The Goos-Hänchen effects are investigated for a monochromatic Gaussian beam totally reflected by a photonic crystal with a negative effective index. By choosing an appropriate thickness for the homogeneous cladding layer, a giant negative GH lateral shift can be obtained and the totally reflected beam retains a single beam of good profile even for a very narrow incident beam. The GH lateral shift can be very sensitive to the change of the refractive index of the cladding layer, and this property can be utilized for e.g. the switching applications.
©2006 Optical Society of America
The Goos-Hänchen (GH) shift refers to a lateral shift between the centre of a reflected beam and that of the incident beam when a total reflection occurs at the interface between two media. The GH shift effect has been studied both theoretically and experimentally for many years [1, 2]. The interest in the study of GH shift renews after the recent predictions of large or giant GH shifts (i.e., defined as a situation when the absolute value of the GH shift is equal to or lager than the waist of the incident beam ) at the surfaces of some special media or structure such as left-handed materials (LHMs) [3, 4], one-dimensional photonic crystal (PC) with a defect , and some absorbing media [6, 7]. However, in these situations, the reflected beam may split into two beams for a narrow incident beam [3, 4] or the reflectance is small [6, 7], when a giant GH shift occurs.
GH shifts for a PC at a frequency inside or outside a bandgap have also been discussed [8, 9]. However, they are only positive and not giant. Since negative refraction index can enable backward waves , which may give a giant negative GH shift, the giant negative GH shifts may also occur for a beam totally reflected from a PC with a negative effective refractive index. In the present paper, we will show that a giant negative GH shift can be achieved if there is a homogeneous cladding layer of appropriate thickness. Furthermore, the profile of the totally reflected beam may remain nearly the same as that of the incident beam, even when a giant GH shift is achieved for a narrow incident beam.
2. Calculation and analysis
We study the same two-dimensional (2D) PC structure of negative refraction as the one considered in Ref . The 2D PC is formed by a triangular lattice of air holes (with radius 0.4a; a is the lattice constant) in GaAs background (n =3.6). The air-PC interface is normal to the Γ - M direction. To achieve a giant negative GH shift, the GaAs background is terminated in such a way that there is a homogeneous cladding layer of thickness d over the PC half-space [see Fig. 1(a)]. For E-polarization (i.e., the electric field is perpendicular to the plane of the holes), the effective refraction index (neff) of the PC is nearly isotropic and has a negative value in a frequency window ranging from 0.29 (2πc/a) to 0.345 (2πc/a) [see the inset of Fig.1(a)]. The incident beam impinges on the PC structure from air. To achieve a total reflection from the PC structure (besides a giant negative GH shift), we will only consider the situation when -1 < neff < 0. If we regard this PC as a homogeneous medium, the condition for the total internal reflection (from air to the PC) at an air-PC interface [i.e., with d=0 in Fig. 1(a)] is (from Snell’s law): sin(θi) ≥ (∣neff∣/nAir). From Fig. 1(b) one sees that the total reflection condition calculated by sin(θi) ≥ (∣neff/nAir|) is consistent with the numerical result calculated by a 2D layer-KKR (Korringa-Kohn-Rostoker) method [11, 12]. Here the incident angle is indicated through the x component of the wave vector in air, i.e., kx = sin(θi)ω/c.
The Gaussian beam incident on the PC can be expressed as follows (see e.g. ),
where A(kx, w) = 0.5wπ -1/2exp[-w 2(kx-k 0sinθi)2/4], w is the waist of the Gaussian beam, θi is the mean angle of incidence of the Gaussian beam, and k 0 is the wave number in air. For simplicity, in this paper we only consider the situation when λ 0 > 2a (λ 0 is the free space wavelength in vacuum), for which the reflection from the PC to air contains only the zero order of diffraction. We simulate the reflection of the Gaussian beam from the interface of the PC by using a layer-KKR method [11, 12]. Since the center of the incident beam is at x=0, the GH shift (the displacement between the centers of the incident and reflected beams) can be calculated by
Figure 2 shows the GH lateral shifts as the mean incident angle increases at several different frequencies when the Gaussian beams (with beam width w = 25a) are totally reflected by the PC with d = 0 (i.e., no homogeneous cladding layer). The GH lateral shifts are negative for the incident beams of low incident angles or low frequencies [ω ≤ 0.325(27πc/a) in Fig. 2.]. The negative GH shifts are caused by the backward (total) energy flux flow of the evanescent wave [13, 14] or leaky surface wave [15, 16]. However, the lateral shifts are small (less than a; the beam waist is 25a) when the GH shift is negative.
The GH shift can be enhanced greatly by a homogeneous cladding layer due to the excited leaky or surface waves [15–18] which transfer the energy of the incident beam along the interface. These surface (or leaky) waves may be backward or forward [3, 15], and a giant negative or positive GH lateral shift may occur if some appropriate surface or leaky wave is excited in the PC structure of Fig. 1(a). Figures 3(a) and 3(b) show the GH lateral shifts and the width of the reflected beam (which is calculated from the beam profile) as the thickness of the cladding layer increases when the parameters for the incident beam are chosen as ω = 0.335(2πc/a), w = 25a and θi = 45c. As expected, the giant negative GH beam shift occurs at some special values of d. The corresponding (normalized) field intensity profiles of the reflected beams are shown in the insets (the unit for the lateral axis is a) in Fig. 3. From insets (2) and (3) of Fig. 3(a) one can see that the reflected beam has double peaks. This indicates a backward leaky wave and a forward wave are exited simultaneously when a Gaussian beam is incident on such a PC structure. The main peak has a giant negative lateral shift due to the excited backward waves, while the small peak of positive lateral shift is due to the forward waves. While the small peak of positive lateral shift is due to the forward waves. After optimization (by eliminating or suppressing the excitation of any forward wave), our PC structure can give a giant negative GH shift and the peak with a positive shift due to some forward wave almost disappears [see insert (3) in Fig. 3(b)].
The PC structure in air can be considered as a 3-layered structure: the air layer, the cladding layer and the PC layer, as shown in Fig. 4(a). For the configuration associated with Fig. 3, when the light (transmitted through the air-cladding interface) is incident on the PC layer, two beams of diffraction orders [i.e., the 0th and (-1)-th] are reflected at the cladding-PC interface [see the right part of the ray diagram in Fig. 4(a)]. However, only the reflected ray of 0th diffraction order can transmit into the air, and that of the (-1)-th diffraction order is totally reflected (internally) at the air-cladding interface since k -1,x =2π/a - k 0 (k -1,x is the x component of the wave vector for the ray of (-1)-th diffraction order). The light beams of (-1)-th and 0th reflection order are reflected at the two interfaces, and the corresponding leaky waves can be excited when one of them satisfies the following self-consistent condition :
where i = -1,0 correspond to the beams of (-1)-th and 0th diffraction orders, respectively, kn is the wave number in the cladding layer, φi,A is the phase shift for the corresponding diffraction order beam reflected back to the cladding layer at the air-cladding interface, and φi,B is for the beam reflected by the PC layer. The resonance of the light of (-1)-th diffraction order [the three white rays in Fig. 4(a)] forms a backward leaky mode (in a zig-zag way from the right to the left), for which the energy transfers backward along the cladding layer. The resonance of the light of 0th diffraction order gives a forward leaky mode. These leaky modes (in the cladding layer) will be coupled with the leaky surface waves on the PC-cladding interface, and enhance either the forward or the backward surface waves [16, 19]. Both the surface waves and the leaky waves contribute to the GH effects. If our PC structure has a thick cladding, the direction of the GH shift mainly depends on the energy flux of the leaky modes excited in the cladding layer, since the energy flux of the surface wave is small as compared with that of the leaky modes. However, if our PC structure has a thin cladding, the situation will be quite different, and it is difficult to predict analytically whether the total energy flux of the leaky waves in the whole PC structure is forward or backward.
The negative GH shifts become large for the PC structure with a thin cladding layer [corresponding to the tips in Fig. 3(a)] when Eq. (3) for the 0th diffraction order is satisfied. This can be explained as follows. When Eq. (3) for the 0th diffraction order is satisfied, the phase difference between ray (2) (direct reflection of the incident ray at the air-cladding interface) and ray (3) [transmission (into air) of the reflected wave (at the cladding-PC interface) of the 0th order diffraction] is exactly equal to (2p+1)π (i.e., out of phase; p is an integer). Such a destructive interference of rays (2) and (3) prevents the energy of ray (1) (incident ray) from being directly reflected to air at the air-cladding interface, and also makes backward surface waves at the PC-cladding interface more difficult to leak into air. Consequently, the energy flowing backward along the surface of the PC structure increases, which then enhances the negative GH shift. On the contrary, for the case of constructive interference between rays (2) and (3), most energy of the incident light is directly reflected to air at the air-cladding interface and thus contributes little to the GH shift. Consequently, the negative GH shift effect becomes small even a backward leaky mode in the cladding layer is excited.
When Eq. (3) is satisfied or almost satisfied for both the (-1)-th and 0th diffraction orders, a backward leaky mode is excited in the cladding layer. Thus, the backward energy flux increases greatly, and the negative GH shift becomes giant [as for the cases of insets (1)-(3) in Fig. 3] or almost giant [marked with stars in Fig. 3(a)]. The resonance condition for the (-1)-th diffraction order is essential for a giant negative GH shift. We found that when the refractive index of the cladding layer is reduced to n clad=2.0 so that only the diffraction beam of the 0th order can be transmitted into the cladding layer, the giant negative GH disappears [see Fig. 4(b)].
A narrower beam, which has a wider spatial spectrum [3, 4], is easier to excite simultaneously the backward and forward leaky waves. Thus, the reflected beam may split into two (or even more) peaks and the width become much wider if we do not optimize our PC structure (e.g. the thickness of the cladding layer). After optimization, our PC structure can give a giant negative GH shift and the profile of the reflected beam remains nearly the same (i.e., a Gaussian beam without any noticeable side lobe). Figure 5 shows the simulation result (with the FDTD method ) for the distribution of the electric fields of such a case. The parameters for the Gaussian beam are ω =0.342(2πc/a), w = 10a, θi = 57° and the cladding thickness is d = 0.113a. From this figure one sees that the Gaussian profile of the totally reflected beam remains well and the negative lateral shift of the center of the main peak is giant [the GH lateral shift calculated by the layer-KKR is about 11a, which is larger than 10a (the waist of the incident beam)].
When the light in the cladding layer contains other high orders of diffraction (such as the +1-th diffraction order), the GH shift effects will become more complicated. Then the GH lateral shift can be giant positive or giant negative, and sensitive to the thickness of the cladding layer or a small change in the refractive index of the cladding layer at some special values of d. Figure 6 shows the GH lateral shift as the refractive index n clad of the cladding layer varies slightly (from 3.578 to 3.606). The parameters for the incident Gaussian beam are ω = 0342(2πc/a), w = 25a and θi =27.73°. The thickness of the cladding layer is kept as d=0.75 a. The profiles of the field intensity of the reflected beams at three different values of nclad are also shown in the insets of Fig. 6. The center of the totally reflected Gaussian beam (the profile remains nearly the same) can shift laterally from a positive position to a negative position by over 35a (larger than the waist of the beam) when there is a small deviation of 0.02 in the refractive index of the cladding layer (the refractive index for the host medium of the PC remains unchanged). This extraordinary property has potential applications as e.g. an optical switch (the small change of the refractive index can be induced by an applied voltage or temperature change) , a modulator, and a sensor.
In the present paper we have studied the Goos-Hänchen lateral shift effects for the total reflection upon a structure of PC with negative effective refraction index. The mechanism of our PC structure is based on both the backward waves and the grating effect, whereas the mechanism of an LHM structure is based on only the backward waves (no grating effect). For example, in our PC structure, the (-1) diffraction order (due to the grating effect) plays an important role in achieving a giant negative GH shift. By choosing an appropriate thickness d of the cladding layer, the totally reflected beam can give a giant negative GH lateral shift while keeping a single beam of good profile even for a very narrow incident beam. For an appropriately designed thickness of the cladding layer, the GH lateral shift can be very sensitive to a small change of the refractive index of the cladding layer, and this property can be utilized for the applications of switching, modulating and sensing.
The authors are grateful to Jun She of Zhejiang University for some helps in the calculation. This work is partially supported by the National Basic Research Program (No. 2004CB719802) and an additional support from the Science and Technology Department of Zhejiang Province. S. L. He's email address is email@example.com.
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