In order to conform the relation of Eq. (4.5), we use the FDTD method to study the refraction phenomenon in material. As plane waves in a nonconductioin medium the phase velocity of the wave is
n c k
v= w= =
µε
1 . The light propagation form can be described with
phase velocity directly. This is the conventional refraction phenomenon. The Fig. 15 is shown the EM wave propagation through the glass prism. We can find the normal refraction angle various from different refractive index of glass. So, I can find the light cannot propagate through the glass prism. Because the glass prism and air are all isotropic medium, the propagation of EM wave can be defined as the normal direction of the phase wave front directly. The amplitude distributions can rightly display the refraction phenomenon also. When the index is 2 , the all-refractive angle is 450. This is to be what we show. Other conditions, including the refractive index are 1.2 & 1.3, are displayed.
Fig. 15 The EM wave propagates through the glass prism. The refractive phenomenon are shown as the refractive index are 1.2, 1.3, and 1.414. When the index is 2, the all-refractive angle is 450.
We know the phase velocity is shown as the velocity of the propagation of an equi-phase surface. This velocity has a definite meaning, for example, for plane waves and spherical waves for which the equi-phase surface can be defined without ambiguity. In the photonic crystal, however, the equi-phase surface cannot be defined rigorously, since its eigenfunction is a superposition of plane waves. This means that the phase velocity cannot be defined appropriately in the photonic crystal. As to former, the light propagates with the block wave form in the periodic structure; the refraction index cannot be described with the former EM wave theory all.
The phase velocity and group velocity, the both of all, directly involves the refraction phenomenon.
We observe the negative refraction phenomenon with the experiment result already as the EM wave propagate through the photonic crystal prism in negative index state. Now, we use the FDTD algorithm to simulate this phenomenon.
In order to understand our former experimental result that the light propagation in photonic crystals is due to highly anisotropic dispersion surfaces derived by photonic band structure, we have numerically analyzed experiment of a negative index of refraction that the EM wave is refracted by the photonic crystal prism with the finite-difference time domain (FDTD) algorisms [20]. In this simulation, we focus on the structure as shown in the experimental setup. The structure with the photonic crystal prism as our experimental setup is modeled and performed FDTD numerical simulatio ns with perfectly matched layer boundary regions. We show that EM waves, in a frequency range of microwave, undergo negative refraction in a two-dimensional (2D) prism-shaped PC, a photonic crystal prism (PCP). From the refraction angle s of EM waves propagating through a PCP, we deduced the indices of negative refraction according to the Snell’s law. We choice a plane wave as the incident source, because the wave propagate incident the PCP former plat was planar wave fronts in our real experimental situation. The results are shown in Fig. 16 and Fig. 17 that shows wave fronts with fixed frequency in phase space. It is clear that the incident E-polarized EM wave was refracted by the photonic crystal prism. As to the phase velocity, we use the FDTD method in phase space to find out the phase wave fronts through the photonic structure.
The Fig. 16 show that the plan wave incident the PCP and the discussion surrounding are in the phase domain. The phase wave fronts were refracted with the interface between the air and the photonic crystal. We can use the wave fronts to define the phase refractive index. The phase refraction phenomenon was deduced form the phase velocity which differs from anisotropic and isotropic medium. In our simulation, we can find the refraction differ form varies wavelengths.
The negative refraction phenomenon in our simulation problem can correspond to the photonic band structure. It occurs as the fixed frequency in the negative state band. However, the fixed frequency in the three and fourth band regime exist two modes, phase wave fronts display the singularity point. With the Snell’s law we defined the refractive index directly. The Fig. 16 show the refractive angle is -180 at 16GHz. In the forward direction, we can still find there is the EM wave propagation also. However, the plane wave run through the photonic crystal prism was refracted to the negative angle. In fact, the phase distribution in our discussion domain can be defined as the phase refraction index explicitly. With the Snell’s law, we can get the phase refraction index. The direct direction was the phenomenon of the
Fourier transform form real space to phase space of the FDTD, even if the propagation amplitude in direct direction is not significant.
Fig. 16 This shows the refractive angle is -180 at 16GHz
14GHz
15.5GHz
17.0GHz
Fig. 17
This shows the refractive angle is different form the different wavelengths
In the negative refraction medium, the phase velocity and the group velocity is anti-parallel to each other, one is not able to tell which way is the forward direction since the wave vector has been folded into the first Brilluoin zone through translation symmetry. Therefore, strictly speaking, only the product of phase index and group index has the physical meaning, thus the phase index and the group index are inseparable. The product is negative in this medium and one can not determine which one of them is negative. By consulting the band structure, the maximum of the fourth band is a result of two non-crossing bands coupling to each other, therefore, the one-band model may be not adequate and two-band model is needed to describe this problem. It implies that the product of phase index of the Bloch wave and the group index
n n p g
λ λ is equivalent to the refractive index of the photonic crystal. It implies that the product of phase index of the Bloch wave and the group index λ λ is equivalent to the refractive index of np gn
the photonic crystal.
If we define the phase velocity for the Bloch wave of wave vector k as vp = n
( )
k s
ω k e and
the group velocity as vg =∇sωn
( )
k , the phase velocity and the group velocity produced to each other is the same to the dispersion relationship of the k•p theorem. The form of λ λnp gn is considered the folding effect of the periodic structure. This physical equation includes the wave vector folding to the first Brilluoin zone. That is must be considered at the photonic band structure. We can directly describe the light beam propagate thorough the negative refraction medium with this form.The Fig. 19 show s the λ λ match to the experiment data. Although the two refracted mode np gn
propagation in the medium, the phase front was not clearly defined. We still can find the experiment data was corresponded to mixed mode well.
14.0 14.5 15.0 15.5 16.0 16.5 17.0 -4.5
-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Refractive Index
Frequency (GHz)
Phase Refraction Index Group Refraction Index
Fig. 18 The phase refraction indices are defined with the refraction angle of FDTD simulation results. the group refraction indices are derived from the photonic band structure.
14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 -1.4
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
Refractive Index
Frequency (GHz)
Fig. 19 Comparison between the experimental refractive indices and the theoretical effective refracive indices. The refractive indices of dash-dotted curve are defined with the product of the FDTD simulation results with the Snell’s law and the group indices.
We have observed the negative refraction phenomenon with the FDTD algorism. The phase fronts can be refracted as the EM wave through the photonic crystal prism. We can define the phase refraction index with the Snell’s law. From the photonic crystal band gap, we can get the group velocity with various wave lengths. Whereas, we show that only the product of phase index and group index has the physical meaning. The calculated product of phase index and group index derived from the FDTD algorism fit reasonably well with the experimental data. And, according to the extended k•p theory, the group velocity of EM waves in PC can be defined as the ratio of momentum to relativistic effect mass of a quasi-particle. The EM wave propagating in PC behaves like a massive quasi-particle. The phase refraction index is the same to the refraction factor with it. The simulation with the FDTD method told me this fact.