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Abstract
In this paper, the relation between gain and resolution of an ideal analog optical differentiator in two different cases and their fundamental limits are investigated. Based on this relation, a figure of merit for comparison of the designed differentiators in recent papers is proposed. The differentiators are optimized using this figure of merit, and they are compared with each other to determine the best one. Also, a new differentiator is presented based on the dielectric slab waveguide in which the trade-off between its gain and resolution is easily controllable, and its best operating point is determined.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Analog optical computing proposes real-time response, high-throughput and low-power consumption operation in special-purpose computers in comparison with digital computers that cause high-power consumption and non-instantaneous response time due to the generality of their operations. In addition, relatively large size and slow response of electronic or mechanical analog computers are important restrictions that the analog optical computing concept has solved. These mentioned advantages have made this concept an attractive research topic in recent years [1].
In general, analog optical computing is divided into two major classes according to the domain in which the computations are performed: temporal or spatial domain. A lot of researches have been done on analog optical computing in the temporal domain, including compact differentiator [2] and differential equation solver with constant coefficients based on microring resonators [3,4], computation system for solving differential equations based on optical intensity differentiator [5], terahertz-bandwidth differentiator based on a silicon-on-isolator (SOI) directional coupler [6], photonic integrator for all-optical computing using an all-fiber active filter [7] and arbitrary-order differentiator based on fiber Bragg gratings [8].
On the other hand, two different methods named "metasurface (MS) approach" and "Green's function (GF) approach" have been proposed recently by Silva et al. [9] to perform spatial analog optical computing in the scale of a single wavelength or even sub-wavelength in contrast to using the traditional bulky system of lenses and filters. In these approaches, metamaterials are designed to manipulate the wavefront of an impinging wave as it propagates through the structure so that the output wave has a wavefront proportional to the signal calculated by performing the desired mathematical operation on the input wavefront. The first approach needs two extra sub-blocks to perform Fourier and inverse Fourier transform because it implements the Green's function of the desired operator in the spatial domain. But the second approach directly implements the desired operator in the spatial Fourier domain and it is easier for fabrication and miniaturization [9].
Differentiation is one of the basic mathematical operations having a lot of applications in engineering. For example, spatial differentiation can be used for edge detection in image processing [10,11]. Edge detection is an important step in image processing that provides information about the boundaries of objects. This information is required for object detection and features classification steps in image processing [12]. Time-consuming computation for high-throughput edge detection is a key challenge in real-time image processing [10]. Furthermore, it is needed that the edge detector has a high gain to provide the possibility of the implementation of edge detection in practice. On the other hand, the resolution is another important feature of an edge detector that determines the minimum resolvable distant of the edges in the image [10]. It should be noted that there is a trade-off between gain and resolution, and an investigation of this relation has great importance in the design procedure of appropriate edge detectors.
In this paper, we firstly investigate the relation between gain and resolution for an ideal differentiator. Then, we examine the ideal case of typical optical differentiators and propose a figure of merit to compare recently presented analog optical differentiators that can be used for edge detection. We optimize these systems based on the proposed figure of merit to work at their optimum operating point for having the best trade-off between resolution and gain. Next, a new design for performing edge detection with controllable resolution and gain is proposed and investigated here. Finally, all the investigated systems are compared using the proposed figure of merit and the best structure is determined.
2. Formulation
Differentiation is a mathematical operator that can be used for detecting the edges of an image. The systematic schematic of a mathematical edge detector working based on the differentiation operator $\partial /\partial {x}$ is shown in Fig. 1. The frequency response of this ideal system is $H_1(k_x)=jG'k_x/k_0$ where $G=\vert {G'}\vert$ can be defined as the gain (normalized to $1/k_0$) of the differentiator and $k_0$ is the wavenumber of the free space. The coefficient $G$ determines the strength of the output signal, and this is the reason why we name it the gain. Generally, it does not mean that the system actually amplify the input. If a rectangular pulse beam $p(x)=\sqcap (x/(2x_0))$ is assumed as the input of this system, the output of the system $q_1(x)$ consists of two impulse function at the locations of the edges in the input pulse as following:
where $\delta (\cdot )$ is the impulse function. However, in the real world, every physical system has limited bandwidth. In this section, we investigate the gain and resolution of two ideal, yet physical systems with different transfer functions. These systems are exact differentiators in a specific spatial bandwidth but they are different outside the band. The only limitation to these systems is that the magnitude of their transfer function should be always smaller than unity, which is inevitable in a passive system working with propagating waves.
Fig. 1. Systematic demonstration of an ideal edge detector performance with $H_1(k_x)=jG'k_x/k_0$. This system can detect edges of its input with zero resolution.
2.1 Case 1
In the first system, we limit the bandwidth by adding an ideal low pass filter with the bandwidth of $\Delta {k}$ to the differentiation operator as shown in Fig. 2. It should be noted that when we are working with propagating waves the maximum allowable bandwidth for this low pass filter is $\Delta {k}=k_0$. The frequency response of the low-pass filter is $H_2(k_x)=\sqcap (k_x/(2\Delta {k}))$. Also, we aim to design a passive system which imposes that the maximum magnitude of the frequency response is limited to unity. Therefore, we select the maximum possible bandwidth for the low pass filter as follows:
As a result, the total frequency response of this new system is obtained as shown in Fig. 3. Now, the output of the new system $q_2(x)$ is calculated as:Fig. 2. Systematic demonstration of an ideal limited-bandwidth edge detector performance with $H_1(k_x)=jG'k_x/k_0$ and $H_2(k_x)=\sqcap (k_x/(2\Delta {k}))$. This system can detect edges of its input with the resolution of $\pi /(\lambda _0\Delta {k})$. The maximum allowable value for $\Delta {k}$ is $k_0$.
Fig. 3. The ideal frequency response of the passive differentiator with limited-bandwidth: a) Magnitude b) Phase.
2.2 Case 2
In optics, most of the differentiators that can be used for performing edge detection have a frequency response similar to that is shown in Fig. 4. To investigate this case, we assume a beam propagating in free space with the wave-number of $k_z$ in the z-direction which has an electric field in the x-direction as $E_x^i(x,z)=p(x)e^{-jk_zz}$ where $p(x)$ is the profile of the beam and $k_z$ is propagation constant. The beam can be expressed as a superposition of plane waves:
where $P(k_x)$ is the spatial spectra of the beam determining the amplitude of a plane wave with the wave-vector $\mathbf {k}=k_x\hat {x}+k_z\hat {z}$ and $k_x=\sqrt {k_0^2-k_z^2}$. $P(k_x)$ is obtained using spatial Fourier transform: If a pulse-shaped profile is assumed for the wave as $p(x)=\sqcap (x/(2x_0))$, we have: Now, the spatial spectra of the output is calculated as $Q(k_x)=P(k_x)H(k_x)$ and $q(x)$ is as follows: and the electric field of the wave after passing through the system is $E_x^o(x,z)=q(x)e^{-jk_zz}$. In Eq. (8), the limits of the integral are from $-k_0$ to $k_0$ instead of $\pm \infty$ since the waves outside this limit are evanescent. After obtaining $q(x)$, the resolution is calculated based on the Rayleigh criterion described above. The results are plotted in Fig. 5. In this case, the relation between resolution and gain is modified as follows: that shows better resolution for a specific gain in comparison with the previous case. Therefore, a figure of merit (FOM) can be defined as:Fig. 5. Comparison of the relationship between resolution and gain in two different investigated cases. In case $1$, the transfer function of the edge detector system is the one plotted in Fig. 3 and in case $2$, the transfer function of the edge detector system is the one plotted in Fig. 4.
in each system to compare them with another. The lower the FOM the better the performance and the lower band for FOM is $0.345$ based on our hypothesized ideal, yet physical edge detector. In edge detection, the high gain and low resolution are always needed but they have a linear dependence as shown. Therefore, we cannot control the gain and the resolution independently and our proposed FOM can determine the best operating point in view of both gain and resolution simultaneously for analog optical edge detection. In the following, we will compare the systems used in previous works for edge detection based on this proposed criterion.
3. Results and discussions
In this section, we investigate some previous works which are lithography free (thus easy to fabricate) and they can be used for edge detection and compare their resolution and gain.
3.1 Brewster differentiator
In [14], the Brewster effect has been used to design a differentiator by just an interface between air and another dielectric with the refractive index $n$. The reflection coefficient of an incident wave with TM polarization has a zero at the Brewster angle defined as
that is used for the design of a differentiator with the normalized gain of [14] This means that the gain and the resolution of the differentiator can be controlled by the refractive index $n$ as shown in Fig. 6. We change $n$ from $2$ to $10$ and calculate the resolution and gain in each case to find the best refractive index. Figure 7 shows the resolution versus the gain. The minimum ratio of $\vert \left (\Delta {x}-0.5\right )/\left (G-1\right )\vert$ (FOM) determines the optimum refractive index. The FOM is decreased by increasing the refractive index as shown in Fig. 8. Therefore, based on the FOM versus the refractive index (from $0$ to $10$) shown in Fig. 8, the maximum investigated refractive index ($n=10$) in which the minimum ratio of $\vert \left (\Delta {x}-0.5\right )/\left (G-1\right )\vert$ (FOM) occurs should be selected for having the best trade-off between gain and resolution in such differentiator. But most of the materials in nature have refractive indices lower than 5. Therefore, we select a typical refractive index, for example $n=4$ (germanium), as the refractive index of optimized Brewster differentiator with $FOM=41.35$.Fig. 6. Green’s function of the differentiator based on Brewster effect for various refractive indices.
Fig. 7. The resolution versus gain of the Brewster differentiator for different refractive indices (n=2,3,…,10) of second medium. The first medium is assumed air. The incident angle for each refractive index is the Brewster angle $\theta _B=\tan ^{-1}(n)$.
Fig. 8. FOM versus different refractive indices (n=2,3,…,10) for the second medium in Brewster differentiator to determine the optimum operating point for Brewster differentiator used in edge detector. The first medium is assumed air. The incident angle for each refrective index is the Brewster angle $\theta _B=\tan ^{-1}(n)$.
3.2 Dielectric slab waveguide differentiator
In [15], a spatial integrator has been designed using a dielectric slab waveguide as shown in Fig. 9. In this work, prism coupling [16] is used to excite the mode of the dielectric slab. At an incident angle in which the mode of the structure is excited, the reflection coefficient of the dielectric slab has a zero. The reflection coefficient near the zero is similar to the Fourier-Green’s function of the spatial differentiator. Also, the trade-off between resolution and gain of the differentiator can be controlled by changing the distance between the dielectric slab and the prism $d$. Therefore, we investigate a controllable differentiator using this structure in our work. The tunability of this structure with $d$ is shown in Fig. 10 in which the Green’s function of the structure has plotted for different values of $d$. We assume $n_1=1.5$ (the approximate refractive index of $SiO_2$ at optical range), $n_2=3.4$ (the approximate refractive index of $Si$ at optical range), and $h=0.4\,\mu {m}$.
Fig. 9. The structure of the proposed differentiator using a dielectric slab waveguide that has the feature of tunability of the gain and resolution.
Fig. 10. Green’s function of the proposed tunable differentiator for different values of parameter $d$ (the distance between the dielectric slab and the prism). a) TE polarization by incident angle of $67^\circ$ and b) TM polarization by incident angle of $59.7^\circ$.
Figure 11 shows the resolution versus gain for different values of $d$ for both TE and TM polarizations. FOM has been plotted in Fig. 12 for both polarizations that determines the optimum value to have the best trade-off between gain and resolution is $d=0.15\,\mu {m}$ with TM polarization for which the ratio of $\vert \left (\Delta {x}-0.5\right )/\left (G-1\right )\vert$ is minimum and $FOM=0.385$. This value is very close to the minimum achievable FOM i.e. 0.345.
Fig. 11. The resolution versus gain of the proposed tunable differentiator for different values of parameter $d$ (the distance between the dielectric slab and the prism) for both TE and TM polarization with $\theta _i=67^\circ$ and $\theta _i=59.7^\circ$, respectively.
Fig. 12. FOM of the proposed tunable differentiator for different values of parameter $d$ (the distance between the dielectric slab and the prism) for both TE and TM polarization with $\theta _i=67^\circ$ and $\theta _i=59.7^\circ$, respectively.
3.3 Half-wavelength slab differentiator
In [17], half-wavelength slabs have been used to perform analog optical differentiation. The structure is a dielectric slab with a length of $L$ and a refractive index of $n_2$ between two semi-infinite dielectric media of the refractive index of $n_1$. The wave incidents on the slab with an angle of $\theta _1$ and refracts with the angle of $\theta _2=sin^{-1}(n_1\sin \theta _1/n_2)$ based on the Snell’s law. Selecting the length
where $k_0$ is the wavenumber of the free space, the reflection coefficient of this structure is zero for both TE and TM polarizations and this is used for applying differentiation on the profile of the incident wave and detecting its edges. If we change the parameter $n_2$ and $\theta _1$, we can control the resolution and gain of this edge detector as shown in Fig. 13 for both TE and TM polarizations.Fig. 13. Green’s function of the half-wavelength slab differentiator for different refractive indices and incident angles. a) TE polarization and b) TM polarization.
Figure 14 shows the ratio of $\vert \left (\Delta {x}-0.5\right )/\left (G-1\right )\vert$ for different values of $n_2$ and $\theta _1$ for TE polarization. Therefore, the optimum point of TE polarization is obtained by selecting $n_2=2.85$ and $\theta _1=21.72^\circ$ having the minimum of $FOM=19.9$.
Fig. 14. FOM of the half-wavelength slab differentiator for TE polarization for different values of $n_2$ and $\theta _1$.
3.4 Plasmonic differentiator
A single layer of a thin metal film in the Kretschmann configuration as the simplest surface plasmonic structure has been used to demonstrate optical spatial differentiation in [18] and its application for edge detection in image processing in [10]. The input wave encounters the metal layer from the glass side with an incident angle $\theta$. At the incident angle in which the phase matching condition is satisfied, the input wave excites the surface plasmon polariton (SPP) wave at the boundary between metal and air. This results in a dip of the spectral transfer function $H(k_x)$ at $k_x=0$ that is the reflection coefficient of the structure. If the radiative leakage rate of the SPP and the intrinsic material loss rate are equal, namely at the critical coupling condition, the spectral transfer function is approximated as
near $k_x=0$, which is the transfer function of a first-order spatial differentiator with the gain of $k_0/B$. Selecting the appropriate thickness for the metal layer satisfying the critical coupling condition and the type of metal used in the structure affects the gain of the differentiator. In [10], silver has been deposited on a BK7 glass substrate. The thickness of the silver film is $50\,nm$ and the dielectric constant of the silver is $\epsilon _{Ag}=-11.51-0.55j$. In this case, the parameter $B$ is determined as $B=0.0075k0$ where $k_0$ is the wavevector of the free space. Therefore, the gain for this structure with the silver film is $G=133.3$. Now, we use this plasmonic differentiator as the system shown in Fig. 1 to determine its resolution.If we change the metal to the one that has a different loss, we can control the trade-off between gain and resolution. Using a metal with more loss results in a better resolution and lower gain. The spatial spectral transfer function of this plasmonic differentiator is calculated and shown in Fig. 15 for the metal layer as gold and aluminum in comparison with the silver case. As we expected, when the gold or aluminum is used instead of silver, the gain of the differentiator decreases and its resolution increases. The gains for the case of the gold and aluminum are $G=7.5$ and $G=50$, respectively. The resolutions are obtained $\Delta {x}=62.45$, $\Delta {x}=12.9$ and $\Delta {x}=31.8$ for silver, gold and aluminum cases, respectively. Therefore, comparison of these three different cases shows that using silver in the structure results in the minimum ratio of $\vert \left (\Delta {x}-0.5\right )/\left (G-1\right )\vert$ and corresponds to ($FOM=0.47$) which is near to the optimum achievable FOM.
3.5 Photonic Spin Hall Effect differentiator
When a beam of light propagates at an interface, in the reflected or transmitted beam its spin components (different polarization components of the beam) split perpendicular to the plane of incidence. This phenomenon is known as the Photonic Spin Hall Effect (PSHE) [19]. Based on PSHE, Zhu et al. in [20] showed that a single planar interface can compute spatial differentiation to paraxial coherent beams under oblique incidence. The three most important distinctions of this research rather than others are that firstly differentiation of optical beam occurs in any oblique incident angle, secondly the differentiation happens for any interface, for example, air-glass or air-gold interface, thirdly because of optic nature (the PSHE is a non-dispersive effect) this non-resonant differentiator will work at any optical frequency. The authors proved the generality of their proposed spatial differentiator by some experiments.
In [20], the transfer function of the system under paraxial approximation has been simplified as (see Fig. 1 of [20]):
and with the assumption of $\delta \vert {k_y}\vert \ll 1$, it has been approximated as where $r_p$ and $r_s$ are the Fresnel’s reflection coefficients of the p and s polarizations for the beam incident angle $\theta _1$, $\delta =2\cot \theta _1/k_0$ and . The detailed derivation of the transfer function has been given in the Supplemental Material of [20]. As can be seen from the above equation, the transfer function is equal to a first-order y-directional differentiator. The parameter $k_0\delta \left (r_s+r_p\right )/2$ is equal to the gain of the differentiator, and the bandwidth of the differentiator is limited by The bandwidth and gain of this system can be controlled by changing the refractive index of the second medium and the incident angle. Figure 16 shows the transfer function of the system for different interfaces which work as a differentiator. Also, Fig. 17 shows the FOM for different edge detectors at different incident angles and refractive indices. It should be noted here that the first medium is assumed to be air. Based on this figure, the optimum operating point for x-polarized incident wave is $n=5.21$ and $\theta _1=48.43^\circ$ that result in $FOM=0.52$. Because of the symmetry, this point is the optimum point for y-polarized incident wave, too.Fig. 16. The spatial spectral transfer function of differentiator based on photonic spin hall effect for different materials as the second medium and different angles of incidence. The first medium is assumed to be air.
Fig. 17. The FOM of differentiator based on photonic spin Hall effect for different refractive indices and incident angles for x-polarized incident wave.
3.6 Comparison
Now, we can summarize the obtained result about the performance of the systems that can be used for edge detection. FOM for different investigated systems is shown in Table 1. Based on this FOM, the dielectric slab waveguide differentiator used for implementing differentiator in this work to perform edge detection has the best performance in TM polarization mode and its FOM is very close to the 0.345 which is the best achievable FOM obtained for an ideal case. It is because of that its Green’s function is the nearest function to the considered ideal case 2 between investigated structures. The structure based on photonic spin Hall effect has a second rank due to the fact that its Green’s function is the nearest function to that considered in case 1.
Table 1. FOM for selected structures that can be used for edge detection
4. Conclusion
The concepts of resolution and gain in optical differentiator were explained. The trade-off between these features and fundamental limits of their relation were investigated. The relation between resolution and gain was calculated numerically for an ideal passive differentiator in two cases. A figure of merit was proposed based on the obtained relation to providing a way for the comparison of various differentiators presented in recent papers. Then, some of the recently presented differentiators were optimized to work at their best operating point in terms of the proposed FOM. Finally, these differentiators were compared with each other and the best one were determined.
Funding
Iran National Science Foundation (96008035).
Disclosures
The authors declare no conflicts of interest.
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