**3. Electromagnetic wave propagation in PPM electrodynamics**

Here we follow the previous method (Ni et al., 1991; Ni, 1998), and separate the electric field and the magnetic induction field into the wave part (small compared to external part) and external part as follows:

$$\mathbf{E} = \mathbf{E}^{\text{wave}} + \mathbf{E}^{\text{ext}} \tag{29}$$

$$\mathbf{B} = \mathbf{B}^{\text{wave}} \mathbf{e} + \mathbf{B}^{\text{ext}}.\tag{30}$$

We use the following expressions to calculate the displacement field **D**wave [= (*D*wave*α*) = (*D*wave*1*, *D*wave*2*, *D*wave*3*)] and the magnetic field **H**wave [= (*H*wave*α*) = (*H*wave*1*, *H*wave*2*, *H*wave*3*)] of the electromagnetic waves:

$$D^{\rm wave}v\_a = D\_a - D^{\rm ext}v\_a = (4\text{m})[(\partial \mathcal{L}\_{\rm FTM}/\partial \mathcal{E}\_a)\_{\rm EdB} \ \ \ \ (\partial \mathcal{L}\_{\rm FTM}/\partial \mathcal{E}\_a)\_{\rm ext}],\tag{31}$$

$$H^{\rm wave}{}\_{a} = H\_{a} - H^{\rm ext}{}\_{a} = \text{ - (4\text{m})}[(\partial L\_{\rm FTM}/\partial \mathbf{B}\_{\rm u})\_{\rm EdB} - (\partial L\_{\rm FTM}/\partial \mathbf{B}\_{\rm u})\_{\rm ext}] \tag{32}$$

where (…) **E**&**B** means that the quantity inside paranthesis is evaluated at the total field values **E** & **B** and (…)ext means that the quantity inside paranthesis is evaluated at the external field values **E**ext & **B**ext.

Since both the total field and the external field satisfy Eqs. (23)-(26), the wave part also satisfy the same form of Eqs. (23)-(26) with the source terms subtracted:

$$\text{curl } \mathbf{H}^{\text{wave}} = \begin{pmatrix} \mathbf{1}/\mathbf{c} \end{pmatrix} \partial \mathbf{D}^{\text{wave}} / \partial t,\tag{33}$$

$$\text{div } \mathbf{D} \text{wave} = \mathbf{0},\tag{34}$$

$$\text{curl } \mathbf{E^{wave}} = \text{-}(1/\text{c}) \, \partial \mathbf{B^{wave}} / \partial t,\tag{35}$$

$$\text{div}\,\mathbf{B}^{\text{wave}} = 0.\tag{36}$$

After calculating *D*wave*α* and *H*wave*α* from Eqs. (31) & (32), we express them in the following form:

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 51

*-*2 *+* 4*η*3*B1B2Bc*

*-*2 *+* 2*η*3(*B1*2*-B2*2)*Bc*

2]1/2} (*B1*2*+B2*2)*Bc*

*η1 = η*2, *η*3 = 0, and no constraint on *ξ* (55)

*-*2, (52)

*-*2. (53)

*-*2. (54)

*-*2 (**E**wave ║**B***ext*), (56)

*-*2 (**E**wave ┴ **B***ext*). (57)

*-*2, (58)

*-*2, (59)

*-*2, (60)

*-*2, (61)

*-*2, (62)

*-*2. (63)

*-*2. (64)

*-*2 (**E**wave║**E***ext*), (65)

*-*2 (**E**wave ┴ **E***ext*). (66)

*-*<sup>2</sup> − 4*η*2*B*22*Bc*

*-*<sup>2</sup> − 4*η*2*B*1*B*2*Bc*

The condition of no birefringence in Eq. (54) means that [*(η1−η*2*)2 +η*32] vanishes, i.e.,

This shows that Eq. (55) is a necessary condition for no birefringence. For **E**ext = 0, the refractive indices in the transverse external magnetic field **B**ext for the linearly polarized lights whose polarizations are parallel and orthogonal to the magnetic field, are as follows:

For **B**ext = 0, we derive in the following the refractive indices in the transverse external electric field **E**ext for the linearly polarized lights whose polarizations (electric fields) are parallel and orthogonal to the electric field. First, we use (39)-(41) & (43)-(46) to obtain

*-*<sup>2</sup> − 4*η*2*E*22*Bc*

*-*<sup>2</sup> − 4*η*2*E1*2*Bc*

The condition of no birefringence in (64) is the same as (55), i.e., that [*(η1−η*2*)2 +η*32] vanishes. For **B**ext = 0, the refractive indices in the transverse external magnetic field **E**ext for the linearly polarized lights whose polarizations are parallel and orthogonal to the magnetic

The magnetic field near pulsars can reach 1012 G, while the magnetic field near magnetars can reach 1015 G. The astrophysical processes in these locations need nonlinear electrodynamics to model. In the following section, we turn to experiments to measure the

*-*2 + 4*η*2*E*1*E*2*Bc*

εαβ=δαβ[1+2*η*1**E**2*Bc*

*λ*αβ=δαβ[φ+*η*3**E**2*Bc*

*n± =* 1 + {(*η*1*+η*2*) ±* [*(η1−η*2*)2 +η*<sup>3</sup>

*J1 =*-4*η*1*E1*2*Bc*

*J2 =*-4*η*1*E2*2*Bc*

*J =*-4*η*1*E*1*E*2*Bc*

Using (47), we obtain the indices of refraction for this case:

 *n*║*=* 1 + {(*η*1*+η*2*) +* [*(η1−η*2*)2 +η*<sup>3</sup>

 *n*┴ *=* 1 + {(*η*1*+η*2*) -* [*(η1−η*2*)2 +η*<sup>3</sup>

parameters of the PPM electrodynamics.

field, are as follows:

2]1/2} (**B***ext*)2*Bc*

2]1/2} (**B***ext*)2*Bc*

*-*2]+4*η*1*E*α*E*β*Bc*

*-*2] +2*η*3*E*α*E*β*Bc*

2]1/2} (**E***ext*)2*Bc*

2]1/2} (**E***ext*)2*Bc*

*-*2]-4*η*2*E*α*E*β*Bc*

*-*<sup>2</sup> *−* 4*η*3*E1E2Bc*

*-*2 *+* 4*η*3*E1E2Bc*

*-*2 *+* 2*η*3(*E1*2*-E2*2)*Bc*

2]1/2} (*E1*2*+E2*2)*Bc*

*J2 =*-4*η*1*B1*2*Bc*

*J =* 4*η*1*B*1*B*2*Bc*

 *n*║*=* 1 + {(*η*1*+η*2*) +* [*(η1−η*2*)2 +η*<sup>3</sup>

 *n*┴ *=* 1 + {(*η*1*+η*2*) -* [*(η1−η*2*)2 +η*<sup>3</sup>

(μ-1)αβ=δαβ[1+2*η*1**E**2*Bc*

Using Eq. (47), we obtain the indices of refraction for this case:

*n± =* 1 + {(*η*1*+η*2*) ±* [*(η1−η*2*)2 +η*<sup>3</sup>

$$D^{\rm wave}{}\_{a} = \Sigma\_{\not p-1}{}^{3} \, \varepsilon\_{a\not p} \, E^{\rm wave}{}\_{\not p} + \Sigma\_{\not p-1}{}^{3} \, \lambda\_{a\not p} \, B^{\rm wave}{}\_{\not p} \tag{37}$$

$$H^{\rm wave}{}\_{a} = \Sigma\_{\rho \star 1}{}^{3} \left(\mathrm{J}^{-1}\right)\_{a\sharp} B^{\rm wave}{}\_{\rho}{}^{\rho} \sim \Sigma\_{\rho \star 1}{}^{3} \lambda\rho\_{a} E^{\rm wave} \rho\_{\prime} \tag{38}$$

where

$$\varepsilon\_{\text{eff}} = \delta\_{\text{eff}} [1 + 2\eta\_1 (\mathbf{E} \cdot \mathbf{B}^2) B\_c \cdot ^2 + 2\eta\_3 (\mathbf{E} \cdot \mathbf{B}) B\_c \cdot ^2] + 4\eta\_1 \mathbf{E}\_a E\_{\beta} B\_c \cdot ^2 + 4\eta\_2 B\_a B\_{\beta} B\_c \cdot ^2 + 2\eta\_3 (\mathbf{E}\_a B\_{\beta} + E\_{\beta} B\_a) B\_c \cdot ^2 \quad \text{(39)}$$

$$(\mu \cdot \mathbf{1})\_{a\beta} = \delta\_{a\beta} [1 + 2\eta\_1 (\mathbf{E} \cdot \mathbf{B}^2) B\_c \cdot ^2 + 2\eta\_3 (\mathbf{E} \cdot \mathbf{B}) B\_c \cdot ^2] + 4\eta\_1 B\_a B\_{\beta} B\_c \cdot ^2 + 4\eta\_2 E\_a E\_{\beta} B\_c \cdot ^2 + 2\eta\_3 (E\_a B\_{\beta} + E\_{\beta} B\_a) B\_c \cdot ^2 \quad \text{(40)}$$

$$\lambda\_{a\beta} = \delta\_{a\beta} [\xi p + 4\eta\_2 (\mathbf{E} \cdot \mathbf{B}) B\_c \cdot ^2 + \eta\_3 (\mathbf{E} \cdot \mathbf{B}) B\_c \cdot ^2] - 4\eta\_1 \mathbf{E}\_a B\_{\beta} B\_c \cdot ^2 + 4\eta\_2 B\_a E\_{\beta} B\_c \cdot ^2 + 2\eta\_3 (E\_a E\_{\beta} + B\_a B\_{\beta}) B\_c \cdot ^2, \tag{41}$$

$$\text{where } \mu \text{ are the same and the same division.} \quad \text{(42)} \quad \text{(43)} \quad \text{Also, that the} \quad \text{sech} \quad \text{(44)}$$

and we have dropped the upper indices 'ext' for simplicity. Note that the coefficients of *B*wave*β* in Eq. (37) is the negative transpose of the coefficients of *E*wave*β* in Eq. (38) and vice versa. This is a property derivable from the existence of Lagrangian. It is a reciprocity relation; or simply, action equals reaction.

Using eikonal approximation, we look for plane-wave solutions. Choose the *z*-axis in the propagation direction. Solving the dispersion relation for *ω*, we obtain

$$\alpha\_{\mathbb{R}^2} \, \_k \{ 1 + (1/4) \, \left[ (\mathbb{I}\_{\mathbb{I}} + \mathbb{I}\_{\mathbb{Z}}) \pm \left[ (\mathbb{I}\_{\mathbb{I}} - \mathbb{I}\_{\mathbb{Z}})^2 + 4 \right]^2 \right] \} \, \tag{42}$$

where

$$J\_1 \equiv (\mu^{-1})\_{22} - \varepsilon\_{11} - 2\lambda\_{12} \tag{43}$$

$$J\_2 \equiv (\mu^{-1})\_{11} - \varepsilon\_{22} + 2\lambda\_{21} \tag{44}$$

$$J \equiv -\varepsilon\_{12} - (\mu^{-1})\_{12} + \lambda\_{11} - \lambda\_{22}.\tag{45}$$

Since the index of refraction *n* is

$$m = k \not\!\!/ \alpha \,,\tag{46}$$

we find

$$m\_{\pm} \equiv 1 \text{ - (1/4) } \{ (\mathbf{I}\_1 + \mathbf{I}\_2) \pm [(\mathbf{I}\_1 - \mathbf{I}\_2)^2 + 4\mathbf{I}\_1^2]^{1/2} \}. \tag{47}$$

From this formula, we notice that "no birefringence" is equivalent to *J1=J2* and *J=*0. A sufficient condition for this to happen is *η1 = η*2, *η*3 = 0, and no constraint on *ξ*. We will show in the following that this is also a necessary condition. The Born-Infeld electrodynamics satisfies this condition and has no birefringence in the theory.

For **E**ext = 0, we now derive the refractive indices in the transverse external magnetic field **B**ext for the linearly polarized lights whose polarizations (electric fields) are parallel and orthogonal to the magnetic field. First, we use Eqs. (39)-(41) & Eqs. (43)-(46) to obtain

$$
\varepsilon\_{\rm a\beta} = \delta\_{\rm a\beta} [1 \text{-} 2\eta\_1 \mathbf{B} \, ^\circ \mathbf{B}\_{\rm c} \, ^\circ \mathbf{I}] + 4\eta\_2 B\_{\rm a} B\_{\beta} B\_{\rm c} \mathbf{B}\_{\rm c} \, ^\circ \mathbf{r} \tag{48}
$$

$$(\mu^{\circ 1})\_{a\beta} \equiv \mathbb{S}\_{a\beta} [1 \text{--} 2\eta \text{ } 1 \text{--} 2\eta \text{ } 2 \text{--} 1 \text{-} 4\eta \text{ } 1 \text{-} a\_i B\_b B\_i \text{-} 2] \tag{49}$$

$$
\lambda\_{a\beta} = \delta\_{a\beta} [\mathbf{q} \cdot \eta\_3 \mathbf{B}^2 B\_c{}^2] + 2\eta\_3 B\_a B\_\beta B\_{c'}{}^2,\tag{50}
$$

$$\mathbf{J}\_1 = \mathbf{-4}\eta\_1 \mathbf{B}\_2 \mathbf{2} \mathbf{B}\_c \mathbf{2} - \mathbf{4}\eta\_2 \mathbf{B}\_1 \mathbf{2} \mathbf{B}\_c \mathbf{2} - \mathbf{4}\eta\_3 \mathbf{B}\_1 \mathbf{B}\_2 \mathbf{B}\_c \mathbf{2} \tag{51}$$

$$\mathcal{J}\_2 = 4\eta\_1 B\_1 \mathcal{Z} B\_c{}^2 - 4\eta\_2 B\_2 \mathcal{Z} B\_c{}^2 + 4\eta\_3 B\_1 B\_2 B\_c{}^2 \tag{52}$$

$$J = 4\eta\_1 B\_1 B\_2 B\_c{}^2 - 4\eta\_2 B\_1 B\_2 B\_c{}^2 + 2\eta\_3 (B\_1 \text{?} \, B\_2 \text{?}) B\_c{}^2. \tag{53}$$

Using Eq. (47), we obtain the indices of refraction for this case:

50 Trends in Electromagnetism – From Fundamentals to Applications

 *D*wave*<sup>α</sup>* = Σβ=13 εαβ *E*wave*β* + Σβ=13 *λαβ B*wave*β*, (37)

 *H*wave*<sup>α</sup>* = Σβ=13 (μ-1)αβ *B*wave*β* - Σβ=13 *λβα E*wave*β*, (38)

*-*2]+4*η*1*E*α*E*β*Bc*

*-2*]-4*η*1*B*α*B*β*Bc*

*-*2]-4*η*1*E*α*B*β*Bc*

and we have dropped the upper indices 'ext' for simplicity. Note that the coefficients of *B*wave*β* in Eq. (37) is the negative transpose of the coefficients of *E*wave*β* in Eq. (38) and vice versa. This is a property derivable from the existence of Lagrangian. It is a reciprocity

Using eikonal approximation, we look for plane-wave solutions. Choose the *z*-axis in the

From this formula, we notice that "no birefringence" is equivalent to *J1=J2* and *J=*0. A sufficient condition for this to happen is *η1 = η*2, *η*3 = 0, and no constraint on *ξ*. We will show in the following that this is also a necessary condition. The Born-Infeld electrodynamics

For **E**ext = 0, we now derive the refractive indices in the transverse external magnetic field **B**ext for the linearly polarized lights whose polarizations (electric fields) are parallel and

*-*2]+4*η*2*B*α*B*β*Bc*

*-*2] +2*η*3*B*α*B*β*Bc*

*-*2]-4*η*1*B*α*B*β*Bc*

*-*<sup>2</sup> *−* 4*η*3*B1B2Bc*

orthogonal to the magnetic field. First, we use Eqs. (39)-(41) & Eqs. (43)-(46) to obtain

*-*<sup>2</sup> − 4*η*2*B*12*Bc*

εαβ=δαβ[1-2*η*1**B**2*Bc*

*λ*αβ=δαβ[φ-*η*3**B**2*Bc*

*J1 =*-4*η*1*B2*2*Bc*

*-*2+4*η*2*B*α*B*β*Bc*



*ω± = k* {1 + (1/4) [(*J1+J2) ± [(J1−J2)2 + 4J*2]1/2]}, (42)

*J1 ≡ (μ−1)22* − *ε<sup>11</sup> − 2λ12*, (43)

*J2 ≡ (μ−1)11* – *ε22 + 2λ21*, (44)

*n = k/ω,* (46)

*-*2, (48)

*-*2, (49)

*-*2, (50)

*-*2, (51)

*J ≡* – *ε<sup>12</sup> - (μ−1)12* + *λ11* – *λ22*. (45)

*n± =* 1 - (1/4) {(*J1+J2) ±* [*(J1−J2)2 + 4J*2]1/2}. (47)

*-*2+2*η*3(*E*α*B*β+*E*β*B*α)*Bc*


*-*2+2*η*3(*E*α*E*β+*B*α*B*β)*Bc*

*-*2, (39)

*-*2, (40)

*-*2, (41)

*-*2+2*η*3(**E**·**B**)*Bc*

*-*2+*η*3(**E**2-**B**2)*Bc*

propagation direction. Solving the dispersion relation for *ω*, we obtain

satisfies this condition and has no birefringence in the theory.

(μ-1)αβ=δαβ[1-2*η*<sup>1</sup> **B**2*Bc*

*-*2+2*η*3(**E**·**B**)*Bc*

where

where

we find

εαβ=δαβ[1+2*η*1(**E**2-**B**2)*Bc*

(μ-1)αβ=δαβ[1+2*η*1(**E**2-**B**2)*Bc*

*λ*αβ=δαβ[ξφ+4*η*2(**E**·**B**) *Bc*

Since the index of refraction *n* is

relation; or simply, action equals reaction.

$$1 \cdot n\_{\pm -1} \cdot 1 + \{ (\eta\_1 + \eta\_2) \pm \{ (\eta\_1 - \eta\_2)^2 + \eta\_3 2 \} 1/2 \} \text{ ( $B\_1 2 \mp B\_2 2$ )} B\_{\hat{c}} \cdot 2. \tag{54}$$

The condition of no birefringence in Eq. (54) means that [*(η1−η*2*)2 +η*<sup>3</sup> 2] vanishes, i.e.,

$$
\eta\_1 = \eta\_2, \ \eta\_3 = 0, \text{and no constraint on } \xi \tag{55}
$$

This shows that Eq. (55) is a necessary condition for no birefringence. For **E**ext = 0, the refractive indices in the transverse external magnetic field **B**ext for the linearly polarized lights whose polarizations are parallel and orthogonal to the magnetic field, are as follows:

$$n\_{\parallel} = 1 + \left\{ (\eta\_1 + \eta\_2) + \left[ (\eta\_1 - \eta\_2)^2 + \eta\_3^2 \right] 1/2 \right\} \text{ (B} \text{crt} \text{)} 2 \text{\textdegree 2} \quad \text{(E} \text{ wave} \parallel \text{B} \text{crt)},\tag{56}$$

$$n\omega = 1 + \left\{ (\eta\_1 + \eta\_2) \cdot \left[ (\eta\_1 - \eta\_2)^2 + \eta\_3^2 \right]^{1/2} \right\} \left( \mathbf{B}^{\text{ext}} \right) 2B\_c^{-2} \quad \text{(E^{wave} \to \mathbf{B}^{\text{ext}})}.\tag{57}$$

For **B**ext = 0, we derive in the following the refractive indices in the transverse external electric field **E**ext for the linearly polarized lights whose polarizations (electric fields) are parallel and orthogonal to the electric field. First, we use (39)-(41) & (43)-(46) to obtain

$$
\varepsilon\_{a\beta} = \delta\_{a\beta} [1 + 2\eta\_1 E^2 B\_c{}^2] + 4\eta\_1 E\_a E\_\beta B\_c{}^2 \tag{58}
$$

$$(\mu \cdot 1)\_{a\#} = \mathbb{S}\_{a\#}[1 + 2\eta\_1 E 2B\_c \cdot 2] \cdot 4\eta\_2 E\_a E\_b B\_c \cdot 2 \tag{59}$$

$$
\lambda\_{\text{a}\beta} = \mathsf{S}\_{\text{a}\beta} [\mathsf{q} + \eta\_{\beta} \mathsf{E}^{2} \mathsf{B}\_{\text{c}} \mathsf{z}^{2}] + 2\eta\_{\beta} \mathsf{E}\_{\text{a}} \mathsf{E}\_{\beta} \mathsf{B}\_{\text{c}} \mathsf{z}^{2},\tag{60}
$$

$$\mathbf{J}\_1 = 4\eta\_1 \mathbf{E}\_1 \mathbf{2} \mathbf{B}\_c \mathbf{2} - 4\eta\_2 \mathbf{E}\_2 \mathbf{2} \mathbf{B}\_c \mathbf{2} - 4\eta\_3 \mathbf{E}\_1 \mathbf{E}\_2 \mathbf{B}\_c \mathbf{2} \tag{61}$$

$$I\_2 = 4\eta\_1 E\_2 2B\_c{}^2 - 4\eta\_2 E\_1 2B\_c{}^2 + 4\eta\_3 E\_1 E\_2 B\_c{}^2 \tag{62}$$

$$\mathcal{J} = 4\eta\_1 E\_1 E\_2 B\_c \cdot 2 + 4\eta\_2 E\_1 E\_2 B\_c \cdot 2 + 2\eta\_3 (E\_1 \text{--} E\_2 \text{2}) B\_c \cdot 2. \tag{63}$$

Using (47), we obtain the indices of refraction for this case:

$$n\_{\pi^\*-1} \mathbf{1} + \{ (\eta\_1 + \eta\_2) \pm [(\eta\_1 - \eta\_2)^2 + \eta\_3^2]^{1/2} \} \text{ ( $E\_12 + E\_22$ )} B\_c \text{-2.} \tag{64}$$

The condition of no birefringence in (64) is the same as (55), i.e., that [*(η1−η*2*)2 +η*32] vanishes. For **B**ext = 0, the refractive indices in the transverse external magnetic field **E**ext for the linearly polarized lights whose polarizations are parallel and orthogonal to the magnetic field, are as follows:

$$n\_{l} = 1 + \left\{ (\eta\_{1} + \eta\_{2}) + \left[ (\eta\_{1} - \eta\_{2})^{2} + \eta\_{3} \varepsilon^{2} \right]^{1/2} \right\} (\mathbf{E}^{\text{ex}})^{2} B\_{l} \cdot^{2} \quad \text{(\textbf{E}^{\text{wave}} \left\| \mathbf{E}^{\text{ex}} \right\|)},\tag{65}$$

$$n\omega = 1 + \left\{ (\eta\_1 + \eta\_2) \cdot \left[ (\eta\_1 - \eta\_2)^2 + \eta\_3 \mathbb{1} \right] 1/2 \right\} (\mathbf{E}^{\text{ext}})^2 B\_c^{-2} \quad (\mathbf{E}^{\text{wave}} \stackrel{\text{L}}{=} \mathbf{E}^{\text{ext}}).\tag{66}$$

The magnetic field near pulsars can reach 1012 G, while the magnetic field near magnetars can reach 1015 G. The astrophysical processes in these locations need nonlinear electrodynamics to model. In the following section, we turn to experiments to measure the parameters of the PPM electrodynamics.

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 53

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

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-

permanent magnet in the middle of the photo.
