**4. Measuring the parameters of the PPM electrodynamics**

There are four parameters *η1*, *η2, η3*, and ξ in PPM electrodynamics to be measured by experiments. For the QED (Quantum Electrodynamics) corrections to classical electrodynamics, *η1* = α/(45π) = 5.1x10-5, *η*2 = 7α/(180π) = 9.0 x10-5, *η3* = 0, and ξ = 0. There are three vacuum birefringence experiments on going in the world to measure this QED vacuum birefringence – the BMV experiment (Battesti et al., 2008), the PVLAS experiment (Zavattini et al.. 2008) and the Q & A experiment (Chen et al., 2007; Mei et al., 2010). The birefringence Δ*n* in the QED vacuum birefringence in a magnetic field **B**ext is

$$
\Delta \mathbf{u} = \boldsymbol{n}\_{\parallel} \cdot \boldsymbol{n} \omega = 4.0 \times 10^{\cdot 24} \,\mathrm{(B^{ext}/1T)^2}.\tag{67}
$$

For 2.3 T field of the Q & A rotating permanent magnet, Δ*n* is 2.1 x 10-23. This is about the same order of magnitude change in fractional optical path-length that ground interferometers for gravitational-wave detection aim at. Quite a lot of techniques developed in the gravitational-wave detection community are readily applicable for vacuum birefringence detection (Ni et al., 1991).

Fig. 2. Principle of vacuum dichroism and birefringence measurement.

The basic principle of these experimental measurements is shown as Fig. 2. The laser light goes through a polarizer and becomes polarized. This polarized light goes through a region of magnetic field. Its polarization status is subsequently analyzed by the analyzer-detector subsystem to extract the polarization effect imprinted in the region of the magnetic field. Since the polarization effect of vacuum birefringence in the magnetic field that can be produced on earth is extremely small, one has to multiply the optical pass through the magnetic field by using reflections or Fabry-Perot cavities. An already performed experiment, the BFRT experiment (Cameron et al., 1993) used multiple reflections; PVLAS, Q & A, BMV experiments all use Fabry-Perot cavities. For polarization experiment, Fabry-Perot cavity has the advantage of normal incidence of laser light which suppressed the part of polarization due to slant angle of reflections. With Fabry-Perot cavity, one needs to control the laser frequency and/or the cavity length so that the cavity is in resonance. With a finesse of 30,000, the resonant width (FWHM) is 17.7 pm for light with 1064 nm wavelength; when rms cavity length control is 10 % of this width, the precision would be 2.1 pm. Hence, one needs a feedback mechanism to lock the cavity to the laser or vice versa. For this, a commonly used scheme is Pound-Drever-Hall method (Drever et al., 1983). Vibration introduces noises in the Fabry-Perot cavity mirrors and hence, in the light intensity and light polarization transmitted through the Fabry-Perot cavity. Since the analyzer-detector subsystem detects light intensity to deduce the polarization effect, both intensity noise and polarization noise will contribute to the measurement results. Gravitational-wave community has a long-standing R & D on this. We benefit from their research advancements.

52 Trends in Electromagnetism – From Fundamentals to Applications

There are four parameters *η1*, *η2, η3*, and ξ in PPM electrodynamics to be measured by experiments. For the QED (Quantum Electrodynamics) corrections to classical electrodynamics, *η1* = α/(45π) = 5.1x10-5, *η*2 = 7α/(180π) = 9.0 x10-5, *η3* = 0, and ξ = 0. There are three vacuum birefringence experiments on going in the world to measure this QED vacuum birefringence – the BMV experiment (Battesti et al., 2008), the PVLAS experiment (Zavattini et al.. 2008) and the Q & A experiment (Chen et al., 2007; Mei et al., 2010). The

For 2.3 T field of the Q & A rotating permanent magnet, Δ*n* is 2.1 x 10-23. This is about the same order of magnitude change in fractional optical path-length that ground interferometers for gravitational-wave detection aim at. Quite a lot of techniques developed in the gravitational-wave detection community are readily applicable for vacuum

The basic principle of these experimental measurements is shown as Fig. 2. The laser light goes through a polarizer and becomes polarized. This polarized light goes through a region of magnetic field. Its polarization status is subsequently analyzed by the analyzer-detector subsystem to extract the polarization effect imprinted in the region of the magnetic field. Since the polarization effect of vacuum birefringence in the magnetic field that can be produced on earth is extremely small, one has to multiply the optical pass through the magnetic field by using reflections or Fabry-Perot cavities. An already performed experiment, the BFRT experiment (Cameron et al., 1993) used multiple reflections; PVLAS, Q & A, BMV experiments all use Fabry-Perot cavities. For polarization experiment, Fabry-Perot cavity has the advantage of normal incidence of laser light which suppressed the part of polarization due to slant angle of reflections. With Fabry-Perot cavity, one needs to control the laser frequency and/or the cavity length so that the cavity is in resonance. With a finesse of 30,000, the resonant width (FWHM) is 17.7 pm for light with 1064 nm wavelength; when rms cavity length control is 10 % of this width, the precision would be 2.1 pm. Hence, one needs a feedback mechanism to lock the cavity to the laser or vice versa. For this, a commonly used scheme is Pound-Drever-Hall method (Drever et al., 1983). Vibration introduces noises in the Fabry-Perot cavity mirrors and hence, in the light intensity and light polarization transmitted through the Fabry-Perot cavity. Since the analyzer-detector subsystem detects light intensity to deduce the polarization effect, both intensity noise and polarization noise will contribute to the measurement results. Gravitational-wave community has a long-standing R & D on this. We benefit from their research

Δ*n* = *n*║ - *n*┴ = 4.0 x 10-24 (**B**ext/1T)2. (67)

**4. Measuring the parameters of the PPM electrodynamics** 

birefringence Δ*n* in the QED vacuum birefringence in a magnetic field **B**ext is

Fig. 2. Principle of vacuum dichroism and birefringence measurement.

birefringence detection (Ni et al., 1991).

advancements.

Now we illustrate with our Q & A experiment. Since 1991 we have worked on precision interferometry --- laser stabilization schemes, laser metrology and Fabry-Perot interferometers. With these experiences, we started in 1994 to build a 3.5 m prototype interferometer for measuring vacuum birefringence and improving the sensitivity of axion search as part of our continuing effort in precision interferometry. In 2002, we finished Phase I of constructing the 3.5 m prototype interferometer and made some Cotton-Mouton coefficient and Verdet coefficient measurements with a 1T electromagnet (Wu et al., 2002). The two vacuum tanks shown on the left photo of Fig. 3 house the two 5 cm-diameter Pabry-Perot mirrors with their suspensions; the 1T electromagnet had been in place of permanent magnet in the middle of the photo.

Fig. 3. Photo on the left-hand side shows the Q & A apparatus for Phase II experiment; photo on the right-hand side shows the Q & A apparatus for Phase III experiment.

Starting 2002, we had been in Phase II of Q & A experiment until 2008. The results of Phase II on dichroism and Cotton-Mouton effect measurement had been reported (Chen et al., 2007; Mei et al. 2009). At the end of Phase II, our sensitivity was still short from detection of QED vacuum birefringence by 3 orders of magnitude; so was the PVLAS experiment and had been the BFRT experiment. In 2009, we started Phase III of the Q & A experiment to extend the 3.5 m interferometer to 7 m with various upgrades. Photo on the left of Fig. 3 shows the apparatus for Phase II; photo on the right side of Fig. 3 shows the apparatus for Phase III, with the big (front) tank moved further to the front (out of the photo). The laser has been changed to 532 nm wavelength and is located next and beyond the front tank. We have installed a new 1.8 m 2.3 T permanent magnet (in the middle to bottom of right side photo) capable of rotation up to 13 cycles per second to enhance the physical effects. We are working with 532 nm Nd:YAG laser as light source with cavity finesse around 100,000, and aim at 10 nrad(Hz)-1/2 optical sensitivity. With all these achieved and the upgrading of vacuum, for a period of 50 days (with duty cycle around 78 % as performed before) the vacuum birefringence measurement would be improved in precision by 3-4 orders of magnitude, and QED birefringence would be measured to 28 % (Mei et al., 2010). To enhance the physical effects further, another 1.8 m magnet will be added in the future.

All three ongoing experiments – PVLAS, Q &A, and BMV – are measuring the birefringence Δ*n*, and hence, *η1−η*2 in case *η3* is assumed to be zero. To measure *η1* and *η2* separately, one-

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 55

To measure *η1, η2* and *η3*, we could do the following three experiments to determine them:

by a strong microwave cavity or wave guide to determine *η1-η2*; (ii) to measure the

by a strong magnet to determine *η3* with *η1-η2* determined by (i); (iii) to measure *η1* and *η<sup>2</sup>* separately using two-arm interferometer with the paths in two arms in magnetic fields with

As to the term *ξΦ* and parameter *ξ*, it does not give any change in the index of refraction. However, as we will see in section 7 and section 8, it gives a polarization rotation and the effect can be measured though observations with astrophysical and cosmological

In the earth laboratory, where variation of gravity is small, we can use standard Maxwell equations together with Lorentz force law for ordinary measurements and experiments. However, in precision experiments on earth, in space, in the astrophysical situation or in the cosmological setting, the gravity plays an important role and is non-negligible. In the remaining part of this chapter, we address to the issue of electromagnetism in gravity and more empirical tests of electromagnetism and special relativity. The standard way of including gravitational effects in electromagnetism is to use the comma-goes-to-semicolon rule, i. e., the principle of equivalence (the minimal coupling rule). This is the essence of Einstein Equivalence Principle (EEP) which states that everywhere in the 4-dimensional spacetime, locally, the physics is that of special relativity. This guarantees that the 4 dimensional geometry can be described by a metric *gij* which can be transformed into the Minkowski metric locally. In curved spacetime, *ηij* is replaced by *gij* with partial derivative (comma) replaced by the covariant derivative in the *gij* metric (semi-colon) in the Lagrangian density for a system of charged particles. When this is done the Lagrangian

*LI = -* (*1*/(*16*π))*χGRijkl Fij Fkl - Ak jk* (*-g*)(1/2) *-* Σ*I mI* (*dsI*)/(*dt*) *δ*(*x*-*xI*), (79)

and *g* is the determinant of *gij*. In general relativity or metric theories of gravity where EEP

are the metric tensor in their respective frames. According to the definition of locally inertial frame, we have *gAB* = *ηAB* + O((Δ*xC*)2). Therefore, in the locally inertial system near *P*, special relativity holds up to the curvature ambiguity, and the definition of rods and clocks is the same as in the special relativity including local quantum mechanics and

} is an arbitrary coordinate system, {*xA*} is a locally inertial frame, and *gij* & *gAB*

*χGRijkl =* (*-g*)1/2 [(1/2) *gik gjl -* (1/2) *gil gkj*], (80)

= *gAB dxA dxB* = [*ηAB* + O((Δ*xC*)2)] *dxA dxB*, (81)

**5. Electromagnetism in curved spacetime and the Einstein equivalence** 

where the GR (General Relativity) constitutive tensor *χGRijkl* is given by

holds, the line element near a world point (event) *P* is given by

*-*2 of light with the external field provided

*-*2 of light with the external magnetic field provided

(i) to measure the birefringence Δ*n =* 2(*η1-η2*)*B*2*Bc*

different strengths (or one with no magnetic field).

birefringence Δ*n =* 2[(*η1-η2*)2*+η3*2]1/2*B*2*Bc*

propagation of electromagnetic waves.

**principle** 

density becomes

where {*xi*

electromagnetism.

 *ds*2 = *gij dxi dxj*

arm common path polarization measurement interferometer is not enough. We need a twoarm interferometer with the paths in two arms in magnetic fields with different strengths (or one with no magnetic field).

To measure *η3* in addition, one needs to use both external electric and external magnetic field. One possibility is to let light goes through strong microwave cavity and interferes. Suppose light propagation direction is the same as the microwave propagation direction which is perpendicular to the microwave fields. Let's choose *z*-axis to be in the propagation direction, *x*-axis in the **E**ext direction and *y*-axis in the **B**ext direction, i.e., **k** = (0, 0, *k*), **E**ext = (*E*, 0, 0) and **B**ext = (0, *B*, 0). We calculate the indices of refraction using Eqs. (39)-(47) without first assuming *E = B* and obtain the following

$$
\varepsilon\_{\text{eff}} \colon \varepsilon\_{11} = 1 + 2\eta\_1 (E^2 \cdot B^2) B\_c \cdot 2 + 4\eta\_1 E^2 B\_c \cdot 2; \; \varepsilon\_{22} = 1 + 2\eta\_1 (E^2 \cdot B^2) B\_c \cdot 2 + 4\eta\_2 B^2 B\_c \cdot 2;
$$

$$
\varepsilon\_{33} = 1 + 2\eta\_1 (E^2 \cdot B^2) B\_c \cdot 2; \; \varepsilon\_{12} = \varepsilon\_{21} = 2\eta\_3 E B B\_c \cdot 2; \; \varepsilon\_{13} = \varepsilon\_{23} = \varepsilon\_{31} = \varepsilon\_{32} = 0,\tag{68}
$$

$$(\mathfrak{h}^{\cdot 1})\_{\mathfrak{a}\mathfrak{f}} .^{\circ} (\mathfrak{h}^{\cdot 1})\_{11} = 1 + 2\eta\_1 (E \text{\textquotedbl} B^2) B\_{\mathfrak{c}} \text{\textquotedbl} \text{\textquotedbl} B\_{\mathfrak{c}} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquotedbl} \text{\textquoted$$

$$(\mathfrak{h}^{\perp})\_{33} = \mathfrak{l} + 2\eta\_1 (E \cdot B^{\perp}) B\_{\mathfrak{c}} \cdot \mathfrak{z} \\ \cdot (\mathfrak{h}^{\perp})\_{12} = (\mathfrak{h}^{\perp})\_{21} = 2\eta\_3 E B B\_{\mathfrak{c}} \cdot \mathfrak{z} \\ \cdot (\mathfrak{h}^{\perp})\_{13} = (\mathfrak{h}^{\perp})\_{23} = (\mathfrak{h}^{\perp})\_{13} = (\mathfrak{h}^{\perp})\_{23} = 0,\qquad \text{(69)}$$

$$\lambda\_{\mathfrak{a}\mathfrak{g}} \colon \lambda\_{11} = \mathfrak{\xi}\mathfrak{p} + \eta\_{\mathfrak{z}}(E^{\mathfrak{z}} \cdot B^{\mathfrak{z}})B\_{\mathfrak{c}}{}^{\mathfrak{z}} + 2\eta\_{\mathfrak{z}}E\_{\mathfrak{z}}B\_{\mathfrak{c}}{}^{\mathfrak{z}};\\\lambda\_{22} = \mathfrak{\xi}\mathfrak{p} + \eta\_{\mathfrak{z}}(E^{\mathfrak{z}} \cdot B^{\mathfrak{z}})B\_{\mathfrak{c}}{}^{\mathfrak{z}} + 2\eta\_{\mathfrak{z}}B^{\mathfrak{z}}B\_{\mathfrak{c}}{}^{\mathfrak{z}};$$

$$
\lambda\_{33} = \xi \mathfrak{g} + \eta \eta \text{(}E^2 \text{-} B^2 \text{)} \\
\lambda\_{12} = 4\eta \mathfrak{g} \text{(}E \text{BB}\_{\mathfrak{c}} \text{-} ^2 \text{)} \\
\lambda\_{21} = 4\eta \mathfrak{g} \text{(}E \text{BB}\_{\mathfrak{c}} \text{-} ^2 \text{)} \\
\lambda\_{13} = \lambda\_{23} = \lambda\_{31} = \lambda\_{32} = 0,\tag{70}
$$

$$I\_1 \equiv -4\eta\_1 (E^2 + B^2) B\_c{}^2 + 4\eta\_1 E B B\_c{}^2 \tag{71}$$

$$I\_2 \equiv -4\eta\_2(E^2 + B^2)B\_c{}^2 + 4\eta\_2 EBB\_c{}^2,\tag{72}$$

$$\mathbf{J} \equiv \mathfrak{D} \eta\_{\mathfrak{N}} (\mathbf{E} \mathbf{\bar{z}} \mathbf{B} \mathbf{\bar{z}}) B\_{\mathbf{c}} \mathbf{\bar{z}} \tag{73}$$

$$\eta\_{\pm} = 1 + (\eta\_{\parallel} + \eta\_{\supseteq})(E^2 + B^2 \text{-}EB)B\_{\varepsilon} \cdot ^2 \pm [(\eta\_{\parallel} \cdot \eta\_{\supseteq})^2 (E^2 + B^2 \text{-}EB)^2 + \eta\_{\supseteq} \cdot ^2 (E^2 \cdot B^2)]^{1/2} \, B\_{\varepsilon} \cdot ^2. \tag{74}$$

As a consistent check, there is no birefringence in Eq. (74) for *η1 = η*2, *η*3 = 0.

Now, we consider two special cases for Eq. (74): (i) *E=B* as in the strong microwave cavity, the indices of refraction for light is

$$m\_{\pm} = 1 + (\eta\_1 + \eta\_2)B^2 B\_c{}^2 \pm (\eta\_1 \cdot \eta\_2)B^2 B\_c{}^2 \prime \tag{75}$$

with birefringence Δ*n* given by

$$
\Delta n = \mathcal{Z}(\eta\_1 \text{-} \eta\_2) B^2 B\_c \text{-} \text{'} \tag{76}
$$

(ii) *E=*0, *B*≠0, the indices of refraction for light is

$$
\mu\_{\pm} = 1 + (\eta\_1 + \eta\_2) B^2 B\_c^{-2} \pm [(\eta\_1 \cdot \eta\_2)^2 + \eta\_3^2]^{1/2} B^2 B\_c^{-2},\tag{77}
$$

with birefringence Δ*n* given by

$$
\Delta \eta = 2[(\eta\_1 \cdot \eta\_2)^2 + \eta\_3 \cdot 2]^{1/2} B^2 B\_c{}^2. \tag{78}
$$

Equation (77) agrees with (54) derived earlier.

54 Trends in Electromagnetism – From Fundamentals to Applications

arm common path polarization measurement interferometer is not enough. We need a twoarm interferometer with the paths in two arms in magnetic fields with different strengths (or

To measure *η3* in addition, one needs to use both external electric and external magnetic field. One possibility is to let light goes through strong microwave cavity and interferes. Suppose light propagation direction is the same as the microwave propagation direction which is perpendicular to the microwave fields. Let's choose *z*-axis to be in the propagation direction, *x*-axis in the **E**ext direction and *y*-axis in the **B**ext direction, i.e., **k** = (0, 0, *k*), **E**ext = (*E*, 0, 0) and **B**ext = (0, *B*, 0). We calculate the indices of refraction using Eqs. (39)-(47) without

*-*2; ε22=1+2*η*1(*E*2-*B*2)*Bc*

*-*2; *λ*22=ξφ+*η*3(*E*2-*B*2)*Bc*


*-*2+4*η*1*EBBc*

*-*2+4*η*2*EBBc*

*-*2*±*(*η1-η2*)*B*2*Bc*

*-*2*±*[(*η1-η2*)2*+η3*2]1/2*B*2*Bc*

*-*2*±* [(*η1-η2*)2(*E*2+*B*2-*EB*)2*+η3*2(*E*2-*B*2)]1/2 *Bc*

*-*2; (μ-1)22=1+2*η*1(*E*2-*B*2)*Bc*

*-*2+4*η*2*B*2*Bc*

*-*2; ε13=ε23=ε31=ε32=0, (68)

*-*2; (μ-1)13=(μ-1)23=(μ-1)13=(μ-1)23=0, (69)

*-*2+2*η*3*B*2*Bc*

*-*2-4*η*1*B*2*Bc*

*-*2;

*-*2; *λ*13=*λ*23=*λ*31=*λ*32=0, (70)

*-*2, (71)

*-*2, (72)

*-*2, (75)

*-*2, (77)

*-*2. (78)

*-*2; (76)

*-*2, (73)

*-*2;

*-*2. (74)

*-*2;

*-*2+4*η*1*E*2*Bc*

*-*2; ε12=ε21=2*η*3*EBBc*

*-*2-4*η*2*E*2*Bc*

*-*2; (μ-1)12=(μ-1)21=2*η*3*EBBc*

*-*2+2*η*3*E*2*Bc*

*-*2; *λ*12=-4*η*1*EBBc*

As a consistent check, there is no birefringence in Eq. (74) for *η1 = η*2, *η*3 = 0.

*n± =* 1 + (*η1+η2)B*2*Bc*

*n± =* 1 + (*η1+η2)B*2*Bc*

Now, we consider two special cases for Eq. (74): (i) *E=B* as in the strong microwave cavity,

Δ*n =* 2(*η1-η2*)*B*2*Bc*

Δ*n =* 2[(*η1-η2*)2*+η3*2]1/2*B*2*Bc*

one with no magnetic field).

first assuming *E = B* and obtain the following

(μ-1)αβ: (μ-1)11=1+2*η*1(*E*2-*B*2)*Bc*

(μ-1)33=1+2*η*1(*E*2-*B*2)*Bc*

εαβ: ε11=1+2*η*1(*E*2-*B*2)*Bc*

*λ*αβ: *λ*11=ξφ+*η*3(*E*2-*B*2)*Bc*

 *J1 ≡* -4*η*1(*E*2+*B*2)*Bc*

 *J2 ≡* -4*η*2(*E*2+*B*2)*Bc*

(ii) *E=*0, *B*≠0, the indices of refraction for light is

Equation (77) agrees with (54) derived earlier.

 *J ≡* 2*η*3(*E*2-*B*2)*Bc*

*λ*33=ξφ+*η*3(*E*2-*B*2)*Bc*

 *n± =* 1 + (*η1+η2)*(*E*2+*B*2-*EB*)*Bc*

the indices of refraction for light is

with birefringence Δ*n* given by

with birefringence Δ*n* given by

ε33=1+2*η*1(*E*2-*B*2)*Bc*

To measure *η1, η2* and *η3*, we could do the following three experiments to determine them: (i) to measure the birefringence Δ*n =* 2(*η1-η2*)*B*2*Bc -*2 of light with the external field provided by a strong microwave cavity or wave guide to determine *η1-η2*; (ii) to measure the birefringence Δ*n =* 2[(*η1-η2*)2*+η3*2]1/2*B*2*Bc -*2 of light with the external magnetic field provided by a strong magnet to determine *η3* with *η1-η2* determined by (i); (iii) to measure *η1* and *η<sup>2</sup>* separately using two-arm interferometer with the paths in two arms in magnetic fields with different strengths (or one with no magnetic field).

As to the term *ξΦ* and parameter *ξ*, it does not give any change in the index of refraction. However, as we will see in section 7 and section 8, it gives a polarization rotation and the effect can be measured though observations with astrophysical and cosmological propagation of electromagnetic waves.
