectrometers, integrated into chips, could enable applications like environmental monitoring, biomedical sensors, remote detection, etc. The related topics, such as the robustness of such spectrometers against disorders introduced by technological limits, the coupling efficiency between the spectrometer and the waveguides, and the resolution limit of such design, are waiting for further study.