Circular dichroism spectroscopy and chiral sensing in optical fibers

Somaye Kaviani Dezaki; Amir Nader Askarpour; Abdolali Abdipour

doi:10.1364/OE.426239

1. Introduction

Chirality is an important phenomenon in nature and plays an essential role in life [1,2]. Many biomolecules like the essential amino acids are chiral and have two structures with opposite handedness that are called right and left-hand enantiomers [3–6]. The enantiomers differ only in their handedness while other physical properties are the same [3,7,8]. It is known that the biological functionality of chiral bio-molecules, as well as synthesized chiral molecules in drugs, is strongly influenced by their chirality [2,9]. Drugs, that have a single enantiomer, are often more efficient than their racemic mixtures. The more important subject is that while one chirality forms a powerful medicament, the other one may cause serious side effects [10–14]. As a result, discrimination of enantiomers is very important in biology and the pharmaceutical industry in terms of safety and production costs [2,13].

The chirality of materials can only be explored by interactions with other chiral objects [3]. Chiroptical spectroscopies are efficient optical methods that are used to characterize the configuration of chiral molecules [2,13,15–18]. Circular dichroism (CD) measurement, one of the most popular chiroptical methods, utilizes differential absorption of chiral molecules corresponding to two opposite circular polarization (CP) in the ultraviolet and visible spectrum [7,9,16,19–21]. However, due to the large scale mismatch between the wavelength of CP lights and size of the molecules, the interaction is generally weak and a long optical path is often required to show noticeable differences for left circularly polarized (LCP) and right circularly polarized (RCP) lights [2–5,12,16,22]. Therefore, amplification of CD signals is a challenging issue in chiroptical spectroscopy [7,16].

In recent years, to overcome the issue of small chirality response in natural materials, there has been a great deal of interest in studying optical chirality (OC) and creating superchiral electromagnetic fields [1,5,23–28]. Standing wave [29], metamaterials and nanostructures such as chiral plasmonic metasurfaces and metamaterials [3,5,7,15,30–32], achiral plasmonic [8,9,21,24], silicon and metal nanospheres [9,12,24,33], nanodisks [6,34], dielectric nanostructures [4,16,35], photonic crystals [36] and twisted optical metamaterials [12,13,37] are used to improve OC and enhance weak CD effects for chiral molecule detection. The CD response of a chiral molecule can be significantly increased when the resonance frequency of a plasmonic nanosphere and the resonance of the chiral transition are matched [5,38].

Chiral nanostructures enhance not only the near-field OC but also lead to electromagnetic fields with a high energy density [39]. In some of the previous works, the dissymmetry factor is not considered while it is more important than the OC, since the dissymmetry factor is more strongly related to the CD response [40]. The dissymmetry factor is defined as the ratio of optical chirality to the energy density. Achiral structures are introduced to eliminate the disadvantages of chiral nanostructures such as additional noise in the measurement and the lack of enantiomorphic fields. In addition to metamaterials, more sophisticated fields such as Bessel beams provide additional freedom to create chiral fields [41]. Most previous research consider local OC enhancement and use the evanescent waves in the near-field region. These regions should be located in such a way that they are easily accessible by the chiral analyte [5].

Up to now, methods based on the waveguide are less investigated, while waveguides with chiral geometry [42–46] and chiral material [47] have been studied in several papers. Also, the chirality and helicity have been considered in fibers [48–51]. In this paper, we use an optical fiber to design a sensor to identify the handedness of chiral materials. The chiral analyte that is located in the core of the fiber, lifts the degeneracy of modes and the cut-off frequency of modes depend on the handedness of the analyte. We define a $ {{\textrm{CD}}_{\textrm{waveguide}}}( {\textrm{CD}}_{\textrm{W}})$ parameter that is similar to the CD parameter of free-space. The sign of this parameter determines the handedness. In addition to characterizing the handedness, this method can obtain the value of the chirality parameter of the material under test.

This paper is organized as follows. In Section 2., the cylindrical waveguide or optical fiber with a chiral core is examined in summary and its dispersion equation is obtained. Here, we use the term chiral-core optical fiber to denote an optical fiber whose core is a chiral medium. By solving the presented relation, the dispersion plots are illustrated in Section 3. Lifting the degeneracy is demonstrated in this section. This means that while cut-off frequencies of the degenerate modes in a normal (achiral) optical fiber are the same, they become different in chiral-core optical fibers. In the following section, we obtain a simplified relation for the difference between these cut-off frequencies. To validate the theoretic results, chiral-core optical fibers are simulated in a commercial full-wave solver (COMSOL Multiphysics) using the finite-element method (FEM) and the results are reported in Section 5. A general method for a practical measurement setup is mentioned in Section 6. and eventually, the paper will be concluded in Section 7.

2. Optical fiber with a chiral core

The Maxwell equations in the frequency domain, with the time-harmonic convention $e^{j\omega t}$ are written as [52]

(1)$$\begin{aligned} &{\nabla} \times \overrightarrow{E}={-}j\omega \overrightarrow{B}\\ &{\nabla} \times \overrightarrow{H}=j\omega \overrightarrow{D} \end{aligned},$$

where $\overrightarrow {E}$, $\overrightarrow {B}$, $\overrightarrow {D}$ and $\overrightarrow {H}$ are the electric field, magnetic field, electric displacement field, and magnetic field strength respectively. Also, $\omega$ is the angular frequency. The constitutive relations in bi-isotropic medium are [52]

(2)$$\begin{aligned} &\overrightarrow{D}=\epsilon \overrightarrow{E}+{{\chi-j\kappa}\over{c}}\overrightarrow{H}\\ &\overrightarrow{B}=\mu \overrightarrow{H}+{{\chi+j\kappa}\over{c}}\overrightarrow{E} \end{aligned},$$

where $\epsilon (=\epsilon _0\epsilon _r)$, $\mu (=\mu _0\mu _r)$ and $c$ ($1/\sqrt {\epsilon _0 \mu _0}$) are the permittivity and permeability of the medium and speed of light in vacuum respectively. The chirality parameter $\kappa$ measures the handedness of the material. Chirality parameters of two enantiomers with mirror images have opposite signs [5]. In this paper, we consider reciprocal chiral medium (the Tellegen’s parameter $\chi =0$). By solving the wave equation, two eigen-polarization, right-hand and left-hand polarizations, are obtained. These eigen-polarizations are derived from two eigen refractive indices $n_+$ and $n_-$, respectively.

(3)$$n_+{=}n+\kappa , \ \ \ \ n_-{=}n-\kappa, \ \ \ \ n=\sqrt{\epsilon _r \mu_r}.$$

The chiral medium acts as an equivalent isotropic medium with respective medium parameters $n_+$ and $n_-$, for each of the eigen-polarizations. So, for bi-isotropic or chiral material, we first decompose the electric and magnetic fields into the right- and left-hand wavefields. A given electromagnetic field $\overrightarrow {E}$ or $\overrightarrow {H}$ is described by [52]:

(4)$$\overrightarrow{E}=\overrightarrow{E_+}+\overrightarrow{E_-} \ \ \ \ \ \ \overrightarrow{H}=\overrightarrow{H_+}+\overrightarrow{H_-} ,$$

(5)$$\overrightarrow{E_+}={-}j\eta _+\overrightarrow{H_+} \ \ \ \ \ \ \overrightarrow{E_-}={-}j\eta _-\overrightarrow{H_-} \ \ \ \ \ \ \eta_{{\pm}}=\sqrt{{\mu}\over{\epsilon_{{\pm}}}}.$$

The plus and minus wavefields are related to right-hand and left-hand polarization respectively. Also, $\epsilon _{\pm }$ are equal to $n_{\pm }^2$. In the following, we consider an optical fiber that its core is filled by a chiral analyte. The geometry of the structure is shown in Fig. 1.

Fig. 1. An optical fiber chirality sensor. The chiral material with chirality $\kappa$ fills the core with radius $a$. $n_1$ and $n_2$ are the refractive indices of the core and cladding respectively.

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To solve the problem and obtain propagating modes in the core, according to Eq. (4), the fields are decomposed into the right and left-hand wavefields, and then wavefields are divided into longitudinal and transversal components. By describing the transverse wavefield components in terms of the longitudinal parts and eliminating the transverse components, Helmholtz equation is obtained for the longitudinal components in the chiral core [52]:

(6)$$({\nabla_t}^2 +k_{t1\pm}^2){E_{z\pm}^\textrm{core}}=0,$$

where $k_{t1\pm }=\sqrt {k_{1\pm }^2-\beta ^2}$, $k_{1\pm }$ ($=\omega \sqrt {\epsilon _{1\pm }\mu _1}$), and $\beta$ are the transverse wavenumber, wavenumber in a homogeneous chiral medium, and the propagation constant along the waveguide. Subscripts $1$ and $2$ are assigned to core and cladding parameters, respectively. This equation can be written for cladding as

(7)$$({\nabla_t}^2 +k_{t2}^2){E_{z\pm}^\textrm{cladd}}=0.$$

Since the fields in the cladding must be evanescent and decay in the radial direction, $k_{t2}$ is equal to $jq=\sqrt {k_2^2-\beta ^2}$ where the $k_2$ ($=\omega \sqrt {\epsilon _2\mu _2}$) is the wavenumber in the cladding. Eventually, solutions of Helmholtz equation in the core are obtained as

(8)$${E_{z\pm}^\textrm{core}}=A_{{\pm}}J_v(k_{t1\pm}\rho)e^{{-}jv\phi},$$

where $J_v$ is the Bessel function of the order $v$. For the cladding, electric fields are equal to

(9)$${E_{z\pm}^\textrm{cladd}}=B_{{\pm}}K_v(q\rho)e^{{-}jv\phi},$$

where $K_v$ is the modified Bessel function of the integer order $v$. Satisfying the boundary conditions at $\rho =a$ gives the dispersion equation as following [52]:

(10)$$\begin{aligned} &{{\beta ^2 v^2}\over{a^2}}\left ({1\over{k_{t1+}^2}}+{1\over{q^2}}\right )\left({1\over{k_{t1-}^2}}+{1\over{q^2}}\right)\\ &-\beta v \left[{{k_{1-}}\over{k_{t1+}^2}}S_{v-}-{{k_{1+}}\over{k_{t1-}^2}}S_{v+}+{{k_{1-}}\over{q^2}}S_{v-}-{{k_{1+}}\over{q^2}}S_{v+}-T_vk_2{{\eta_1^2+\eta_2^2}\over{2\eta_1\eta_2}}\left({1\over{k_{t1+}^2}}-{1\over{k_{t1-}^2}}\right)\right]\\ &-k_{1+}k_{1-}a^2S_{v+}S_{v-}-k_2^2a^2T_v^2+T_vk_2{{\eta_1^2+\eta_2^2}\over{2\eta_1\eta_2}}a^2(k_+S_{v+}+k_-S_{v-})=0 ,\end{aligned}$$

where $S_{v\pm }$ and $T_{v}$ are defined as ${{J_v^{'}(k_{t1\pm }a)}\over {k_{t1\pm }aJ_v(k_{t1\pm }a)}}$ and $-{{K_v^{'}(qa)}\over {qaK_v(qa)}}$, respectively. Also, $\eta _{1}(\eta _{2})$ is equal to $\sqrt {\mu _{1}/\epsilon _1}(\sqrt {\mu _{2}/\epsilon _2})$. If $\kappa =0$, Eq. (10) coincides with the dispersion relation of an isotropic optical fiber. Degenerate modes are obtained for $v=\pm m$ ($m$ is a positive integer) in a normal optical fiber. However, the propagation constants in the same frequencies are different for $\pm m$ in a chiral-core optical fiber. In fact, the degeneracy in a bi-isotropic optical fiber is lifted and modes $M$ (modes are obtained for $v=\pm m$) are converted to mode $M_+$ and mode $M_-$, which correspond to $n_{1+}=n_1+\kappa$ and $n_{1-}=n_1-\kappa$, respectively.

Note that the defined functions are symmetric relative to the integer order $v$. In other words, $S_m=S_{-m}$ and $T_m=T_{-m}$. Also, when the sign of $\kappa$ changes, $S_{v+}$ and $k_+$ are converted to $S_{v-}$ and $k_-$ respectively (and vice versa). So, when the sign of $v$ and $\kappa$ change simultaneously, Eq. (10) does not change. This means that mode $M_+$ ($M_-$) for material with $\kappa _0$ is the same as mode $M_-$ ($M_+$) for a material with $-\kappa _0$.

3. Numerical results

Equation (10) is a transcendental equation and it is not straight forward to study its solutions analytically. Before we use approximation techniques to obtain solutions for Eq. (10), we plot the dispersion results numerically and observe the lifting of the degeneracy. The dispersion diagrams of an isotropic optical fiber are usually plotted in terms of the normalized propagation constant $b$, which is defined as ${{{{\beta }\over {k_0}}-n_2}\over {n_1-n_2}}$ and the normalized frequency $V=k_0 a\sqrt {n_1^2-n_2^2}$. Here, $k_0$ is the wavenumber of the free-space. For illustration purposes, we use nondispersive refractive indices $n_1=\sqrt {1.5}$, $n_2=\sqrt {1.4}$ and $\kappa =\pm 0.001$, and $a=500$ nm. Figure 2(a) represents the dispersion diagram of $\textrm {HE}_{v1}$ that are plotted for $v=\pm 1,\pm 2,\pm 3$. The solid curves in the figure present the degenerated modes of the achiral fiber. So, note that every solid curve is dedicated to two modes. When the chiral material fills the core of the fiber, these solid curves are replaced by dashed and dotted lines (for all $m$). As we observe in this figure, the cutoff frequencies of the modes in chiral-core optical fiber are different for $|m|>1$.

Fig. 2. (a) Dispersion plots for the optical fiber with chiral core. (b) Separation of cut-off frequencies of the degenerated modes for $v=\pm 2$. $V_\textrm {cutoff}$ of the $\textrm {HE}_{21}$ modes in isotropic (achiral) optical fiber is $2.432$, but $V_\textrm {cutoff}$ of the $\textrm {HE}_{21}$ modes in chiral-core optical fiber are shifted to $2.185$ and $2.785$.

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By increasing the chirality parameter to $0.01$ and $v=\pm 2$, the above observations are more clear. However, this chirality parameter is too high and for illustration purpose only. The separation of the cut-off frequencies for degenerate modes is observed in Fig. 2(b). The mode $M_+$ for $v=m(-m)$ and $M_-$ for $v=-m(m)$ are the same. This can be concluded from Eq. (10), too. Then, for two chirality parameters with same absolute values and different signs, the curve of Mode $M_+$ is replaced with curve of Mode $M_-$ and vice versa.

In the next step, to get closer to reality, we add imaginary part to the refractive index and the chirality parameter of the core. The new parameters are $n_1=\sqrt {1.5}-0.01j$ and $\kappa =\pm 0.001(1-j)$. In this case, the dispersion plot is obtained as Fig. 3.

Fig. 3. (a) Dispersion plots of $\textrm {HE}_{21}$ modes. (b) If the real part plot is magnified, It can be seen that $V_\textrm {cutoff}=2.548$ and $2.586$.

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According to Fig. 3, the main idea to identify the handedness of the chiral material in our work is degeneracy lifting. Same as the previous figures, the degenerated modes have been separated. But in this case, the imaginary part of $b$, that causes different absorptions, is nonzero. To compare the proposed method to the conventional CD measurement methods, we define a waveguide-based CD factor as a comparison metric that is similar to the CD factor of circular dichroism spectroscopy in the free-space. The CD for free-space follows as [2]

(11)$$\textrm{CD}_i={{\textrm{exp}(2 \textrm{Im}\lbrace\kappa\rbrace k_0 l)-\textrm{exp}({-}2 \textrm{Im}\lbrace\kappa\rbrace k_0 l)}\over{\textrm{exp}(2\textrm{Im}\lbrace\kappa\rbrace k_0 l)+\textrm{exp}({-}2 \textrm{Im}\lbrace\kappa\rbrace k_0 l)}},$$

where $\textrm {Im}\lbrace \kappa \rbrace$ and $l$ are the imaginary part of $\kappa$ and length of the chiral material. So, the defined factor for waveguide-based CD is

(12)$$\textrm{CD}_w={{\textrm{exp}(2 \textrm{Im}\lbrace\beta_{v=m}\rbrace l)-\textrm{exp}(2 \textrm{Im}\lbrace\beta_{v={-}m}\rbrace l)}\over{\textrm{exp}(2 \textrm{Im}\lbrace\beta_{v=m}\rbrace l)+\textrm{exp}(2 \textrm{Im}\lbrace\beta_{v={-}m}\rbrace l)}},$$

where $l$ is the length of the optical fiber. Indeed, we measure the transmitted power by the $M_+$ and $M_-$. $\beta _{v=m}$ is the propagation constant of the mode with $v=m$. When the first mode is excited, at frequencies below the cut-off frequency of the second mode, the transmitted power will only belong to this mode. So depending on which mode is excited earlier, the CD factor will be $1$ or $-1$ in the interval between the cut-off frequencies of two modes. But after propagation of two modes, the CD factor decreases. In the following, we present the CD plots of the free-space and optical fiber for lossless material and real chirality parameter ($\textrm {Im}\lbrace \kappa \rbrace =0$). It is obvious that the CD factor of free-space is 0 in this situation. However, as seen in Fig. 4 and according to the given description, when there is one of the modes, waveguide-based CD will be nonzero. The reported $\textrm {CD}_w$ in this paper has been obtained for $\textrm {HE}_{21}$ modes.

Fig. 4. $\textrm {CD}_i$ is the result of circular dichroism in the free-space and according to Eq. (11), is 0 for the real chirality parameter. $\textrm {CD}_w$ is obtained from Eq. (12). Between the cut-off frequencies, this parameter is equal to $1$ and $-1$ for negative $\kappa$ and positive $\kappa$ respectively. However, after the excitation of two modes, $\textrm {CD}_w$ decreases to 0. Interval $\Delta V$ with $|\textrm {CD}_w|=1$ is equal to $0.059$ ($n_1=\sqrt {1.5}$ and $n_2=\sqrt {1.4}$).

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In the next step, we consider the complex chirality parameter and compare CD of free-space with CD of optical fiber. The peak in the previous plots is seen here, too. As seen in Figs. 5 and 6, after the excitation of two modes, the absolute value of $\textrm {CD}_w$ decrease and then increase again, such that this parameter converges to CD of free-space at the high frequencies. It must be noted that the peak is independent of chiral material length, but the CD at other frequencies depends on the length of the chiral material. According to these figures, we can determine the sign of real and imaginary parts of the chirality parameter and its handedness. Also, the intervals $\Delta V$ with $|\textrm {CD}_w|=1$ are the same for all situations in these figures.

Fig. 5. $\textrm {CD}_w$ for the complex chirality. The sign of $\textrm {CD}_w$ at the frequencies between 2 cut-off frequencies of the modes and the sign of the $\textrm {CD}_w$ after the excitation of 2 modes are opposite, If the sign of the real and imaginary part of $\kappa$ are opposite. Interval $\Delta V$ with $|\textrm {CD}_w|=1$ is equal to $0.059$ ($n_1=\sqrt {1.5}$ and $n_2=\sqrt {1.4}$).

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Fig. 6. $\textrm {CD}_w$ for the complex chirality. The sign of $\textrm {CD}_w$ at the frequencies between two cut-off frequencies of the modes and the sign of the $\textrm {CD}_w$ after the excitation of two modes are the same, if the sign of real part and imaginary part of the chirality parameter are the same. Interval $\Delta V$ with $|\textrm {CD}_w|=1$ is equal to $0.059$ ($n_1=\sqrt {1.5}$ and $n_2=\sqrt {1.4}$).

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Finally, we consider the CD of the chiral materials with complex refractive index $n_1$. In this case, $V_\textrm {cutoff}$ for achiral optical fiber is obtained as $2.57$. As seen in Figs. 7–9, unlike previous cases, the intervals $\Delta V$ are different.

Fig. 7. $\textrm {CD}_w$ at the high frequencies tend to 0 because $\kappa$ is real. Interval $\Delta V$ with $|\textrm {CD}_w|=1$ is equal to $0.063$ ($n_1=\sqrt {1.5}-0.01i$ and $n_2=\sqrt {1.4}$).

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Fig. 8. Interval $\Delta V$ with $|\textrm {CD}_w|=1$ is equal to $0.036$. In comparison to Fig. 5, $\Delta V$ has been decreased ($n_1=\sqrt {1.5}-0.01i$ and $n_2=\sqrt {1.4}$).

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Fig. 9. Interval $\Delta V$ with $|\textrm {CD}_w|=1$ is equal to $0.095$. In comparison to Fig. 6, $\Delta V$ has been increased ($n_1=\sqrt {1.5}-0.01i$ and $n_2=\sqrt {1.4}$).

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4. Simplified relation

The results of the previous section show that we can determine the handedness of the chiral medium. However, it remains to show that the chirality parameter can be derived from the CD results. In other words, what is the relationship between the frequency interval with $\textrm {CD}_w=1$ and the chirality parameter $?$ To answer these questions, the relation between the frequency interval and the chirality parameter and other parameters of the chiral-core optical fiber must be obtained and simplified. The frequency interval ( $\Delta V$ or $\Delta k$) is the difference between two cut-off frequencies of the $M_-$ and $M_+$ modes. To obtain and simplify this relation, we use an approach based on perturbation. We consider the dispersion relation of the isotropic optical fiber and calculate the cut-off frequency of its degenerate modes. Then, we find the change in the cut-off frequency by adding a small chirality parameter to the equations. By setting the propagation constant to the wavenumber of the cladding ($\beta =k_0 n_2$) at the cut-off frequency, the dispersion equation of an achiral fiber can be written as

(13)$${{k_{c0} ^2 n_2^2 v^2}\over{a^2}}\left({1\over{k_{c0}^2(n_1^2-n_2^2)}}+{1\over{q^2}}\right)^2-k_{c0}^2a^2 n_1n_2S_{v}^2-k_2^2a^2T_v^2+T_v S_v k_{c0}^2 n_1 n_2a^2{{\eta_1^2+\eta_2^2}\over{\eta_1\eta_2}}=0.$$

Solving Eq. (13) results in the wavenumber $k_{c0}$ ($k_{c0}={{2\pi }\over c}f_\textrm {cutoff}$) of the cut-off frequency in an isotropic fiber. We assume that the wavenumber in the chiral-core optical fiber with a small chirality parameter $\kappa$ is equal to $k_\textrm {chiral}=k_{c0}+\Delta k_0$. By applying $\beta =(k_{c0}+\Delta k_0)n_2$ in the dispersion relation of the chiral-core optical fiber [Eq. (10)], $q$ will be 0. Then we can approximate $T_v$ with the Taylor series for small values of $q$. This approximation for $v>0$ is

(14)$$T_v={-{K_v^{'}(qa)}\over{qaK_v(qa)}}\overset{q\rightarrow 0}{\longrightarrow} \begin{cases} {1\over{2(v-1)}}+{v\over{q^2 a^2}} \quad \quad \quad \quad \quad \ \ v>1 \\ 0.116+\textrm{log}(qa)+{1\over{q^2a^2}} \quad v=1 \end{cases}.$$

Using Eq. (14), the dispersion relation for $v>1$ becomes

(15)$$\begin{aligned} &(\eta_1+\eta_2)^2{{n_2^2 v^2}\over{(n_1+\kappa)^2-n_2^2}}-(\eta_1-\eta_2)^2{{n_2^2 v^2}\over{(n_1-\kappa)^2-n_2^2}}\\ &+(\eta_1+\eta_2)^2(k_{c0}+\Delta k_0)^2 a^2 n_2(n_1+\kappa)v S_{v+}-(\eta_1-\eta_2)^2(k_{c0}+\Delta k_0)^2a^2n_2(n_1-\kappa)S_{v-}\\ &-2\eta_1\eta_2(k_{c0}+\Delta k_0)^2a^2n_2^2{v\over{v-1}}=0, \end{aligned}$$

for the chiral-core optical fiber and

(16)$$2 {{n_2^2 v^2}\over{n_1^2-n_2^2}}+{{\eta_1^2+\eta_2^2}\over{\eta_1 \eta_2}}k_{c0}^2n_1n_2vS_v-k_{c0}^2n_2^2{v\over{v-1}}=0,$$

for the isotropic optical fiber. The refractive indices and chirality parameter are assumed to be real in obtaining these equations. The result of subtracting Eqs. (15) from (16) is simplified under the conditions $\kappa <<1$, ${{\Delta k_0}\over {k_{c0}}}<<1$ and $\Delta ={{n_1-n_2}\over {n_1}}<<1$. Finally the relation between $\Delta k$ and $\kappa$ is obtained as

(17)$${{\Delta k_0}\over{k_{c0}}}={{\Delta V_0}\over{V_0}}={-}{{\kappa}\over{2n_1\Delta}}={-}{{\kappa}\over{2(n_1-n_2)}} \ \ \ \ \ v>1.$$

However, for $v=1$, log($qa$) tend to infinity when $q\rightarrow 0$ and this term appears in the denominator of the relation of $\Delta k_0$, thus

(18)$${{\Delta k_0}\over{k_{c0}}}\approx 0 \ \ \ \ \ \ \ \ v=1.$$

A similar expression for negative values of $v$ can be obtained. Finally, we consider $v=0$. $\textrm {TE}_{01}$ and $\textrm {TM}_{01}$ modes in an achiral optical fiber have the same dispersion plots. In this case, the dispersion relation of an achiral optical fiber is solved by $S_v=T_v$. Since $T_v\rightarrow \infty$ at the cut-off frequency, the roots of Bessel function ($J_v$) are the solutions. Also, the dispersion relation of a chiral-core optical fiber for $v=0$ can be written as

(19)$$-k_{1+}k_{1-}S_{v+}S_{v-}-k_2^2t_v^2+T_vk_2{{\eta_1^2+\eta_2^2}\over{2\eta_1 \eta_2}}(k_{1+}S_{v+}+k_{1-}S_{v-})=0.$$

The solutions to this equation are the roots of the denominator of $S_{v+}$ and $S_{v-}$. After some calculations, $\Delta k_0$ is obtained as

(20)$$\Delta k_0={\mp} {{k_{c0} \kappa}\over{2 n_1 \Delta}}$$

Eventually, the relation of $\Delta k_0 (=k_\textrm {chiral}-k_{c0})$ for all relevant cases is

(21)$$\Delta k_0\cong \begin{cases} -{{k_{c0}\kappa}\over{2n_1\Delta}}\textrm{sign}[v] \quad \quad v\neq 0,\pm 1 \\ 0 \quad \quad \quad \quad \quad \quad \quad v={\pm} 1 \\ \mp{{k_{c0}\kappa}\over{2n_1\Delta}} \quad \quad \quad \quad \quad v= 0 \end{cases},$$

and the difference between the cut-off frequencies of $M_-$ and $M_+$ ($\Delta k =k_\textrm {chiral}|_{\nu <0}-k_\textrm {chiral}|_{\nu >0}=-2\Delta k_0$) is equal to:

(22)$$\Delta k\cong \begin{cases} {{k_{c0}\kappa}\over{n_1-n_2}} \quad \quad v\neq{\pm} 1, \\ 0 \quad \quad \quad \quad v={\pm} 1 \end{cases}.$$

(23)$$\Delta V\cong \begin{cases} {{V_0\kappa}\over{n_1-n_2}} \quad \quad v\neq{\pm} 1 \\ 0 \quad \quad \quad \quad v={\pm} 1 \end{cases}$$

The relative frequency interval ($\Delta k/k_{c0}$) is independent on $v$ and $k_{c0}$. Also, it is noteworthy that the similarity of the cut-off frequencies of $M_-$ and $M_+$ for $v=1$ does not mean that the degeneracy remains. According to Fig. 2(a), the modes for $v=1$ are separated at frequencies higher than the cut-off. We consider the first mode with $v=\pm 2$ in the following ($\textrm {HE}_{21}$).

According to Figs. 5 and 6, the $\Delta V$ with $|\textrm {CD}_w|=1$ is equal to $0.059$. This parameter ($\Delta V$) is calculated by Eq. (23) as $0.058$. From these figures, it is observed that the $\Delta V$ for the real and complex chirality parameters are the same. So, when (1) the refractive indices are real or (2) imaginary parts of $n_1$ and $n_2$ are equal, Eq. (23) is converted to

(24)$$\Delta k\cong{{k_0 \textrm{Re}\lbrace\kappa\rbrace}\over{n_1-n_2}} \quad \quad v\neq{\pm} 1.$$

If the chirality parameter is real and refractive indices are complex, $\Delta k$ is calculated by

(25)$$\Delta k\cong{{k_0 \kappa}\over{\textrm{Re}\lbrace n_1-n_2\rbrace}} \quad \quad v\neq{\pm} 1.$$

Note that $k_{c0}$ is obtained for the isotropic optical fiber with the same parameters of the chiral-core optical fiber. ($k_{c0}$ for the complex $n_1$ is larger than $k_{c0}$ for the real $n_1$.) According to Fig. 7, the $\Delta V$ with $|\textrm {CD}_w|=1$ is equal to $0.063$ and this parameter ($\Delta V$) is calculated by Eq. (25) as ${{2.57\times 0.001}\over {\sqrt {1.5}-\sqrt {1.4}}}=0.062$. Equations (24) and (25) have negligible errors. However, according to Figs. 8 and 9, when all parameters are complex, we can use the approximate relation

(26)$$\Delta k\cong \begin{cases} {{k_0|\kappa|}\over{\textrm{Re}\lbrace(n_1-n_2)\rbrace}} \quad \quad \quad \quad \\ {{k_0\textrm{Re}\lbrace\kappa\rbrace}\over{\textrm{Re}\lbrace(n_1-n_2)\rbrace}}{{\textrm{Re}\lbrace\kappa\rbrace}\over{|\kappa|}} \quad \end{cases}.$$

According to Figs. 8 and 9, the $\Delta V$ with $|\textrm {CD}_w|=1$ are equal to $0.036$ and $0.095$, respectively. These parameters are calculated by Eq. (26) as $0.044$ and $0.088$. The observed error is reduced by decreasing the imaginary part of $(n_1-n_2)$ relative to its real part.

Note that $n_1$ is always larger than $n_2$ in optical fibers, so, the denominators of Eqs. (21)–(26) are never zero.

5. Simulation results

To verify the theoretic results, an optical fiber with a chiral core is simulated using the finite element method. At the first step, we perform a 2-dimensional simulation for mode analysis and obtain the dispersion plot for the isotropic optical fiber.

According to Fig. 10, the cut-off frequencies for $\textrm {TE}_{01}$, $\textrm {TM}_{01}$ and $\textrm {HE}_{21}$ are approximately at $V=2.41$. In general case, $\textrm {HE}{m1}$ for $m>1$ includes two degenerated modes for $v=\pm m$ that (first mode:) $\textrm {HE}{m1}(1)$ and (second mode:)$\textrm{HE}{m1}(2)$ are dedicated to $v=-m$ and $v=+m$ respectively [indeed, $\textrm {HE}{m1}(1)$ and $\textrm {HE}{m1}(2)$ are $\textrm {HE}{-m1}$ and $\textrm {HE}{m1}$]. As we observe in Fig. 10, mode $\textrm {HE}{m1}(1)$ and mode $\textrm {HE}{m1}(2)$ for $m=0,1,2$ have the same dispersion plots, but the electric and magnetic fields of these modes are different.

Fig. 10. Dispersion plots of the isotropic optical fiber ($n_1=\sqrt {1.5},n_2=\sqrt {1.4}$). In this figure, mode $\textrm {TETM}_{01}$ and Mode $\textrm {TETM}_{02}$ are mode $\textrm {TE}_{01}$ and $\textrm {TM}_{01}$ respectively.

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In the second step, to verify the Fig. 2(b), $\kappa =\pm 0.01$ is applied to the simulation ($n_1=\sqrt {1.5},n_2=\sqrt {1.4}$) and results are presented in Fig. 11. According to these figures, the degenerate modes have been separated and the cut-off frequencies of $\textrm {HE}_{21}(1)$ and $\textrm {HE}_{21}(2)$ are shifted from $V=2.432$ (approximately) to $V=2.785$ and $V=2.185$ respectively for $\kappa =0.01$ and vice versa for $\kappa =-0.01$ ($\Delta V=0.59$). In addition, $\textrm {TE}_{01}$ and $\textrm {TM}_{01}$ that have the same cut-off frequency, are converted to modes with different cut-off frequencies. We call these modes as $\textrm {HE}{01}(1)$ and $\textrm {HE}{01}(2)$, because both $E_z$ (the electric field component that is in line with the direction of wave propagation) and $H_z$ (the magnetic field component that is in line with the direction of wave propagation) are nonzero in these modes. It is noteworthy that $\Delta V$ is obtained $0.58$ from Eq. (23).

Fig. 11. Dispersion plots of the chiral-core optical fiber with $\kappa =0.01$.

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In the following, the simulated $\textrm {CD}_w$ plots are presented in Fig. 12 for one case ($n_1=\sqrt {1.5}-0.01j, n_2=\sqrt {1.4}$). As we observe in the figure, the plot conforms to the results of the theoretical solution. According to these figures, the theoretical and simulation results have the small difference. The absolute error between these results have been plotted in Fig. 13 for different cases. The absolute error is less than 1% for all cases, that is the negligible error. Curves of the theoretical results have been plotted in Section 3 (Figs. 4–9).

Fig. 12. Simulation results. The results of simulation and theory are consistent with negligible error. The absolute error between the theoretical and simulation results have been plotted in the next figure.

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Fig. 13. Error plots for different cases between the theoretical and simulation results.

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6. Measuring the CD factor

To implement the proposed method and measured CD plots, we operate similar to the circular dichroism spectroscopy in the free-space. At first, the mode with $v=m>0$ is excited in the optical fiber and the transmitted power is measured. Then, this is repeated for the mode with $v=-m$ at the frequency spectrum. The optical fiber must be designed such that the cut-off frequencies of the intended modes with $v=\pm m$ are located at a frequency spectrum that the chirality parameter of the core medium is not equal to 0. For excitation of the special modes in the optical fiber, we can connect the proposed waveguide to the chiral-core optical fiber that can excite the right and left circular polarization [42–44]. The general and simple schematic of the proposed test setup is presented in Fig. 14.

Fig. 14. Sample setup of the proposed CD spectroscopy.

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7. Conclusion

To measure the handedness of enantiomers in small quantities is an important problem especially in the pharmaceutical industry. Therefore, designing the sensors and methods with a better performance than the usual approaches is desired. In this paper, we proposed a method based on waveguide to detect the handedness of chiral material. The main concept in this work is the degeneracy lifting in an optical fiber with a bi-isotropic or chiral core. An adapted definition of CD ($\textrm {CD}_w$) is presented for this method. In addition to detecting the handedness of the chiral medium (the sign of the chirality parameter) and improving the response of the conventional circular dichroism (CD) spectroscopy, the proposed approach can approximate the chirality parameter by measuring the shift in the cut-off frequency.

Funding

Iran National Science Foundation.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Circular dichroism spectroscopy and chiral sensing in optical fibers

Abstract

1. Introduction

2. Optical fiber with a chiral core

3. Numerical results

4. Simplified relation

5. Simulation results

6. Measuring the CD factor

7. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (14)

Equations (26)

Optics Express