The Design of Optical Diodes

In this paper, the optical diodes have been designed by one-dimensional function photonic crystals (FPCs), which the refractive indexes of media A and B are not constant, but it is the functions of space coordinate. By calculation the transmissivity and electric ﬁeld distribution, we found the positive incident light can pass the function photonic crystals, but the negative incident light can not pass through it, the function photonic crystals (FPCs) can be made into optical diodes.


Introduction
Electrical diodes are basic electronic devices and essential building blocks of modern electronics. Such as p-n junction diodes and Schottky diodes. But they have too many limitations. So we want to overcome the limitations posed by electronic devices, and to satisfy the need for optical elements in integrated photonic circuits, Asymmetric light transmission is essential for fabricating such an all-optics based system. Optical diodes enable asymmetric transmission or one-way transmission of light. So, numerous optical diodes have been manufactured using the magneto-optic effect, the acousto-optic effect, photonic crystals, second harmonic generators, the thermo-optic effect, dynamically modulated ring resonator structures or absorbing multilayer systems, [1][2][3][4][5][6][7] in the past.
There has been rapidly growing interest in optical-diode phenomenon, we can employ magneto-optical materials, [18,19] nonlinear media, [20] or spatial-temporal modulations of refractive indices [21,22]. All the three approaches could implement an ideal optical diodes that can transmit and block any spatial mode in the opposite two directions. It's characterized by cheap, don't need require large magnetic fields and it can be used to miniaturized optical circuits straightforward [23,24].
An important feature of the photonic crystal is that there are allowed and forbidden ranges of frequencies, and the forbidden frequency range known as photonic band gap (PBG). In Refs. [25][26][27][28][29][30], we have proposed the FPCs, which the medium refractive index is the function of space position, which can be realized by the electro-optic effect or the Kerr effect. On the basis, we have designed optical diodes using one-dimensional FPCs (AB) N , which the refractive indexes of media A and B are not constant, but the functions of space coordinate, such as n a (x) = n a (0) + na(a)−na(0) a x, and n b (x) = n b (0) x, We calculate the transmissivity and electric field distribution of the negative incident and reverse incident of light in onedimensional FPCs respectively. By calculating, we found the positive incident light can pass the function photonic crystals, this is the characteristic of optical diodes. The function photonic crystals (FPCs) can be made into optical diodes. In Ref. [30], we have designed the optical triode with the one-dimensional FPCs. For the convention photonic crystals, the refractive indexes of media A and B are constant, because its transmissivity and electric field distribution are the same for the positive incident and negative incident, it can not be made into optical diodes.
2. The transfer matrix and transmissivity of one-dimensional function photonic crystals (FPCs) In Refs. [25][26][27][28][29][30], we have given the transfer matrices of media A and B, they are where In one period, the transfer matrix M is The form of the FPCs transfer matrix M is more complex than the conventional PCs. The characteristic equation of FPCs is Where N is the period number.
With the transfer matrix M , we can obtain the transmission and reflection coefficient t and r, and the transmissivity and reflectivity T and R, they are and Where η 0 = η N +1 = ε0 µ0 , E 0i and E 0r are incident and reflection electric fields, and E 1 = E 0i + E 0r . In the following, we give the electric field of light in the one-dimensional FPCs, it is shown in FIG. 1. The optical diode model is shown in FIG. 2, the light positive incident (red line) and negative incident (blue line) to the FPCs, there is where d 1 and d 2 are the thickness of first and second medium, respectively, E(z) and H(z) are the electric field and magnetic field in the second medium, when position z changes we can obtain the electric field and magnetic field in other medium. The E tN +1 and H tN +1 are the output electric field and magnetic field. From Eqs. (10)(11)(12)(13)(14), we get and then

Numerical result
In this section, we report our numerical results of transmissivity and electric field distribution. The refractive indexes of media A and B are the linearity functions as space coordinates, they are n a (z) = n a (0) + n a (a) − n a (0) a z, 0 ≤ z ≤ a,    FIG. 3 (a). For the negative incident, the endpoints of media A and B are n a (0) = 2.56, n a (a) = 2.52 and n b (0) = 1.29, n b (b) = 1.25, it is shown in FIG. 3(b). The center frequency ω 0 = 1.215 × 10 15 Hz, λ 0 = 1.55 × 10 −6 nm corresponding to center wavelength, and the structure of FPCs is (BA) 16 .
By calculation, we can obtain the transmissivity of the one-dimensional FPCs for the positive and negative incident of light. The positive incident figure (red line) and the negative incident figure (blue line) is shown in FIG. 4. In the range of ω/ω 0 = 7.692 ∼ 9.0, 12.04 ∼ 13.98, 16.64 ∼ 18.71 and 21.29 ∼ 23.01, the positive incident light can pass through the one-dimensional FPCs, but the negative incident light cannot pass through, because the negative incident light is in the forbidden band. So, the one-dimensional FPCs with structure and parameters have the characteristics of optical diodes in the range of frequency above.
Take the frequency range of 16.64 ∼ 18.71 as an example, we give the relationship between the transmissivity and the wavelength of positive incident (red line) and negative incident (blue line), it is shown in FIG. 5. We can find in the wavelength range of λ/λ 0 = 0.055 ∼ 0.060, the light can pass through the positive direction but can not pass through from reverse incident, this characteristic enable one-way transmission of light.
In addition, we also calculated the electric field distribution. In the forbidden bands ω/ω 0 = 7.692 ∼ 9.0 and 16.64 ∼ 18.71, we take frequency ω/ω 0 = 8 and ω/ω 0 = 17.5, respectively, and obtained the electric field distribution for positive incident (red line) and negative incident (blue line), they are shown in FIG. 6  and FIG. 7. We can find that the positive incident light can pass, but the negative incident light can not pass through, i.e., the light can only be transmitted one way in the one-dimensional FPCs, which can be made into optical diodes.

. Conclusion
In summary, We have theoretically investigated the function photonic crystals (FPCs), which the refractive index of medium is the function of space position. We choosed the line refractive index functions for two media A and B. By calculation the transmissivity and electric field distribution, we can find that the positive incident light can pass the function photonic crystals, but the negative incident light can not pass through it, the function photonic crystals (FPCs) can be made into optical diodes.

Back matter
Funding. Thanks to Research Innovation of Jilin Normal University (no. 201937) and Doctoral research project of Jilin Normal University (no. 2017005) for the funding support.
Acknowledgements. Thanks are due to Professor wu for assistance with the theory, and this work was supported by the Doctoral research project of Jilin Normal University (no. 2017005) and Research Innovation of Jilin Normal University (no. 201937).
Disclosures. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data availability. No data were generated or analyzed in the presented research. Figure 1 The input and output electric eld and magnetic eld in the FPCs.

Figure 2
The light positive incident (red line) and negative incident (blue line) to the FPCs.  The relation between transmissivity and the incident light frequency, red line correspond to positive incident and blue line correspond to negative incident.

Figure 5
The relationship between the transmittance and the wavelength of positive (red line) and negative (blue line) incident light Figure 6 The electric eld distribution of the positive incident (red line) and negative incident (blue line) with ω/ω0 = 8.

Figure 7
The electric eld distribution of the positive incident (red line) and negative incident (blue line) with ω/ω0 = 17.5.