Optical Wave Guiding and Spectral Properties of Micro/Nanofibers Used for Quantum Sensing and Quantum Light Generation

Abstract

Subwavelength optical micro/nanofibers have been widely used as basic building blocks in the field of quantum sensing and quantum light source by virtue of their properties which include pronounced evanescent field, large surface area, and small optical mode area. This paper presents theoretical studies on the propagation properties of the guided optical wave and the spectral properties of entangled photons from spontaneous four-wave mixing in micro/nanofibers. We first analyze numerically single-mode propagation, field distribution, fraction of power, and group-velocity-dispersions by solving Maxwell’s equations with boundary conditions in cylindrical coordinates. Then, optical wave guiding properties of micro/nanofibers are applied to estimate the spectral properties such as central wavelengths and bandwidths of the created photons via spontaneous four-wave mixing that can be tailored by controlling diameter and length of micro/nanofibers. This theoretical work provides useful guidelines to design micro/nanofiber-based quantum sensing and quantum light sources for quantum technologies.

1. Introduction

Optical micro/nanofibers (MNFs) are usually referred to one-dimensional optical waveguide with the diameter close to or smaller than the wavelength of guided light, which confines and guides the beam by the total internal reflection at the silica(SiO${}_2$)-air interface [1,2,3,4]. Recently, MNFs have been extensively studied for quantum optics and photonic quantum information because of their extraordinary properties which include low transmission loss, high-nonlinearity, large evanescent fields, and small bending radius [5,6,7,8]. The strong radial confinement much longer than the Rayleigh range of a focused Gaussian beam has made it possible to efficiently couple atoms to the fiber mode via the evanescent field [9,10,11,12,13]. With these waveguiding properties, MNFs, so far, have been used for broad application in optical sensing [14,15,16,17,18,19,20,21,22,23], optical trapping [24,25], evanescent field coupling [26], and quantum optics [27,28]. Also, the flexibility and mechanical strength of MNFs have been able to offer great advantage to demonstrate various microstructures as sensing head, such as loop, coil, and optical coupler [29,30,31]. In addition, thanks to the strong field confinement and tunable dispersion, MNFs have been applied to nonlinear optics [32,33,34,35,36]. Furthermore, MNFs have been used as quantum light sources based on spontaneous four-wave mixing (SFWM) process, that are suitable for directly connecting to other fiber-based system with negligible coupling loss [37,38,39,40,41]. Their small optical mode areas provide low-threshold optical nonlinear effect, so that MNFs significantly enhance photon-pair generation in a short length of MNF. Compared to the area of normal single-mode fiber about 100 $\mathsf{\mu }$m${}^2$, an MNF has 1 $\mathsf{\mu }$m${}^2$. A few years ago, the generation of correlated photon-pairs via SFWM was experimentally demonstrated in a micro/nanofiber [42]. By using 15 cm long MNF with the diameter of 900 nm and the laser centered at 1040 nm, correlated photon-pairs were generated at 1310 nm and 850 nm, respectively, that are compatible with the telecom-band fiber system and the high efficiency single photon detector (SPD) for the near IR region. In a very recent study, photon-pairs were generated in a 12 cm long MNF with a 615 nm diameter [43]. For the 725 nm pump beam, the signal and idler photons were created at 632 and 850 nm, respectively, which can be detected by Si-based SPDs with high quantum efficiency.

In this work, we theoretically investigate optical wave guiding properties and the spectral properties of entangled photon pairs via SFWM process in MNFs. Thanks to the extraordinary properties such as pronounced evanescent field and small mode area, MNFs have become powerful tools for quantum sensing, detection, and photon-pair generation. In particular, by controlling the waveguide dimension, photon-pairs are designed at one’s desired wavelengths. Here, based on the exact solutions of the Maxwell’s equations, we first numerically calculate the propagation constants for guided optical modes through a circular silica-air optical fiber with a subwavelength-diameter, and then evaluate field distributions, fractional power, effective diameters, group velocity, and group-velocity-dispersions according to the changes of the fiber diameter and the wavelength of the light. Finally, optical wave guiding properties based on numerical caculations are applied to estimate spectral properties such as central wavelengths and bandwidths of the created photons via spontaneous four-wave mixing in MNFs.

This paper is organized as follows: Section 2 is devoted to the analysis of the Maxwell’s equation in the step-index cylindrical wire model, and from the exact solutions of the Maxwell’s equation, Section 3 presents propagation constants and fundamental modes in MNFs. Based on the mode properties, Section 4 and Section 5 describe the fractional power, effective diameter, group velocity, and group velocity dispersion that are essential parameters to estimate spectral properties of entangled photon-pair via spontaneous four-wave mixing (SFWM). In Section 6, the SFWM in an MNF is introduced with phase-matching conditions, and finally spectral properties of created photons are shown in Section 7. It is summarized in Section 8.

2. Light Propagation in Step-Index Cylindrical Wires

We first consider the geometry of a step-index optical fiber with a circular cross-section as shown in Figure 1a. An MNF has a silica core of radius a with refractive index $n_1$ and an infinite air-clad of refractive index of $n_2=1$. The corresponding refractive index profile can be written as

$$n\left(r\right)=\left\{\begin{array}{cc}{n}_1,\hfill & \mathrm{if} r<a\hfill \\ n_2,\hfill & \mathrm{if} r>a\hfill \end{array}\right.,$$

(1)

where $n_1>n_2$. According to the Sellmeier equation, the refractive index for fused silica at room temperature can be given by [44]

$$n^2-1=\frac{0.6961663{\lambda }^2}{{\lambda }^2-{\left(0.0684043\right)}^2}+\frac{0.4079426{\lambda }^2}{{\lambda }^2-{\left(0.1162414\right)}^2}+\frac{0.8974794{\lambda }^2}{{\lambda }^2-{\left(9.896161\right)}^2}.$$

(2)

Since a short length of the MNF is sufficient for microphotonic scales, we assume that both media are non-absorbing and have a magnetic permeability equal to the vacuum permeability ${\mu }_0$. The corresponding Maxwell’s equations can be reduced to Helmholtz equations [1,3]

$$\begin{array}{ccc}({\nabla }^2+n^2{k}^2-{\beta }^2)·\mathbf{e}& =& 0,\\ ({\nabla }^2+n^2{k}^2-{\beta }^2)·\mathbf{h}& =& 0,\end{array}$$

(3)

where $k=2\pi /\lambda$, $\lambda$ is the wavelength of the light, $\beta$ is the propagation constant, $\mathbf{e}$ and $\mathbf{h}$ are the electric field vector and the magnetic field vector, respectively. The longitudinal propagation constant $\beta$ of various modes can be determined by the eigenvalue equations of Equation (3) [1,3,45]:

$$\frac{J_{l-1}\left(ha\right)}{ha{J}_l\left(ha\right)}=\left(\frac{n_1^2+n_2^2}{2n_1^2}\right)\frac{K_{l-1}\left(qa\right)+K_{l+1}\left(qa\right)}{2qa{K}_l\left(qa\right)}+\frac{l}{ha}\pm R$$

(4)

with

$$R={\left[{\left(\frac{n_1^2-n_2^2}{2n_1^2}\right)}^2{\left(\frac{K_{l-1}\left(qa\right)+K_{l+1}\left(qa\right)}{2qa{K}_l\left(qa\right)}\right)}^2+{\left(\frac{l\beta }{n_1{k}_0}\right)}^2{\left(\frac{1}{{\left(qa\right)}^2}+\frac{1}{{\left(ha\right)}^2}\right)}^2\right]}^{1/2},$$

(5)

where $h={(n_1^2{k}_0^2-{\beta }^2)}^{1/2}$ and $q={({\beta }^2-n_2^2{k}_0^2)}^{1/2}$ characterize the fields inside and outside the fiber. These two parameters are directly related to the V-number, $V=\sqrt{{\left(ha\right)}^2+{\left(qa\right)}^2}=2\pi a/\lambda \sqrt{n_1^2-n_2^2}$ which determines the fraction of the optical power in a certain mode confined to the fiber core. $J_l$ and $K_l$ are the Bessel functions of the first kind and the modified Bessel functions of the second kind, respectively. There are different solutions depending on the value l in Equation (4). For $l=0$, the ± signs stand for the TM $(+)$ and TE $(-)$ modes that are transversal magnetic and electric modes, respectively. For $l=1$, the ± signs lead to two different states of modes, the HE $(+)$ and EH $(-)$ which imply $E_z$ being larger and smaller than $H_z$, respectively [45,46].

Figure 2 shows the solutions of the TM and TE modes of Equation (4) that the magnetic field (electric field) is perpendicular to the fiber axis. Both modes are placed at $V>2.405$. That means that any non-fundamental mode is not allowed to propagate inside the fiber below the lowest cut-off value of 2.405. Therefore, the single mode condition for the silica-air waveguide is $V<2.405$. Figure 3 shows the solutions of the EH and HE modes at $V=11.03$. There is a lower limit in V-number for the EH modes so that the EH modes are not allowed below the single mode condition. The HE${}_{11}$ mode is only the fundamental mode which can always propagate inside the fiber since it has no cut-off value.

Figure 4 shows the single mode condition of the silica-air waveguide as a function of the wavelengths and wire diameters. A silica-air MNF with a diameter less than 550 nm will always have a single-mode propagation at the wavelength of 780 nm. For the telecom-band wavelength of 1.5 $\mathsf{\mu }$m, a diameter of MNF should be less than 1.1 $\mathsf{\mu }$m to be a single-mode waveguide.

3. Propagation Constant $\mathbf{\beta }$ and Fundamental Mode HE${}_{\mathbf{11}}$

In order to investigate the propagation constant $\beta$ as a function of a diameter d, the Equation (4) can be rewritten in terms of $\beta$ and d,

$$\begin{array}{cc}\hfill {\left(\frac{4l\beta k_0(n_1^2-n_2^2)}{d^2({\left(k_0{n}_1\right)}^2-{\beta }^2)({\beta }^2-{\left(k_0{n}_2\right)}^2)}\right)}^2& =\left(\frac{J_{l-1}\left(ha\right)-\frac{l}{ha}{J}_l\left(ha\right)}{ha{J}_l\left(ha\right)}-\frac{K_{l-1}\left(qa\right)+K_{l+1}\left(qa\right)}{2qa{K}_l\left(qa\right)}\right)\hfill \\ \hfill & \times \left(\frac{n_1^2(J_{l-1}\left(ha\right)-\frac{l}{ha}{J}_l\left(ha\right))}{ha{J}_l\left(ha\right)}-\frac{n_2^2(K_{l-1}\left(qa\right)+K_{l+1}\left(qa\right))}{2qa{K}_l\left(qa\right)}\right).\hfill \end{array}$$

(6)

Figure 5a presents the propagation constant (or effective refractive index) as a function of V-number for the first seven modes. The first two excited modes TM${}_{01}$ and TE${}_{01}$ lie at $V\ge 2.405$ so that the single-mode and multi-mode regimes are clearly distinguished. When V-number decreases, $\beta /k_0$ tends to be close to $n_1$. It means that the field will propagate more into the surrounding medium while decreasing the diameter. Figure 5b shows the dependence of the propagation constant as a function of a diameter at the wavelength of 780 nm. The HE${}_{11}$ mode only propagates for $d\le$ 550 nm.

We now study a fundamental mode HE${}_{11}$ with rotating polarization. In the cylindrical coordinates ($r,\varphi ,z$), for $r<a$ (inside the fiber), the solutions of Maxwell’s equations for the electric field $\mathit{E}$ can be given by [1,45,47]

$$\begin{array}{ccc}\hfill E_r(r,\varphi ,z,t)& =& -iA\frac{\beta }{2h}[(1-s)J_0\left(hr\right)-(1+s)J_2\left(hr\right)]e^{i(\omega t\pm \varphi -\beta z)},\hfill \\ \hfill E_{\varphi }(r,\varphi ,z,t)& =& \pm A\frac{\beta }{2h}[(1-s)J_0\left(hr\right)+(1+s)J_2\left(hr\right)]e^{i(\omega t\pm \phi -\beta z)},\hfill \\ \hfill E_z(r,\varphi ,z,t)& =& A{J}_1\left(hr\right)e^{i(\omega t\pm \phi -\beta z)}.\hfill \end{array}$$

(7)

For $r>a$,

$$\begin{array}{ccc}\hfill E_r(r,\varphi ,z,t)& =& -iA\frac{\beta }{2q}\frac{J_1\left(ha\right)}{K_1\left(qa\right)}[(1-s)K_0\left(qr\right)+(1+s)K_2\left(qr\right)]e^{i(\omega t\pm \varphi -\beta z)},\hfill \\ \hfill E_{\varphi }(r,\varphi ,z,t)& =& \pm A\frac{\beta }{2q}\frac{J_1\left(ha\right)}{K_1\left(qa\right)}[(1-s)K_0\left(qr\right)-(1+s)K_2\left(qr\right)]e^{i(\omega t\pm \varphi -\beta z)},\hfill \\ \hfill E_z(r,\varphi ,z,t)& =& A\frac{J_1\left(ha\right)}{K_1\left(qa\right)}{K}_1\left(qr\right)e^{i(\omega t\pm \varphi -\beta z)},\hfill \end{array}$$

(8)

where

$$\begin{array}{cc}\hfill s=\left[\frac{1}{{\left(ha\right)}^2}+\frac{1}{{\left(qa\right)}^2}\right]& {\left[\frac{J_0\left(ha\right)-\frac{1}{ha}{J}_1\left(ha\right)}{ha{J}_1\left(ha\right)}-\frac{K_0\left(qa\right)+K_2\left(qa\right)}{2qa{K}_1\left(qa\right)}\right]}^{-1}.\hfill \end{array}$$

(9)

The +(−) sign in Equations (7) and (8) accounts for clockwise (counterclockwise) rotation of the polarization around the z-axis. The coefficient A is determined by the normalization condition. Figure 6 shows vector plots of the electric field component transversal to the z-axis at time $t=0,\pi /4\omega ,\pi /2\omega$, and $3\pi /4\omega$. We perform numerical calculations for a silica-air fiber with a radius $a=250$ nm at a wavelength of $\lambda =780$ nm. A red circle describes the fiber surface. With increasing time, the real part of the transversed electric field rotates along the circular or elliptical with respect to the fiber axis.

To apply for sensing and trapping of atoms by the optical force of an evanescent wave around an MNF [10,11,12], it is necessary to calculate the optical potential which is proportional to the total intensity of the electric field. From the electric field in the HE${}_{11}$ mode with rotating polarization, we can easily calculate the total intensity ${\left|E\right|}^2$ which is given by

$$\begin{array}{ccc}\hfill {\left|E\right|}_{in}^2& =& \frac{A^2{\beta }^2}{2h^2}[{(1-s)}^2{J}_0^2\left(hr\right)+{(1+s)}^2{J}_2^2\left(hr\right)+2\frac{h^2}{{\beta }^2}{J}_1^2\left(hr\right)],\hfill \\ \hfill {\left|E\right|}_{out}^2& =& \frac{A^2{\beta }^2}{2q^2}\frac{J_1^2\left(ha\right)}{K_1^2\left(qa\right)}[{(1-s)}^2{K}_0^2\left(qr\right)+{(1+s)}^2{K}_2^2\left(qr\right)+2\frac{q^2}{{\beta }^2}{K}_1^2\left(qr\right)].\hfill \end{array}$$

(10)

Since the total intensity ${\left|E\right|}^2$ is independent of $\varphi$, the distribution of ${\left|E\right|}^2$ is cylindrically symmetric as shown in Figure 7. Since the polarization of the field breaks the symmetry, in general, the electric field propagating in the fiber is not cylindrically symmetric. However, for the rotating polarization, averaging over a period washes out this symmetry breaking. The regions of $r<a$ and $r>a$ can be clearly identified due to the discontinuity at the boundary $r=a$. This strong discontinuity arises from the large refractive index difference between the bulk and the surrounding medium as well as from the strong radial confinement of the field when $\lambda >a$.

We now present the field equations of the fundamental HE${}_{11}$ mode with quasi-linear polarization. In the cylindrical coordinates ($r,\varphi ,z$), for $r<a$ (inside the fiber), the solutions of Maxwell’s equations for the Cartesian components of the electric field can be obtained by [1,45,47]

$$\begin{array}{ccc}\hfill E_x(r,\varphi ,z,t)& =& A\frac{\beta }{2h}[(1-s)J_0\left(hr\right)cos{\varphi }_0-(1+s)J_2\left(hr\right)cos(2\varphi -{\varphi }_0)]e^{i(\omega t-\beta z)},\hfill \\ \hfill E_y(r,\varphi ,z,t)& =& A\frac{\beta }{2h}[(1-s)J_0\left(hr\right)sin{\varphi }_0+(1+s)J_2\left(hr\right)sin(2\varphi -{\varphi }_0)]e^{i(\omega t-\beta z)},\hfill \\ \hfill E_z(r,\varphi ,z,t)& =& iA{J}_1\left(hr\right)cos(\varphi -{\varphi }_0)e^{i(\omega t-\beta z)}.\hfill \end{array}$$

(11)

For $r>a$,

$$\begin{array}{cc}\hfill E_x(r,\varphi ,z,t)& =A\frac{\beta }{2q}\frac{J_1\left(ha\right)}{K_1\left(qa\right)}[(1-s)K_0\left(qr\right)cos{\varphi }_0+(1+s)K_2\left(qr\right)cos(2\varphi -{\varphi }_0)]e^{i(\omega t-\beta z)},\hfill \\ \hfill E_y(r,\varphi ,z,t)& =A\frac{\beta }{2q}\frac{J_1\left(ha\right)}{K_1\left(qa\right)}[(1-s)K_0\left(qr\right)sin{\varphi }_0+(1+s)K_2\left(qr\right)sin(2\varphi -{\varphi }_0)]e^{i(\omega t-\beta z)},\hfill \\ \hfill E_z(r,\varphi ,z,t)& =iA\frac{J_1\left(ha\right)}{K_1\left(qa\right)}{K}_1\left(qr\right)cos(\varphi -{\varphi }_0)e^{i(\omega t-\beta z)}.\hfill \end{array}$$

(12)

The angle ${\varphi }_0$ accounts for the orientation axis of the polarization of the electric field. Two sets of the solutions corresponding to ${\varphi }_0=0$ and $\pi /2$ imply x-polarization and y-polarization, respectively. On the contrary to the rotational polarization, the polarization of transversed $\mathit{E}$ field does not depend on time but it is position-dependent as shown in Figure 8. The total intensity of the electric field can be given by

$$\begin{array}{cc}\hfill {\left|E\right|}_{in}^2& =\frac{A^2{\beta }^2}{4h^2}{[(1-s)}^2{J}_0^2\left(hr\right)+{(1+s)}^2{J}_2^2\left(hr\right)+2\frac{h^2}{{\beta }^2}{J}_1^2\left(hr\right)+2\left(\frac{h^2}{{\beta }^2}{J}_1^2\left(hr\right)-(1-s^2)J_0\left(hr\right)J_2\left(hr\right)\right),\hfill \\ & \times cos\left(2(\varphi -{\varphi }_0)\right)],\hfill \\ \hfill {\left|E\right|}_{out}^2& =\frac{A^2{\beta }^2}{4q^2}\frac{J_1^2\left(ha\right)}{K_1^2\left(qa\right)}{[(1-s)}^2{K}_0^2\left(qr\right)+{(1+s)}^2{K}_2^2\left(qr\right)+2\left(\frac{q^2}{{\beta }^2}{K}_1^2\left(qr\right)+(1+s^2)K_0\left(qr\right)K_2\left(qr\right)\right)\hfill \\ & \times cos\left(2(\varphi -{\varphi }_0)\right)].\hfill \end{array}$$

(13)

Figure 9a presents the total intensity of the electric field in a fundamental HE${}_{11}$ mode with quasi-polarization. We choose the x-axis as the major orientation of polarization (${\varphi }_0=0$). A discontinuity of the field intensity takes place at the fiber surface because of the boundary condition for the radial component of the electric field. It is dramatically induced by the large gap between the refractive indices $n_1$ and $n_2$. The spatial distribution of the field intensity is not cylindrically symmetric. The $cos\left(2(\varphi -{\varphi }_0)\right)$ terms in Equation (13) break the symmetry and make much larger evanescent field in the regions, where the field component is transversal to the fiber surface.

Figure 9b shows the spatial distribution of the field along the different polarization axis. The z-component of the electric field disappears while the transversed component is parallel to the fiber surface, and it becomes the maximum value while the transversed component is perpendicular to the fiber surface. For the HE${}_{11}$ mode with quasi-linear polarization, the evanescent field on the surface of an MNF can reach the maximum intensity at $a/\lambda =0.24$. In fiber-based sensing application, the diameter-dependent power density at the fiber-air interface is important to design possible strong interaction outside the media [48].

4. Fraction of Power Propagating Inside and Outside a Micro/Nanofiber

In this section, we investigate the fractional power and effective diameter of the fundamental mode guided in an MNF. In the case of propagation in an optical fiber, the average flux transmitted through the fiber is written by the z-component of the cycle-averaged Poynting vector [49]

$$〈\overrightarrow{S}〉=\frac{1}{2}\mathrm{Re}\left[\overrightarrow{E}\times {\overrightarrow{H}}^{*}\right].$$

(14)

For the HE${}_{11}$ mode, the z-components of Poynting vector are obtained by [1,3]

$$\begin{array}{cc}\hfill {〈\overrightarrow{S_z}〉}_{in}& =\frac{1}{2}{\left(\frac{{ϵ}_0}{{\mu }_0}\right)}^{1/2}\frac{k_0{n}_1^2}{\beta J_1^2\left(ha\right)}[a_1{a}_3{J}_0^2\left(hr\right)+a_2{a}_4{J}_2^2\left(hr\right)+\frac{1-F_1{F}_2}{2}{J}_0\left(hr\right)J_2\left(hr\right)cos\left(2\right(\varphi )],\hfill \\ \hfill {〈\overrightarrow{S_z}〉}_{out}& =\frac{1}{2}{\left(\frac{{ϵ}_0}{{\mu }_0}\right)}^{1/2}\frac{k_0{n}_1^2}{\beta K_1^2\left(qa\right)}\frac{h^2}{q^2}[a_1{a}_5{K}_0^2\left(qr\right)+a_2{a}_6{K}_2^2\left(qr\right)+\frac{1-2\Delta -F_1{F}_2}{2}{K}_0\left(qr\right)K_2\left(qr\right)\hfill \\ & \times cos\left(2\right(\varphi )],\hfill \end{array}$$

(15)

where

$$\begin{array}{c}{F}_1={\left(\frac{a^{2}hq}{V}\right)}^2[b_1+(1-2\Delta )],F_2={\left(\frac{V}{a^{2}hq}\right)}^2\frac{1}{b_1+b_2},b_1=\frac{1}{2ah}\left\{\frac{J_0\left(ha\right)}{J_1\left(ha\right)}-\frac{J_2\left(ha\right)}{J_1\left(ha\right)}\right\},\hfill \\ b_2=-\frac{1}{2aq}\left\{\frac{K_0\left(qa\right)}{K_1\left(qa\right)}+\frac{K_2\left(qa\right)}{K_1\left(qa\right)}\right\},a_1=\frac{F_2-1}{2},a_2=\frac{F_2+1}{2},a_3=\frac{F_1-1}{2},a_4=\frac{F_1+1}{2},\hfill \\ a_5=\frac{F_1-1+2\Delta }{2},a_6=\frac{F_1+1-2\Delta }{2},\Delta =\frac{n_1-n_2}{n_1}.\hfill \end{array}$$

(16)

The radial and azimuthal components transversal to the fiber axis, $〈\overrightarrow{S_r}〉$ and $〈\overrightarrow{S_{\varphi }}〉$, describe the fraction of the energy flow of the electromagnetic field that does not propagate. The fraction of power propagating inside and outside the fiber can be given by $\eta =\frac{P_{in}}{P_{in}+P_{out}}$, where $P_{in}={\int }_0^{2\pi }d\varphi {\int }_0^a{〈\overrightarrow{S_z}〉}_{in}rdr$ and $P_{out}={\int }_0^{2\pi }d\varphi {\int }_a^{\infty }{〈\overrightarrow{S_z}〉}_{out}rdr$[3]. The fractional power $\eta$ is related to the diameter-dependent group velocity and waveguide dispersion. Figure 10 shows the fraction of power for the fundamental mode inside the core as a function of the diameter at $\lambda =780$ nm and $\lambda =1.5\mathsf{\mu }$m. At $\lambda =780$ nm ($1.5\mathsf{\mu }$m), the 90% energy inside the core is confined by $d=700$ nm ($1.36\mathsf{\mu }$m) of an MNF whereas the rest of the energy is confined by 270 nm (515 nm) of it. Because of the highly confined optical modes in high-index contrast structures, a tight bending radius of less than 5 $\mathsf{\mu }$m can be achieved, that results in compact guided optical components while giving more flexibilities in the photonic circuit layout [50,51,52,53]. Now, let us define the effective radius, $a_{\mathrm{eff}}$, which is a hypothetic radius for possessing 86.5% (=1$-e^2$) of the total power [3]. $a_{\mathrm{eff}}$ can be obtained from $\frac{P_{in}}{P_{in}+P_{out}}=86.5\%$ for $a_{\mathrm{eff}}\le a$ and $\frac{P_{in}+P_{\mathrm{eff}}}{P_{in}+P_{out}}=86.5\%$ for $a_{\mathrm{eff}}>a$, where $P_{\mathrm{eff}}={\int }_0^{2\pi }d\varphi {\int }_a^{a_{\mathrm{eff}}}{〈\overrightarrow{S_z}〉}_{in}rdr$. As the diameter decreases, d${}_{\mathrm{eff}}(=2a_{\mathrm{eff}})$ is getting larger so that it leads most of light energy to propagate in air. In Figure 11, the major power can be confined within $d=500$ nm at $\lambda =780$ nm and $d=1.1\mathsf{\mu }$m at $\lambda =1.5\mathsf{\mu }$m.

5. Group Velocity and Group Velocity Dispersion (GVD)

The knowledge of group velocity and dispersion property of MNFs are very useful for designing fiber-based quantum sensing and quantum light generation. They have dependences of the dimension of a diameter and the wavelength of an optical wave. Group velocity of the MNF can be given as [1,3]

$$v_g=\frac{c\beta }{n_1^2{k}_0[1-2\Delta (1-\eta )]}.$$

(17)

Diameter-dependent group velocities of the fundamental modes are shown in Figure 12a. As the diameter decreases, group velocity, $v_g$ approaches the speed of light in vacuum because most of the light energy propagates in air. When the diameter becomes large enough, $v_g$ approaches $c/n_1$ which is the group velocity of a plane wave in the optical fiber. For a large value of the wavelength compared to the diameter, $v_g$ approaches the speed of light, c, as shown in Figure 12b. Based on the group velocity, we can obtain diameter- and wavelength-dependent group velocity dispersion (GVD) which is written by [3,54]

$$GVD=\frac{d\left(v_g^{-1}\right)}{d\lambda }.$$

(18)

The optical dispersions of MNFs are affected by both the material dispersion and the waveguide dispersion. In particular, the waveguide dispersion is strongly modified by the waveguide dimension. Therefore, when we determine the optimum diameter for generating entangled photon pairs, it is useful to estimate GVD parameter of MNFs according to the changes of the diameter and the wavelength as depicted in Figure 13. As the MNF diameter and the wavelength increase, zero GVDs are up-shifted.

6. Spontaneous Four-Wave Mixing (SFWM)

The SFWM process is mediated by a ${\chi }^{\left(3\right)}$ nonlinear susceptibility, wherein two pump photons are annihilated by the medium, and then create two different photons, called signal and idler. MNFs can be used as quantum light sources generating entangled photon pairs via SFWM process. The small mode area of the MNF strongly enhances nonlinear coefficient which is roughly 100 times larger than that of the standard single-mode fiber. The SFWM process must satisfy both energy conservation and phase-matching condition which are given by

$$\begin{array}{ccc}\hfill 2{\omega }_p& =& {\omega }_s+{\omega }_i,\hfill \\ \hfill 2{\beta }_p& =& {\beta }_s+{\beta }_i+2\gamma P_p,\hfill \end{array}$$

(19)

where ${\beta }_j$ and ${\omega }_j$ are propagation constants and angular frequencies, respectively. The last term, $2\gamma P_p$, refers to the contribution of the self-phase modulation. The nonlinear coefficient, $\gamma$, can be given as $2\pi n_2/\left({\lambda }_p{A}_{\mathrm{eff}}\right)$, where $n_2=2\times {10}^{-20}$m${}^2$/W is the second-order nonlinear index of silica, $P_p$ is the peak power of the pump laser, and $A_{\mathrm{eff}}$ is the effective mode area. When we consider the reasonable pump power, i.e., 0.5 mW-average power, the self-phase modulation term leads the phase-matching condition to have two different regions in the normal dispersion regime. Figure 14a shows numerically calculated phase-matching curves of the signal and idler photons for various MNF diameters as a function of the pump wavelength. As a diameter increases, overall phase-matching curves are shifted toward the long wavelength. In the region where the phase-matched wavelengths are far-detuned from the pump wavelength, the wavelengths of created photon pair are weakly power-dependent. On the contrary, the region which is close to the pump wavelength has strong dependence of the pump power. As the pump power increases, the wavelength difference between signal and idler photons becomes large as shown in Figure 14b. Here, in order to consider pump power dependences in numerical calculations, we assume that a femto-second pulsed laser is operated at 78 MHz repetition rate. When a spectral filter with the full width at half-maximum of 1 nm is used, for the sech${}^2$-shape pulse, the pulse duration and peak power are obtained. From the conditions of Figure 14a,b, we can design from visible to NIR photon-pair sources working in the silicon-based single photon detector (SPD) [43]. In Figure 14c, one of the entangled photon-pair is generated at a telecom-band wavelength which is compatible with the existing fiber-based system, while the other photon has a NIR wavelength which can be detected in high efficiency by using the silicon-based SPD. Moreover, due to the large detuning between the pump and signal (idler) photons, Raman noise can be significantly suppressed at the room temperature [42]. Therefore, by the proper diameter control and the pump wavelength selection, photon-pair can be created at one’s desired wavelengths.

We now present a theoretical study of the two-photon state generated by SFWM with degenerated pump photons. The interaction Hamiltonian of SFWM process can be written as

$$H_{int}\left(t\right)={ϵ}_0{\chi }^{\left(3\right)}{\int }_{-L/2}^{L/2}{\widehat{E}}_{p_1}^{(+)}{\widehat{E}}_{p_2}^{(+)}{\widehat{E}}_s^{(-)}{E}_i^{(-)}dz+\mathrm{H}.\mathrm{C}.,$$

(20)

where ${\chi }^{\left(3\right)}$ is a non-linear susceptibility that characterizes the non-linear process, and the effective length of the crystal is L. The pump beam can be expressed with plane wave, ${\widehat{E}}_p^{(+)}(z,t)=\int E_p\left({\omega }_p\right){\mathrm{e}}^{i(k_{p}z-{\omega }_{p}t)}d{\omega }_p$. The signal and idler photons are quantized as

$${\widehat{E}}_{s,i}^{(+)}(z,t)={\int }_0^{\infty }d{\omega }_{s,i}\sqrt{\frac{\hslash {\overline{\omega }}_{s,i}}{2{ϵ}_{0}A}}\widehat{a}\left({\omega }_{s,i}\right){\mathrm{e}}^{i(k_{s,i}z-{\omega }_{s,i}t)},$$

(21)

where ${\omega }_j(j=s,i)$ and $k_j(j=s,i)$ represent the frequencies and wave vectors of signal (s) and idler (i), respectively. A is the single-mode cross-section area. ${\widehat{a}}_s^{†}$ and ${\widehat{a}}_i^{†}$ are the creation operators for signal and idler photons. The Hamiltonian becomes

$$\begin{array}{cc}\hfill {\widehat{H}}_{int}\left(t\right)=& \frac{{\chi }^{\left(3\right)}\hslash \sqrt{{\omega }_s{\omega }_i}L}{2A}{\int }_0^{L/2}dz\int d{\omega }_{p_1}\int d{\omega }_{p_2}\int d{\omega }_s\int d{\omega }_i{E}_{p_1}\left({\omega }_{p_1}\right)E_{p_2}\left({\omega }_{p_2}\right)\hfill \\ \hfill & \times {\mathrm{e}}^{-i\frac{\Delta kL}{2}}\mathrm{Sinc}\left(\frac{\Delta kL}{2}\right)a_s^{†}\left({\omega }_s\right)a_i^{†}\left({\omega }_i\right)+\mathrm{H}.\mathrm{C}.,\hfill \end{array}$$

(22)

where $\Delta k=k_{p_1}\left({\omega }_{p_1}\right)+k_{p_1}\left({\omega }_{p_2}\right)-k_s\left({\omega }_s\right)-k_i\left({\omega }_i\right)$ is the phase-mismatching condition. Hence we can obtain the two-photon state as

$$\begin{array}{cc}\left|\psi \right.〉\hfill & =-\frac{i}{\hslash }\int dt{\widehat{H}}_{int}\left(t\right)\left|0\right.〉,\hfill \\ & =\frac{{\chi }^{\left(3\right)}\sqrt{{\omega }_s{\omega }_i}L}{2iA}\int \int \int \int d{\omega }_{p_1}d{\omega }_{p_2}d{\omega }_{s}d{\omega }_i{E}_{p_1}\left({\omega }_{p_1}\right)E_{p_2}\left({\omega }_{p_2}\right){\mathrm{e}}^{-i\frac{\Delta kL}{2}}\mathrm{Sinc}\left(\frac{\Delta kL}{2}\right)a_s^{†}\left({\omega }_s\right)a_i^{†}\left({\omega }_i\right)\left|0\right.〉,\hfill \\ & =\int \int d{\omega }_{s}d{\omega }_{i}F({\omega }_s,{\omega }_i)a_s^{†}\left({\omega }_s\right)a_i^{†}\left({\omega }_i\right)\left|0\right.〉,\hfill \end{array}$$

(23)

where $F({\omega }_s,{\omega }_i)$ refers to the joint spectral amplitude (JPA) that consists of the pump envelope function and the phase-matching function.

7. Joint Spectrum

The joint spectral amplitude, $F({\omega }_s,{\omega }_i)$, is an important physical quantity to show the spectral property and directly influence on the amount of entanglement in a photon-pair. Now, it is convenient to apply the integration transformation, $\Omega ={\nu }_p-\frac{{\nu }_s+{\nu }_i}{2}$, in Equation (23). ${\nu }_j$ (with $j=p,s,i$) represents the frequency detuned from the center frequency, $\overline{{\omega }_j}$. From the phase-matching condition $\Delta k$, each $k_j\left({\omega }_j\right)$ (with $j=p,s,i$) can be extended into a Taylor series around the center frequency which is given by $\Delta k=k_{p_1}\left({\overline{\omega }}_{p_1}+\frac{{\nu }_s+{\nu }_i}{2}+\Omega \right)+k_{p_2}\left({\overline{\omega }}_{p_2}+\frac{{\nu }_s+{\nu }_i}{2}-\Omega \right)-k_s\left({\overline{\omega }}_s+{\nu }_s\right)-k_i\left({\overline{\omega }}_i+{\nu }_i\right)$. For the degenerate pump beams, ${\omega }_p={\omega }_{p_1}={\omega }_{p_2}={\overline{\omega }}_p+{\nu }_p$. As a result, the joint spectrum of the generated photon pair can be given as

$$\begin{array}{cc}\hfill F({\omega }_s,{\omega }_i)& =B\int \int d{\omega }_{p_1}d{\omega }_{p_2}{\mathrm{e}}^{-{\left(\frac{{\omega }_{p_1}-{\overline{\omega }}_{p_1}}{{\sigma }_{p_1}}\right)}^2}{\mathrm{e}}^{-{\left(\frac{{\omega }_{p_2}-{\overline{\omega }}_{p_2}}{{\sigma }_{p_2}}\right)}^2}{\mathrm{e}}^{-i\frac{\Delta kL}{2}}\mathrm{Sinc}\left(\frac{\Delta kL}{2}\right),\hfill \\ \hfill F({\nu }_s,{\nu }_i)& =C{\mathrm{e}}^{-{\left(\frac{{\nu }_s+{\nu }_i}{\sqrt{2}{\sigma }_p}\right)}^2}\mathrm{Sinc}\left[\frac{{\tau }_s{\nu }_s+{\tau }_i{\nu }_i}{2}\right]{\mathrm{e}}^{-i\frac{{\tau }_s{\nu }_s+{\tau }_i{\nu }_i}{2}},\hfill \\ \hfill & =C{\mathrm{e}}^{-i\frac{{\tau }_s{\nu }_s+{\tau }_i{\nu }_i}{2}}\alpha ({\nu }_s,{\nu }_i)\Phi ({\nu }_s,{\nu }_i),\hfill \end{array}$$

(24)

where B and C are normalized constants which do not contribute to the joint spectrum property, ${\sigma }_{p_{1,2}}$ are the bandwidths of pump beams, and ${\tau }_j=\left(\frac{1}{v_p}-\frac{1}{v_j}\right)L$ (with $j=s,i$) represents the group delay between the pump and signal ($j=s$) or idler ($j=i$). v indicates the group velocity. If the group velocity of pump photon is the same as that of signal photon, both exist in the medium at the same time such that ${\tau }_s=0$. If they have different group velocities, they gradually become temporally separated in the fiber which is called temporal walk-off. $\alpha ({\nu }_s,{\nu }_i)$ is the pump envelope function which is governed by the energy conservation and depends on the spectral envelope of the pump beam. $\mathsf{\Phi }({\nu }_s,{\nu }_i)$ is called the phase-matching function which accounts for the momentum conservation and depends on the dispersion property in the fiber. For example, Figure 15 represents a JSA whose formula is given in Equation (24). Here, the pump beam centered at 725 nm and a 12-cm long MNF with 615 nm-diameter are used for numerical calculation. For the degenerate pump beams at 725 nm, temporal walk-off between pump beams is negligible in which they are overlapped for the length of the fiber. In Ref. [43], the signal and idler photons are generated at 632 nm and 850 nm, respectively. There are central and side lobes which are related to the wings of the sinc function. Due to relatively narrow bandwidth of pump beam ($\Delta {\lambda }_p=1$ nm) and long fiber length, bandwidths of signal and idler photons are about 2 nm and 5 nm, respectively. Another example is shown in Figure 16. For the pump beam centered at ${\lambda }_p=1064$ nm and $\Delta {\lambda }_p=3$ nm bandwidth, signal and idler are generated at 800 nm and $1.5\mathsf{\mu }$m, by using a 5 cm-long and 950 nm-diameter MNF. As a result, relatively broader bandwidth of pump beam and shorter fiber length than the previous case, lead signal and idler photons to have broad bandwidths about 5 nm and 15 nm.

In Figure 15 and Figure 16, the JSA describes strong spectral correlation between the signal and idler photons because of energy and momentum conservation constraints. In other words, if the signal (idler) photon is detected to herald the production of the idler (signal) one, the heralded photon is not in a pure state but in a mixed state. This is not suitable for the Linear Optics Quantum Computing (LOQC) protocol which is based on frequency-uncorrelated photons, allowing direct heralding of single photons in pure state. Generally, in photonic quantum computing, the error rates of logic gates increase with photon state impurity which decreases the Hong-Ou-Mandel (HOM) dip visibility [55]. In the heralded process, the detection of one photon in a twin-photon source indicates that the other photon is projected into a single-photon state. Photon-pairs created in SFWM are usually frequency-entangled due to energy and momentum conservation constraints [56,57]. Therefore, in this case, the heralding process projects the twin photon into a mixed state [58]. Even though this entanglement can be eliminated by using spectral and spatial filters, it degrades the production rate of created photons and heralding efficiency [59]. Here, we show that frequency-entangled states can be factorable states by controlling pump bandwidth and group delay. In the JSA, $\left|F({\nu }_s,{\nu }_i)\right|$ given in Equation (24), $\alpha ({\nu }_s,{\nu }_i)$ always exhibits an anti-diagonal slope in frequency space, since the maximum value is governed by the relation ${\nu }_s+{\nu }_i=0$. On the other hand, the slope of $\mathsf{\Phi }({\nu }_s,{\nu }_i)$ is determined by the condition ${\tau }_s{\nu }_s+{\tau }_i{\nu }_i=0$, so that the angle of the slope in frequency space depends on the signs of the group velocities of pump, signal, and idler. We now present mathematical condition for generating factorable states by using Gaussian approximation, $\mathrm{Sinc}\left[\frac{{\tau }_s{\nu }_s+{\tau }_i{\nu }_i}{2}\right]\simeq \mathrm{Exp}\left[-r\frac{{({\tau }_s{\nu }_s+{\tau }_i{\nu }_i)}^2}{4}\right]$ with $r=0.193$. Consequently, JSA can be given as [60,61]

$$\begin{array}{cc}\hfill F({\nu }_s,{\nu }_i)& =C{\mathrm{e}}^{-i\frac{{\tau }_s{\nu }_s+{\tau }_i{\nu }_i}{2}}\mathrm{Exp}\left[-\frac{{({\nu }_s+{\nu }_i)}^2}{2{\sigma }_p^2}\right]\mathrm{Sinc}\left[\frac{{\tau }_s{\nu }_s+{\tau }_i{\nu }_i}{2}\right],\hfill \\ \hfill & \simeq C{\mathrm{e}}^{-i\frac{{\tau }_s{\nu }_s+{\tau }_i{\nu }_i}{2}}{\mathrm{e}}^{\left[-\frac{{\nu }_s^2}{2}\left(\frac{1}{{\sigma }_p^2}+\frac{r{\tau }_s^2}{2}\right)\right]}{\mathrm{e}}^{\left[-\frac{{\nu }_i^2}{2}\left(\frac{1}{{\sigma }_p^2}+\frac{r{\tau }_i^2}{2}\right)\right]}{\mathrm{e}}^{\left[-{\nu }_s{\nu }_i\left(\frac{1}{{\sigma }_p^2}+\frac{r{\tau }_s{\tau }_i}{2}\right)\right]}.\hfill \end{array}$$

(25)

In order to make a factorable state in the JSA, the frequency entangled term between the signal and idler, $\mathrm{Exp}\left[-{\nu }_s{\nu }_i\left(\frac{1}{{\sigma }_p^2}+\frac{r{\tau }_s{\tau }_i}{2}\right)\right]$, should be eliminated. Therefore, the factorable condition can be given by

$$\frac{1}{{\sigma }_p^2}+\frac{r{\tau }_s{\tau }_i}{2}=\frac{1}{{\sigma }_p^2}+\frac{r{L}^2}{2}\left(\frac{1}{v_p}-\frac{1}{v_i}\right)\left(\frac{1}{v_p}-\frac{1}{v_s}\right)=0.$$

(26)

There are two conditions for factorizing photon-pair state. If, for the broad pump width, ${\sigma }_p\gg {\tau }_j$, either $v_p=v_s$ or $v_p=v_i$ is satisfied, generated photons are factorable. But, since the pump wavelength is very close to the wavelength of either signal or idler photon, it is difficult to suppress the pump noise. The second condition is that the two terms $1/{\sigma }_p^2$ and $r{\tau }_s{\tau }_i/2$ are canceled each other. In this case, group delays ${\tau }_s$ and ${\tau }_i$ have different signs and the phase-matching function exhibits a positive slope. The most conventional way to reduce the correlation between the signal and idler photons is to use spectral filters. By choosing narrowband signal and idler photons, they consist of a single spectral mode that makes them uncorrelated. However, using a filter with narrow bandwidth brings about a reduction in the probability that the photon transmits through it, and reduces the efficiency of generation. Figure 17 illustrates the possibility of generating experimentally realizable uncorrelated photon-pairs without employing spectral filters. Here, for 1164 nm pump, a 5 cm-long MNF with 810 nm-diameter can generate uncorrelated photon-pairs with 825 nm and 1.55 $\mathsf{\mu }$m. In this condition, idler photons at 1.55 $\mathsf{\mu }$m are compatible with the telecom-band fiber-based system, while the signal photons at 825 nm have high detection efficiency in a Si-based single photon detector that makes it suitable to build heralded single photon sources in telecom-band with high efficiency. Also, the Raman noise can be significantly suppressed due to the large detuning between pump and signal/idler photons. As a result, in Figure 17c, uncorrelation between photon-pairs is not completely observed due to the sinc-shaped phase-matching function. The secondary lobes along the vertical axis always cause a small amount of correlation between signal and idler, although the main peak is not correlated. In this case, we are able to suppress the secondary peaks of the sinc function by using a spectral filter. Because of the side lobes which are less than 10% of the total intensity, the photon flux suffers from a little reduction in a high peak signal. It is on the contrary to the correlated main peak cases in Figure 15 and Figure 16.

8. Conclusions

We demonstrated theoretical study about the single-mode guiding properties and the spectral properties of entangled photon pairs via SFWM process in MNFs. Numerical simulations based on the exact solutions of the Maxwell’s equations manifested characteristics of essential parameters, such as the propagation constants for guided optical modes, field distributions, fraction of power, and effective diameters. These parameters are important in quantum sensing and detection implementing the light-matter interaction via evanescent field. In addition to the pronounced evanescent field, small optical area significantly enhances entangled photon-pair generation via the SFWM process in MNFs. Based on the obtained parameters, group velocity and GVD are used for investigating spectral properties of entangled photon-pairs. As a result, we can estimate phase-matching condition and JSA depending on the pump wavelength and tapered diameters of MNFs in NIR and telecom-band regimes, and show the possibility of uncorrelated photon-pair generation via MNFs that are suitable for LOQC protocol. Our theoretical work provides useful guidelines to design micro/nanofiber-based quantum sensing and quantum light source used for quantum technologies.