1. Introduction

It has been predicted [1] that the totally reflective frustrated Gires-Tournois interferometer exhibits a negative group delay time for an incident angle that is larger than the critical angle. Contrary to the conditions where the negative group delays and superluminal velocities are associated with signal envelope reshaping caused by transmission and reflection probabilities less than unity [2], the negative group delay time in the frustrated Gires-Tournois interferometer contradicts the relativistic causality by considering its 100% reflectivity. To solve this causality paradox, Resch et al. [2] indicate that the Goos-Hänchen shift contributes an extra positive time that is referred to as Goos-Hänchen time. This time is considered to be always large enough to make the total time delay positive in the frustrated Gires-Tournois interferometer. But recently, it is shown [3] that, if the Goos-Hänchen time is taken into account, the sum of the group delay time and the Goos-Hänchen time can become negative for total reflection of a plane TM wave from vacuum upon an ideal nonabsorbing plasma mirror. And now, we seem to be in a dilemma [3] that the total reflection on the frustrated Gires-Tournois interferometer contradicts causality if the Goos-Hänchen time is not included, and the total reflection of a plane TM wave upon an ideal nonabsorbing plasma mirror violates causality if the Goos-Hänchen time is included.

In the present paper, we plan to solve this contradiction with the help of a slab optical waveguide constituted by two ideal nonabsorbing plasma mirrors. We will show that only taking the Goos-Hänchen time into account, the ray model, [4] which is used to illustrate the characteristics of a propagation mode in a waveguide, can then coincide with the electromagnetic field theory, and the time associated with total reflection is exactly the sum of the group delay time and the Goos-Hänchen time. Based on this concept, relativistic causality is preserved in the total reflection upon the frustrated Gires-Tournois interferometer. Furthermore, by considering the negative Goos-Hänchen shift, the causality paradox does not exist in the total reflection of a plane TM wave from vacuum upon an ideal nonabsorbing plasma mirror as well.

2. Confirmation of the Goos-Hänchen time

In order to test whether the Goos-Hänchen time exists or not in total reflection, we consider a symmetric slab waveguide where the guiding layer is vacuum and two cladding layers are ideal nonabsorbing plasma mirrors. Let ω_p be the plasma frequency and u = ω/ω_p where ω is the angular frequency of the light. The refractive index of the plasma is n_p =i(1-u ²)^1/2/u with 0 < u < 1. Both the electromagnetic theory and the ray model can be used to illustrate the characteristics of the propagation modes in the waveguide. In electromagnetic theory, the field of a guided mode can be expressed as E(x, y, t)= E(y)exp[i(βx - ωt)] where E(y) is the field amplitude and β is the propagation constant. In the ray model, a simple zigzag-ray picture [4] is used to describe mode propagation as total reflections of optical rays from vacuum on two plasma mirrors. Figure 1 shows the slab waveguide and zigzag-propagation of the rays for a guided mode with TE and TM polarizations.

 

Fig. 1. Symmetric slab waveguide with both cladding layers constituted by ideal nonabsorbing plasma mirrors and zigzag-propagation of the rays: (a) TE guided mode; (b) TM guided mode.

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In the ray model of the slab waveguide, the phase shift φ associated with total reflection of optical rays from vacuum upon two plasma mirrors with an angle of incidence θ is given by

According to Eq. (1), the group delay time [3] defined by t_g = ∂φ/∂ω is

And the Goos-Hänchen time [2] defined by

is written as

For this case of total reflection, we can see from Eq. (4) that the Goos-Hänchen time is always positive for TE wave and is always negative for TM wave. The positive and negative Goos-Hänchen times are caused by the positive and negative Goos-Hänchen shifts [2] Δx that are

where k ₀ is the wave number of the light in vacuum.

We consider a guided mode in the slab waveguide with an effective index of N ≡ β/k ₀ = sinθ. The length l between A and B illustrated in both (a) and (b) of Fig. 1, which corresponds to one period propagation of the zigzag-rays in the guiding layer, is the sum of 2htanθ and two Goos-Hänchen shifts at two boundaries of the guiding layer, that is

where h is the thickness of the guiding layer and h _eff is the effective thickness [4,5] of the guided mode with the form of

From Eq. (6) and Fig. 1, we can see that, for both TE and TM polarizations, the rays behave [4, 5] as if they were propagating in a guide of effective thickness. Since the Goos-Hänchen shifts are negative for TM polarization, h _eff < h holds true for TM mode, whereas h _eff > h holds true for TE mode because of the positive Goos-Hänchen shifts.

The propagation time of the guided mode in length l can be calculated by two methods: (1), the ray model takes account of Goos-Hänchen times and group delay times that occur at the guiding layer boundaries, and (2), the electromagnetic theory determines the propagation time by dividing the length l with the group velocity. Since the group velocity of the guided mode is derived from the rigid electromagnetic theory, if the times calculated from two methods are exactly the same, the existence of Goos-Hänchen time is confirmed. Dispersion relation of the guided mode is

where κ = k ₀(1-N ²)^1/2 is the component of wave vector normal to the interface in guiding layer, m is the mode order, and φ defined by Eq. (1) can be rewritten as

The group velocity v_g is defined by

where c is the velocity of light in vacuum and ∂N/∂ω can be calculated by performing partial derivative of Eq. (8) with respect to the angle frequency. The partial derivative of κ with respect to the angular frequency is

For TE mode in the slab waveguide, the partial derivative of φ with respect to ω is

where ∂$n_p^2$/∂ω=2(1-$n_p^2$) ω. By using Eqs. (8), (11), and (12), we have

From Eq. (10), the group velocity of TE guided mode is

And then, by considering N = sinθ and Eqs. (6), (7), (2), and (4), the total propagation time of the TE guided mode in length l illustrated in Fig. 1(a) is

It is obvious that τ_total includes three parts: the propagation time of the rays in guiding layer 2h/(ccosθ), two Goos-Hänchen times, and two group delay times occurring at two boundaries of the guiding layer.

For TM mode in the slab waveguide, the partial derivative of φ with respect to ω is

According to Eqs. (16), (11), (8) and (7), we have

From Eq. (10), we have

And then, the total propagation time of the TM guided mode in length l shown in Fig. 1(b) is

With the help of Eqs. (2) and (4), Eq. (19) can be rewritten as

For the TM guided mode, we can see that τ_total also includes three parts: the propagation time of the rays in guiding layer 2h/(ccosθ), two Goos-Hänchen times, and two group delay times occurring at two boundaries of the guiding layer. Therefore, according to Eqs. (15) and (20) and the ray model of the guided mode demonstrated in Fig. 1, the total time delay associated with total reflection is exactly the sum of the group delay time and the Goos-Hänchen time for both TE and TM polarizations. Although the Goos-Hänchen time is negative for TM polarization, the existence of this time coincides well with the rigorous frequency-domain electromagnetic theory of the waveguide. The existence of Goos-Hänchen time also indicates that the causality paradox does not exist in the frustrated Gires-Tournois interferometer [2].

3. Total reflection of a plane TM wave upon a plasma mirror

In the case of total reflection of a plane TM wave from vacuum upon an ideal nonabsorbing plasma mirror, the negative Goos-Hänchen time can result in negative total time delay [3], which seems to violate relativistic causality. In fact, this is not the case. The negative Goos-Hänchen time, which must be associated with the negative Goos-Hänchen shift, has its profound physical meaning. From the point of view of electromagnetic theory, because of $n_p^2$ < 0, the direction of the time-averaged Poynting vectors and its flux lines in the two cladding layers of the slab waveguide shown in Fig. 1(b) is opposite to the propagation direction of the guided mode [6], resulting in the power flow of the guided mode concentrating in the range of effective thickness [5] that is less than the thickness of the guiding layer. In the ray model, this effect can be explained as that total reflections occur at two boundaries of the effective thickness (O _1p in Fig. 1(b)). This consideration is also supported by a calculation [6] that shows time-averaged Poynting vectors and its flux lines in the two media for a Gaussian beam in the case of a negative Goos-Hänchen shift. It is indicated that the incoming flux lines do not pass through the interface between two media at all, and the closed-loop flux lines exist around the interface. It is also shown [6] that the intersection of the incident beam axis and the reflected beam axis lies in front of the interface, and is located exactly at O _1p shown in Fig. 1(b). Therefore, the negative Goos-Hänchen shift and time are reasonable by considering that the points labeled as O in Fig. 1(b) are selected as the reference points, whereas total reflections occur at the points labeled as O _1p. From this point of view, we can rewrite Eq. (20) as

The first term in the right hand of Eq. (21) is the propagation time of the rays in a guide of effective thickness in length l. The third term is the sum of two group delay times. The second term, which is the sum of the two times associated with two negative Goos-Hänchen shifts, is obtained by considering that total reflections occur at the points O _1p. Therefore, for total reflection of a plane TM wave from vacuum upon an ideal nonabsorbing plasma mirror, the time associated with the negative Goos-Hänchen shift should be

which is always positive. By employing t _GH=n ₁Δx sin θ/c[2], Eq. (22) can also be expressed as

which is exactly the sum of three times: two propagation times of rays in O _1p O and in O′O _1p, and the Goos-Hänchen time t _GH at the interface. Expression (23) is then the general form of the time caused by a negative Goos-Hänchen shift if calculated it at the location where total reflection occurs, and is always positive.

In this paper, the frequency-domain analysis is used for the incident light. Since the fields exist for all time and amplitudes of all fields are invariable, energy transfer can not occur between the closed-loop flux lines around the interface and the incident and reflected flux lines (see Ref. [6]). Therefore, although the light is reflected before it reaches the interface, it is reasonable.

4. Conclusion

We have demonstrated that, for both TE and TM polarizations, the total time delay upon total reflection is the sum of the group delay time and the Goos-Hänchen time. For the case of total reflection with a negative Goos-Hänchen shift, we show that the time is still positive if taking the location where total reflection exactly occurs into account. Therefore, the causality paradox does not exist both in the frustrated Gires-Tournois interferometer case and in the case of total reflection of a plane TM wave upon an ideal nonabsorbing plasma mirror.