Detection of Vacuum Birefringence with Intense Laser Pulses
Andre N. Luiten
We propose a novel technique that promises hope of being the first to directly detect a polarization in the quantum electrodynamic (QED) vacuum. The technique is based upon the use of ultra-short pulses of light circulating in low dispersion optical resonators. We show that the technique circumvents the need for large-scale liquid helium cooled magnets, and more importantly avoids the experimental pitfalls that plague existing experiments that make use of these magnets. Likely improvements in the performance of optics and lasers would result in the ability to observe vacuum polarization in an experiment of only a few hours duration.
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\begin{document}
\bibliographystyle{apsrev}
\title{Detection of Vacuum Birefringence with Intense Laser Pulses}
\author{Andre N. Luiten}
\email{andre@physics.uwa.edu.au}
\author{Jesse C. Petersen}
\altaffiliation{Now with Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada.}
\affiliation{\uwa}
\date{\today}
\begin{abstract}
We propose a novel technique that promises hope of being the first to directly  detect   a polarization in the quantum electrodynamic (QED) vacuum. The technique is based upon the use of ultra-short pulses of light circulating in low dispersion optical resonators.   We show that the technique circumvents the need for large-scale liquid helium cooled magnets, and more importantly avoids the experimental pitfalls that plague  existing experiments that make use of these magnets. Likely improvements in the performance of optics and lasers would result in the ability to observe vacuum polarization in an experiment of only a few hours duration.

\end{abstract}
\pacs{42.50.Xa,12.20.Fv,42.62.Eh,42.25.Lc,41.20.Jb}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Introduction and Overview}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


It was predicted almost seventy years ago that virtual positron-electron pairs in the quantum electrodynamic vacuum provide a means for interaction  between photons~\cite{euler, heisenberg, weisskopf,schwinger}. As yet this effect has not been observed directly in any laboratory experiment although high energy scattering experiments have shown indirect evidence of  such interactions~\cite{wilson,burke}.  It is believed that these processes play an important role in  extreme astrophysical environments~\cite{heyl}. 

The QED mediated interaction predicts  scattering of real photons from virtual photons in an electromagnetic field as well as direct photon-photon scattering mechanism.  For achievable fields, the principal effect of these processes is  a polarization dependent change of the phase velocity of the interacting photons,  in other words, a laser beam propagating through a region of strong electromagnetic field will observe a birefringent and refractive vacuum, in which both polarization states have a phase velocity that differs from that in a field-free vacuum.  
%A number of physical effects of great interest manifest themselves as an optically detectable birefringence arising under the application of strong electromagnetic fields. One example is the small degree of magnetically induced birefringence that arises in the Cotton-Mouton effect~\cite{Muroo,muroo2,cameron}. In this case light traversing  a medium exposed to a strong transverse magnetic field experiences a different refractive index for polarization states parallel and perpendicular to the magnetic field direction.  An analogous, but even smaller field-induced birefringence is predicted to occur in vacuum because of corrections to Maxwell's equations arising under quantum electrodynamics (QED)~ \cite{euler,heisenberg,weisskopf,schwinger}. 
%
For example, light traversing  a  transverse magnetic field region will experience differing refractive indices for polarization parallel and perpendicular to the magnetic field, with the  birefringence  being of the order of ${\Delta n \sim 10^{-21}}$ for realistic  laboratory magnetic fields ($5 - 10$\,T).  % For a magnetic field of a metre in length this means that the relative phase shift between the two polarisation states will be of order $10^{-18}$\.rad.

The conventional route to searching for such field-induced birefringence is to couple intense magnetic fields, produced by  superconducting coils, to an optical delay line or a Fabry-P\'erot resonator which forces the light to traverse the strong field many times~\cite{cameron,pvlas1,bmv,lee}.  In contrast, here we propose a novel method for the experimental detection of \vb\, which  dispenses with the need for any large static magnetic fields, and instead makes use of a combination of frequency-stabilized mode-locked lasers~\cite{diddams,jones-lock} and low dispersion optical resonators~\cite{jones-dco,me-pulse}. %This approach has been made possible because of recent  developments in mode-locked laser frequency stabilization techniques~\cite{diddams}. 
We propose a search for the vacuum birefringence induced by the extremely high fields that exist within a focussed femtosecond duration pulse of light.  This technique holds the promise of  improved sensitivity while   using only room-temperature apparatus that is both reliable and relatively inexpensive. 

%%% because of the nonlinear coupling between electromagnetic fields in vacuum is polarization-dependent. 
$\Re_{the man} x^{2} \frac{1}{2}$ 
%The QED-mediated interaction between a polarized field and a photon gives rise to a photon polarization-dependent optical refractance of the vacuum. 
To produce  \vb\ with an optical field, a linearly polarized `pump' beam must interact with a counter-propagating `detection' beam.  %The detection beam can then be regarded as moving in the mean field  of the pump beam. 
 The refractive indices of the vacuum for detection light polarized parallel and perpendicular to the polarization of the pump beam are denoted as \npar\ and \nperp, and given by \cite{alek,vpol}:
\eq{nn}{\npar=1+\frac{16}{45}\frac{\alpha^2U}{U_e};\;\;\;\;\;\nperp=1+\frac{28}{45}\frac{\alpha^2U}{U_e},}
where $\alpha$ is the fine structure constant, $U$ is the energy density in the optical field and
${U_e={m_e^4c^5}/{\hbar^3}\approx 1.44\times 10^{24}}$
is the Compton energy density of the electron ($m_e$ is the electron rest mass).  From Eq.~(\ref{nn}) we observe a birefringent vacuum of magnitude:
\eq{b}{\Delta n=\frac{4}{15}\frac{\alpha^2U}{U_e}=\frac{4}{15}\frac{\alpha^2\Iav}{c U_e}.}
where \Iav\ is the intensity of the pump field.  These expressions are valid for  infinite plane waves, although they give a birefringence of the correct order for laser beams when the beams are well-collimated over the interaction region.  %Substituting the maximum average intensity which can be applied along the axis of a pulsed RPI from Eq. (\ref{Iav4}) gives an estimate of the expected  birefringence.

%The pulsed output of these lasers are stored in high Finesse cavities which has two key benefits: it allows an increase in the coherent interaction length and also allows a frequency metrology approach to the measurement of the effect.  These twin advantages allow a resolution that is significantly higher than 


%To date most actual and proposed experiments to detect weak field-induced birefringence make use of high intensity static or low-frequency oscillating magnetic fields supplied by extremely powerful superconducting magnets~\cite{iacopini,cameron, pvlas1,lee,taiwan1,bmv, taiwan2, boer,hall,muroo2,Muroo}. The magnetic field induces an ellipticity in a linearly polarised light beam that is traversing the field. The magnets can supply extremely intense fields ($5 -  25$\,T) but have a number of key limitations.  First, the most obvious of these disadvantages is  that the magnet systems are both large and operationally expensive and can only be modulated at low frequencies. The limitations on modulation frequency means that any birefringence signals of interest can easily be buried in the low frequency noise of the detection technique unless more elaborate modulation schemes are implemented~\cite{cameron, pvlas1,taiwan1}. Perhaps even more importantly for sensitive birefringent measurements are the consequences of the high forces that come from the intense and large volume fields generated by the magnets. The forces can result in movement of the optical elements in  the detection system which can masquerade as a birefringence signal~\cite{cameron,itnoise}. In addition, the unconfined nature of the magnetic field makes it difficult to properly shield the polarizing field from the detection apparatus. This is problematic since low levels of residual field can directly act  on the detection system components to generate a false  birefringence signal~\cite{lee,bmv,cameron,itnoise}.  In existing searches for a vacuum birefringence the experiments were limited by these types of false signals.




%An attractive alternative to the application of intense and macroscopic magnetic fields could be the use of optical fields to generate the polarizing fields necessary for the experiment. 
% The degree of vacuum polarization induced by either a magnetic field or an optical field is proportional to the the energy density of the two fields.  
%The generation of the polarizing field by focussed continuous-wave (cw) laser radiation has been considered, however,  it is easy to show that the energy density of these optical fields is extremely small in comparison with that of the superconducting magnet generated fields~\cite{vpol,alek}.  
In this letter we propose the use of short pulses of intense laser  radiation to generate the high fields necessary to polarize the vacuum. It is by this technique that one can generate average field intensities comparable with those produced by   superconducting magnets. As an extreme example, the peak magnetic field within a 1\,J, 50\,fs pulse that is focussed into  1 $\mu$m$^2$    can be of the order of $10^{5}$\,T~\cite{lee}.  The high confinement of the optical field means that while the peak fields are very high, the total energy stored in the field is much smaller than a static magnetic field that produces an equivalent birefringence signal. The pulsed light technique thus has twin benefits:   it  eliminates large forces from the experiment, and also  make shielding of the detection apparatus from stray fields very simple. In existing searches for vacuum birefringence, spurious signals arising from stray magnetic field modifying the detection system components, or stray forces moving  the detection system components, are responsible for limiting the sensitivity~\cite{lee,bmv,cameron,itnoise}.  
 
 The obvious disadvantage of the pulsed approach is that high intensity fields only persist for a short period of time in any particular location, and over a very small volume. This requires a detection technology with a high temporal and spatial resolution so as not to average the signal away.   We propose a novel synchronous detection \tec\ that satisfies both of these requirements. %Our technique will simultaneously resonate the strong polarizing field and the fields that detect the resulting birefringence. This approach has the advantage of allowing highly intense fields together with  a high interaction rate.  
 % The combination of a highly sensitive detection \tec\ and high level of vacuum polarization    potentially puts  detection of vacuum polarization within the grasp of an all-optical tabletop experiment using existing technology.  

%% From fairbanks the typical B field of 10T can make 4 x 10^-22 type shifts. In the case of the femtosecon dhigh energy pulses that they propose (10Hz, 1J per pulse, 50fs pulse length) they get a mag field of 2.3 10^5 T which will make a shift of about 2.5 10^-11.From his suggestion does this make sense?  
%   Alternative: optical field inside an optical pulse.  



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Resonant Polarization Interferometry}%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%J. Hall et al have reported an experimental technique capable of measuring the integrated birefringence inside a Fabry-Perot resonator with great precision~\cite{hall}.  We will refer to the device, illustrated schematically in Figure 1, as a Resonant Polarization Interferometer (RPI).  The basis of the technique is to frequency lock two continuous-wave (cw) and orthogonally-polarized lasers to the same longitudinal mode of the resonator using the Pound-Drever-Hall technique~\cite{pdh,black,hils}.  To first order the fractional frequency difference between the stabilized laser frequencies is equal to the fractional difference in the optical path length of the resonator measured in the two polarization states:

Our detection system is a modification of a previously reported technique which is capable of measuring  birefringence with extremely high precision~\cite{hall}.  The basis of the original technique %, illustrated schematically in Figure 1, 
  is to frequency lock two continuous-wave (cw), orthogonally-polarized lasers to the same longitudinal mode of a resonator using the Pound-Drever-Hall technique~\cite{pdh,black,hils}.  Our proposal is to use a laser pulse stream from a mode-locked laser  rather than cw lasers to excite the resonator.  The advantages of this approach will become apparent below, but we commence with a description of the cw device as this is valid for both cases.

The fractional frequency difference between the two orthogonally-polarized lasers  is equal to the fractional difference in the optical length of the resonator as measured in the two polarization states:
\eq{fracfreqdiff}{\frac{\fperp-\fpar}{\nu_0} = \frac{\lpar-\lperp}{l_0}.}
where $\nu_0$ is the average frequency of the two modes and $l_0$ is the average length of the resonator.
 A path length difference will arise from any birefringence in the cavity in addition to that coming from any intrinsic birefringence of the cavity mirror coatings~\cite{hall}:
 \eq{fracbirefringence}{\fperp-\fpar  \sim   \frac{ \npar-\nperp}{n_{0}} \nu_0 +\frac{c}{2 n_0 L}  \frac{\delta \phi}{2 \pi}.}
where ${c}/({2 n_0 L})$ is the longitudinal mode spacing of the resonant cavity, $\delta \phi$ is the difference in the reflection phase for the two polarisations, and $n_0$ is the average refractive index in  the resonator.  The laser frequency difference, $\fperp-\fpar$, can be extracted by detecting the beat-note between the lasers  and measuring its frequency with   a conventional high precision frequency counter. 

%\makefig{fig1}{fpbi}{A Resonant Polarization Interferometer following that described in reference \cite{hall}.}%%%   <=== [FIGURE 1]

%From Eq.~(\ref{fracbirefringence}) it can be seen that an enormous gain in sensitivity is possible with this approach since a small difference in the refractive index is multiplied up by the optical frequency $\nu_0$ ($\sim 3$~x~$10^{14}$ Hz). 
We note that cavity length fluctuations that arise from vibration or temperature fluctuations will be common to both polarizations and hence do not appear in the measured frequency difference signal.  This avoids the need for high quality vibration isolation or temperature control of the detection  resonator. 
 
A number of previous experiments have shown that   over a certain frequency band it is feasible to suppress all technical noise sources that afflict frequency locking systems~\cite{day,bondu,uehara}.  In this case the residual frequency instability is limited by the inherent quantum noise of the detected light itself (shot noise).  
%
%reduced, then the key residual fluctuations in the frequency difference signal will be due to the inherent noise in the frequency locking systems.  With sufficient servo gain and high modulation frequencies, the dominant noise source is photon shot noise. 
%
A simple estimate shows that this will limit the accuracy of each locked laser frequency to~\cite{hall,black}:  \eq{dshot}{\delta_{\mathrm{shot}} \sim \frac{c}{2LF} \sqrt{\frac{h\nu}{\Pd \ti}}.}
% check the above equation and use the proper one (in terms of the photocurrent).  In addition check that this is really true for the limit of the PDH servos.
where $h$ is Planck's constant, $\nu$ is the laser frequency, \Pd\ is the power falling on the feedback photodiode,  $\ti$ is the integration time, and  where we use  an optical resonator of length $L$ with finesse $F$. An experiment using 800\,nm laser light and a detected optical power of a few milliwatts allows stabilization of the  laser  to $10^{-8}$ of the  cavity bandwidth after 1\,s of integration time.   For a measurement of the difference between two mode frequencies, the expected sensitivity is equal to the residual frequency instability of one laser multiplied by $\sqrt{2}$ (because a comparison is being made between two uncorrelated and equally noisy signals).  %This gives a measurement sensitivity (as a fraction of the laser frequency) of
%\eq{dnurel}{\delta\nu_\mathrm{rel}&=&\sqrt{2}\delta_{\mathrm{shot}}\frac{\delta\nu_{\frac{1}{2}}}{\nu}\\
%&\sim&\sqrt{\frac{h}{2\Pd\tau\nu}}\frac{c}{LF}.}
Thus with 800\,nm lasers we get a fractional frequency (or length) sensitivity of:
\eq{Ndnurel}{\delta\nu_\mathrm{rel}&\approx&6\times 10^{-19}\sqrt{\frac{1\unit{s}}{\ti}}\brac{\frac{4\unit{m}}{L}}\brac{\frac{10^5}{F}}\sqrt{\frac{5\unit{mW}}{\Pd}}}
In order to access this level of sensitivity we modulate the expected \vb\ effect so that  the useful signal falls in the shot-noise limited part of the sensitivity spectrum.


%In practice, to attain a shot-noise-limited measurement sensitivity it is necessary to modulate the birefringence  at a judicious frequency that is well-removed from electrical or mechanical interference and which is high enough to negate seismic and other  low frequency electrical  noise sources.  Although it is unlikely to expect shot-noise limited sensitivity at all frequencies it is certainly experimentally feasible to achieve this over a limited frequency band~\cite{day,bondu,sal,uehara, kawamura}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Measuring Optically-Induced Birefringence}%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To generate \vb\  we use an auxiliary linearly polarized `pump' laser beam to interact with the two detection beams.  In order to exploit the polarization dependence of the coupling between the pump and orthogonally-polarized detection beams we align the  pump field polarisation with the polarization of one of the detection beams.  In order to maximize the interaction between the detection and pump beams they need to be exactly counter-propagating and coaxial~\cite{alek}. % check that this is really the right reference. 
Unfortunately, coaxial pump and detection beams will overlap on the resonator mirrors and it has been shown that a photo-refractive interaction between the beams in the mirror coatings can generate spurious birefringence signals~\cite{hall}. 
%The detection beams do not act to  produce  birefringence upon themselves because all photon-photon scattering processes need a non-zero momentum in the  centre of mass frame of the collision~\cite{partovi}.  It would be in principle possible to use a coaxial but counter-propagating pair of pump and detection beams and thus use a single set of mirrors for both the detection and pump beams.  However, it has been shown that this approach is potentially unsafe since dielectric mirrors can exhibit  a  strong photo-refractive effect, which  can masquerade as a spurious birefringence signal by providing a means for the detection and pump beams to directly interact~\cite{hall}. 
%
Thus we use a second optical resonator to enhance the power of the pump beam, as illustrated in Fig.~\ref{dualcavities}, which lies at an angle, $\theta$,  to the axis of the detection resonator.    The resonators are of identical length $L$, and the separation between the resonator axes of $x$ at the cavity mirrors of radius, $a$. 
\makefig{fig1}{dualcavities}{Measurement scheme for optically-induced birefringence.}  % <== [FIGURE 1]
The crossed cavity design potentially leads to a limited interaction zone between the pump and detection beams. 
 %As we will show below induced \vb\ is proportional to the intensity of the field creating it. Simply %shortening both resonant cavities could raise the average intensity but the resonance linewidth would %increase by an equal fraction.
A  key suggestion of this letter is that if both detection and pump beams are pulsed  rather than continuous-wave (cw) signals and are synchronized so that the detection and pump pulses meet head on at $C$ (see Fig.~\ref{dualcavities})~\cite{ma,joneshol}, and in addition, each of the pulses is short enough to completely pass through the other before the beam axes begin to separate, then all of the light circulating in the detection cavity will interact with all of the light circulating in the pump cavity on every pass.  Furthermore, the pulses interact   while the beams are tightly focussed, and therefore where they are most intense.  %Although  the use of  pulsed lasers will complicate the experimental arrangement there is, in principle, no reason that a mode-locked laser signal can not be frequency locked with the same accuracy as cw signal.
  Mode-locked lasers have already been frequency-locked to a resonator with relatively high precision~\cite{jones-lock}, and low-dispersion  cavities can allow even very short pulses to be coupled into high-finesse resonators with low power loss and relatively little broadening of the circulating pulse with respect to the input pulse~\cite{jones-dco,me-pulse}. 

%An additional advantage of this pulsed-RPI approach is that we have put energy into many successive longitudinal modes of the detection and pump cavities. This circumvents the strong low frequency interaction that is possible between the two cw detection beams that was seen in the  experiments of Hall et al~\cite{hall}.  The pulsed-RPI scheme automatically implements the more complex detection strategies proposed by Hall et al and Lee et al~\cite{hall,lee2}. These detection strategies all require frequency shifting of the  detection beams in order to prevent interaction via the optical coatings.



To determine the sensitivity of this approach we 
%will consider the case of a birefringence effect which is proportional to the intensity of the local optical field. This is true for both the Cotton-Mouton effect and the predicted QED vacuum polarization.  First it is necessary to 
calculate the the average intensity seen by a pulse circulating in the detection cavity, which is equal to
		\eq{Iav}{\Iav&=&\frac{1}{L}\Lint{I(z)}}
where $I(z)$ is the intensity as a function of longitudinal position in the cavity.  When short pulses are used, it is only  the small region approximately half the pulse length either side of the beam crossing point that contributes significantly to the above integral.  %This interaction region is approximately half the spatial length of the pulses themselves. %extending  a small distance 
	%	\eq{zodash}{z_0'=\frac{c\tau}{4}.}
%either side of point $C$ on Fig.~\ref{dualcavities}. % where $\tau$ is the full width at half maximum pulse duration. 
If  the separation, $\rho(z)$, between the beam axes remains significantly less than the beam radii in the interaction region, then the beams can be treated as approximately coaxial when calculating \Iav.  

%The separation between the beam axes, in terms of beam radii, can be expressed as:
%		\eq{rorel}{\frac{\rho(z)}{w(z)}&\apx&\frac{z \,  \theta}{w_0\sqrt{1+\brac{\frac{\lam z}{\pi %w_0^2}}^2}}.}
%where we are only interested in a range of $z$ that falls within the interaction region as given by Eq.~\ref{zodash}, $w_0$ is the beam waist size, and $\lambda$ is the wavelength of the stored radiation.
%%% need to ask Jesse whether this is what he meant
The minimum angle by which the beams must be skewed is set by the mirror spacing, $x$, %as shown on Fig.~\ref{dualcavities}
 which is determined by the size of  the cavity mirrors. The mirror radius, $a$,  is set by the need for the mirrors to be larger than the laser spot radius evaluated at the mirror position, $w(L/2)$, by a factor $\alpha$. The value of $\alpha$  is determined by the extent to which aperture losses can be tolerated for a particular application. %This combination of factors  dictates a minimum separation $x$ given by
	%\eq{x1}{x=2a=2\alpha  \,\, w\brac{{L/2}}.}
%Since we wish to maximize the induced \vb\ we choose detection and pump cavity configurations which are close to the concentric stability limit~\cite{sieg},  so as to minimize the waist size in the cavity, and hence maximize the pump energy density. 
In the limit of a small beam waist size, $w_0$,  the spot size at the mirrors can be calculated as
 		${x\approx  {\alpha L\lam}/{\pi w_0}}$.
%so that
%		\eq{roz}{\rho(z)\approx\theta z}
%
%which  determines $\theta$, the angle between the beams as
%		\eq{theta}{\theta=\frac{2\alpha\lambda}{\pi w_0}.}
%For reasonable assumptions of a pulse duration  $\leq 200$fs, a wavelength $\lam\geq 500 \, $nm and $w_0\geq 5\lam$, then we find that the pulse length is less than one Rayleigh range, $\pi w_0^2/%\lambda$, and we can simplify    
		%%\eq{ass}{\brac{\frac{\lam c\tau}{\pi w_0^2}}^2\ll1.}
%Eq.~(\ref{rorel}) and use Eqs.~(\ref{zodash}) and  (\ref{theta}) to give the following approximation for the maximum relative separation:
%	\eq{}{\frac{\rho(z)}{w(z)}\apx\frac{2 z \alpha\lam}{\pi w_0^2},}
%	which is a maximum at the beginning and end of the pulse, $z_0' \sim c \tau/4$, 
%		\eq{rorel2}{\frac{\rho(z_0')}{w(z_0')}\apx\frac{c\tau\alpha\lam}{2\pi w_0^2}.}
%If this maximum relative separation is small, the beams may be treated as coaxial over the entire region.  As an example, if $\rho(z_0')/w(z_0')=0.3$, then $I(z_0')$ is only about 6\% less on the probe axis than it is on the pump axis.  The reduction in average intensity when integrated over the entire interaction region is even smaller than this figure.  To give a rough criterion for the minimum waist radius that can be used without encountering significant beam separation inside the interaction region, we set equation %(\ref{rorel2}) equal to 0.3 and rearrange, obtaining
%		\eq{wo1}{
%			w_0\gtrsim 10\micron
%			\brac{\frac{\tau}{200\mathrm{\,fs}}}^{\frac{1}{2}}
%			\brac{\frac{\alpha}{4}}^{\frac{1}{2}}
%			\brac{\frac{\lam}{800\mathrm{\,nm}}}^{\frac{1}{2}}.}

%We note that if the interaction region is smaller than the Rayleigh range of the beam, the beams will be of approximately constant radius as the pulses pass through each other.  A waist radius which is too small, though, will cause the beams to begin to diverge while still inside the interaction region and reduce \Iav.  Equating the Rayleigh range to $z_0'$ as given in (\ref{zodash}) and rearranging yields the following expression which must be satisfied in order to prevent significant beam divergence inside the interaction region.
%		\eq{wo2}{w_0\gtrsim 2\micron\brac{\frac{\tau}{200\mathrm{fs}}}^{\frac{1}{2}}\brac{\frac{\lam}{800\mathrm{nm}}}^{\frac{1}{2}}.}
%For realistic values of $\alpha$, adhererence to (\ref{wo1}) automatically satisfies (\ref{wo2}).

%So long as the inequalities in Eq. (\ref{wo1}) and (\ref{wo2}) hold,  calculation of \Iav\ is straightforward.  Each time a probe pulse passes through the interaction region, it sees a burst of light which carries the effectively the entire energy \Ep\ of the pulse circulating in the pump cavity.   During the entire interaction time the pulses are approximately coaxial, with beam radii equal to that at  the waist.  Thus
%\eq{Iav2}{\Iav&\apx&\recip{L}\frac{\log 2}{\pi w_0^2}\Lint{P(z)}\\
%&\apx&\frac{c}{L}\frac{\log 2}{\pi w_0^2}\Ep.}
	%
	Using these results we have shown elsewhere~\cite{andjess} that %under a reasonably relaxed set of criteria  i.e. a pulse duration of $\leq 200$fs, a wavelength $\lam\geq 500 \, $nm and $w_0\geq 5\lam$, then
	 the beam waist has to obey the following inequality:
	\eq{wo1}{
 			w_0\gtrsim 10\micron
 			\sqrt{\frac{\tau}{200\mathrm{\,fs}}}^{\frac{1}{2}}
 			\brac{\frac{\alpha}{4}}^{\frac{1}{2}}
 			\brac{\frac{\lam}{800\mathrm{\,nm}}}^{\frac{1}{2}}.}
	to ensure  that the beams do not significantly separate within the interaction zone.  In addition, under this same limit  the Rayleigh range of the interacting beams is significantly  longer than the spatial extent of the pulses and thus the beams are also essentially collimated in the interaction zone. Under these conditions the detection beam sees all of the pump energy on each pass through the interaction zone.%check this later statement is true.

	
The circulating pump pulse energy, \Ep\, is determined by the average input power \Pav , the repetition rate, $R$, of the input pulse train, resonator finesse $F$, and an efficiency factor, $k_\mathrm{cav}$, which allows for imperfect mode matching, impedance matching and dispersion related losses~\cite{me-pulse}:
\eq{ep}{\Ep=k_\mathrm{cav} \frac{F}{\pi}\frac{\Pav}{R}.}
Since we wish the circulating pulse to be efficiently reinforced on each pass by the incident pulse train, the free spectral range of the cavity must be identical to the repetition rate of the laser, $R$~\cite{jones-dco,me-pulse}.
%\eq{R}{R=\frac{c}{2L}.}
%	In a frequency domain view this is equivalent to setting the frequency comb spacing of the input pulse train equal to the free spectral range of the cavity.
In this case we find, that:
\eq{Iav3}{\Iav\apx\frac{2 k_\mathrm{cav}\log 2 }{(\pi w_0)^2}F\Pav.}
	
%According to Eq.~(\ref{Iav3}), \Iav\ is determined solely by the pump oscillator average power, the pump resonator finesse and the size of the beam waist.  %The beam waist, in turn, depends on $\lam$, $\alpha$ and $\tau$ via equation (\ref{wo1}).  
	Combining the expressions above, we arrive at   the following indicative numerical expression for \Iav.
\eq{Iav4}{\Iav\apx \frac{F}{52\, \mathrm{x} 10^{3}}\frac{\Pav}{20\unit{W}}\frac{200\unit{fs}}{\tau}\frac{4}{\alpha}\frac{k_\mathrm{cav}}{1}\frac{800\unit{nm}}{\lam}\times1.5\unit{\frac{PW}{m^2}}}
%	
where we have made use of  achievable experimental parameters.  A finesse of 52\,000 corresponds to a reflectance of $99.994\%$ which is available in a custom low dispersion mirror coating~\cite{ltech}, that has sufficiently low dispersion to allow 200\,fs incident laser pulses to be directly coupled into a cavity with near-unity efficiency~\cite{jones-dco,me-pulse}.  A mode-locked laser has been reported with a 200\,fs duration output pulse and  20\,W  average power~\cite{brunner}.% It is likely that there will be further improvements in the output power of mode-locked lasers given the relatively early stage of development of this technology together with the rapidly decreasing cost of pump lasers.  
Thus, using readily available equipment  it should be possible to construct a pump cavity which gives a effective average  intensity in the detection cavity of  1.5\,$\mathrm{PW/m}^2$.  Such high average intensity is only possible because  the arrangement of pulsed and counter-propagating detection and pump beams circumvents the high divergence that would normally afflict tightly focussed beams. In fact, the pulsed beams show the same degree of  interaction as  cw beams that were parallel and nondivergent throughout the cavity, which is of course impossible for tightly focussed, non-coaxial beams.

%The average intensity given by Eq.~(\ref{Iav3}) is independent of the length of the cavity, because the increase in energy per pulse due to the lower repetition rate is cancelled by the decrease in the fractional length of the cavity which falls inside the interaction region.  Of course, that assumes that the average output power of the pump laser is independent of the repetition rate, which is reasonably well followed by commercial laser systems. However, it should be noted that a longer cavity is preferable for the purpose of increased measurement sensitivity because of the associated reduction in bandwidth of the detection cavity modes (see Eq.~(\ref{Ndnurel})\,). 
	 The optimal cavity length in a real experiment relies on constructing sufficiently large mirrors as implied by the beam size calculation above.  For a 3\,m  cavity with $\alpha=4$, the mirrors would need to be 20\,cm in diameter.  Although this presents a significant challenge, it is not insurmountable, as demonstrated by the recent construction of even larger diameter mirrors for gravitational wave interferometers~\cite{mirrors}.
	 
	%%%%%%
	%%%%%
	%
	%
	%One of the challenges of the concentric cavity required for this proposal is its extreme sensitivity to misalignment of the cavity mirrors and pointing fluctuations of the input beam.  One can show that the waist size in a near concentric cavity is given by~\cite{sieg}:
%	\eq{waistsize}{w_0 = \sqrt{\frac{R\,\lambda}{\,\pi}} 
%  {\left( \frac{{\Delta L}}{2\,R - {\Delta L}}
 %     \right) }^{\frac{1}{4}}}
	%
%	where $\Delta L  = 2 R-L \ll R$, and $R$ is the radii of curvature of the two symmetric mirrors.  Thus in order to have a waist size of order 10\,$\mu$m in a cavity of length 3\,m it is necessary to tune the length to within 2~x~$10^{-7}$\,m of the instability limit.  In this near-concentric position the input coupling is highly sensitive to relative angular and translational misalignments of the cavity and the input beam with the beam displacement on mirror 1 or 2  given by~\cite{sieg}:
%	\eq{displace}{{{{\Delta x}}_{(1,2)}} = 
%  \pm \frac{R^2\,\left( -{{\theta }_{1}} + {{\theta }_{2}} \right) }
%   {{\Delta L}}}
%	where $\theta_{(1,2)}$ is the angular rotation of mirror 1 or 2.  In order to restrict translation motion of the mode on the mirrors to less than $1 \%$ of the spot size of the beam it is necessary to limit rotations of the mirrors to below 3~x~$10^{-10}$\,rad. 
	%The mode beam angular rotation is given by~\cite{sieg}:
	%\eq{angle}{\Delta \theta = \frac{R\,\left( -{{\theta }_1} + {{\theta }_2} \right) }
  %{{\Delta L}}}
% The alignment expression is clearly divergent as the cavity condition approaches the concentric limit and thus may cause concern. To meet the challenge of these alignment requirements one can implement a mode-cleaning cavity before the detection and pump cavities~\cite{gossler}, or make use of an auto-alignment system~\cite{sampas}. We note that the requirements for mirror stability stated above are well within the capacity of such alignment systems~\cite{sampas}.
  
 % We now turn our attention to the higher order modes in this concentric cavity system.  The transverse mode spacing, $\Delta \nu_\mathrm{TM}$, in a near-concentric resonator can be found to be~\cite{sieg}: 
%  	\eq{tms}{\Delta \nu_\mathrm{TM} = \frac{c}{4R} \left( 1 - {{}\,{\sqrt{\frac{{2\, \Delta L}}
 %        {R \pi^2 }}}} \right) }
         %
%        Using the example of a 3\,m near-concentric cavity  with a 10\,$\mu$m waist we find an $\sim$8\,kHz frequency splitting between the fundamental mode and the first-order transverse mode.  The higher order modes will be frequency resolved	if the finesse of the cavity is greater than 6,000.  In order that pointing fluctuations do not couple into frequency fluctuations it is necessary to have a finesse higher than this value.  In addition, we note that the transverse mode spacing can provide a useful diagnostic for setting the length of the cavity to achieve a desired waist  size.
	%%%%%%%%%%	
	%%%%%%
	
	
	
%%%%%%%%%%%%%%%%%%%%%
	
%\section{\label{sec4}Comparing pulsed RPI and  conventional Ellipsometry}
%Make specific comparison to standard ellipsometric techniques and discuss the advantages of using a frequency-metrological approach.
	
	%%%%%%%%%%%%%%%%%%

%To place the sensitivity of the proposed pulsed RPI technique in context we should compare it with conventional resonant ellipsometry and with continuous-wave RPI (cw-RPI). In this section we also compare the energy density of an optical pump field  with that obtainable from a large scale static magnetic field.  The local energy density is proportional to the degree of polarization created in a media of interest.  
	
%	Conventional resonant ellipsometry relies on `tuning' the rotation of the cavity mirrors  to set the intrinsic birefringence of the cavity to nearly zero for the linearly polarized input radiation~\cite{chui}. In this case  the ``slow" or ``fast" axis of each of the mirrors  is well aligned with input polarization state  resulting in  limited conversion of the input radiation into   	the other polarization state by the mirrors.  %Eq.~(\ref{fracbirefringence}) being smaller than the bandwidth of each of the polarization modes in the resonator i.e. $\delta \nu_{\frac{1}{2}} \gg \Delta \nu \frac{\delta \phi}{2 \pi}$. In this case both polarization modes are frequency degenerate and a single input beam to the resonator can excite both modes.
%	 If the polarization of this input beam is oriented at $\pi / 4$
 %with respect to the applied magnetic field direction using a high quality polarizer	then it becomes elliptically polarized by the residual birefringence of the cavity.  The change in ellipticity of the beam can be expressed as:
 
 %\eq{ellipticity}{\psi = k L \frac{\npar-\nperp}{2}.}
%
%
%where $k$ is the wavenumber of the input light, and $L$ is the interaction length of the field and the light.  This ellipticity is measured by a polarization analyser at the output of the cavity. The analyzer consists of  a polarizer that has been set to pass radiation with polarization orthogonal to  the input radiation.   
% check that this is the correct config to find ellipticity - in Cameron.
%The most sensitive measurements of ellipticity, $\psi$, using a  single-pass ellipsometry have reported a $\psi$ detection limit of ~$10^{-8}/\sqrt{\ti}$, which is  less than a factor of 2 from the shot-noise imposed limit under their respective conditions~\cite{cameron,Muroo,muroo2}:
	%
	%
	%
	
%	 \eq{ellipNtoS}{\delta \psi_{\rm{ellips}} = \frac{\sqrt{2 +  4\, \sigma^2 / \theta^2}}{\sqrt{P\,
%         \eta_{\rm{PD}}/ (h\,\nu)}} \frac{1}{\sqrt{\ti }}} %% shoved the stupid tau downstairs without a good reason.  I will need to check the maths that it should be 1/root tau
	 %
	%
%	where  ${\eta }_{{PD}}$ is the quantum efficiency of the detection photodiode and $\sigma^2$ is the extinction coefficient of the polarizer and analyzer. In order to linearize the sensitivity of the ellipsometer to small birefringence signals and to shift the signal of interest away from zero frequency it is usual to add a polarization modulation of depth  $\theta$ using a modulator at the output of the ellipsometer cavity. In this case the birefringence signal now appears as sidebands about the modulation signal and can be demodulated using synchronous detection techniques.
%	Unfortunately, the introduction of a resonant cavity or delay line into the ellipsometer in order to increase the length of the interaction between the applied field and detection beams ($L$ in Eq.~\ref{ellipticity}), and hence improve the birefringence sensitivity, results in a significantly worsened single pass phase  sensitivity~\cite{cameron}. The most sensitive birefringence measurements with delay lines or a high finesse cavity incorporated into the ellipsometer  have a birefringence sensitivity  in  the range of $\Delta n \sim 10^{-17} - 10^{-18}$~\cite{cameron,pvlas3}.	
	
%	We now turn our attention to the RPI technique of measuring birefringence. We stated above in Eq.~(\ref{Ndnurel}) that an order of magnitude estimate of a shot-noise limited frequency locking detection showed a birefringence sensitivity of below  $10^{-18}/\sqrt{\ti}$ where $\ti$ is the integration time of the measurement.  A more detailed examination of the the sensitivity limits under Pound-Drever-Hall (PDH) locking to a cavity with perfect impedance and mode matching shows that~\cite{sal,black}:
%	 \eq{PDHNtoS}{ \delta \psi_{\mathrm{PDH}} ={\frac{\sqrt{2}\,\pi \,{n_0}}
%     {8\,{F}\,{\sqrt{\ti }}\,
%       {\sqrt{{P\,{{\eta }_{\rm{PD}}}}
 %          /({h\,\nu })}}}}}
	%  I just put the 1/root tau at the end without doing a proper conversion from sf to time domain.  Might need some pis flying around
	
%	To compare the sensitivity of the ellipsometer and RPI approaches we note that Eq.~(\ref{ellipNtoS}) represents the ellipsometry sensitivity for a single pass through the interaction zone whereas Eq.~(\ref{PDHNtoS}) naturally refers to a resonant measurement in a cavity of Finesse, $F$.  The sensitivity of a resonant   ellipsometer measurement  can be found by adjusting $L$ in Eq.~\ref{ellipticity} for the number of passes through the interaction zone, which for a resonator of finesse, $F$, will be a factor of  $2\,F/\pi $.  In the case where the intentional modulation depth in the ellipsometer is much greater than the extinction of the polarizer-analyzer pair ($\theta \gg \sigma$) then the sensitivity of the two approaches has an identical dependence on  the main experimental parameters with the RPI approach being $2 \sqrt{2}$ more sensitive. It is likely that  subtle technical details will be the ultimate determinant of which technique is optimal. 
	%For example, the reflectivity of the FP cavity on resonance and the extinction ratio of the polarizer-analyzer pair play analogous roles in determining the ultimate power limit of the experiment and hence .
	% As mentioned above, it is necessary that the residual birefringence of the resonator is small enough that both polarisation states are simultaneously resonant - if this is not the case (e.g. $\delta \nu_{\frac{1}{2}} \lesssim \Delta \nu \frac{\delta \phi}{2 \pi}$)  then the ellipsometer performance becomes much more complex to model. 
	
%	As an example of the types of experimental details which are of importance, the above expressions have excluded the effects of amplitude noise in the input laser beams. The two techniques will be sensitive to the amplitude noise in the immediate frequency environment of the modulation frequency. In the case of the ellipsometer this is    the  polarization modulation frequency, while  the phase modulation frequency inherent in a  PDH frequency lock  is the relevant parameter in the other case.  In a suitable resonator (where  $\delta \nu_{\frac{1}{2}} \gg \Delta \nu \frac{\delta \phi}{2 \pi}$) it is however possible  to have both systems deployed simultaneously~\cite{lee2}. 
	%Although it is true that the broad-band sensitivity limits of the two appraoches are ultimately imposed by quantum fluctuations in the light, in the actual experimental 
 
% We note the analysis by Chui et al~\cite{chui} which compares the sensitivity of a continuous-wave(cw) RPI scheme and conventional resonant ellipsometer  to cavity mirror temperature changes.  In both schemes a mirror temperature change gives rise to a false birefringence signal, although, the RPI approach is claimed to be potentially much more sensitive to temperature changes~\cite{chui}.  For the ellipsometer approach it is possible to reduce the sensitivity to temperature changes by $10^6$ times by accurate alignment of the input beam direction with the intrinsic birefringence axis of the mirror surfaces.  In the worst case the cw-RPI technique will require  the intrinsic birefringence of the mirrors to be stable to 1 part in $10^{11}$ during the measurement period, which  corresponds to a temperature stability for the mirrors in the nanoKelvin range.    Although this appears to be an extreme challenge for the RPI approach we point out two  important differences in our scheme in comparison to that considered by those authors. First, it is possible to choose birefringence matched mirrors and align the "slow" axis of one mirror with the "fast" axis of the other mirror in construction of the detection cavity.  In this case the frequency difference between the two polarisation modes of the cavity will be much reduced, and reduce the temperature stability requirement by the same large factor (if the temperature fluctuations of the mirrors are correlated). In addition, as will be pointed out below, in the case of a pulsed RPI system it is possible to modulate the effective 'pump' intensity at a high frequency ($>10$\,kHz) (unlike the magnet produced field of Chui et al. which has modulation frequencies of $\sim$1\,mHz). Thus, slow temperature changes of the mirrors will be strongly suppressed by our modulated measurement technique.  
	
	%We now turn our attention to the magnitude of the vacuum polarizing field (the `pump' beam).
	%A pulse intensity of 1.5\,$\mathrm{PW/m}^2$ corresponds to an averaged energy density of 5\,$\mathrm{MJ/m}^3$, a little lower than the 39\,$\mathrm{MJ/m}^3$ produced by a 10\,T laboratory magnetic field.  %This small decrease in energy density requires an improvement in detection sensitivity   in order to compete with the static magnetic field approaches, or an improvement in resonator mirror quality and mode-locked laser output power.
	% We note in passing that extremely high intensity fields (much higher than can be generated by any macroscopic magnetic field technique) can be created by tightly focussing the output of  a high energy laser pulse amplifier~\cite{lee}.   The difficulty with this approach lies in constructing a detection system with sufficient sensitivity to probe inside these short pulses given the tight temporal and spatial restrictions~\cite{lee}.  
	
	A very important  advantage of our particular approach  is the ability to modulate the effective strength of the interacting fields without varying the energy load or distribution on the mirror surfaces.  This enables detection of the birefringence signal in a frequency domain where there is minimal noise interference without giving rise to spurious signals. We achieve this effective power modulation by temporally delaying or advancing the pump pulse with respect to the detection pulse and thus varying the degree of energy overlap at the crossing point of the two cavities.  The absence of modulation  in the thermal load on the mirrors  eliminates many potential spurious effects that could otherwise masquerade as the effect of interest.  This technique can be implemented as part of the control system that synchronizes the detection and pump pulses~\cite{ma,joneshol}.
	
		
%%%%%%%%%%%%%%%%
%\section{Detecting Vacuum Birefringence}
%%%%%%%%%%%%%%%
		
%A birefringence effect of significant interest at this time is that arising from a scattering of photons from a static electric or magnetic field, or even from other real photons. Although it was predicted almost seventy years ago that virtual positron-electron pairs in the quantum electrodynamic vacuum could mediate interactions between photons~\cite{euler, heisenberg, weisskopf,schwinger}, this effect is still to be observed directly in the laboratory.   
%It is believed that these types of processes play an important role in extreme astrophysical environments such as would exist at the surfaces of pulsars~\cite{heyl}. 
%\makefig{fdiagram}{fd}{caption}



Various challenging technical issues must be addressed in order to implement this experiment. For example, the pump and detection beams must be synchronized so that the pulses meet where their axes cross~\cite{ma,joneshol}. This implies that the offset frequency and repetition rate of the pulse trains from the  lasers must be controlled so as to match the cavity resonance frequencies and free spectral range of both resonators~\cite{jones-lock}.  The final hurdle to the detection of vacuum birefringence will be the duration of the experiment observation time in order to unambiguously detect  the effect.  We calculate these integration times by equating the expression for shot-noise limited measurement sensitivity in Eq.~(\ref{Ndnurel}) and the expected vacuum birefringence signal in Eq.~(\ref{b})  and present them in Table~\ref{tab1}. 
%We have substituted with the  experimentally feasible values as suggested by the expansions of Eq.~(\ref{Ndnurel}) and Eq.~(\ref{}).
 The first two lines predict the performance available using  low dispersion mirrors: the first line shows the capability of the best ``off-the-shelf'' commercially available low dispersion mirrors while the second line shows the capability of the best custom built mirrors~\cite{ltech}.  The final line of the table predicts that performance that could be achieved  if   low dispersion mirrors were to have a reflectivity equal to that of the best commercially-available super-mirrors. %We note that the predicted integration times are very much in line with that necessary for conventional vacuum birefringence experiments.


%\begin{table}[tb]
%\centering
%\begin{tabular}{|c|cc|}
%\hline
%$R$, \%&$F$&$\tau_{int}$\\
%\hline
%99.8&1600&5200 years\\
%99.98&1.6\ee{4}&204 days\\
%99.99&3.1\ee{4}&12.8 days\\
%99.997&1.0\ee{5}&2.5 hours\\
%\hline
%\end{tabular}
%\caption{Resonator finesse $F$ and required integration times for the detection of vacuum birefringence with a 20-W, 200-fs pump laser in a 5-meter-long resonator, tabulated as a function of the resonator reflectance (of the mirrors in both the pump and probe cavities).}
%\label{tab1}
%\end{table}

\begin{table}[tb]
\centering
\begin{tabular}{|c|c|c|}
\hline
$R$, \%&$F$&$\tau_\mathrm{int}$\\
\hline
99.97&1.0\ee{4}&2.6 years\\
99.994&5.2\ee{4}&1.7 days\\
99.997&1.0\ee{5}&2.5 hours\\
\hline

\end{tabular}
\caption{Required integration times  for the detection of vacuum birefringence as a function of resonator mirror reflectivity and resonator finesse, $F$.  The other experimental parameters are explained in the text.}
\label{tab1}
\end{table}
  
 The measurement time required to detect \vb\ scales with the inverse fourth power of the finesse since the finesse affects both the measurement sensitivity and the average intensity in our proposal approach.  Competing techniques that rely on a macroscopic magnetic field to create the vacuum polarization have  an integration period that decreases as the square of the finesse of the detection cavity. %Thus improvements in mirror technology will result in the pulsed RPI technique soon outpacing competing strategies. 
  If low dispersion mirrors could be improved to the point that 99.997\% reflectivity mirrors became available (only as good as existing super-mirrors) then  the corresponding increase in finesse would allow vacuum birefringence to be detected in just a few hours.  Any increases in  available laser power would also reduce the required measurement time.
 
% Despite the difficulties that should be expected in operating an optical system based on resonant cavities of such a large size, an all-optical device should be smaller, cheaper, easier to operate and more reliable than systems using helium-cooled superconducting magnets.  In addition, there are a couple of extremely important benefits accruing from the use of an optical pump field. First, there is the possibility to modulate the effective strength of the pump field at high rates as mentioned above, without changing the thermal load on the mirror system. Second,  the low forces and power necessary for generation of high intensity optical fields, combined with the high confinement of optical fields enables the elimination of many effects in the detection system which masquerade as a \vb\ signal in current experiments~\cite{cameron, itnoise}. Finally, we would suggest that there will be a rapid development of optical and laser technology over the next few years, especially in low dispersion mirrors with high reflectance and the development of higher average power  mode-locked lasers. These developments will directly feed into an improvement in the performance of \vb\ detection system based on these types of technology.  We would not expect the same rate of development in superconducting magnetic field technology.


%\section{Conclusion}

We have proposed a completely new approach to the experimental detection of very low levels of field-induced birefringence.   The system is based on the intersection of two  high-finesse short-pulse resonant cavities, one of which pumps the QED vacuum to produce the birefringence, while the other detects this induced birefringence using highly sensitive frequency metrology techniques. Current limits in mode-locked laser technology and low-dispersion mirrors  already allow detection of the predicted vacuum nonlinearity with a measurement period of just a few days. Readily forseeable advances in laser output power and mirror coating technology should reduce the required integration time for such an experiment to just hours. This approach has eliminated many of the defects that limit the performance of alternative techniques that use superconducting magnet systems.
 
 \section{Acknowledgements}
 
 We thank the Australian Research Council (ARC) for financial support of this research. We acknowledge all members of the Frequency Standards and Metrology Group  which make it such a pleasant and stimulating environment in which to work. % In particular, we would like to thank John Winterflood, John McFerran and Sam Dawkins for reading the manuscript and providing useful feedback to the authors.
 
 
 
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\end{thebibliography}