1. Introduction

The field of transformation optics began with two papers published back-to-back in Science written by J. Pendry [1] and U. Leonhardt [2]. The latter paper is related to transformation optics with isotropic and hence polarization-insenstive materials, for example to carpet cloaking proposed by J. Pendry [3] and demonstrated by some groups later [4–7]. A new application of transformation optics is the transmutation of singularities [8], demonstrated by U. Leonhardt et al [9]. Recently, broadband, non-Euclideam transformation optics was developed [10], and a primer on transformation optics also appeared [11]. Using transformation optics it became possible to implement optical devices that were previously difficult to achieve using classical optics, such as the invisibility cloak [1,12,13], the field rotating device [14] and the field concentrator [15]. In 2008, Rahm et al proposed an embedded coordinate transformation approach [16], which permits a discontinuity at the outer boundary of the transformed spatial zone. Many important functional elements, such as reflectionless beam shifters/splitters [16], cylindrical-to-planar wave converters [17], wave collimators [18,19], beam bends [18,20], beam compressors/expanders [20], beam splitters and polarization rotators [21], can all be designed based on this idea. However, these designs are neither perfect [11] nor unified [16–21]. For instance, both the splitting angle and the transmission of the beam shifter/splitter in Ref [21]. cannot be flexibly adjusted. In this work, the authors suggest a universal transformation formula based on the embedded coordinate transformation approach. Through the use of this function, several physical processes can be controlled flexibly, and a series of functions for the above elements, with the exception of the polarization rotator, can be achieved easily.

2. Theoretical basis

In the two-dimensional case, the behavior of TE/TM waves can be controlled individually by the material parameters $\mu /\epsilon$. This is the basis for design of functional elements based on the embedded coordinate transformation approach.

Let zone A be the free space, and zone B be the transformation zone (see Fig. 1 ). The beam passing through zone B will be transformed, and the transformed coordinates are given by:

 

Fig. 1 Schematic diagram of the embedded coordinate transformation for a polarization controller.

Download Full Size | PDF

For the TE wave, $\Psi ={\mu }_r$, ${\epsilon }_r=1$;for the TM wave, $\Psi ={\epsilon }_r$, ${\mu }_r=1$. Then, a universal transformation formula is introduced:

In Eq. (2), a₂₁ and a₂₂ are independent of x and y for linear transformation, which means that the transformation is not related to the sign (plus or minus) of x and y. However, for nonlinear transformation, x and y are included in a₂₁ and a₂₂, i.e., this kind of transformation is related to the sign (plus or minus) of x and y. In this letter, for simplicity, only cases where x and y are both positive was discussed. Some functional elements such as those discussed in Ref [10]. are thus not demonstrated here.

3. Numerical simulation and analysis

The behavior of the beam after the transformation determined by Eq. (3) is here. For simplicity, only the TM wave is discussed. The discussion for the TE wave is similar to that for the TM wave.

Assuming that the incident beam is a Gaussian beam with a frequency of 5GHz, and the beam waist diameter is 0.1m, the center of the beam waist is located at the center of the transformation zone B, i.e., at the point (0.5, 0.5) in the zone of $0.4\le x\le 0.6$ and $0\le y\le 1$ shown in Fig. 2.

3.1 Linear transformation

Let $m=n=1$, and $s=t=0$ or $k_3=0$, and Eq. (3) becomes $f\left(x,y\right)=k_{1}x+k_{2}y$. This is a linear transformation. Figure 2 shows the transformation results for change in the value of $k_1$, while keeping $k_2$ unchanged. It can be seen that the role of $k_1$ is to control the deviation angle of the beam in the transformation zone. When $k_1>0$, the beam will deviate upwards. When $k_1<0$, the beam will deviate downwards. Thus, beam displacement at the exit surface of the transformation zone is induced by change in the value of $k_1$.

 

Fig. 2. Field distribution of linear transformation.(a) $k_1=1.8,k_2=1$(b) $k_1=1.2,k_2=1$.

Download Full Size | PDF

 

Fig. 3. Energy distribution of linear transformation. (a) $k_1=1.2,k_2=1$. (b) $k_1=1.2,k_2=0.5$.

Download Full Size | PDF

Figure 3 shows the transformation results for change in the value of $k_2$, while keeping $k_1$ unchanged. It can be seen that the change in $k_2$ does not change the deviation angle of the beam in the transformation zone, but changes the transmitted energy (or the coupling efficiency between zone A and zone B in Fig. 1). This means that the transformation can control the transmitted energy of the TE/TM waves. The reason for this is that, when $\left|k_2\right|\ne 1$, the permittivity in the y direction (see Eq. (2)) of the transformation zone B and that of the free space A are different. This results in reflection at the interface between A and B ($x=0.4,0.6$), and thus varies the transmitted energy. Calculations show that the transmitted energy is independent of $k_1$, and is related only to the value of $k_2$. The coupling efficiency reaches its maximum around $k_2=1$. Therefore, $k_1$ can be regarded as the controlling parameter for the deviation angle, and $k_2$ can be regarded as the controlling parameter for the coupling efficiency of the transmitted energy between zone A and zone B.

3.2 Nonlinear transformation without cross product term

Let$s=t=0$ or $k_3=0$, Eq. (3) becomes$f\left(x,y\right)=k_1{x}^m+k_2{y}^n$. When $n=1,m=2$, $k_1=1.8,k_2=1$, the transformation result is as shown in Fig. 4(a), and when $n=1,m=5,k_1=1.8,k_2=1$, the transformation result is as shown in Fig. 4(b). It can be seen that the bigger the value of m, the less sensitive the deviation angle becomes to the value of$k_1$. For cases where $n=1,m<0$, the transmission result is as shown in Fig. 4(c). It can be seen that when m is negative, the deviation angle can approach $90°$for small values of $k_1$. When m is negative and $k_1$ is larger than a certain critical value, the deviation angle of the beam can be larger than $90°$ and can achieve total reflection; this phenomenon is shown in Fig. 4(d).

 

Fig. 4. Field distribution of nonlinear transformation without cross product term. ($m\ne 1$, n = 1)(a) $m=2,k_1=1.8,k_2=1$.(b) $m=5,k_1=1.8,k_2=1$. (c) $m=-2,k_1=0.3,k_2=1$. (d)$m=-2,k_1=3,k_2=1$.

Download Full Size | PDF

The features of the transmission controlled by the second term in Eq. (3) are discussed as follows. Because it is known that $k_1$controls only the deviation angle, $k_1$ is assumed to be 0, i.e. $f\left(x,y\right)=k_2{y}^n$ to show more clearly the rule of the coordinate transformation when $n\ne 1$. The results are shown in Fig. 5. It can be seen that the light beam deviates upwards when $n>1$, and downwards when $n<1$. This function can be used to control the deviation of the light beams.

 

Fig. 5. Field distribution without cross product term for various values of n when .$k_1=0$..(a) $n=2,k_1=0,k_2=1$. (b) $n=3,k_1=0,k_2=1$. (c) $n=0.5,k_1=0,k_2=1$.

Download Full Size | PDF

3.3 Nonlinear transformation with only cross product term

Let $k_1=0,k_2=0$, and the transformation formula becomes $f\left(x,y\right)=k_3{x}^s{y}^t$. Certain special cases are analyzed as follows.

The transmission results are shown in Fig. 6. They demonstrate that the beam deviates upwards when $k_3>0$, and downwards when $k_3<0$, i.e., the bigger the value of $k_3$, the larger the deviation angle, and when $k_3$ is a large enough value, total reflection with reflectivity of nearly 1 can be achieved. In this case, $k_3$ also influences transmission.

 

Fig. 6. Nonlinear transformation containing only cross product term. ($s=1,t=1$) (a) and (b): k₃ = 1. (c) and (d): k₃ = 80. (a) and (c): Field distribution. (b) and (d): Energy distribution.

Download Full Size | PDF

The transformation results are shown in Fig. 7. It can be seen that the function of the order of the x transformation is to change the propagation route of the light beam in transmission zone B, or to change the wavefront of the beam and change the Gaussian beam into a spherical wave.

 

Fig. 7. Nonlinear transformation containing only cross product term. ($s\ne 1,t=1$). (a) and (b): $s=2,t=1,k_3=2$. (c) and (d): $s=-2,t=1,k_3=0.3$. (a) and (c): Field distribution. (b) and (d): Energy distribution.

Download Full Size | PDF

The transformation results are shown in Fig. 8. It can be seen that although the change of the order of y transformation may change the propagation route in zone B, its control over the transmitted beam is obviously different from that of the x transformation mentioned above.

 

Fig. 8. Field distribution of nonlinear transformation containing only cross product term. ($s=1,t\ne 1$) (a) s = 1, t = 0.2,k₃ = 1. (b) s = 1, t = 2, k₃ = −2.5. (c) s = 1, t = 2, k₃ = −1. (d) s = 1, t = 2, k₃ = 1.5.

Download Full Size | PDF

4. Discussion and applications

The above analysis discussed the roles of the various parameters in transmission formula (3) related to the TM wave only. In this section, it is considered together with the transformation of the TM and TE waves, and its possible applications are also discussed.

Let the transformation formulas for TM and TE waves be$f\left(x,y\right)=k_1{x}^m+k_2{y}^n+k_3{x}^s{y}^t$ and $f^{\prime }\left(x,y\right)={k^{\prime }}_1{x}^{m^{\prime }}+{k^{\prime }}_2{y}^{n^{\prime }}+{k^{\prime }}_3{x}^{s^{\prime }}{y}^{t^{\prime }}$, respectively.

The first application is a tunable polarization beam splitter.

(1) A linear polarization beam splitter can be implemented when $f\left(x,y\right)$ and $f^{\prime }\left(x,y\right)$ are both linear functions. The transformation results are shown in Fig. 9. The function of this splitter is to split the incident beam into parallel transmitted TE and TM waves, and the position and intensity of the output are adjustable.

 

Fig. 9. Linear polarization beam splitter with adjustable angle and transmission. (a) and (b): Field transformation results for angle controlled linear polarization beam splitter. (a): $k_1=1$,${k^{\prime }}_1=1.8$,$k_2={k^{\prime }}_2=1$. (b): $k_1=-0.5$, ${k^{\prime }}_1=1.8$, $k_2={k^{\prime }}_2=1$. (c) and (d): Energy transformation results for transmission controlled linear polarization beam splitter. (c): $k_1={k^{\prime }}_1=1.8$, $k_2=1$, ${k^{\prime }}_2=0.5$. (d): $k_1={k^{\prime }}_1=1.8$, $k_2=1$, ${k^{\prime }}_2=3$.

Download Full Size | PDF

(2) A mixed linear and nonlinear transformation polarization beam splitter can be implemented when $f\left(x,y\right)$ is a linear function (or nonlinear function) and $f^{\prime }\left(x,y\right)$ is a nonlinear function (or linear function). Figure 10 shows the transmission results when $f\left(x,y\right)$ is the linear function ($k_1=1.8,k_2=1,m=n=1$, $k_3=0$) and $f^{\prime }\left(x,y\right)$ is the nonlinear function. It can be seen that not only are angle and transmission controlled polarization beam splitters achieved, but also one or both of the split TE and TM beams, can be changed into spherical waves which are different from the incident light.

 

Fig. 10. Field distribution and energy distribution of mixed linear and nonlinear transformation polarization beam splitters. (a):$k_1=1.8,k_2=1,m=n=1$,$k_3=0$,${k^{\prime }}_1=5$, ${k^{\prime }}_2=1$, $m^{\prime }=5$, $n^{\prime }=1$, ${k^{\prime }}_3=0$. (b):$k_1=1.8,k_2=1,m=n=1$, $k_3=0$, $k_1^{\text{'}}=k_2^{\text{'}}=0$,$k_3^{\text{'}}=5$,$s\text{'}=t\text{'}=1$. (c): $k_1=1.8,k_2=1,m=n=1$, $k_3=0$, $k_1^{\text{'}}=k_2^{\text{'}}=0$, $k_3^{\text{'}}=-0.55$, $s\text{'}=-0.5$, $t\text{'}=1$.

Download Full Size | PDF

(3) A nonlinear transformation polarization beam splitter can be achieved when$f\left(x,y\right)$ and$f^{\prime }\left(x,y\right)$ are both nonlinear functions. Figure 11 shows the typical transformation results. Not only can a polarization beam splitter with controllable angle and transmission be obtained, but the split TE/TM polarized beams can also be made to have different propagation routes.

 

Fig. 11. Nonlinear transformation polarization beam splitter. (a) and (b): $k_1=3,k_2=1,m=4,n=1,k_3=0$, $k_1^{\text{'}}=3,k_2^{\text{'}}=1,m\text{'}=4,n\text{'}=1,k_3^{\text{'}}=0$. (c) and (d): $k_1=0$,$k_2=1$,$k_3=0$,$n=3$,$k_1^{\text{'}}=k_2^{\text{'}}=0$,$k_3^{\text{'}}=3$,$s\text{'}=t\text{'}=1$. (a) and (c): Field distribution. (b) and (d): Energy distribution.

Download Full Size | PDF

The second application is a polarizer.

The polarizer can be implemented using either a linear or a nonlinear transformation, i.e. there is only a TE or a TM wave in the transmitted light after passing through the transformation region, and the position, direction and intensity of the polarized beam are all controllable. Based on the above discussions, a polarizer could be implemented using any one of the following methods:

(1) For a linear transformation, adjust $k_2$ to make the coupling efficiency of the polarized beam zero;

(2) For a nonlinear transformation containing no cross product term, when m is negative, adjust $k_1$ to make the deviation angle larger than 90°, as shown in Fig. 12.

(3) For a nonlinear transformation containing only a cross product term, adjust $k_3$to produce a reflectivity value of 1.

Any one of these methods could be selected according to the user’s need.

 

Fig. 12. A linear and nonlinear mixed transformation TM wave polarizer. (a) and (b): $k_1=1.8,k_2=1,k_3=0,m=n=1$,$k_1^{\text{'}}=3,k_2^{\text{'}}=1,k_3^{\text{'}}=0,m\text{'}=-2,n\text{'}=1$. (a) Field distribution. (b): Energy distribution.

Download Full Size | PDF

The third application is a “cloak” based on polarization splitting.

Elements with numerous functions can be obtained by arranging or combining a number of transmission regions. For example, a “cloaking” effect based on polarization splitting could be achieved by using a combination of two transmission regions to cause the output wave to resume the pre-incident state. Let $f_i\left(x,y\right)$ and ${f^{\prime }}_i\left(x,y\right)$ (i = 1, 2) be the transformation functions of the TM and TE waves in the ith transformation region, respectively, and the thicknesses of the two regions are $d_1$and $d_2$, respectively. Many transformation phenomena could be obtained by adjusting $f_i\left(x,y\right)$, ${f^{\prime }}_i\left(x,y\right)$ (i = 1, 2), $d_1$and $d_2$. In Fig. 13, the upper and lower rows show field and intensity transformations, respectively. The field and intensity distributions at the incident plane of the first transformation region and the exit surface of the second transformation region can be made exactly the same by adjusting the values of the parameters of the two functions. This is a “cloak”. The circular region in Fig. 13 is the region of invisibility. This cloak is proposed for the first time by the present authors, and is different from the type of cloak which has been widely discussed recently. In this cloak, the TE and TM components of the incident beam separate and pass around an object and then combine again, to make the object “invisible”. The diameter of the cloak region in Fig. 13 is 0.05m. If the beam is obliquely incident (Fig. 13(c)), then the angle of invisibility will be limited by the deviation angle, and the sizes of the transformation region and the cloak region. For example, in a linear transformation, when $k_1=k_1^{\text{'}}=1.5$, the range of the angle of invisibility is about [–20°~20°].

 

Fig. 13. Typical examples of cloak based on polarization splitting. (a) Combination of two linear transformations regions ($d_1=d_2$). For the first transformation region $k_1={k^{\prime }}_1=1.8$, $k_2={k^{\prime }}_2=1$, $m=m^{\prime }=1$, $n=n^{\prime }=1$, $k_3={k^{\prime }}_3=0$. For the second transformation region $k_1={k^{\prime }}_1=-1.8$, $k_2={k^{\prime }}_2=1$, $m=m^{\prime }=1$, $n=n^{\prime }=1$, $k_3={k^{\prime }}_3=0$. (b) Combination of two nonlinear transformation regions ($d_1=d_2$). For the first transformation region $k_1={k^{\prime }}_1=0.9$, $k_2={k^{\prime }}_2=1$, $m=m^{\prime }=-0.5$, $n=n^{\prime }=1$, $k_3={k^{\prime }}_3=0$. For the second transformation region $k_1={k^{\prime }}_1=-1.7$, $k_2={k^{\prime }}_2=1$, $m=m^{\prime }=-0.5$, $n=n^{\prime }=1$, $k_3={k^{\prime }}_3=0$. (c) Oblique incidence (incident angle is 6°, $d_1=d_2$). For the first transformation region $k_1={k^{\prime }}_1=1.5$, $k_2={k^{\prime }}_2=1$, $m=m^{\prime }=1$, $n=n^{\prime }=1$, $k_3={k^{\prime }}_3=0$. For the second transformation region $k_1={k^{\prime }}_1=-1.5$, $k_2={k^{\prime }}_2=1$, $m=m^{\prime }=1$, $n=n^{\prime }=1$, $k_3={k^{\prime }}_3=0$. (d) Combination of two regions with different thickness ($1.5d_1=d_2$). For the first transformation region $k_1={k^{\prime }}_1=0.25$, $k_2={k^{\prime }}_2=1$, $m=m^{\prime }=-1$, $n=n^{\prime }=1$, $k_3={k^{\prime }}_3=0$. For the second transformation region $k_1={k^{\prime }}_1=2.5$, $k_2={k^{\prime }}_2=1$, $m=m^{\prime }=9$, $n=n^{\prime }=1$, $k_3={k^{\prime }}_3=0$.

Download Full Size | PDF