Enhancement of the Faraday and Other Magneto-Optical Effects in Magnetophotonic Crystals

Abstract

It is shown that for existent natural materials the Faraday rotation is far below the theoretical limit [Steel et al. in J. Lightwave Technol. 18:1297, 2000]. Under this condition the value of the Faraday rotation is primarily determined by the Q-factor, while the low group velocity value, multipass traveling and energy concentration in magneto-optical material play a secondary role. A comparative analysis of the efficiency of different schemes employing defect modes, Tamm surface states, the Borrmann effect and plasmon resonance is presented.

Appendices

Appendix A: Relationship Between the Transmission Coefficient and the Faraday Rotation Angle

We assume that the distance between a given resonance and other ones is much larger than the resonance width. In principle, if the resonance has a large Q-factor, a small magnetization is sufficient for obtaining the required value of splitting of resonances. But, irrespective of the Q-factor, one obtains the same dependence of the transmission coefficient |T|² on the Faraday rotation angle ϑ (Fig. 1.11). This is due to the fact that the frequency range in the vicinity of the resonance, at which the amplitude, |T|², and the phase, argT, change significantly, is the same for both these quantities. More precisely, for $T( k_{0} ) = \frac{\alpha}{k_{0} - \tilde{k}_{0}}$ with a real frequency, k ₀, and complex values of residue, α, and the pole position, $\tilde{k}_{0} = k_{r} - i\varGamma$, one can obtain the expression $|T|^{2} = T_{0}^{2}\cos^{2}\vartheta$ ($T_{0}^{2}$ is the resonant transmission), which is independent of the Q-factor. Indeed, designating a deviation of the frequency from the resonant one as δk ₀=k ₀−k _r , one has $T( \delta k_{0} ) = \frac{\alpha}{\delta k_{0} + i\varGamma }$ and

$$ \big| T( \delta k_{0} ) \big|^{2} = \frac{|\alpha |^{2}}{\delta k_{0}^{2} + \varGamma ^{2}}. $$

(1.20)

Presuming that the presence of the magnetization shifts the peak without changing its form, one obtains ϑ=(argT(δk ₀)−argT(−δk ₀))/2, from which it follows that

$$ \tan \vartheta = - \frac{\delta k_{0}}{\varGamma } . $$

(1.21)

Eliminating δk ₀ from expressions (1.20) and (1.21), one obtains $|T|^{2} = \frac{|\alpha |^{2}}{\varGamma ^{2}}\cos^{2}\vartheta$. After denoting the transmission at the resonance as $T_{0}^{2} = \frac{|\alpha |^{2}}{\varGamma ^{2}}$, the required result arises:

$$ |T|^{2} = T_{0}^{2}\cos^{2}\vartheta . $$

(1.22)

Fig. 1.11

Universal dependence of the transmission coefficient on the Faraday rotation angle for a single pole (solid line) and the same dependence for the optical Tamm state resonance (dashed line). The deviation is caused by resonances neighboring to the Tamm state

The comparison between the one-resonance model system meeting condition (1.22) and results obtained from calculations of the Tamm state (Fig. 1.11) shows that a difference appears when the frequency moves away from the resonance, to the region in which the resonances next to the Tamm state become visible.

From this analysis one can conclude that for any lossless system with separated resonances, a large rotation angle is possible only for a small transmission, and their mutual dependence is the same for any Q-factor. A large Q-factor is needed only for obtaining the desirable value of splitting of peaks which is equal to 2δk ₀.

Appendix B: Rate of the Phase Change at the Resonance Frequency

In this appendix we show that at the resonance, the absolute value of the quantity $d\varphi /d\omega|_{\omega = \omega _{r}}$ in (1.15), is equal to the inverse width of the resonance. Near the resonance, the transfer function can be approximated as

$$T( \omega) = \frac{\alpha}{\omega - \omega _{r} - i\varGamma } , $$

where ω _r and Γ are the frequency and the half-width of the resonance, respectively. By differentiating the relation T(ω)=|T(ω)|exp(iφ(ω)), one obtains

$$\frac{\partial \varphi}{\partial \omega} = \frac{i}{|T|}\frac{\partial |T|}{\partial \omega} - \frac{i}{T}\frac{\partial T}{\partial \omega} . $$

Calculating derivatives

$$\frac{\partial |T|}{\partial \omega} = \frac{\partial}{\partial k}\frac{| \alpha |}{\sqrt{( \omega - \omega _{r} )^{2} + \varGamma ^{2}}} = - |T|\frac{\omega - \omega _{r}}{( \omega - \omega _{r} )^{2} + \varGamma ^{2}} $$

and

$$\frac{\partial T}{\partial \omega} = - \frac{T}{\omega - \omega _{r} - i\varGamma } , $$

we find

$$\frac{\partial \varphi}{\partial \omega} = - \frac{\varGamma }{( \omega - \omega _{r} )^{2} + \varGamma ^{2}}. $$

It immediately follows that at the resonance

$$\frac{d\varphi ( \omega )}{d\omega} = - \frac{1}{\varGamma } . $$