## Abstract

The helical crystal structure in optically active media acts as the natural micro-solenoids for the electromagnetic waves passing through them, producing the longitudinal magnetic field in the direction of the axis of helices. Magnetic flux through the helical structure is quantized. The Berry phase is induced by rotation of the electrons around the helical structure. Optical rotation is related to the difference in the accumulative Berry phase between the right-, and the left-circularly polarized waves, which is proportional to the magnetic flux through the helical structure, according to the Aharonov-Bohm effect. The optical activity is the natural Faraday effect and the natural Aharonov-Bohm effect.

© 2008 Optical Society of America

## 1. Introduction

In optically active media, the plane of vibration of light is found to undergo a continuous rotation as the incident light propagates along the optic axis [1]. Since the linear polarization can be mathematically represented as a superposition of the right (*r*)-, and the left (*l*)-circularly polarized waves, the two components in optically active media are suggested to propagate at different speeds, and thus with circular birefringence [1]. Optically active media have the helical and dissymmetric crystal structure, which constrains the motions of the electrons to a helical path under the influence of the incident electric field [2–4]. The analogous quantum mechanical model of optical activity is the one-electron theory, in which the electrons in optically active substances are constrained to move along twisting paths [2]. The charge flow *eυ* along the helices with the radius *r* induces a magnetic field *B* in the direction of the axis of helices [3,4], here -*e* and *υ* are the charge and the velocity of an electron moving along the helical path. The radius is equal to the crystallographic radius of helical structure, or its multiplications [4]. Therefore, the helical structure acts as natural micro-solenoids for the electromagnetic waves passing through them. The induced longitudinal magnetic field causes the optical rotation of the plane-polarized wave through optically active medium, according to the Faraday effect. In this sense, optical activity is a natural Faraday effect, and the rotatory power is proportional to the Verdet constant for optically active media, with the ratio equal to the induced magnetic induction [3,4].

Optical rotation of a plane-polarized wave, when passing through a transparent medium, is due to the phase difference between the *r*-, and the *l*-waves. On the other hand, when the electron is transported along twisting paths and around a closed contour, the Berry phase is induced in addition to the conventional dynamical phase factor [5]. The Berry phase is proportional to the magnetic flux through the closed contour, or helical structure, according to the Aharonov-Bohm (AB) effect [6]. Therefore, there exists a relationship between the Berry phase and the optical activity, and the optical activity is related to the AB-effect.

## 2. The Berry phase and the AB-effect on the motion of an electron around a closed contour

In an uniform magnetic field *B*, the motion of the electron around a contour *C* is affected by the Lorentz force, *eυB*, and the centrifugal force, *mυ*
^{2}/*r*, as that schematically shown in Fig. 1. Suppose the radius *r* of the contour is perpendicular to the magnetic field vector *B*. The direction of the charge flow can be found from Lenz’s law [3]. The moving electron has both linear moment *p* (*p*=*mυ*) and the angular moment *L*, with *L*=*r*×*p*, where *m* is the electron mass. In quantum mechanics, the angular momentum *L* is quantized, that is [3,4],

where *j* is an integer, and *ħ* is Planck’s constant. In the equilibrium state, the Lorentz and the centrifugal forces are balanced, namely *eυB*=*mυ*
^{2}/*r*. Under these circumstances, the magnetic field is proportional to the angular momentum, and is hence quantized [3,4]

The magnetic flux, *Φ*, through the closed contour is equal to *πr*
^{2}
*B*, which is expressed by

The magnetic flux is related to the angular momentum. The quantum flux is *πħ*/*e*, being exactly equal to the quantum magnetic flux trapped in hollow superconducting cylinders [7–9] and in mesoscopic normal metal rings [10–12]. The flux periodicities of *πħ*/*e* and 2*πħ*/*e* have been observed in single normal-metal rings [10–12], corresponding to the quantum numbers of *j*=*1* and *j*=2, respectively.

Classically, an electromotive force (emf) is related to the time derivative of the magnetic flux, according to Faraday’s law of induction [13]. From Eq. (3) the electromotive force, *ε*, acting on moving electrons is given by

where *τ* is the torque, which is related to an electric field, *E*, by *τ*=-*2reE*. Notice ∮_{C}
*E*·*ds*=*ε*, and *E*=*ε*/(*2πr*).

Slow transport of the electrons round the closed contour is the condition of a balance of the Lorentz force, *eυB*, and the centrifugal force, *mυ*
^{2}/*r*. Under this circumstance, the magnetic flux is the function of the angular momentum, *L*, as depicted by Eq. (3). On the other hand, a quantal system in an eigenstate, slowly transported round the closed contour by varying parameters *R* in its Hamitonian *H*(*R*), will acquire a geometrical phase factor in addition to the conventional dynamical phase factor [5]. The geometrical phase factor is known as the Berry phase [5,14], *γ*. Particularly, when the electron is transported around the closed contour, the Berry phase *γ* is equal to the phase shift of the AB-effect [6,15], which is proportional to the magnetic flux through the contour, namely

Combining Eqs. (3) and (5) we have

The Berry phase is proportional to the angular momentum. For rotation of the electrons around the closed contour, the quantum of the Berry phase is *π*.

## 3. The rotatory power and the Berry phase

Optical rotation of a plane-polarized wave, when passing through a transparent medium, is due to the difference between the indices of refraction *n*
_{l} and *n*
_{r} for the *l*-, and *r*-waves. After passing through a distance *d*, the rotation angle *θ* of the plane of polarization is [1–4]

where λ is the wavelength of the incident light in vacuum. When optical rotation is caused by optical activity, we have [1–4]

where *ρ* is the rotatory power. When the rotation is induced by a magnetic field *B*, known as the Faraday effect, the rotation angle is expressed by [1–4]

where *V* is referred to as the Verdet constant.

Optically active media have the helical and dissymmetric crystal structure, which constrains the motions of the electrons to a helical path under the influence of the incident electric field [1–4]. Thus the propagation of the light waves is accompanied with rotations of the electrons around the helices. The helical structure acts as natural micro-solenoids for the electromagnetic waves passing through them. The induced longitudinal magnetic field causes the optical rotation of the plane-polarized wave through optically active medium, according to the Faraday effect. In this sense, we see that optical activity is a natural Faraday effect [3,4]. The rotatory power *ρ* is proportional to the Verdet constant *V* for optically active medium, with the ratio equal to the induced magnetic induction *B*, namely *ρ*=*BV*. Using Eq. (2), we have [3,4]

Equation (10) indicates that the rotatory power is a quantized quantity, with the quantum of *ħV*/(*er*
^{2}). The validity of Eq. (10) has been experimentally confirmed by the results of the rotatory power and the Verdet constant of α-quartz at different wavelengths of the incident light [3,4,16–18]. Figure 2 shows the experimental results of the rotatory power and the Verdet constant of α-quartz in the wavelength range from 0.19 to 2 µm [17,18]. The rotatory power is proportional to the Verdet constant in a wide wavelength region. The evaluated *B* is 82.98 Tesla. Recently, high number (*j*=3) of quanta has been experimentally observed in α-quartz [4]. With the determined Verdet constant and the rotatory power, the evaluated radius *r* of the helical structure in α-quartz is shown to be *6a*
_{0}, where *a*
_{0} is the cell parameter [4]. Therefore, the radius is equal to a multiple of the cell parameter.

The *r*-, and the *l*-waves propagate in the medium with the wavelengths of *λ*/*n*
_{r}, and *λ*/*n*
_{l}, respectively. The *r*-wave packet induces a charge flow in a sub-solenoid of the radius *r* and the length *λ*/*n*
_{r}, whereas the *l*-wave packet induces a charge flow in a sub-solenoid of the radius *r* and the length *λ*/*n*
_{l}, as that schematically shown in Fig. 3. Note *λ* is much larger than *r*. The magnetic field *B*
_{r} induced by the *r*-waves is in the opposite direction of *B*
_{l} induced by the *l*-waves. Therefore, the total macroscopic magnetic field induced by the incident light waves is zero. In the equilibrium state, we have: *eυ*
_{r}
*B*
_{l}=*mυ*
^{2}
_{r}/*r*, and *eυ*
_{l}
*B*
_{r}=*mυ*
^{2}
_{l}/*r*, according to Lenz’s law, where *υ*
_{r} and *υ*
_{l} are the velocities of the electrons driven by the *r*-, and *l*-waves, respectively. Hence optical rotation vanishes when the plane-polarized light is reflected back in optically active medium. Conversely, optical rotation is doubled when the light beam is reflected back in the traditional Faraday effect.

Consider the optical rotation in the direction of optic axis (*z*-axis). Slow transport of the electrons round a sub-solenoid induces a Berry phase *γ* [5]. After passing through a distance, *d*, the numbers of sub-solenoids are *n*
_{r}
*d*/*λ*, and *n*
_{l}
*d*/*λ* for the *r*-, and the *l*-waves, respectively. Therefore, the difference in the accumulative Berry phase between the *r*-, and the *l*-waves is *γ* (*n*
_{r}-*n*
_{l}) d/λ, which is equal to the rotation angle *θ*, namely

Comparing Eq. (8) with (11), we have

The rotatory power is related to the Berry phase. Combining Eqs. (6) and (12) yields the relation of

When *j*=*1*, Eq. (13) leads to a classical result [1–4]: *ρ*=*π*(*n*
_{r}-*n*
_{l})/*λ*. This result indicates that optical rotation is associated with the difference in the accumulative Berry phase between the *r*-, and the *l*-waves. Comparing Eq. (10) with (13), the Verdet constant is given by

where *A*=*πr*
^{2}, is the area of the helix. Therefore, the Verdet constant depends on the dimensions of the helical structure.

## 4. Discussion

The helical structure in optically active media acts as natural micro-solenoids for the electromagnetic waves passing through them. The propagation of the plane-polarized light induces the rotations of the electrons along the helical paths, which produces the longitudinal magnetic field. In this sense, the optical rotation in optically active media is the natural Faraday effect. The rotatory power is proportional to the Verdet constant at different wavelengths of the incident light, as depicted by Eq. (10). However, the magnetic field *B*
_{r} induced by the *r*-waves is in the opposite direction of *B*
_{l} induced by the *l*-waves. Therefore, the total macroscopic magnetic field induced by the incident light waves is zero. Hence optical rotation vanishes when the plane-polarized light is reflected back in optically active medium. In the equilibrium state, slow transport of the electrons around the closed contour induces the Berry phase, which is equal to the phase shift of the AB-effect. Therefore, the Berry phase is proportional to the magnetic flux through the helical structure. Balancing the Lorentz and the centrifugal forces acting on a moving electron around the contour, the magnetic flux through the helical structure is quantized. The difference in the accumulative Berry phase between the *r*-, and the *l*-waves is equal to the rotation angle. Therefore, optical rotation is related to the difference in the accumulative Berry phase. The optical activity is found to be the natural Faraday effect and the natural AB-effect. The helical structure in optically active media works as molecular microstructure for the electromagnetic waves passing through them.

Because of the induced longitudinal magnetic field in the *z*-direction, the electromagnetic force on a moving electron is -*e*(*E*+*υ*×*B*), where *E* (*E*
_{x}, *E*
_{y}) is the electric field of the incident light [13]. In classical electrodynamics, the motion of the electrons is generally described by a differential equation of *d*
^{2}
*s*/*dt*
^{2}+*ω*
_{0}
^{2}
*s*=-*e*(*E*+*υ*×*B*)/*m*, where *s* (*s*
^{2}=*x*
^{2}+*y*
^{2}) is the displacement of the electron from its equilibrium position, *ω*
_{0} is the resonance frequency, and *υ*=*ds*/*dt*=[(*dx*/*dt*)^{2}+(*dy*/*dt*)^{2}]^{1/2} denotes the velocity of the forced motion [13,19,20]. Notice the equilibrium condition in the direction perpendicular to *E* (and hence to s): *eυB*=*mυ*
^{2}/*r*. The equation of motion is then changed into the following forms:

The sign “±” depends on the direction of *B* with respect to the *z*-axis. If *B* is in the direction of *x*×*y*, the sign in Eq. (15a) is “+”, and in (15b) is “-”. Conversely, if *B* is in the direction of *y*×*x*, the sign in Eq. (15a) is “-”, and in (15b) is “+”. These are nonlinear differential equations. However, because *r* is much larger than *s*, (in α-quartz, *r*=*6a*
_{0} [4]), it may be experimentally difficult to probe the nonlinear effect induced by the term, (*ds*/*dt*)^{2}/*r*.

## 5. Conclusions

The propagation of the plane-polarized light induces the rotations of the electrons along the helical paths, which produces the longitudinal magnetic field in optically active media. Balancing the Lorentz and the centrifugal forces acting on a moving electron around a closed contour, the magnetic flux through the helical structure is quantized. Slow transport of the electron around the helical structure induces the Berry phase proportional to the magnetic flux, according to the AB-effect. Optical rotation is related to the difference in the accumulative Berry phase between the *r*-, and the *l*-waves. The optical activity is the natural Faraday effect and the natural AB-effect.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (under grant 60578033).

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