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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 19 — Sep. 15, 2008
  • pp: 14675–14682
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The Berry phase and the Aharonov-Bohm effect on optical activity

C. Z. Tan  »View Author Affiliations


Optics Express, Vol. 16, Issue 19, pp. 14675-14682 (2008)
http://dx.doi.org/10.1364/OE.16.014675


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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

1. E. Hecht, Optics (Addison Wesley, New York, 2002).

]. 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

1. E. Hecht, Optics (Addison Wesley, New York, 2002).

]. 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

2. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937). [CrossRef]

4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]. 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

2. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937). [CrossRef]

]. The charge flow along the helices with the radius r induces a magnetic field B in the direction of the axis of helices [3

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

,4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

], 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

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]. 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

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

,4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

].

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, 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

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

]. The moving electron has both linear moment p (p=) 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

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

,4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

],

L=mrυ=jħ,
(1)

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= 2/r. Under these circumstances, the magnetic field is proportional to the angular momentum, and is hence quantized [3

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

,4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]

B=Ler2=jħer2.
(2)
Fig. 1. The rotating electron of the charge -e around a contour C of the radius r in an uniform magnetic field B, with its velocity vector υ perpendicular to B. The motion of the electron is affected by the Lorentz force, eυB, and the centrifugal force, 2/r. In the equilibrium state, the two opposite forces are balanced. The magnetic flux Φ through the contour and the Berry phase γ are proportional to the angular momentum L. An electromotive force ε is related to a torque τ.

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

Φ=πr2B=πeL=jπħe.
(3)

ε=Φt=πedLdt=πeτ,
(4)

Slow transport of the electrons round the closed contour is the condition of a balance of the Lorentz force, eυB, and the centrifugal force, 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

5. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45–57 (1984). [CrossRef]

]. The geometrical phase factor is known as the Berry phase [5

5. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45–57 (1984). [CrossRef]

,14

14. F. Plastina, G. Liberti, and A. Carollo, “Scaling of Berry’s phase close to the Dicke quantum phase transition,” Europhys. Lett. 76, 182–188 (2006). [CrossRef]

], 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

6. Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959). [CrossRef]

,15

15. M. P. Silverman, More Than One Mystery, Explorations in Quantum Interference (Springer-Verlag, Berlin, 1995). [CrossRef]

], which is proportional to the magnetic flux through the contour, namely

γ=eħΦ.
(5)

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

γ=πħL=jπ.
(6)

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

1. E. Hecht, Optics (Addison Wesley, New York, 2002).

4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]

θ=π(nlnr)dλ,
(7)

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

1. E. Hecht, Optics (Addison Wesley, New York, 2002).

4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]

θ=ρd,
(8)

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

1. E. Hecht, Optics (Addison Wesley, New York, 2002).

4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]

θ=VBd,
(9)

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

1. E. Hecht, Optics (Addison Wesley, New York, 2002).

4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]. 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

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

,4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]. 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

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

,4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]

ρ=BV=jħVer2.
(10)

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

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

,4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

,16

16. P. Van Den Keybus and W. Grevendonk, “Comparison of optical activity and Faraday rotation in crystalline SiO2,” Phys. Status Solidi B 136, 651–659 (1986). [CrossRef]

18

18. D. N. Nikogosyan, Properties of Optical and Laser-Related Materials A Handbook (John Wiley & Sons, New York, 1997).

]. 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

17. J. L. Dexter, J. Landry, D. G. Cooper, and J. Reintjes, “Ultraviolet optical isolators utilizing KDP-isomorphs,” Opt. Comm. 80, 115–118 (1990). [CrossRef]

,18

18. D. N. Nikogosyan, Properties of Optical and Laser-Related Materials A Handbook (John Wiley & Sons, New York, 1997).

]. 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

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]. 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

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]. Therefore, the radius is equal to a multiple of the cell parameter.

Fig. 2. Relationship between the rotatory power ρ (in degree/mm) and the Verdet constant V (in degree/(tesla·mm)) of α-quartz in the wavelength range from 0.19 to 2 µm. Experimental results (symbols) are taken from Refs. [17,18]. The rotatory power is found to be proportional to the Verdet constant at different wavelengths of the incident light. This is the experimental confirmation of Eq. (10).

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: r B l= 2 r/r, and l B r= 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.

Fig. 3. The sub-solenoids of the radius r in optically active media. The lengths of the sub-solenoids are λ/n r, and λ/n l for the r-, and the l-waves, respectively. The magnetic field B r induced by the r-waves is in the opposite direction of B l induced by the l-waves. In the equilibrium state, the Lorentz and the centrifugal forces are balanced: r B l= 2 r/r, and l B r= 2 l/r.

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

5. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45–57 (1984). [CrossRef]

]. 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

θ=γ(nlnr)dλ.
(11)

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

ρ=γ(nlnr)λ.
(12)

ρ=jπ(nlnr)λ.
(13)

When j=1, Eq. (13) leads to a classical result [1

1. E. Hecht, Optics (Addison Wesley, New York, 2002).

4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]: 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

V=eA(nlnr)ħλ,
(14)

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

4. Discussion

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

13. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1999).

]. 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

13. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1999).

,19

19. C. Z. Tan, “Piezoelectric lattice vibrations at optical frequencies,” Solid State Commun. 131, 405–408 (2004). [CrossRef]

,20

20. C. Z. Tan, H. Li, and L. Chen, “Generation of mutual coherence of eigenvibrations in α-quartz at infrared frequencies by incidence of randomly polarized waves,” Appl. Phys. B 86, 129–137 (2007). [CrossRef]

]. Notice the equilibrium condition in the direction perpendicular to E (and hence to s): eυB= 2/r. The equation of motion is then changed into the following forms:

d2xdt2±1r(dydt)2+ω02x=emEx,
(15a)
d2ydt21r(dxdt)2+ω02y=emEy.
(15b)

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

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

]), it may be experimentally difficult to probe the nonlinear effect induced by the term, (ds/dt)2/r.

5. Conclusions

Acknowledgments

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

References and links

1.

E. Hecht, Optics (Addison Wesley, New York, 2002).

2.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937). [CrossRef]

3.

C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B 82, 633–636 (2006). [CrossRef]

4.

C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. 32, 2936–2938 (2007). [CrossRef] [PubMed]

5.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45–57 (1984). [CrossRef]

6.

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959). [CrossRef]

7.

B. S. Deaver and W. M. Fairbank, “Experimental evidence for quantized flux in superconducting cylinders,” Phys. Rev. Lett. 7, 43–46 (1961). [CrossRef]

8.

R. Doll and M. Näbauer, “Experimental proof of magnetic flux quantization in a superconducting ring,” Phys. Rev. Lett. 7, 51–52 (1961). [CrossRef]

9.

W. L. Goodman and B. S. Deaver Jr., , “Detailed measurements of the quantized flux states of hollow superconducting cylinders,” Phys. Rev. Lett. 24, 870–873 (1970). [CrossRef]

10.

D. Yu. Sharvin and Yu. V. Sharvin, “Magnetic-flux quantization in cylindrical film of a normal metal,” JETP Lett. 34, 272–275 (1981).

11.

R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” Phys. Rev. Lett. 54, 2696–2699 (1985). [CrossRef] [PubMed]

12.

A. D. Stone and Y. Imry, “Periodicity of the Aharonov-Bohm effect in normal-metal rings,” Phys. Rev. Lett. 56, 189–192 (1986). [CrossRef] [PubMed]

13.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1999).

14.

F. Plastina, G. Liberti, and A. Carollo, “Scaling of Berry’s phase close to the Dicke quantum phase transition,” Europhys. Lett. 76, 182–188 (2006). [CrossRef]

15.

M. P. Silverman, More Than One Mystery, Explorations in Quantum Interference (Springer-Verlag, Berlin, 1995). [CrossRef]

16.

P. Van Den Keybus and W. Grevendonk, “Comparison of optical activity and Faraday rotation in crystalline SiO2,” Phys. Status Solidi B 136, 651–659 (1986). [CrossRef]

17.

J. L. Dexter, J. Landry, D. G. Cooper, and J. Reintjes, “Ultraviolet optical isolators utilizing KDP-isomorphs,” Opt. Comm. 80, 115–118 (1990). [CrossRef]

18.

D. N. Nikogosyan, Properties of Optical and Laser-Related Materials A Handbook (John Wiley & Sons, New York, 1997).

19.

C. Z. Tan, “Piezoelectric lattice vibrations at optical frequencies,” Solid State Commun. 131, 405–408 (2004). [CrossRef]

20.

C. Z. Tan, H. Li, and L. Chen, “Generation of mutual coherence of eigenvibrations in α-quartz at infrared frequencies by incidence of randomly polarized waves,” Appl. Phys. B 86, 129–137 (2007). [CrossRef]

OCIS Codes
(160.4760) Materials : Optical properties
(260.5430) Physical optics : Polarization
(270.0270) Quantum optics : Quantum optics

ToC Category:
Quantum Optics

History
Original Manuscript: July 10, 2008
Revised Manuscript: August 13, 2008
Manuscript Accepted: August 13, 2008
Published: September 3, 2008

Citation
C. Z. Tan, "The Berry phase and the Aharonov-Bohm effect on optical activity," Opt. Express 16, 14675-14682 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14675


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References

  1. E. Hecht, Optics (Addison Wesley, New York, 2002).
  2. E. U. Condon, "Theories of optical rotatory power," Rev. Mod. Phys. 9, 432-457 (1937). [CrossRef]
  3. C. Z. Tan, "Quantum magnetic flux through helical molecules in optically active media," Appl. Phys. B 82, 633-636 (2006). [CrossRef]
  4. C. Z. Tan and L. Chen, "Quantum effects in the optical activity of ?-quartz," Opt. Lett. 32, 2936-2938 (2007). [CrossRef] [PubMed]
  5. M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. Lond. A 392, 45-57 (1984). [CrossRef]
  6. Y. Aharonov and D. Bohm, "Significance of electromagnetic potentials in the quantum theory," Phys. Rev. 115, 485-491 (1959). [CrossRef]
  7. B. S. Deaver, Jr., and W. M. Fairbank, "Experimental evidence for quantized flux in superconducting cylinders," Phys. Rev. Lett. 7, 43-46 (1961). [CrossRef]
  8. R. Doll and M. Näbauer, "Experimental proof of magnetic flux quantization in a superconducting ring," Phys. Rev. Lett. 7, 51-52 (1961). [CrossRef]
  9. W. L. Goodman and B. S. Deaver, Jr., "Detailed measurements of the quantized flux states of hollow superconducting cylinders," Phys. Rev. Lett. 24, 870-873 (1970). [CrossRef]
  10. D. Yu. Sharvin and Yu. V. Sharvin, "Magnetic-flux quantization in cylindrical film of a normal metal," JETP Lett. 34, 272-275 (1981).
  11. R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, "Observation of h/e Aharonov-Bohm oscillations in normal-metal rings," Phys. Rev. Lett. 54, 2696-2699 (1985). [CrossRef] [PubMed]
  12. A. D. Stone and Y. Imry, "Periodicity of the Aharonov-Bohm effect in normal-metal rings," Phys. Rev. Lett. 56, 189-192 (1986). [CrossRef] [PubMed]
  13. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1999).
  14. F. Plastina, G. Liberti, and A. Carollo, "Scaling of Berry???s phase close to the Dicke quantum phase transition," Europhys. Lett. 76, 182-188 (2006). [CrossRef]
  15. M. P. Silverman, More Than One Mystery, Explorations in Quantum Interference (Springer-Verlag, Berlin, 1995). [CrossRef]
  16. P. Van Den Keybus and W. Grevendonk, "Comparison of optical activity and Faraday rotation in crystalline SiO2," Phys. Status Solidi B 136, 651-659 (1986). [CrossRef]
  17. J. L. Dexter, J. Landry, D. G. Cooper, and J. Reintjes, "Ultraviolet optical isolators utilizing KDP-isomorphs," Opt. Comm. 80, 115-118 (1990). [CrossRef]
  18. D. N. Nikogosyan, Properties of Optical and Laser-Related Materials A Handbook (John Wiley & Sons, New York, 1997).
  19. C. Z. Tan, "Piezoelectric lattice vibrations at optical frequencies," Solid State Commun. 131, 405-408 (2004). [CrossRef]
  20. C. Z. Tan, H. Li, and L. Chen, "Generation of mutual coherence of eigenvibrations in ?-quartz at infrared frequencies by incidence of randomly polarized waves," Appl. Phys. B 86, 129-137 (2007). [CrossRef]

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