# Faraday Effect in High- Whispering-Gallery Mode Optical Cavities

**ABSTRACT** The general theory of Faraday rotation in high- Q whispering-gallery (WG) mode optical cavities is investigated. We have derived a matrix equation for the transmitted optical field through a WG mode optical cavity in a time-independent magnetic field. Using the corresponding mathematical framework, we have studied the behavior of the output power spectrum as a function of the magnetic field strength for a typical WG mode optical cavity. The effects of input polarization, optical quality factor, and cavity diameter on the magnetic response are studied.

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**ABSTRACT:**We have measured the Faraday effect in silica standard optical fibers in the wavelength range 458-1523 nm. An effective Verdet constant Vef that exhibits a linear dependence on the square of the optical frequency ν is defined: V(ef) = (0.142 ± 0.004) × 10(-28) ν(2) rad T(-1) m(-1). We demonstrate that the negative effects of a small linear birefringence can be minimized by adjustment of the input polarization to an optimum state.Applied Optics 02/1996; 35(6):922-7. · 1.69 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**First Page of the ArticleIEEE Journal of Quantum Electronics 10/1979; · 2.11 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Electromagnetic wave propagation through materials that possess both Faraday rotation and birefringence is analyzed. A matrix equation is developed which relates the amplitude and relative phase of the electric vectors between any two points along the propagation direction. It is shown that the presence of birefringence can drastically affect the behavior of wave propagation and that it is considerably different from pure Faraday rotation. Methods of measuring the material parameters are also described. Criteria for viewing domains in this type of material are established. It is shown that the thickness of the sample plays a great role in determining the contrast between domains and at some thicknesses no contrast at all can be obtained. It is also shown that the method using elliptical analyzers gives greater contrast over the plane analyzers. Photographs of domain patterns in a wedge of ytterbium orthoferrite are presented and they verify the calculated results.Journal of Applied Physics 07/1969; · 2.19 Impact Factor

Page 1

Faraday Effect in High-Q Whispering-Gallery

Mode Optical Cavities

Volume 3, Number 5, October 2011

Shoufeng Lan

Mani Hossein-Zadeh

DOI: 10.1109/JPHOT.2011.2168514

1943-0655/$26.00 ©2011 IEEE

Page 2

Faraday Effect in High-Q

Whispering-Gallery Mode

Optical Cavities

Shoufeng Lan and Mani Hossein-Zadeh

Department of Electrical and Computer Engineering, Center for High Technology

Materials (CHTM), University of New Mexico, Albuquerque, NM 87131 USA

DOI: 10.1109/JPHOT.2011.2168514

1943-0655/$26.00 ?2011 IEEE

Manuscript received July 26, 2011; revised September 3, 2011; accepted September 6, 2011. Date

of publication September 19, 2011; date of current version October 4, 2011. This work was

supported by the Optoelectronic Research Center, Center for High Technology Materials, University

of New Mexico under Grant FA9550-09-1-0202 from the Air Force Office of Scientific Research.

Corresponding author: M. Hossein-Zadeh (e-mail: mhz@chtm.unm.edu).

Abstract: The general theory of Faraday rotation in high-Q whispering-gallery (WG) mode

optical cavities is investigated. We have derived a matrix equation for the transmitted optical

field through a WG mode optical cavity in a time-independent magnetic field. Using the

corresponding mathematical framework, we have studied the behavior of the output power

spectrum as a function of the magnetic field strength for a typical WG mode optical cavity.

The effects of input polarization, optical quality factor, and cavity diameter on the magnetic

response are studied.

Index Terms: Optical cavities, nonlinear optical effects, light-material interactions, theory.

1. Introduction

High-quality factor (high-Q) optical Whispering-Gallery (WG) microcavities have been the subject of

intensive research during the past decade [1]. The small mode volume, large resonant power build-

up, long photon lifetime, and ultranarrow bandwidth have made these cavities the ideal platforms for

studying nonlinear optical effects [2]–[4] and detection of small changes in the effective refractive

index (due to presence of external particles or fields in the vicinity of the optical mode) [5]. Using

these properties low threshold lasers, high-sensitivity molecule detectors, and electrooptic

modulators have been demonstrated [6]–[8].

A well-known optical phenomenon is the Faraday effect or polarization rotation due to a

longitudinal external magnetic field. Although the Faraday effect has been studied in a large

variety of different configurations [9]–[11], its study in optical resonators has been mainly limited to

Fabry–Perot (FP) type optical cavities [12], [13]. It has been shown that by placing the Faraday

medium in an optical resonator, one can amplify the Faraday rotation by resonant enhancement of

the interaction length. In an alternative approach nonresonant fiber coils have been used in order

to reduce the volume of the Faraday rotator without sacrificing the interaction length [14]. In these

structures, magnetic field interacts with the optical beam propagating along a helical fiber

waveguide. Given the high-Q, compactness, and simple structure of WG optical microresonators,

it seems natural to consider them as sensitive Faraday rotators as well. The long photon lifetime in

these cavities results in a large interaction length between the optical wave and the magnetic field.

In this case, the optical resonator is also the Faraday medium as opposed to large FP cavities

Vol. 3, No. 5, October 2011Page 872

IEEE Photonics JournalFaraday Effect in High-Q WG Optical Cavities

Page 3

where the Faraday medium is placed inside the cavity (formed by mirrors) and only partially

overlaps the optical path.

Previously, the polarization conversion in ring resonators and WG microcavities was

investigated in spherical WG mode optical microcavities mainly in the absence of a constant

magnetic field [15], [16]. In [17], using quantum theory and the Hamiltonian approach, it was

shown that an alternating magnetic field with a frequency close to the difference between the

frequencies of the transverse electric (TE) and transverse magnetic (TM) modes can alter the

population difference between resonant TE and TM photons in an optical cavity. However, to our

knowledge, a comprehensive study of the transmitted power spectrum of a waveguide coupled

circular optical resonator in the presence of constant magnetic field does not exist. Here, we

investigate the Faraday rotation in WG-mode optical cavities using a theoretical framework that

combines the Faraday rotation, birefringence, and optical resonance in the context of classical

electromagnetic theory. Using this model, we explore the magnetic sensitivity of the transmitted

optical power through a waveguide coupled with a silica sphere (as a typical WG optical cavity). In

the presence of a time-independent magnetic field, we study the behavior of the optical output

spectrum from a WG microresonator as a function of input polarization, optical quality factor,

diameter, and magnetic field strength using silica microsphere as the main platform. We identify

two operational regimes that can be used for magnetic field (B-field) sensing applications. Our

study shows that even in high-Q WG-mode cavities, the sensitivity of the optical spectrum to the

magnetic field is significantly reduced due to the large magnitude of geometrical birefringence. A

previously shown fiber coil optical B-field sensor also suffers from the same problem (birefringence

induced by fiber bending), but matching the birefringence half-beat-length to the circumference of the

coilsolvestheproblem[14].WeshowthatinaWG-modeopticalcavity,thepolarization birefringence

half-beat-length cannot be matched to the circumference of cavity. In spite of this problem, using a

mixed TE–TM input polarization, the appropriate level of sensitivity for certain applications can be

obtained, even in a high-Q spherical optical cavity made of silica (that has a small Verdet constant).

Clearly, fabrication of high-Q WG mode cavities using materials with large Verdet constant can

significantly improve the sensitivity. Moreover, using naturally birefringent materials can counterbal-

ance the geometrically induced birefringence resulting in WG cavities with very small birefringence.

Thisisapreliminarystudybasedonsilicasphere;however,themodelandoutcomesmayinitiatenew

directions in the WG optical cavity application toward all-dielectric, compact, and sensitive magnetic

field sensors that are immune to electromagnetic interference. Beyond its possible application for

magnetic field sensing, the Faraday rotation in WG mode microcavities provides a means for

controlling the circulating optical field using an external field. This is an important degree of freedom

specially for high-Q optical microcavities made of nonelectrooptic materials that are insensitive to

electric fields.

2. Theory

Using the general theory of electromagnetic propagation in the presence of Faraday rotation and

birefringence[18]–[21] combinedwiththetheoryofwaveguide-resonator system[22]–[24], wederive

a general relation between the input and transmitted optical field components for a waveguide

coupledWGcavityinatime-independentmagneticfield.Usingthistheoreticalframework,weidentify

two specific input polarizations that can translate the presence of an external magnetic field to optical

out power variations.

2.1. General

Fig. 1 shows the schematic diagram of a WG optical cavity side-coupled to a waveguide. The

constant magnetic field can be uniform ðBPÞ or a circular field ðBCÞ that is always tangential to the

circulatingopticalpath(forexample,thefieldgeneratedbyawireperpendiculartotheWGmodeplane

and passing through the resonator center). ? and t are the optical field coupling and transmission

IEEE Photonics JournalFaraday Effect in High-Q WG Optical Cavities

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Page 4

coefficientsforthecouplingjunctionbetweenthecavityandthewaveguide.Here,weassumethatthe

coupling is lossless. We define

? ¼4?2neffR

?

?

a ¼exp

?2neff?2R

?Q0

?

astheroundtripopticalphaseshiftandroundtriplossfactor,respectively.neffistheeffectiverefractive

indexofthecirculatingopticalwave,?isthelaserwavelength,andQ0istheintrinsic(unloaded)optical

quality factor of the cavity. In the absence of birefringence and Faraday rotation, the transmitted

electric field ðETÞ can be written as [23]

ET¼?a þ te?i?

Note that even if the resonator is made of isotropic material (with a single bulk refractive index), the

geometrically induced birefringence in a WG resonator results in a different round trip optical phase

for the TE and TM polarized modes [24]. The geometrical birefringence that is caused by the curved

boundary condition between the cavity and the surrounding medium may be quantified as

?ng¼ neff;TE? neff;TM, where neff;TEand neff;TMare the effective optical refractive index of the TE

and TM polarized WG modes. ?ng can be calculated by solving the Maxwell equations and the

boundary conditions associated with a spherical dielectric cavity [24]. In the absence of a coupling

mechanism between TE and TM polarized modes, (1) is valid for both TE and TM (with different neffs)

and results in independent resonant dips in the optical transmission spectrum for each polarization

state.InthepresenceofmechanismssuchastheFaradayeffectthatcanrotatethepolarizationofthe

optical E-field vector inside the resonator and, therefore, couple TE and TM modes, the situation is

more complicated, and a matrix treatment is required. In what follows, we use the matrix equation

developed for nonresonant Faraday rotation in birefringent crystals [18]–[21] to derive a general

matrix equation for the optical wave circulating in WG optical cavity.

?at?þ e?i?:

(1)

2.1.1. Input With Arbitrary Polarization

In the presence of magnetic field and, therefore, Faraday effect, the propagation of the x (TE) and y

(TM) components of the optical E-field inside the cavity can be described by a matrix equation [14],

[18]–[21]. In a traveling wave circular optical resonator (with a radius R), the matrix equation can be

usedtocalculatetheroundtripevolutionoftheresonantopticalfields.Intheabsenceofloss,therelation

between amplitude and relative phase for the mth round trip propagation starting from the coupling

junction can be written as

?

Ex

Ey

m

m

?

¼ Mð?Þ½?

Ex

Ey

m?1

m?1

??

(2)

Fig. 1. Schematic diagram of a WG mode optical cavity side coupled to a waveguide.

IEEE Photonics JournalFaraday Effect in High-Q WG Optical Cavities

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Page 5

where

Mð?Þ

? ?2

½ ? ¼

cosð?Þ ? i??

2F

?sinð?Þ

??

2

?sinð?Þ

?2F

?sinð?Þ

cosð?Þ þ i??

?sinð?Þ

?ðnTE? nTMÞ ¼2?

"

?

#

?

2

¼

?2

þF2

? ¼ ??R

?? ¼2?

??ng:

?? represents the birefringence in the optical cavity (in the case of isotropic medium geometrically

or bending induced). F is the Faraday rotation angle per unit length of the optical path. For a circular

magnetic field (BCin Fig. 1) that is always tangential to the optical path, F ¼ BC:V. For a uniform

field parallel to the waveguide (BPin Fig. 1), F ¼ Bp:V:Cosðs=RÞ, where s is the arc length from the

point where the magnetic field is parallel to the optical propagation direction [14]. For the sake of

simplicity, here, we consider F ¼ B:V for both cases where B is the effective average magnetic field

parallel to pointing vector of the circulating optical field (WG mode). For the circular magnetic field,

B ¼ BC, while for a uniform magnetic field, B ¼ 2BP=? or the integral over tangential B-field

component around the circle. (This is applicable to the general case of the uniform field parallel to

the waveguide-resonator plane and not necessarily the waveguide itself.) In our calculations, we

study the behavior of the optical transmission spectrum as a function of B, which can be applied to

both cases. Using (2), it can be shown that after m roundtrip, the input and output-polarization states

are related through

Ex

Ey

m

m

??

¼ Mðm?Þ½?

Ex

Ey

0

0

??

(3)

or

Ex

m¼Ex

0cosðm?Þ ? i??

?

?sinðm?Þ

??

?

? Ey

0

2F

?sinð?Þ

2F

?sinð?Þ:

(4)

Ey

m¼Ey

0cosðm?Þ þ i??

?sinðm?Þþ Ex

0

(5)

Note that the above equations do not include the net propagation phase shifts. To obtain the net

propagation phase shift for each component, we have to multiply (4) and (5) by a constant

phase factor corresponding to the roundtrip phase shift. For a ring resonator, the Ex(TE) and

Ey(TM) components of the total transmitted electric field through the coupling junction can be

written as

Ex

Ey

out¼tEx

out¼tEy

0þ a?ð???Þexpð?i?ÞEx

0þ a?ð???Þexpð?i?ÞEy

1þ a2ðt?Þ?ð???Þexpð?i2?ÞEx

1þ a2ðt?Þ?ð???Þexpð?i2?ÞEy

2...

(6-a)

2...

(6-b)

where ? ¼ 4n?2R=?, a ¼ expð?2n?2R=?Q0Þ, and n ¼ ðnTEþ nTMÞ=2.

For infinite number of roundtrips (resonant build-up), (6-a) and (6-b) converge into finite

values, and a closed-form matrix equation can be obtained for the total transmitted elec-

tric field

h

?j?j22F

Ex

Ey

out

out

??

¼

t ? j?j2fð?;?Þ ? i??

?gð?;?Þ

i

j?j22F

h

?gð?;?Þ

?gð?;?Þ

t ? j?j2fð?;?Þ þ i??

?gð?;?Þ

i

2

4

3

5Ex

0

Ey

0

??

(7)

IEEE Photonics JournalFaraday Effect in High-Q WG Optical Cavities

Vol. 3, No. 5, October 2011Page 875

Page 6

where

fð?;?Þ ¼

ae?i?cosð?Þ ? a2e?i2?

ðae?i?t?Þ2? 2ae?i2?t?cosð?Þ þ 1

ae?i?sinð?Þ

ðae?i?t?Þ2? 2ae?i2?t?cosð?Þ þ 1:

gð?;?Þ ¼

2.1.2. Special Case: X-Polarized Input

When the input electric field is linearly polarized along the x-direction E0¼ ½1;0?T, (3)–(5) can be

simplified as

?

Ex

?sinðm?Þ

Ex

Ey

m

m

?

m¼cosðm?Þ ? i??

¼ Mðm?Þ½?

1

0

? ?

Ey

m¼2F

?sinðm?Þ

and the total transmitted electric fields can be written as

Ex

out¼t þ ? ? ð???Þ

X

amðt?Þðm?1Þe?im?2F

1

m¼1

amðt?Þðm?1Þe?im?cosðm?Þ ? i??

?

?sinðm?Þ

??

Ey

out¼? ? ð???Þ

X

1

m¼1

?sinðm?Þ

?

or

Ex

out¼t ?1

2aj?j2

1 ? i??

1 ? at?e?ið???Þ

e?ið???Þ

1 ? at?e?ið???Þ?

?

??

e?ið???Þ

þ

1 þ i??

1 ? at?e?ið?þ?Þ

e?ið?þ?Þ

1 ? at?e?ið?þ?Þ

?

??

e?ið?þ?Þ

0

@

1

A

(8-a)

Ey

out¼?F

?

aj?j2

!

:

(8-b)

3. Results

Equations (7) and (8) can be used to study the behavior of output optical E-field components

as a function of the external magnetic field and optical cavity parameters (quality factor,

diameter). In our study, we use spherical silica optical WG cavities with a Verdet constant of

V ¼ 5:6 ? 10?5degree/G.m. The theoretical outcomes are in good agreement with previous

experimental results for B ¼ 0 ½15;16?.

3.1. TE-Polarized Input

If the input is polarized along the x-direction (TE polarized), the output optical field can be

calculated using (8-a) and (8-b). To understand the general behavior of the WG modes in the

presence of an external magnetic field, the optical output spectrum is first evaluated for relatively

large magnetic fields and for a spherical silica cavity with a radius of 0.5 mm. Fig. 2(a) shows the

transmission spectrum of the x and y-polarized output intensities (Ix and Iy) normalized to input

intensity for magnetic field strengths B ¼ 0, 5 ? 105and 8 ? 105Gauss. The B-field is uniform and

parallel to the waveguide that feeds the resonator. In the presence of the Faraday rotation, in

addition to the primary resonant dip at ?0, a secondary resonant dip appears at ?0þ ??B in the

transmission spectrum of the x-polarized component. Meanwhile, two peaks (at ?0and ?0þ ??B)

with almost equal magnitudes ð?Iy;maxÞ appear in the transmission spectrum of the y-polarized

component (note that Iyis zero in the absence of the B-field). Ideally, ??Band ?Iy;maxcan be used

for B-field sensing applications or controlling the circulating optical power by an external field.

IEEE Photonics Journal Faraday Effect in High-Q WG Optical Cavities

Vol. 3, No. 5, October 2011Page 876

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However, in the presence of birefringence, the sensitivity to the magnetic field is relatively small. In

Fig. 2(a), the value of geometrically induced birefringence ð?ngÞ has been estimated using the

characteristic equations for TE and TM polarized WG modes in a spherical resonator [24]. To show

the impact of the magnitude of ?ngon B-field sensitivity, in Fig. 2(b), the normalized spectrum of

Ix and Iy are plotted for a hypothetical case where ?ng¼ 0. As is evident from the spectrums if

?ng¼ 0, the original resonance symmetrically splits into two resonant transmission dips, and the

presence of the magnetic field strongly modifies the y-polarized component. When ?ng¼ 0 at

B ¼ 5 ? 105G, ?Iy;max is saturated ð¼1Þ, while for the actual value of ?ng¼ 3:54 ? 10?4,

?Iy;max¼ 0:0015. More details about the impact of ?ngand its dependence on sphere radius are

presented in Section 3.3.

The magnitude of ?Iy;max also depends on optical coupling factor and is maximized in the

overcoupled regime. We found that for a silica sphere with a radius of 0.5 mm, a coupling coefficient

ð?Þ of 0.141 (corresponding to t ¼ 0:99) results in the largest ?Iy;max. Fig. 3(a) shows ?Iy;maxplotted

against the B-field for different values of intrinsic quality factor ðQ0Þ and t ¼ 0:99. In Fig. 3(b), ?Iy;max

is plotted against Q0for ?ng¼ 0 and ?ng¼ 3:54 ? 10?4to show the saturation behavior, as well

as the impact of a nonzero ?ng. As expected, a larger Q0results in a larger ?Iy;maxand eventually

reaches a saturated value for a given B-field strength. For a given value of Q0, ?Iy;maxalso saturates

at large B-field strengths, as shown in Fig. 3(c).

3.2. Mixed Polarization Input

If both x and y components of the input optical field are nonzero (but in-phase), the behavior of

the transmission spectrum is more complicated. Fig. 4(a)–(e) shows the optical transmission

spectrum of the y-component intensity for different polarization angles ð?Þ of the input optical field

measured relative to the x-axis. As the polarization angle of the input optical field changes from

0 ðE0y¼ 0Þ to larger values ðE0x6¼ 0;E0y6¼ 0Þ, the background transmission of the y-polarization

grows; the primary transmission peak turns into a dip-to-peak transition, and the secondary-peak

turns into a dip.

Here, we define ?Iy;max as the maximum normalized output intensity change of the y-

component after the B-field is applied. In Fig. 4(f), ?Iy;max is plotted against polarization angle,

showing a maximum at 45?. A comparison between ?Iy;maxwith ? ¼ 45?, and ?Iy;max(x-polarized

input) for a given B-field strength shows that ?Iy;maxis two to three orders of magnitude larger and

is therefore a better parameter for B-field sensing applications. Therefore, using a mixed polarized

optical input field with E0x¼ E0y¼ E0=p2, the output intensity will have the maximum sensitivity to

B-field magnitude. Fig. 5(a) shows ?Iy;max (for ? ¼ 45?) plotted against the B-field for different

Fig. 2. Normalizedtransmittedpowerspectrumthroughawaveguidecoupledtoasilicasphericalcavityfor

TE ðxÞ and TM ðyÞ polarization. (a) For the real case where ?ng¼ 3:54 ? 10?4. (b) In the absence of

birefringence,?ng¼ 0(hypotheticalcase),Here,thesilicasphereradiusðRÞ ¼ 0:5 mm,thetransmission

coefficient ðtÞ ¼ 0:95, neff¼ 1:4317, Q0¼ 1 ? 107, and Verdet constant ðVÞ ¼ 5:6 ? 10?5degree/G.m.

IEEE Photonics JournalFaraday Effect in High-Q WG Optical Cavities

Vol. 3, No. 5, October 2011Page 877

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values of intrinsic quality factor ðQ0Þ and t ¼ 0:99. In Fig. 5(b), ?Iy;max is plotted against Q0 for

?ng¼ 0 and ?ng¼ 3:54 ? 10?4to show the saturation behavior, as well as the impact of a

nonzero ?ng. Similar to the x-polarized case, ?Iy;max saturates around Q0¼ 108. For a given

value of Q0, ?Iy;maxsaturates at large B-field strengths, as shown in Fig. 5(c).

3.3. Impact of Geometrically Induced Birefringence

As shown in Sections 3.1 and 3.2, independent of input polarization, the geometrically induced

birefringence significantly quenches the Faraday rotation and the corresponding changes in Iy

spectrum similar to what has been observed in the context of fibers and fiber coils [11], [14].

Fig. 6(a) shows both ?Iy;maxand ?Iy;maxrapidly decay as ?ngincreases. In Fig. 6(b), ?ngis plotted

against radius for a silica spherical optical cavity. As is evident from the graph, although the obvious

advantage of high-Q microcavities is their small size, there is still a tradeoff between size and

sensitivity mainly due to bending induced geometrical birefringence ð?ngÞ. Using the information in

Fig. 6(a) and (b), one can estimate the minimum radius of the cavity based on the photodetector

sensitivity and the magnitude of the magnetic field under test.

In Fig. 6(c), we have calculated the minimum detectable B-field as a function of silica sphere

radius for 1-mW input power polarized at ? ¼ 45?and a photodetector with minimum sensitivity of

Fig. 4. (a)–(e) Normalized transmission spectrum of the y-component intensity ðIyÞ in the presence and

absence of the B-field for nonzero E0x and E0y (the input is linearly polarized at an arbitrary angle

between 15 and 75?relative to the TE polarization). ?Iy;maxplotted against B-field input polarization

angle. Here, R ¼ 0:5 mm, ?ng¼ 3:54 ? 10?4, Q0¼ 107, and t ¼ 0:99.

Fig. 3. (a) ?Iy:max [defined in Fig. 2(a) and (b)] plotted as a function of external magnetic field for

?n ¼ 3:54 ? 10?4. (b) ?Iy:maxplotted against intrinsic optical quality factor ðQ0Þ at B ¼ 3900 G. (c) ?Iy;max

plotted against the B-field for Q0¼ 1 ? 107. Here, the silica sphere radius is 0.5 mm, and t ¼ 0:99. Note

that ?Iy:maxis always normalized to the input optical power.

IEEE Photonics Journal Faraday Effect in High-Q WG Optical Cavities

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1 nW. Note that in a fiber coil, the bending radius can be chosen such that the birefringence half-

beat length ðLp=2 ¼ ?=2?ngÞ is matched to half-circumference (Lp=2 ¼ m?R or R ¼ ?=2? m?ng,

where m is an integer) that results in a large interaction length, in spite of a nonzero ?ng[14]. In a

WG optical cavity, however, the dependence of the geometrical birefringence (and, therefore, half-

beat length) on the sphere circumference cannot satisfy the matching condition. Our calculation

shows for a typical silica sphere with a radius between 0.1 to 5 mm, 1 G Lp=2 G 1:4 ? ?R.

4. Conclusion

We have studied the Faraday rotation in WG optical cavities. The outcomes of the simple

theoretical model presented here clearly show that high-Q optical WG resonance can improve the

efficiency of Faraday rotation in a small volume. However, the amplifying effect of long interaction

length (due to long photon life) is significantly reduced by the geometrically induced birefringence.

To overcome the limitation imposed by birefringence, one can use using larger spherical or disk-

shaped cavities. Although fabricating small microspherical cavities with Q09 109is relatively easy,

obtaining optical quality factors in large spherical cavities requires special fabrication techniques.

One possibility is to reduce the index contrast between the cavity and the surrounding medium

(which will reduce ?ng) by embedding the cavity in a medium with a refractive index larger than 1.

Another solution is fabricating the WG cavity using naturally birefringent materials that can

counterbalance ?ng and reduce the total by a natural ?ntot¼ ?ng? ?nb (?nb: natural

birefringence). Note that as with any optical system, here, the B-field sensitivity can be improved

by increasing the optical input power and the sensitivity of the photodetector. However, in this work,

we have targeted a low-power and low-cost optical B-field sensor with the smallest possible

volume.

Fig. 6. (a)?Iy;max(at ? ¼ 45?) and ?Iy;maxplotted against geometrically induced birefringence ð?ngÞ for

R ¼ 0:5 mm, t ¼ 0:99, and Q0¼ 108. (b) ?ngplotted against radius for a silica spherical optical cavity.

(c) Minimum detectable B-field as a function of silica sphere radius for 1 mW input power and a

photodetector with minimum sensitivity of 1 nW. The intrinsic quality factor ðQ0Þ ¼ 108, t ¼ 0:99, and the

input polarization is 45?.

Fig. 5. (a) ?Iy;max (defined in Fig. 4) for ? ¼ 45?plotted as a function of external magnetic field for

?n ¼ 3:54 ? 10?4. (b) ?Iy;maxplotted against intrinsic optical quality factor ðQ0Þ at B ¼ 3900 G. In all

cases, silica sphere radius R ¼ 0:5 mm, intrinsic quality factor Q0¼ 1 ? 107, and t ¼ 0:99. (c) ?Iy;max

plotted against B-field for Q0¼ 1 ? 107. Here, the silica sphere radius is 0.5 mm, and t ¼ 0:99. Note

that ?Iy:maxis always normalized to the input optical power.

IEEE Photonics Journal Faraday Effect in High-Q WG Optical Cavities

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Page 10

Acknowledgment

S. Lan would like to thank F. Liu and M. Shirazi for useful discussions.

References

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IEEE Photonics Journal Faraday Effect in High-Q WG Optical Cavities

Vol. 3, No. 5, October 2011 Page 880

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