Isotropic Filters in Spatial Frequency Domain by a Photonic Crystal Slab

Abstract

An isotropic imaging filter is provided that includes a photonic crystal slab, where the photonic crystal slab includes a square lattice of air through holes, a dielectric constant, a thickness (d), a through hole radius (r), and a lattice constant (a), where the square lattice of air holes are separated according to a value of the lattice constant, where the thickness is configured according to d=M(a), where the through hole radii is configured according to r=N(a), where the thickness and the hole radii are configured to generate isotropic bands of guided resonances of an incident image.

Claims

1) A method of making an isotropic imaging filter comprising a photonic crystal slab, comprising: designing a photonic crystal slab having a periodic lattice of air through holes, wherein parameters of a photonic crystal slab design include a dielectric constant, a thickness, a through hole radius and a lattice constant, wherein the periodic lattice of air holes are separated according to a value of the lattice constant, wherein the thickness and the hole radius are configured such that the photonic crystal slab generates isotropic bands of guided resonances; and making a photonic crystal slab according to the photonic crystal slab design.

2) The method of claim 1, wherein the guided resonances are selected from the group consisting of an isotropic high-pass filter guided resonance, an isotropic low-pass filter guided resonance, an isotropic band-reject filter guided resonance, and an isotropic band-pass filter guided resonance.

3) The method of claim 2, wherein isotropic low-pass filtering and isotropic band-pass filtering are obtained in reflection.

4) The method of claim 2, wherein isotropic high-pass filtering and isotropic band-reject filtering are obtained in transmission.

5) The method of claim 1, wherein the photonic crystal slab comprises a dielectric material.

6) The method of claim 1, further comprising disposing a uniform dielectric slab proximal to the photonic crystal slab, wherein a thickness of the uniform dielectric slab and an air gap between the uniform dielectric slab and the photonic crystal slab are configured to set a background transmission to unity.

7) The method of claim 1, wherein the hole radius is 0.5 the lattice constant.

8) A method of making an isotropic image filter comprising a photonic crystal slab, comprising: designing a photonic crystal slab having a periodic lattice of air through holes, wherein parameters of a photonic crystal slab design include a dielectric constant, a thickness, a through hole radius and a lattice constant; wherein the periodic lattice of air holes are separated according to a value of the lattice constant; wherein the thickness and the hole radius are configured such that the photonic crystal slab generates isotropic guided resonances; and making a photonic crystal slab according to the photonic crystal slab design.

9) The method of claim 8, wherein the guided resonances are selected from the group consisting of an isotropic high-pass filter guided resonance, an isotropic low-pass filter guided resonance, an isotropic band-reject filter guided resonance, and an isotropic band-pass filter guided resonance.

10) The method of claim 9, wherein isotropic low-pass filtering and isotropic band-pass filtering are obtained in reflection.

11) The method of claim 9, wherein isotropic high-pass filtering and isotropic band-reject filtering are obtained in transmission.

12) The method of claim 8, wherein the photonic crystal slab comprises a dielectric material.

13) The method of claim 8, further comprising disposing a uniform dielectric slab proximal to the photonic crystal slab, wherein a thickness of the uniform dielectric slab and an air gap between the uniform dielectric slab and the photonic crystal slab are configured to set a background transmission to unity.

14) The method of claim 8, wherein the hole radius is 0.5 the lattice constant.

15) A method of making an isotropic image filter comprising a photonic crystal slab, comprising: designing a photonic crystal slab having a periodic lattice of air through holes, wherein parameters of the photonic crystal slab include a dielectric constant, a thickness, a through hole radius and a lattice constant; wherein the periodic lattice of air holes are separated according to a value of the lattice constant; wherein the thickness and the hole radius are configured such that the photonic crystal slab generates isotropic image filtering; and making a photonic crystal slab according to the photonic crystal slab design.

16) The method of claim 15, wherein the isotropic image filtering is selected from the group consisting of isotropic high-pass filtering, isotropic low-pass filtering, isotropic band-reject filtering, and isotropic band-pass filtering.

17) The method of claim 16, wherein the isotropic low-pass filtering and the isotropic band-pass filtering are obtained in reflection.

18) The method of claim 16, wherein the isotropic high-pass filtering and the isotropic band-reject filtering are obtained in transmission.

19) The method of claim 15, wherein the photonic crystal slab comprises a dielectric material.

20) The method of claim 15, further comprising disposing a uniform dielectric slab proximal to the photonic crystal slab, wherein a thickness of the uniform dielectric slab and an air gap between the uniform dielectric slab and the photonic crystal slab are configured to set a background transmission to unity.

21) The method of claim 15, wherein the hole radius is 0.5 the lattice constant.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] FIGS. 1A-1C show (1A) a schematic drawing of the geometry of a single slab device having a photonic crystal slab with a square lattice of air holes, (1B) a schematic drawing of the geometry of a single slab device having a photonic crystal slab with a square lattice of air holes separated from a uniform dielectric slab by an air gap. The slabs have a dielectric constant ε=4.67. The geometry parameters are d=0.50a, r=0.11a, d.sub.s=0.12a, and for FIG. 1B d.sub.g=1.93a, where a is the lattice constant. The coordinate system is shown above the device. The arrow (L) indicates the direction of the incident light. The electric field directions of the S- and P-polarized light are also indicated. (1C) Geometry of the photonic crystal slab differentiator, which shows a photonic crystal slab separated from a uniform dielectric slab by an air gap. For ϵ=12, the geometry parameters are: d=0.55a, r=0.111a, d.sub.s=0.07a, d.sub.g=0.21a. The plane above and below shows the input and output image respectively. The transmitted image (e.g. ring) through the device is the Laplacian of the incident image (e.g. disk) illuminated by a normally incident light with frequency ω.sub.0=0.47656×2πc/a, according to embodiments of the current invention.

[0016] FIGS. 2A-2F show (2A), (2C), and (2D), nearly isotropic photonic band structure of the photonic crystal slab shown in FIGS. 1A-1B near the frequency ω.sub.0=0.77098×2πc/a. (2A) Band dispersions along Γ×X and Γ−M directions. (2C) Constant frequency contours of the lower band. (2D) Constant frequency contours of the upper band. (2B) Scheme of multiple filtering functions of the device. When the light frequency is on resonance at the normal incidence [ω=ω.sub.0, labeled by dark arrows in (2B) and bottom horizontal lines in (2E), (2F)], the device realizes an isotropic high-pass filter (Laplacian) at transmission and low-pass filter at reflection. When the light frequency is detuned from the resonance at the normal incidence [ω=ω.sub.k, labeled by gray arrows in (2B) and top horizontal lines in (2E), (2F)], the device realizes an isotropic band-reject filter at transmission and band-pass filter at reflection. (2E) Transmittance for S-polarized light as a function of ω and |k| near ω.sub.0=0.77098×2πc/a along a general wavevector direction (ω=14°). Due to the isotropic band structure, S light only excites the upper band, and the transmission spectra are almost identical along any wavevector direction φ. |t|=0 when ω=ω.sub.k. (2F) Reflectance |r| for S-polarized light as a function of co and |k| near ω.sub.0=0.77098×2πc/a along a general wavevector direction (φ=14°). The reflection spectra are identical along any wavevector direction. |r| 1 when ω=ω.sub.k. In all of the plots, frequency ω−ω.sub.0 is in units of 10.sup.−4×2πc/a, while |k|, k.sub.x, and k.sub.y are in units of 10.sup.−3×2π/a, according to the current invention.

[0017] FIGS. 3A-3F show isotropic high-pass filter (Laplacian). (3A) Transmittance for S-polarized light |t.sub.s|k.sub.x, k.sub.y at the frequency ω.sub.0=0.77098×2πc/a. (3B) |t.sub.s| as a function of |k| along a general wavevector direction (φ=14°). (3C) Log plot of the Fourier transform of the field profile for the incident image (3E): log|{tilde over (S)}.sub.in|(.sub.x,.sub.y). (3D) Log plot of the Fourier transform of the field profile for the reflected image (3F): log|{tilde over (S)}.sub.out|(.sub.x,.sub.y). The low wavevector components have been filtered out. (3E) Incident image |S.sub.in|.sup.2 of the Stanford emblem. The image size is 5220a×3456a. (3F) Calculated transmitted image |S.sub.out|.sup.2, which shows the edges with different orientations. |k|k.sub.x, and k.sub.y are in units of 10.sup.−2×2π/a, according to the current invention.

[0018] FIGS. 4A-4F show the isotropic low-pass filter performance. (4A) Reflectance for S-polarized light |r.sub.s|(k.sub.x, k.sub.y) at the frequency ω.sub.0=0.77098×2πc/a. (4B) |r.sub.s| as a function of along a general wavevector direction (φ=14°). (4C) Log plot of the Fourier transform of the field profile for the incident image (4E): log|{tilde over (S)}.sub.in|(.sub.x,.sub.y). (4D) Log plot of the Fourier transform of the field profile for the reflected image (4F): log|{tilde over (S)}.sub.out|(.sub.x,.sub.y). The high wavevector components have been filtered out. (4E) Incident image |S.sub.out|.sup.2 of the Stanford emblem corrupted by white noise. The image size is 5220a×3456a. (4F) Calculated reflected image |S.sub.out|.sup.2, which reduces the white noise by image smoothing. |k|k.sub.x, and k.sub.y are in units of 10.sup.−2×2π/a, according to the current invention.

[0019] FIGS. 5A-5F show isotropic band-reject filter results. (5A) Transmittance for S-polarized light |k|k.sub.x, and k.sub.y at the frequency ω.sub.q=0.77130×2πc/a. (5B) |t.sub.s| as a function of |k| along a general wavevector direction (φ=14°). (5C) Log plot of the Fourier transform of the field profile for the incident image (5E): log|{tilde over (S)}.sub.out|(.sub.x,.sub.y). The sinusoidal noise appears as peaks in the spectrum, which lie on an approximate circle around the origin. (5D) Log plot of the Fourier transform of the field profile for the transmitted image (5F): log|{tilde over (S)}.sub.out|(.sub.x,.sub.y). The wavevector components corresponding to the periodic noise have been filtered out. (5E) Incident image |S.sub.in|.sup.2 of the Stanford emblem corrupted by sinusoidal noise. The image size is 5220a×3456a. (5F) Calculated transmitted image |S.sub.out|.sup.2, which eliminates the periodic noise. |k|k.sub.x, and k.sub.y are in units of 10.sup.−2×2π/a, according to the current invention.

[0020] FIGS. 6A-6F show isotropic band-pass filter results. (6A) Reflectance for S-polarized light |r.sub.s|k.sub.x, k.sub.y at the frequency ω.sub.q=0.77130×2πc/a. (6B) |r.sub.s| as a function of |k| along a general wavevector direction (φ=14°). (6C) Log plot of the Fourier transform of the field profile for the incident image (6E): log|{tilde over (S)}.sub.out|(.sub.x,.sub.y). The sinusoidal noise appears as impulses in the spectrum, which lie on an approximate circle around the origin. (6D) Log plot of the Fourier transform of the field profile for the reflected image (6F): log|{tilde over (S)}.sub.out|(.sub.x,.sub.y). (6E) Incident image |S.sub.in|.sup.2 of the Stanford emblem corrupted by sinusoidal noise. The image size is 5220a×3456a. (6F) Calculated reflected image |S.sub.out|.sup.2 which isolates the periodic noise and simplifies its analysis. |k|k.sub.x, and k are in units of 10.sup.−2×2π/a, according to the current invention.

DETAILED DESCRIPTION

[0021] Disclosed herein is a showing that several types of isotropic image filters in the wavevector domain can be implemented with a single photonic crystal slab device, according to the current invention. According to one embodiment, a slab is provided that is configured so that the guided resonance near the F point exhibits an isotropic band structure. Depending on the light frequency and the choice of transmission or reflection mode, the invention realizes isotropic high-pass, low-pass, band-reject, and band-pass filtering in wavevector space. These filter functions are important for various image processing tasks, including edge detection, smoothing, white noise suppression, and suppression or extraction of periodic noises. Further disclosed is a numerical demonstration of these filter functionalities by simulations of a slab structure that is designed to operate in the visible wavelength range. The current invention expands the application of nanophotonics-based optical analog computing for image processing.

[0022] The purpose of filtering in the wavevector domain is to transform an image by modifying its Fourier transformation. In general, for a normally incident light beam along the z axis with a transverse field profile S.sub.in(x, y), the transmitted or reflected beam has a profile S.sub.out(x,y)=.sup.−1[H(.sub.x,k.sub.y){tilde over (S)}.sub.in(.sub.x, .sub.y)], where .sup.−1 is the inverse Fourier transform, {tilde over (S)}.sub.in(.sub.x, .sub.y) is the Fourier transform of the input image, and H(k.sub.x, k.sub.y) is the filter transfer function. If H(.sub.x,.sub.y)=H(|k|) only depends on the magnitude of the wavevector |k|, the filter is isotropic. Here, k=(k.sub.x, k.sub.y) refers to the in-plane wavevector.

[0023] According to one embodiment of the invention, the isotropic filters are realized by a photonic crystal slab device, as shown in FIGS. 1A-1c. The photonic crystal slab of the current invention exhibits a nontrivial isotropic band structure near the F point. The design details for such a photonic crystal slab, see for example the embodiment shown FIG. 1C. The exemplary embodiment shown in FIG. 1C can use either Si, GaAs, or any dielectric, where Si, or GaAs is applicable only in the infrared wavelength range. Further embodiments follow the same principle as the embodiment shown in FIG. 1C to enable a structure that can operate in the visible wavelength range. In one embodiment, the dielectric constant of the material for the slabs is ε=4.67, which approximates that of Si.sub.3N.sub.4 in the visible wavelength range. Provided here is a dielectric constant that is real, as the extinction coefficient of Si.sub.3N.sub.4 at a visible wavelength is negligible. The photonic crystal slab has a thickness of d=0.50a and a square lattice of air holes with radii r=0.11a, where a is the lattice constant. The thickness d and radius r are chosen to realize isotropic bands of guided resonances. A uniform dielectric slab with a thickness d.sub.s=0.12a is placed in the vicinity of the photonic crystal slab. The air gap between the two slabs has a thickness d.sub.g=1.93a. d.sub.s and d.sub.g are chosen to set the background transmission to be unity. As a side note, the two-layer structure discussed herein is quite compact. For a resonant wavelength A=500 nm, a=650 nm, the total thickness is d+d.sub.g+d.sub.s=1.66 μm. Such a structure can be fabricated using, e.g., focused ion beam assisted laser interference lithography.

[0024] The photonic crystal slab in FIGS. 1A-1C hosts a pair of guided resonances, which are degenerate at the Γ point with a frequency ω.sub.0=0.77098×2πc/a. In general, in the vicinity of the Γ point, the band structure is highly anisotropic in the k space, assuming only the C.sub.4v symmetry of the structure. However, remarkably, with the carefully tuned geometry parameters above, both bands of the guided resonances exhibit almost complete circular symmetry in the real part of the eigenfrequencies:

ω.sub.k≈ω.sub.0A|k|.sup.2,  (1)

where A.sub.+=4.35, A.sub.−=−1.41 from fitting the band dispersion, and the upper (lower) sign corresponds to the upper (lower) band. In Eq. (1), and throughout the rest of this disclosure, the wavevector will be in units of 2π/a.

[0025] The nearly isotropic photonic band structure (ω.sub.±()−ω.sub.0) for the structures shown in FIGS. 1B-1C are plotted in FIG. 2A, FIG. 2C, and FIG. 2D. FIG. 2A shows that for both bands, the dispersions along the Γ−M and Γ−X directions have almost identical effective masses. Here, the effective mass tensor is defined as m.sub.ij*=ℏ[∂.sup.2ω()/∂k.sub.i∂k.sub.j].sup.−1. FIG. 2C and FIG. 2D show that the constant frequency contours for both bands are almost circular. It is noted here that the radiative linewidths γ.sub.±(), unlike ω.sub.±() are anisotropic. Nonetheless, (γ.sub.±(−γ.sub.0) are much smaller than (ω.sub.±(−ω.sub.0), thus, they do not affect the circular symmetry of the transfer functions much, as we will show later.

[0026] In general, guided resonances in photonic crystal slabs may induce sharp Fano resonance features in the transmission and reflection spectra. For the specific pair of guided resonances considered here, it has been proved that, due to the isotropic band structure, S-polarized (P-polarized) light can only excite the upper (lower) band for every direction of incidence. This effect is referred to as single-band excitation.

[0027] Due to the single-band excitation effect, as well as the presence of the uniform dielectric slab, which sets the background transmission coefficient to be unity, for this structure shown in FIGS. 1B-1C, the transmission and reflection coefficients are

[00001]$\begin{array}{cc}{t}_{\pm }\left(\omega ,k\right)=\frac{i\left[\omega -{\omega }_{\pm }\left(k\right)\right]}{i\left[\omega -{\omega }_{\pm }\left(k\right)\right]+{\gamma }_{\pm }\left(k\right)},& \left(2\right)\\ r_{\pm }\left(\omega ,k\right)=-e^{i\varphi }\frac{{\gamma }_{\pm }\left(k\right)}{i\left[\omega -{\omega }_{\pm }\left(k\right)\right]+{\gamma }_{\pm }\left(k\right)},& \left(3\right)\end{array}$

where the upper (lower) sign corresponds to S-polarized (P-polarized) light and upper (lower) band; ω is the incident light frequency. Therefore, on resonance,

t.sub.±(ω.sub.±(),)=0,r.sub.±(ω.sub.±(),)=−e.sup.jϕ.  (4)

[0028] The numerically determined transmission and reflection spectra for S-polarized light are plotted in FIG. 2E and FIG. 2F. Due to the effect of single-band excitation, S-polarized light only excites the upper band of guided resonances. Moreover, as expected from Eqs. (1)-(3), the resultant transmission and reflection spectra are isotropic, i.e., the spectra are identical along any wavevector direction as defined by the azimuthal angle φ in FIGS. 1B-1C. When ω=ω(k), the transmittance exhibits sharp dips with |t|=0, while the reflectance exhibits peaks with |r|=1, as expected from Eq. (4).

[0029] Depending on the operating conditions, the structures as shown in FIGS. 1A-1C can be used to perform several very useful image processing functionalities.

[0030] Turning now to the isotropic high-pass filter embodiment, where the inventors have shown the k-dependent transmittance at the frequency ω=ω.sub.0≡ω.sub.±(=0)

[00002]$\begin{array}{cc}.Math.t_{\pm }\left({\omega }_0,k\right).Math.=\frac{.Math.{\omega }_{\pm }\left(k\right)-{\omega }_0.Math.}{\sqrt{{\left[{\omega }_{\pm }\left(k\right)-{\omega }_0\right]}^2+{{\gamma }_{\pm }\left(k\right)}^2}}& \left(5\right)\\ \approx \frac{.Math.{\omega }_{\pm }\left(k\right)-{\omega }_0.Math.}{{\gamma }_0}=\frac{A_{\pm }}{{\gamma }_0}{.Math.k.Math.}^2.& \left(6\right)\end{array}$

[0031] This transmittance realizes the Laplacian, a special isotropic high-pass filter.

[0032] Disclosed herein, it is show that the same device can provide a few other very useful image processing functionalities under different operating conditions.

[0033] Regarding the isotropic low-pass filter embodiment, at the frequency ω=ω.sub.0, if one considers instead the reflected light, the transfer function is

[00003]$\begin{array}{cc}.Math.r_{\pm }\left({\omega }_0,k\right).Math.=\frac{1}{\sqrt{1+{\left[{\omega }_{\pm }\left(k\right)-{\omega }_0\right]}^2/{{\gamma }_{\pm }\left(k\right)}^2}}\approx \frac{1}{\sqrt{1+A_{\pm }^2{.Math.k.Math.}^4/{\gamma }_0^2}}.& \left(7\right)\end{array}$

[0034] This transfer function realizes an isotropic low-pass filter with reflection peak |r.sub.±|=1 at the Γ point.

[0035] Regarding the isotropic band-reject filter, this embodiment is configured to operate away from the frequency ω.sub.0, but at the frequency ω=ω.sub.±,q where q is the amplitude of a non-zero in-plane wavevector, the transfer function then becomes

[00004]$\begin{array}{cc}.Math.t_{\pm }\left({\omega }_{\pm ,q},k\right).Math.=\frac{.Math.{\omega }_{\pm }\left(k\right)-{\omega }_{\pm ,q}.Math.}{\sqrt{{\left[{\omega }_{\pm }\left(k\right)-{\omega }_{\pm ,q}\right]}^2+{{\gamma }_{\pm }\left(k\right)}^2}}.& \left(8\right)\end{array}$

[0036] This transfer function realizes an isotropic band-reject filter with transmission dip=|t.sub.±|=0 at ||=q.

[0037] Turning now to the isotropic band-pass filter, at the frequency ω=ψ.sub.±,q the reflrection to has a transfer function

[00005]$\begin{array}{cc}.Math.r_{\pm }\left({\omega }_q,k\right).Math.=\frac{1}{\sqrt{1+{\left[{\omega }_{\pm }\left(k\right)-{\omega }_{\pm ,q}\right]}^2/{{\gamma }_{\pm }\left(k\right)}^2}}.& \left(9\right)\end{array}$

[0038] This transfer function realizes an isotropic band-pass filter with reflection peak |r.sub.±|=1 at ||=q.

[0039] Therefore, devices according to the current invention achieve multiple filtering functions. As schematically shown in FIG. 2B, when the light frequency is on resonance at normal incidence (ω=ω.sub.0), the devices operate as an isotropic high-pass filter (Laplacian) at the transmission mode and an isotropic low-pass filter at the reflection mode. When the light frequency is detuned a bit from the resonance at normal incidence, the device operates as an isotropic band-reject filter at the transmission mode and an isotropic band-pass filter at the reflection mode, where the rejected or passed wavevector components are determined by the light frequency detuning and polarization. For transmission mode, the transmitted image is the required filtered result. For reflection mode, the reflected image is the required filtered result, which can be separated from the incident image by using a beam splitter.

[0040] The filtering functions of the current invention are numerically demonstrated herein. The isotropic high-pass filter (Laplacian) is disclosed in FIG. 1C. Since the physical structure is different, a brief discussion of the performance of this device is disclosed herein as a Laplacian for completeness, with focus included on the other three isotropic filters.

[0041] In all the numerical demonstrations below, the incident beam is S polarized. The transmitted image is calculated following the standard way in image processing. (1) Compute the Fourier transform {tilde over (S)}i.sub.n(.sub.k,.sub.y) of the incident field profile S.sub.in(x, y). Note the incident image is |S.sub.in(x,y)|.sup.2. (2) Compute the Fourier transform of the output field profile, {tilde over (S)}.sub.out(.sub.x,.sub.y)=H(.sub.x,.sub.y){tilde over (S)}.sub.in(.sub.x,.sub.y), where H(k.sub.x, k.sub.y) is the transfer function. (3) Obtain the output field profile S.sub.out(x, y) by inverse Fourier transform. Calculate the output image |S.sub.out(x,y)|.sup.2.

[0042] Regarding the isotropic high-pass filter (the Laplacian) FIG. 3 shows the isotropic high-pass filter (the Laplacian). FIG. 3A plots the filter transfer function: the transmittance for S-polarized light |t.sub.s|(k.sub.x, k.sub.y) at the frequency ω.sub.0=0.77098×2πc/a. The transfer function is almost isotropic. FIG. 3B plots |t.sub.s| as a function of along a general wave-vector direction (φ=14°), and the fitting result using Eqs. (5) and (6), respectively. The fitting of Eq. (5) is almost perfect in the wavevector range as shown, while the quadratic fitting using Eq. (6) is very good for up to 0.6×10.sup.−2×2π/a. These plots confirm that the device indeed operates as an isotropic high-pass filter (the Laplacian) in this case. In FIG. 3A [and also in FIG. 4A, as shown later], at larger wavevectors, the transfer function exhibits some anisotropy due to the dependency of the radiative linewidth γ(k) on the direction of k.

[0043] The Laplacian enables image sharpening and edge detection. FIG. 3E shows an incident image of the Stanford emblem, while FIG. 3C plots the Fourier transform of the field profile for this incident image. FIG. 3D shows the calculated Fourier spectrum for the transmitted image, which is obtained by a pointwise product of FIG. 3A and FIG. 3C. The low wavevector components have been filtered out. FIG. 3F is the calculated transmitted image, which shows all the edges with different orientations.

[0044] Turning now to the isotropic low-pass filter, FIG. 4 shows the isotropic low-pass filter. FIG. 4A plots the filter transfer function: the reflectance for S-polarized light ∥r.sub.s|(k.sub.x,k.sub.y) at the frequency ω.sub.0=0.77098×2πc/a. The transfer function is almost isotropic. FIG. 4B plots |r.sub.s| as a function of |k| along a general wavevector direction (φ=14°) and the fitting result of Eq. (7). The fitting is almost perfect in the wavevector range as shown. These plots confirm that the device indeed operates as an isotropic low-pass filter in this case.

[0045] The isotropic low-pass filter accomplishes image smoothing, with applications ranging from character recognition in ma-chine perception, preprocessing functions in the printing and publishing industry, to satellite and aerial image processing. Here, one specific application of the low-pass filter is shown in white noise reduction. FIG. 4E shows an incident image of the Stanford emblem corrupted by white noise, while FIG. 4C plots the Fourier transform of the field profile for this incident image. FIG. 4D shows the calculated Fourier spectrum for the reflected image, which is obtained by a pointwise product of FIG. 4A and FIG. 4C. The high wavevector components have been filtered out. FIG. 4F shows the calculated reflected image, where the white noise has indeed been reduced, demonstrating image smoothing.

[0046] Regarding the isotropic band-rejection filter, FIG. 5 illustrates the isotropic band-reject filter. FIG. 5A plots the filter transfer function: the transmittance for S-polarized light |t.sub.s|(k.sub.x,k.sub.y) at the frequency ω.sub.q=0.77130×2πc/a. The transfer function is almost isotropic. FIG. 5B plots |t.sub.s| as a function of |k| along a general wavevector direction (φ=14°), which shows |t.sub.s|=0 at =0.84×10.sup.−2×2π/a. The transmittance curve is fit with Eq. (8) together with Eq. (1) and γk≈γ.sub.0. The fitting is almost perfect in the wavevector range as shown. These plots confirm that the device indeed operates as an isotropic band-reject filter in this case.

[0047] The isotropic band-reject filter can effectively eliminate periodic noise, a common type of noise arising typically from electrical or electromechanical interference during image acquisition. As periodic noise appears as peaks in the Fourier transform at locations corresponding to the wavevectors of the periodic interference, it can be isolated and filtered by band-reject filters. Here, shown is an example of periodic noise reduction with the isotropic band-reject filter. FIG. 5E shows an incident image of the Stanford emblem corrupted by sinusoidal noise, while FIG. 5C plots the Fourier transform of the field profile for this incident image. The periodic noise appears as spectral peaks in the wavevector space, which lie on an approximate circle around the origin. FIG. 5D shows the calculated Fourier spectrum for the transmitted image, which is obtained by a pointwise product of FIG. 5A and FIG. 5C. The spectral peaks corresponding to the periodic noise have been filtered out. FIG. 5F shows the calculated transmitted image, where the periodic noise has indeed been eliminated effectively. Noted here is that the rejected wavevector where the maximum rejection occurs in our band-reject filter can be easily tuned by tuning the light frequency.

[0048] Turning now to the isotropic band-pass filter, FIG. 6 illustrates the isotropic band-pass filter. FIG. 6A plots the filter transfer function: the reflectance for S-polarized light |r.sub.s|(k.sub.x, k.sub.y) at the frequency ω.sub.q=0.77130×2πc/a. The transfer function is almost isotropic. FIG. 6B plots |r.sub.s| as a function of |k| along a general wavevector direction (φ=14°), which shows |r.sub.s|=1 at =0.84×10.sup.−2×2π/a. The reflectance curve is fit with Eq. (9) together with Eq. (1) and γ.sub.k≈γ.sub.0. The fitting is almost perfect in the wavevector range as shown. These plots confirm that the device indeed operates as an isotropic band-pass filter in this case.

[0049] The isotropic band-pass filter performs the opposite operation of the band-reject filter. It is quite useful in isolating the effects on an image caused by selected wavevector bands. Shown here is an example of extracting periodic noise patterns with the isotropic band-pass filter. FIG. 6E shows an incident image of the Stanford emblem corrupted by sinusoidal noise [same as FIG. 5E], while FIG. 6C plots the Fourier transform of the field profile for this incident image. FIG. 6D shows the calculated Fourier spectrum for the reflected image, which is obtained by a pointwise product of FIG. 6A and FIG. 6C. FIG. 6F shows the calculated reflected image, where the periodic noise pattern is isolated and appears more clearly. This is useful because it simplifies the analysis of the noise, largely independent of the image content.

[0050] The design of isotropic wavevector domain image filters using a photonic crystal slab is based on the guided resonances with isotropic band structure. The same idea can extend to other photonic structures that host resonant modes with isotropic band structures. In particular, a phase-shifted Bragg grating can also perform the four filtering functionalities of our device, but with the transmission/reflection mode flipped.

[0051] The present invention has now been described in accordance with several exemplary embodiments, which are intended to be illustrative in all aspects, rather than restrictive. Thus, the present invention is capable of many variations in detailed implementation, which may be derived from the description contained herein by a person of ordinary skill in the art. For example, one may use other photonic devices rather than photonic crystals that incorporate guided resonances with desired band structure, such as metasurfaces. One may use geometry that is different from a square lattice of air holes; other lattices of other shapes of holes are also possible. Moreover, our design can be readily extended to multiple frequencies, by using a stack of multiple layers. Our design principle is not restricted to specific materials; it is general to any dielectrics.

[0052] All such variations are considered to be within the scope and spirit of the present invention as defined by the following claims and their legal equivalents.