## Fresnel coefficients in materials with magnetic monopoles |

Optics Express, Vol. 19, Issue 4, pp. 3742-3757 (2011)

http://dx.doi.org/10.1364/OE.19.003742

Acrobat PDF (793 KB)

### Abstract

Recent experiments have found entities in crystals whose behavior is equivalent to magnetic monopoles. In this paper, we explain some optical properties based on the reformulated “Maxwell” equations in material media in which there are equivalent magnetic charges. We calculate the coefficients of reflection and transmission of an electromagnetic wave in a plane interface between the vacuum and a medium with magnetic charges. These results can give a more extended vision of the properties of the materials with magnetic monopoles, since the phase and the amplitudes of the reflected and transmitted waves, differ with and without these magnetic entities.

© 2011 Optical Society of America

## 1. Introduction

**B**= 0. Later, Dirac argued [2,3

3. P. A. M. Dirac, “The Theory of Magnetic Poles,” Phys. Rev. **74**, 817–830 (1948). [CrossRef]

4. G. ’t Hooft, “Magnetic monopoles in unified gauge theories,” Nucl. Phys. **B79**, 276–284 (1974). [CrossRef]

5. B. Cabrera, “First Results from a Superconductive Detector for Moving Magnetic Monopoles,” Phys. Rev. Lett. **48**, 1378–1381 (1982). [CrossRef]

6. C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature **451**, 42–45 (2008). [CrossRef] [PubMed]

11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. **6**, 359–363 (2010). [CrossRef]

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature **461**, 956–960 (2009). [CrossRef] [PubMed]

12. S. Sondhi, “Wien route to monopoles,” Nature **461**, 888–889 (2009). [CrossRef] [PubMed]

7. X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, “Inducing a Magnetic Monopole with Topological Surface States,” Science **323**, 1184–1187 (2009). [CrossRef] [PubMed]

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature **461**, 956–960 (2009). [CrossRef] [PubMed]

11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. **6**, 359–363 (2010). [CrossRef]

5. B. Cabrera, “First Results from a Superconductive Detector for Moving Magnetic Monopoles,” Phys. Rev. Lett. **48**, 1378–1381 (1982). [CrossRef]

6. C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature **451**, 42–45 (2008). [CrossRef] [PubMed]

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature **461**, 956–960 (2009). [CrossRef] [PubMed]

11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. **6**, 359–363 (2010). [CrossRef]

6. C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature **451**, 42–45 (2008). [CrossRef] [PubMed]

**461**, 956–960 (2009). [CrossRef] [PubMed]

9. D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, “Dirac Strings and Magnetic Monopoles in the Spin Ice Dy_{2}Ti_{2}O_{7},” Science **326**, 411–414 (2009). [CrossRef] [PubMed]

**6**, 359–363 (2010). [CrossRef]

12. S. Sondhi, “Wien route to monopoles,” Nature **461**, 888–889 (2009). [CrossRef] [PubMed]

**451**, 42–45 (2008). [CrossRef] [PubMed]

**461**, 956–960 (2009). [CrossRef] [PubMed]

12. S. Sondhi, “Wien route to monopoles,” Nature **461**, 888–889 (2009). [CrossRef] [PubMed]

13. L. J. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” J. Chem. Phys. **2**, 599–615 (1934). [CrossRef]

17. J. Costa-Quintana and F. Lopez-Aguilar, “Propagation of electromagnetic waves in material media with magnetic monopoles,” Prog. Electromagn. Res. **110**, 267–295 (2010). [CrossRef]

**K**) that in general is a complex vector, as in a conducting material. We make a complementary analysis of Classical Physics to determine the electromagnetic transmittance and reflectance of an incident wave in the separation surfaces of two different media, where one of them can be a spin-ice material. Using the boundary conditions we calculate the magnitudes and polarizations of the different fields

**E**and

**B**, as well as the relationships between the incident electromagnetic wave and the reflected/transmitted ones, i.e., we determine the so-called Fresnel coefficients. The measurement of the reflection coefficient of an electromagnetic wave incident on a material with magnetic monopoles can be decisive in confirming the existence of such magnetic monopoles as well as their optical properties.

## 2. Fields equations with monopoles

17. J. Costa-Quintana and F. Lopez-Aguilar, “Propagation of electromagnetic waves in material media with magnetic monopoles,” Prog. Electromagn. Res. **110**, 267–295 (2010). [CrossRef]

*ρ*is the electric charge density,

_{e}*ρ*the magnetic monopole density,

_{m}**J**

*the electric current density and*

_{e}**J**

*the magnetic current density. The constant*

_{m}*K*can have any value since it defines the unit of magnetic charge. Obviously for

*K*= 0 we have the standard equations of electromagnetism.

*q*) and magnetic (

*g*) value can be written as:

*Q*(sometimes called

*dyon*[18]) has two components, an electric,

*q*, and a magnetic,

*κg*, charge. Similarly, the other magnitudes have two components: one in a subspace that is “electrical”, and another that is “magnetic”.

### 2.1. Duality transformation

**-vector are presented in an electromagnetic double space where the up (down) subspace corresponds to the electric (magnetic) component. This allows us to make a representation of both the field sources (charges and currents) and the fields in the complex plane so that Eq. (5) can be re-written as In this complex representation, the Maxwell equation (6) are given by and the Lorentz force takes the form In this notation, the particles are dyons whose charge is defined by the**

*G**ζ*-angle and its module |

*Q*|. Thus, the electron, negatively charged, has a value of

*ζ*=

*π*, the proton is a dyon with

*ζ*= 0 and for a magnetic monopole

*ζ*= ±

*π*/2. It must be remarked that Eqs. (6) and (7) are totally equivalent to Eqs. (9) and (10). However, in these latter equations of the complex representation, the explanation and the meaning of the duality transformation [14, 15

15. K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rep. Prog. Phys. **69**, 1637–1711 (2006). [CrossRef]

15. K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rep. Prog. Phys. **69**, 1637–1711 (2006). [CrossRef]

*ζ*

_{0}in the dyonic complex plane; the charges represented as

*Q*≡ |

*Q*| exp(

*iζ*) are transformed in

*Q*′ ≡

*Q*exp(

*iζ*

_{0}) = |

*Q*| exp[

*i*(

*ζ*+

*ζ*

_{0})], and the electromagnetic field components, defined as

*G*

_{x,y,z}≡

*E*

_{x,y,z}+

*icB*

_{x,y,z}, are transformed in the same way,

*G′*

_{x,y,z}=

*G*

_{x,y,z}exp(

*iζ*

_{0}). Then, the Maxwell equations are invariant, that is, the equations of the primed field (

**′) are the same as Eq. (9) with the primed sources present [14], since this rotation implies a multiplication of both sides of the equality by the same phase [exp(**

*G**iζ*

_{0})]. Furthermore, the Lorentz force is the same before and after the transformation, because the product

*ρ*

^{*}**does not suffer any change by multiplying by exp(**

*G**−iζ*

_{0}) exp(

*iζ*

_{0}).

**461**, 956–960 (2009). [CrossRef] [PubMed]

12. S. Sondhi, “Wien route to monopoles,” Nature **461**, 888–889 (2009). [CrossRef] [PubMed]

13. L. J. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” J. Chem. Phys. **2**, 599–615 (1934). [CrossRef]

**461**, 956–960 (2009). [CrossRef] [PubMed]

12. S. Sondhi, “Wien route to monopoles,” Nature **461**, 888–889 (2009). [CrossRef] [PubMed]

*ζ*=

*π*), the proton should be a dyon with

*ζ*= 0. A positive charge (

*ζ*= 0) or a magnetic monopole (

*ζ*= ±

*π*/2) could have been assigned to the electron with identical legitimacy but the

*ζ*-angle of the other particles should be consistently changed. This is equivalent to say that if the electron charge is located in a dyonic line (points with the same value of

*ζ*) in the complex plane, then the proton charge should be located in the same line in the symmetric point with respect the origin.

*ζ*=

*π*/2-dyons, with electric charges whose

*ζ*-phases are either 0 or

*π*. This analysis has been encouraged by the recent experimental analysis [8

**461**, 956–960 (2009). [CrossRef] [PubMed]

9. D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, “Dirac Strings and Magnetic Monopoles in the Spin Ice Dy_{2}Ti_{2}O_{7},” Science **326**, 411–414 (2009). [CrossRef] [PubMed]

**6**, 359–363 (2010). [CrossRef]

7. X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, “Inducing a Magnetic Monopole with Topological Surface States,” Science **323**, 1184–1187 (2009). [CrossRef] [PubMed]

### 2.2. Field equations in matter

15. K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rep. Prog. Phys. **69**, 1637–1711 (2006). [CrossRef]

**B**≠ 0 in solid is not sufficiently treated being far from constituting a closed theoretical matter.

**P**

*(*

_{e}**P**

*) is the electric (magnetic) polarization due to a split in the gravity centers of the electric (magnetic) charges, and*

_{m}**M**

*(*

_{e}**M**

*) is the magnetization due to the electric (magnetic) currents. We call*

_{m}

*P**split-charge polarization*and

*M**kinetic polarization*.

### 2.3. Linear responses of the materials

**E**and

**B**, which constitute the so-called kinetic polarization.

#### Conductivity

**461**, 956–960 (2009). [CrossRef] [PubMed]

*σ*, in linear approximation, is a constant characteristic of each material media and the Θ matrix are If there is more than one class of particles (particles with a different value of tan

*ζ*), then, the conductivity becomes the sum of all contributions, i.e., where

*ζ*is the electromagnetic angle in the dyonic complex plane for each class of particles. In the specific case of coexistence of pure electric and magnetic charges (i.e., dyons exclusively located either in the real axis or imaginary axis of the complex plane), the conductivity is:

_{p}#### Split-charge susceptibility

*χ*, relationship between the density of split-charge dipole moment

**and the electromagnetic field,**

*P***: therefore and the permittivity matrix is If there are no magnetic monopoles which obviously reduces to the expression of standard electromagnetism (**

*G**χ*=

_{e}*χ*, is the electric susceptibility). If there are only magnetic monopoles, we have: where,

*χ̄*is the magnetic charge susceptibility equivalent to the electric susceptibility. If there are several kinds of particles of the same class (different particles but with the same value of tan

_{e}*ζ*) one needs to modify only the value of

*χ*. With several class of particles it is also true,

**=**

*d**ɛ*⃡

**, and then, we have the following expression for**

*G**ɛ*⃡:

#### Kinetic susceptibility

**B**. This

**B**produces a magnetic force (

**v**×

**B**) which is able to change the current and the magnetic polarization. The extension of this concept to the dyonic currents requires the consideration of the Lorentz force proportional to the velocity, The changes of the kinetic polarization become: where

*α*′ is a response function which depends on the concrete material. Because this response function depends on the speed, we call it as kinetic polarizability of the material. In linear approach [16]

**is: We can define the permeability matrix,**

*M**μ*⃡,

*ζ*= 0) that is the classical expression, with

*χ*=

_{m}*ξ*/(1 –

*ξ*) the magnetic susceptibility. And in the case in which one has only magnetic monopoles (

*ζ*=

*π*/2)

*ξ*. If one considers several classes of particles, the expression

**=**

*G**c μ*⃡

**, continues being valid, and we have the following expression**

*h*## 3. Plane waves

**f**and

**b**) in matter, and the propagation

**K**-wave vector versus the linear responses of material,

*σ*⃡,

*μ*⃡ and

*ɛ*⃡. This is carried out by considering the continuity conditions of the fields yielded by the extended field equations.

*ɛ*⃡

*μ*⃡ (i.e., det|

*ɛ*⃡

*μ*⃡| ≠ 0), then we have the transversality condition of the fields where it must be remembered that

**f**and

**b**are the electric and magnetic flavors of the

**-vector**

*h**propagation matrix*is Writing the

*x*,

*y*,

*z*-components the Eq. (38) reads This equation can have a solution different from the trivial (

*f*

_{x,y,z}=

*b*

_{x,y,z}= 0) if the corresponding determinant vanishes, i.e., In a general case, the

**K**-vector should have the

**k**+

*i*

**l**-structure, where

**k**is really the propagation vector and

**l**is the electromagnetic wave extinction vector, present in a conductor media. Therefore, having in mind the freedom for choosing a coordinate system, the general expression of the

**K**-wave vector can be written as follows: where

*k*,

_{x}*l*i

_{x}*k*are real numbers. We just choose the vector

_{y}**e**

*in the direction of the imaginary part of*

_{x}**K**and then we make a rotation until

**K**has no

*z*component (i.e.,

*K*= 0 and the

_{z}*x*-component of

**K**is complex). Then Eq. (41) with the

**K**-wave vector of Eq. (42) becomes where Δ ≡

*RU*–

*ST*,

*δ*≡

*S*–

*T*and

**461**, 956–960 (2009). [CrossRef] [PubMed]

13. L. J. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” J. Chem. Phys. **2**, 599–615 (1934). [CrossRef]

## 4. Fresnel coefficients

**k**

_{0}real, that propagates from vacuum to a medium. We consider that the surface separating the two media coincides with the plane

*x*= 0. Obviously, such as it occurs in the standard electromagnetism, there will be a reflected wave

**″ with**

*h***k**″ real, and a transmitted wave

**′ which in a general case, considering the possibility of existing electric and magnetic charge conductivities,**

*h***K**′ is complex. To satisfy the boundary conditions of the

**d**and

**h**-fields [whose deduction can be extended from the standard electrodynamics but considering Eqs. (1), (2), (12) and (13)] at any point of the separation surface between media, must be Since the medium in which there is the initial traveling wave is the vacuum, then

*k*

_{0y}=

*k*″

*=*

_{y}*k*′

*and*

_{y}*k*

_{0z}=

*k*″

*=*

_{z}*k*′

*are real. Besides, we can rotate the coordinates system so that 0 =*

_{z}*k*

_{0z}=

*k*″

*=*

_{z}*k*′

*. In the medium were*

_{z}**′ propagates**

*h***K**′ may be complex, but as

*k*′

*= 0 and*

_{z}*k*′

*are real, it has imaginary component only in the*

_{y}*x*-direction, i.e., we can always choose a system of coordinates in which where

*k*′

*,*

_{x}*l*′

*i*

_{x}*k*′

*are real and*

_{y}*K*′

*complex. Applying the boundary conditions*

_{x}*k*

_{0y}=

*k*

_{0}sin

*θ*=

*k*′

*, where*

_{y}*θ*is the angle of incidence, the wave vector magnitude is and with the help of (45), the wave vector

**K**′ = (

*K*′

*,*

_{x}*k*

_{0}sin

*θ*, 0) is fully determined.

### 4.1. TE polarization

**f**

_{0}has only

*z*-component,

*f*

_{z0}, i.e.,

**f**

_{0}is perpendicular to the plane of incidence, and therefore, parallel to the separation plane between the vacuum and the material. It is called

*s*-polarization or transverse electric (TE) polarization.

*K*being

_{x}*k*

_{0x},

*K*′

*or*

_{x}*K*″

*for Λ, Λ′ or Λ″ (Ξ, Ξ′ or Ξ″). The continuity of the field components which are parallel to the surface (plane*

_{x}*x*= 0) can be written: or, using Eq. (56) and taking into account that if the material media of the incident electromagnetic wave zone is considered the vacuum, then

*β*″ = 0 =

*β*, where obviously, Ξ′(Λ′) and Ξ″(Λ″) are the corresponding Ξ-parameters for the transmitted and reflected waves. In addition, if one considers that Ξ″ = −Ξ

_{0}, Λ″ = −Λ

_{0}, removes for convenience the subscript 0, and recalls that

*f*= 0 (i.e.,

_{y}**f**perpendicular to the plane of incidence), then the result is: The solutions are and

**f**-vector in function of the wave vector components are the following:

*S*′ ≠ 0 and the incidence is perpendicular to the surface, the result is: and

*There is a rotation of an angle α between the polarization plane of the incident and the reflected wave.*This angle, that depends on

*S*′, can have any value which constitutes a novelty with respect to standard electromagnetism in non birefringent materials

#### Diagonal propagation matrix

*S*′ =

*T*′ = 0, then the electric charges can coexist with magnetic charges but there are not charges with mixed charges, i.e., dyons outside of the axes of the dyonic complex plane, and then we have the following relationships between the electric fields of the different waves, the incident, transmitted and reflected electromagnetic waves: The coefficient

*f*″ /

_{z}*f*will be real if

_{z}*K*′

*/*

_{x}*R*′ is real and then the incident and the reflected wave will have the same phase or 180 degrees different. Assuming that then

*f*″

*/*

_{z}*f*will be real if If in a medium with electric charges, one adds magnetic monopoles and does not consider kinetic susceptibility, then the propagation matrix, Eq. (A. 3), is: where

_{z}*σ*is the electron conductivity and

*σ̄*is the monopole conductivity. Then If magnetic monopoles are not present, this imaginary part never vanishes for a conductive medium, but with magnetic monopoles, this imaginary part vanishes if the following condition is satisfied This equality can be possible for certain value of the parameters (frequency, conductivity, dielectric constant, etc.). Therefore, without monopoles there is always a phase change of the reflected wave, but with monopoles there may not be a phase change for a given angle. Thus the measurement of the phase of the reflected wave can be used to detect magnetic monopoles in material media.

*χ*)(1 +

_{e}*χ*)]

_{m}^{1/2}is the refraction index and

*Z*≡ (

*μ*/

*ɛ*)

^{1/2}the impedance of the medium, and then These relationships about transmitted and reflected fields constitute the standard Fresnel coefficients in a standard non conductive medium (see for instance Fowles [19]).

### 4.2. TM polarization

**b**has only

*z*-component,

*b*, i.e.,

_{z}**b**is perpendicular to the plane of incidence, and parallel to the plane of separation between the vacuum and the medium. It is called

*p*-polarization or transverse magnetic (TM) polarization.

*f*

_{y,z}for

*b*

_{y,z}and vice versa (the value of the constants

*β*, Ξ and Λ are also different).

*x*= 0 plane) can be written: and the solutions are (identically to TE) and Substituting the wave vector components

**b**-field components along with those of Eqs. (63) constitute one of the main effective results of this paper, since they are a distinctive signature of the existence of magnetic monopoles.

#### Diagonal propagation matrix

*U*value is and then, in a non conductive material where

*Z*≡ (

*μ*/

*ɛ*)

^{1/2}the impedance of the medium. They are also the standard Fresnel coefficients in a non-conductive medium:

**f**instead of

**b**, we consider Eq. (76) and therefore The Fresnel coefficients read

## 5. Summary and conclusions

*ζ*≠ 0,

*π*/2, even the plane of polarization of the reflected wave would be modified with respect to the polarization of the incident wave. In any case, we want to emphasize that the most obvious change is the phase of the reflected wave, since, in conducting material media with monopoles, the reflected wave may have the same phase as the incident, situation that never occurs without existence of magnetic monopoles.

## A. Propagation matrix with two classes of particles

*ζ*and

*ζ̄*, then the permeability matrix (33) is and the propagation matrix (39) is written as It should be remarked that this matrix is, in general a non-symmetric operator, (

*S*≠

*T*), because ΘΘ̄ is non-symmetric.

*ζ*

*–*

*ζ̄*=

*±π*/2 (from a duality transformation, there are only electric charges and magnetic monopoles, but there are not dyons with both electric and magnetic charges), then, ΘΘ̄ = 0 and sin

^{2}(

*ζ*

*–*

*ζ̄*) = 1, and the propagation matrix is: that is a symmetric matrix (

*S*=

*T*), and consequently

*K*for a conductor medium, when

*σ̄*=

*ξ̄*= 0, i.e., when there are only electric charges and the corresponding electric conductivity

*σ*.

## References and links

1. | P. Peregrinus, |

2. | P. A. M. Dirac, “Quantized Singularities in the Electromagnetic Field,” Proc. R. Soc. Lond. |

3. | P. A. M. Dirac, “The Theory of Magnetic Poles,” Phys. Rev. |

4. | G. ’t Hooft, “Magnetic monopoles in unified gauge theories,” Nucl. Phys. |

5. | B. Cabrera, “First Results from a Superconductive Detector for Moving Magnetic Monopoles,” Phys. Rev. Lett. |

6. | C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature |

7. | X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, “Inducing a Magnetic Monopole with Topological Surface States,” Science |

8. | S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature |

9. | D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, “Dirac Strings and Magnetic Monopoles in the Spin Ice Dy |

10. | M. J. Gingrass, “Observing monopoles in a magnetic analog of ice,” Science |

11. | S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. |

12. | S. Sondhi, “Wien route to monopoles,” Nature |

13. | L. J. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” J. Chem. Phys. |

14. | See for example, J. D. Jackson, |

15. | K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rep. Prog. Phys. |

16. | J. Costa-Quintana and F. Lopez-Aguilar, “Extended classical electrodynamics with magnetic monopoles,” Far East J. Mech. Eng. Phys. |

17. | J. Costa-Quintana and F. Lopez-Aguilar, “Propagation of electromagnetic waves in material media with magnetic monopoles,” Prog. Electromagn. Res. |

18. | Y.M. Shnir, |

19. | G.R. Fowles, |

**OCIS Codes**

(160.3820) Materials : Magneto-optical materials

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 29, 2010

Revised Manuscript: February 3, 2011

Manuscript Accepted: February 6, 2011

Published: February 11, 2011

**Citation**

J. Costa-Quintana and F. López-Aguilar, "Fresnel coefficients in materials with magnetic monopoles," Opt. Express **19**, 3742-3757 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3742

Sort: Year | Journal | Reset

### References

- P. Peregrinus, The Letter of Petrus Peregrinus, translated by B. Arnold (McGraw Publishing Company, 1904).
- P. A. M. Dirac, "Quantized Singularities in the Electromagnetic Field," Proc. R. Soc. Lond. 133, 60 (1931).
- P. A. M. Dirac, "The Theory of Magnetic Poles," Phys. Rev. 74, 817-830 (1948). [CrossRef]
- G. ’t Hooft, "Magnetic monopoles in unified gauge theories," Nucl. Phys. B 79, 276-284 (1974). [CrossRef]
- B. Cabrera, "First Results from a Superconductive Detector for Moving Magnetic Monopoles," Phys. Rev. Lett. 48, 1378-1381 (1982). [CrossRef]
- C. Castelnovo, R. Moessner, and S. L. Sondhi, "Magnetic monopoles in spin ice," Nature 451, 42-45 (2008). [CrossRef] [PubMed]
- X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009). [CrossRef] [PubMed]
- S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009). [CrossRef] [PubMed]
- D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009). [CrossRef] [PubMed]
- M. J. Gingrass, "Observing monopoles in a magnetic analog of ice," Science 326, 375-376 (2009). [CrossRef]
- S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010). [CrossRef]
- S. Sondhi, "Wien route to monopoles," Nature 461, 888-889 (2009). [CrossRef] [PubMed]
- L. J. Onsager, "Deviations from Ohm’s Law in Weak Electrolytes," J. Chem. Phys. 2, 599-615 (1934). [CrossRef]
- See for example,J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).
- K. A. Milton, "Theoretical and experimental status of magnetic monopoles," Rep. Prog. Phys. 69, 1637-1711 (2006). [CrossRef]
- J. Costa-Quintana, and F. Lopez-Aguilar, "Extended classical electrodynamics with magnetic monopoles," Far East J. Mech. Eng. Phys. 1, 19-56 (2010).
- J. Costa-Quintana, and F. Lopez-Aguilar, "Propagation of electromagnetic waves in material media with magnetic monopoles," Prog. Electromagn. Res. 110, 267-295 (2010). [CrossRef]
- Y. M. Shnir, Magnetic monopoles, (Springer-Verlag, 2005).
- G. R. Fowles, Introduction to Modern Optics, (Dover Publications, Inc., 1975).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.