- Email: [email protected]
Design and simulation of polarization transformers using transformation electromagnetics
Accepted Manuscript Title: Design and simulation of polarization transformers using transformation electromagnetics Author: Sayyed Saleh Sayyed Mousavi Mohammad Saeed Majedi Hossein Eskandari PII: DOI: Reference:
S0030-4026(16)31465-6 http://dx.doi.org/doi:10.1016/j.ijleo.2016.11.129 IJLEO 58545
To appear in: Received date: Accepted date:
4-10-2016 24-11-2016
Please cite this article as: Sayyed Saleh Sayyed Mousavi, Mohammad Saeed Majedi, Hossein Eskandari, Design and simulation of polarization transformers using transformation electromagnetics, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.11.129 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Design and simulation of polarization transformers using transformation electromagnetics Sayyed Saleh Sayyed Mousavia,b , Mohammad Saeed Majedib, , Hossein Eskandaria,b a
Communications and Computer Research Center, Ferdowsi University of Mashhad, Mashhad, Iran b
Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Abstract In this paper we design polarization transformers using transformation electromagnetics method. We introduce a new coordinate transformation that by compressing or expanding the space, creates a desired delay on one component of electromagnetic fields and therefore causes the polarization of electromagnetic wave to change. Linear to circular, right (left) hand to left (right) hand circular and linear to linear polarization transformers have been designed and analyzed. The proposed polarization transformers are homogeneous and reflectionless and they can also be non-magnetic. COMSOL Multiphysics software is used to validate our results. Keywords: polarization transformer, transformation electromagnetic, non-magnetic medium, coordinate transformation.
1. Introduction In 2006 an article was published that introduced a systematic methodology capable of controlling the electromagnetic field by defining appropriate permittivity and permeability tensors for the medium [1]. This method is called transformation electromagnetics or transformation optics. The design procedure suggested by transformation electromagnetics is frequency independent. Generally the permittivity and permeability obtained by this method lead to an inhomogeneous and anisotropic medium. Transformation electromagnetics has a variety of applications in design of microwave and optic devices. There are several articles that used transformation electromagnetics in cloaking devices that was realized by metamaterial structures [2-6]. It is also used in planar photonic devices for integrated system including waveguide and photodetectors [7]. Furthermore, devices such as beam bender and expander [8], beam polarization splitter and rotator [9, 10], field concentrators [11, 12], flat focusing lenses [13] and wave collimators [14], have been proposed based on this method. Polarization state is one of the most important properties of any electromagnetic wave. Changing the polarization of wave is a practical issue in microwave engineering and optics. In literature, there are several devices, such as wave retarders, that perform polarization transformation. Retarders exert a delay on one component of the electromagnetic fields and therefore change the polarization. Most popular types of retarders are well known full, half and quarter wave plates [15]. Also, Fresnel rhomb and Babinet compensator are examples of retarders [15]. Furthermore polarization transformation for a plane wave is done by transversely uniaxial chiral medium [16, 17]. There are also structures that composed of waveguides, two phase shifter, TE to TM converter and crystals that are capable of performing arbitrary polarization transformations [18, 19]. In some articles the polarization transformer is implemented by metamaterial structures such as split-ring resonators [20-22] or metallic helices [23].
Corresponding author. Tel.: +98-513-880-5171 E-mail Address: [email protected].
In this paper, we propose a different strategy for polarization transformation based on transformation electromagnetics method. Accordingly, we present a new coordinate transformation that by compressing or expanding the space, creates a desired phase difference between orthogonal components of electromagnetic fields and this leads to the polarization changes. It will be pointed out that a proper linear coordinate transformation can lead to a homogeneous, reflectionless and non-magnetic media if only the TM polarization is modified. Based on this method, linear to circular, right (left) hand to left (right) hand circular and linear to linear polarization transformers are designed and simulated. In the next section, we give a brief summary on transformation electromagnetics method and introduce our proposed coordinate transformation for polarization transformer. Constitutive material parameters for modifying only TE or TM polarization are presented and available tradeoffs are discussed. Also, a classical electromagnetic explanation is given about the device functionality. The third section, includes simulation results for various types of polarization transformation scenarios. 2. Theory 2.1. Transformation Electromagnetics Implementing Transformation electromagnetics method basically includes two steps. The first step is to choose a suitable mapping that results in a desired functionality. The second one is to use the form invariance of Maxwell’s equations to determine permittivity and permeability tensors of medium which realize the desired functionality. ) ( ) be a coordinate transformation that maps ( ) space to ( ). In ( ) Let ( space, which is called virtual space, the behavior of wave is known. So, it is often chosen to be vacuum. The ( ) space is called physical space and is the result of coordinate transformation. Let the electric field and magnetic field in virtual space and physical space be E,H and E',H', respectively. Also and are permittivity and permeability tensors of virtual space and physical space, respectively. Transformation electromagnetics claims that, the relations between the electric and magnetic fields vectors in virtual and physical spaces are as follows: ( )
(1)
( ) J in Eq. (1) is the Jacobian matrix from transformation f and is expressed as:
(2)
[ ] It also claims that the relations between permittivity and permeability tensors of virtual and physical spaces are as follows: ( )
(3)
( ) In a summary, based on this method the desired transformed fields in Eq. (1) satisfy Maxwell equations, if tensors in Eq. (3) are chosen for the medium. 2.2. design of polarization transformer In this section we introduce an electromagnetic device that changes the wave polarization. Let the input wave be composed of two orthogonal components, namely TE and TM. We aim to create a certain amount of phase difference between these two orthogonal components, in order to change the polarization of the wave. To do so, we propose a proper coordinate transformation that compresses or stretches the space. Based on the method of transformation electromagnetics, these perturbation in space will later be realized by calculated constitutive media parameters. Finally these constitutive parameters are modified in such a way that only one component of the wave is affected and
hence the desired phase difference between orthogonal components is achieved. We prove later that affecting only TM polarization can lead to a non-magnetic media that is a great advantage when it comes to fabrication. The proposed coordinate transformation is expressed as Eq. (4). (4)
{ Fig. 1 sketches the above transformation.
Virtual space in Fig. (1) is a square with side d. Dash-dotted line represents the physical space right boundary for expansion while dotted line indicates the right boundary of compressed physical space. Eq. (4) and Fig. (1) indicate that this linear transformation compresses the propagation medium in x direction when and expands it when . If the proposed transformation affects only one component of the wave, then the amount of phase difference between orthogonal components after propagation distance of d' in xˆ direction is obtained from the following equation: (5) ⁄ For example, if the phase difference of π/2 is desired over propagation distance of d'=25λ/4, m=25/24 is required according to Eq. (5). It is useful to note that the proposed transformation is linear and hence the transformation media will be homogeneous. Also, according to the reflectionless criteria in [24] and [25], proposed transformation is continuous at the boundaries and hence the media is reflectionless. (
)
→
Applying Eqs. (2)-(4), the relative permittivity and permeability tensors are calculated as: [
(6)
]
If we investigate Maxwell equations for 2D propagation scenario ( ⁄ ), we conclude that distinct constitutive parameters affect wave propagation for basic orthogonal polarization namely TE and TM. Parameters involved in Maxwell equations for TE and TM polarization are
and
respectively. It is good to recall that TE and TM polarization include respectively ( ) and ( ) components. Therefore the relative permittivity and permeability of medium for affecting only TE polarization are: [
]
[
]
(7)
]
(8)
And for affecting only TM polarization are as follows: [
]
[
Further investigation of Maxwell equations proves that dispersion relation for TM polarization is unchanged if multiplication of and remains constant. Similar rule applies to multiplication of and for TE polarization. So, alternative media parameters for Eq. (8) can be: [
]
[
]
(9)
which result in a non-magnetic medium. The original design procedure results in reflectionless media. However the parameters of Eq. (9) will not guaranty the reflectionless characteristic of the device; since a mismatch in impedance occurs. TM wave Impedance equals for media proposed by Eq. (9). The half space reflection √ ⁄ ⁄ coefficient equals . The total amount of reflection is easily calculated by turning the main problem into the calculation of reflection from a dielectric slab of width d' and impedance m [26]. Total reflection coefficient of the slab is as follows: ( (
⁄
) ⁄
(10) )
It is seen that a proper choice of d' can result in a reflectionless slab. We can choose d' so that the numerator of vanishes and the desired phase difference is obtained, i.e. →
(
( )
⁄
)
(11)
It is worthy to note that without using Eq. (11) for d', the reflection from media is generally negligible since values of m are very close to unity. In next section, a classical explanation of Eq. (9) media is given. 2.3. Classical explanation of derived reduced media The proposed media in Eq. (9) is non-magnetic and hence the principal refractive index tensor is as follows: √
(
)
(12)
Normal surfaces of this media at the boundary is shown in Fig. (2). It consists of a circle and an ellipse connected on y axis. In Fig. (2), circle represents normal surface for TE wave while ellipse shows normal surface for TM mode. As shown in Fig. (2), when incident wave vector is normal to the boundary, transmitted wave vector will also be normal to the boundary and the intersection of vector with wave normal surfaces, determines the refractive indices of polarizations. It is clear that refractive index of media for TE wave is unity while TM wave refractive index equals ⁄ . So phase difference between polarizations after propagating distance of d in media will be ( ⁄ ) that agrees with Eq. (5). 3. Simulation In this section we introduce several applications of the proposed polarization transformer. The operation frequency is set to 12 THz. Simulation domain is a square with side length of . All of our proposed designs are simulated and verified by commercial finite element software COMSOL Multiphysics. Fig (3) represents the geometry of proposed polarization transformer. PEC and PMC boundary conditions are used for different polarizations. Input and output ports are used to launch and absorb plane waves. If Gaussian wave is needed, the Gaussian wave is launched at the input and absorbed by scattering boundary condition at the output port. Effective width of incident Gaussian wave is set to . In all simulations, the fields for different polarizations is plotted along a direct line which connects the middle points of simulation domain sides. In all simulation results, the area covered with transformation media is highlighted with gray color. 3.1. linear to circular polarization transformer In order to create a circular polarization from an oblique linear polarization with both Ey and Ez components, we expand the space for TM polarization only, to achieve a phase difference of π/2 radians between Ey and Ez components. We choose d'=25λ/4 and m=25/24. Eq. (8) is used to calculate the parameters of media for slab. Fig. (4) illustrates the results. It can be seen that after propagating the distance d' inside transformer, TM component gains extra π/2 phase which creates right-hand circular polarization. It is clear that we can also compress the space for TM polarization and change the handedness of polarization. Since d'=25λ/4 meets the condition of Eq. (11), hence exactly the same functionality is expected from non-magnetic material realized by Eq. (9). 3.2. left hand to right hand circular polarization transformer The handedness of circular polarization is altered by applying an extra π radians phase difference between two orthogonal polarizations. Here, parameters m=13/12 and d'=13λ/2 are chosen and Eq. (8) is used in order to expand
the space for TM polarization and create this phase difference. Fig. (5) illustrates the results for right hand to left hand circular polarization transformation. Here d' meets the condition of Eq. (11) too and hence using the non-magnetic material of Eq. (9) results in the same functionality as shown by Fig. (6). 3.3. Vertical to Horizontal Linear polarization transformer In order to apply such a change in polarization of the wave, we rotate the device π/4 radians around x axis and utilize the previous transformation that results in π radians phase shift. It is seen in Fig. (7) that Ez component at input transforms into Ey component at the output. This method can be generalized to transform an arbitrary linear polarization into another arbitrary one. The only important parameter is the amount of rotation of the device and the fact that a π radians phase shift between TE and TM polarization is needed. 4. Conclusion In this paper, we designed and simulated a polarization transformer using transformation electromagnetics. By compressing or stretching the space, this design results in a media that can create arbitrary phase shift between different components of the wave which propagate in it. The proposed media affects only TM component. A nonmagnetic device is then achieved after modification of the media parameters which functions perfectly and is reflectionless at the boundaries. Using the proposed device, a linear to circular polarization, right hand to left hand circular and a linear to linear polarization transformers were designed and simulated using COMSOL Multiphysics.
References [1]
J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," science, vol. 312, pp. 1780-1782, 2006.
[2]
D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. B. Pendry, A. Starr, et al., "Metamaterial electromagnetic cloak at microwave frequencies," Science, vol. 314, pp. 977-980, 2006.
[3]
R. Liu, C. Ji, J. Mock, J. Chin, T. Cui, and D. Smith, "Broadband ground-plane cloak," Science, vol. 323, pp. 366-369, 2009.
[4]
T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, "Three-dimensional invisibility cloak at optical wavelengths," Science, vol. 328, pp. 337-339, 2010.
[5]
W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature photonics, vol. 1, pp. 224-227, 2007.
[6]
L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructure cloak operating at optical frequencies," Nature Photonics, vol. 3, pp. 461-463, 2009.
[7]
Q. Wu, J. P. Turpin, and D. H. Werner, "Integrated photonic systems based on transformation optics enabled gradient index devices," Light: Science & Applications, vol. 1, p. e38, 2012.
[8]
M. Rahm, D. Roberts, J. Pendry, and D. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Optics Express, vol. 16, pp. 11555-11567, 2008.
[9]
D.-H. Kwon and D. H. Werner, "Polarization splitter and polarization rotator designs based on transformation optics," Optics Express, vol. 16, pp. 18731-18738, 2008/11/10 2008.
[10]
T. Zhai, Y. Zhou, J. Zhou, and D. Liu, "Polarization controller based on embedded optical transformation," Optics express, vol. 17, pp. 17206-17213, 2009.
[11]
W. Wang, L. Lin, J. Ma, C. Wang, J. Cui, C. Du, et al., "Electromagnetic concentrators with reduced material parameters based on coordinate transformation," Optics express, vol. 16, pp. 11431-11437, 2008.
[12]
J. Yang, M. Huang, C. Yang, Z. Xiao, and J. Peng, "Metamaterial electromagnetic concentrators with arbitrary geometries," Optics Express, vol. 17, pp. 19656-19661, 2009.
[13]
D.-H. Kwon and D. H. Werner, "Flat focusing lens designs having minimized reflection based on coordinate transformation techniques," Optics Express, vol. 17, pp. 7807-7817, 2009/05/11 2009.
[14]
D.-H. Kwon and D. H. Werner, "Transformation optical designs for wave collimators, flat lenses and rightangle bends," New Journal of Physics, vol. 10, p. 115023, 2008.
[15]
E. Hecht, "Optics," Addison Weley, 2002.
[16]
I. V. Lindell and A. H. Sihvola, "Plane-wave reflection from uniaxial chiral interface and its application to polarization transformation," IEEE Transactions on Antennas and Propagation, vol. 43, pp. 1397-1404, 1995.
[17]
A. Viitanen and I. Lindell, "Uniaxial chiral quarter-wave polarisation transformer," Electronics letters, vol. 29, pp. 1074-1075, 1993.
[18]
F. Heismann, "Integrated-optic polarization transformer for reset-free endless polarization control," IEEE Journal of Quantum Electronics, vol. 25, pp. 1898-1906, 1989.
[19]
R. Alferness and L. Buhl, "Waveguide electro‐optic polarization transformer," Applied Physics Letters, vol. 38, pp. 655-657, 1981.
[20]
Z. Z. Cheng, Y. Z. Cheng, and C. Fang, "Compact asymmetric metamaterial circular polarization transformer based on twisted double split-ring resonator structure," Journal of Electromagnetic Waves and Applications, vol. 28, pp. 485-493, 2014.
[21]
Y. Cheng, Y. Nie, Z. Cheng, and R. Z. Gong, "Dual-band circular polarizer and linear polarization transformer based on twisted split-ring structure asymmetric chiral metamaterial," Progress In Electromagnetics Research, vol. 145, pp. 263-272, 2014.
[22]
Y. Z. Cheng, Y. Nie, Z. Z. Cheng, L. Wu, X. Wang, and R. Z. Gong, "Broadband transparent metamaterial linear polarization transformer based on triple-split-ring resonators," Journal of Electromagnetic Waves and Applications, vol. 27, pp. 1850-1858, 2013.
[23]
M. G. Silveirinha, "Design of linear-to-circular polarization transformers made of long densely packed metallic helices," IEEE Transactions on Antennas and Propagation, vol. 56, pp. 390-401, 2008.
[24]
Y. Wei, Y. Min, R. Zhichao, and Q. Min, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New Journal of Physics, vol. 10, p. 043040, 2008.
[25]
Y. Wei, Y. Min, and Q. Min, "Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background," arXiv:0806.3231, 2008.
[26]
P. Yeh and M. Hendry, "Optical Waves in Layered Media," Phys. Today, vol. 43, pp. 77-78, 2008.
Figure Captions
Figure 1. Geometry of the proposed transformation. (a) Virtual space, (b) Physical space.
Figure 2: normal surfaces for material results from Eq. (9) for m<1
Figure 3: Geometry of polarization transformer.
Figure 4. Oblique-linear to circular polarization transformation
Figure 5. left hand to right hand circular polarization transformation
Figure 6. left hand to right hand circular polarization transformation using non-magnetic material
Figure 7.Vertical to horizontal linear polarization transformation
S0030-4026(16)31465-6 http://dx.doi.org/doi:10.1016/j.ijleo.2016.11.129 IJLEO 58545
To appear in: Received date: Accepted date:
4-10-2016 24-11-2016
Please cite this article as: Sayyed Saleh Sayyed Mousavi, Mohammad Saeed Majedi, Hossein Eskandari, Design and simulation of polarization transformers using transformation electromagnetics, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.11.129 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Design and simulation of polarization transformers using transformation electromagnetics Sayyed Saleh Sayyed Mousavia,b , Mohammad Saeed Majedib, , Hossein Eskandaria,b a
Communications and Computer Research Center, Ferdowsi University of Mashhad, Mashhad, Iran b
Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Abstract In this paper we design polarization transformers using transformation electromagnetics method. We introduce a new coordinate transformation that by compressing or expanding the space, creates a desired delay on one component of electromagnetic fields and therefore causes the polarization of electromagnetic wave to change. Linear to circular, right (left) hand to left (right) hand circular and linear to linear polarization transformers have been designed and analyzed. The proposed polarization transformers are homogeneous and reflectionless and they can also be non-magnetic. COMSOL Multiphysics software is used to validate our results. Keywords: polarization transformer, transformation electromagnetic, non-magnetic medium, coordinate transformation.
1. Introduction In 2006 an article was published that introduced a systematic methodology capable of controlling the electromagnetic field by defining appropriate permittivity and permeability tensors for the medium [1]. This method is called transformation electromagnetics or transformation optics. The design procedure suggested by transformation electromagnetics is frequency independent. Generally the permittivity and permeability obtained by this method lead to an inhomogeneous and anisotropic medium. Transformation electromagnetics has a variety of applications in design of microwave and optic devices. There are several articles that used transformation electromagnetics in cloaking devices that was realized by metamaterial structures [2-6]. It is also used in planar photonic devices for integrated system including waveguide and photodetectors [7]. Furthermore, devices such as beam bender and expander [8], beam polarization splitter and rotator [9, 10], field concentrators [11, 12], flat focusing lenses [13] and wave collimators [14], have been proposed based on this method. Polarization state is one of the most important properties of any electromagnetic wave. Changing the polarization of wave is a practical issue in microwave engineering and optics. In literature, there are several devices, such as wave retarders, that perform polarization transformation. Retarders exert a delay on one component of the electromagnetic fields and therefore change the polarization. Most popular types of retarders are well known full, half and quarter wave plates [15]. Also, Fresnel rhomb and Babinet compensator are examples of retarders [15]. Furthermore polarization transformation for a plane wave is done by transversely uniaxial chiral medium [16, 17]. There are also structures that composed of waveguides, two phase shifter, TE to TM converter and crystals that are capable of performing arbitrary polarization transformations [18, 19]. In some articles the polarization transformer is implemented by metamaterial structures such as split-ring resonators [20-22] or metallic helices [23].
Corresponding author. Tel.: +98-513-880-5171 E-mail Address: [email protected].
In this paper, we propose a different strategy for polarization transformation based on transformation electromagnetics method. Accordingly, we present a new coordinate transformation that by compressing or expanding the space, creates a desired phase difference between orthogonal components of electromagnetic fields and this leads to the polarization changes. It will be pointed out that a proper linear coordinate transformation can lead to a homogeneous, reflectionless and non-magnetic media if only the TM polarization is modified. Based on this method, linear to circular, right (left) hand to left (right) hand circular and linear to linear polarization transformers are designed and simulated. In the next section, we give a brief summary on transformation electromagnetics method and introduce our proposed coordinate transformation for polarization transformer. Constitutive material parameters for modifying only TE or TM polarization are presented and available tradeoffs are discussed. Also, a classical electromagnetic explanation is given about the device functionality. The third section, includes simulation results for various types of polarization transformation scenarios. 2. Theory 2.1. Transformation Electromagnetics Implementing Transformation electromagnetics method basically includes two steps. The first step is to choose a suitable mapping that results in a desired functionality. The second one is to use the form invariance of Maxwell’s equations to determine permittivity and permeability tensors of medium which realize the desired functionality. ) ( ) be a coordinate transformation that maps ( ) space to ( ). In ( ) Let ( space, which is called virtual space, the behavior of wave is known. So, it is often chosen to be vacuum. The ( ) space is called physical space and is the result of coordinate transformation. Let the electric field and magnetic field in virtual space and physical space be E,H and E',H', respectively. Also and are permittivity and permeability tensors of virtual space and physical space, respectively. Transformation electromagnetics claims that, the relations between the electric and magnetic fields vectors in virtual and physical spaces are as follows: ( )
(1)
( ) J in Eq. (1) is the Jacobian matrix from transformation f and is expressed as:
(2)
[ ] It also claims that the relations between permittivity and permeability tensors of virtual and physical spaces are as follows: ( )
(3)
( ) In a summary, based on this method the desired transformed fields in Eq. (1) satisfy Maxwell equations, if tensors in Eq. (3) are chosen for the medium. 2.2. design of polarization transformer In this section we introduce an electromagnetic device that changes the wave polarization. Let the input wave be composed of two orthogonal components, namely TE and TM. We aim to create a certain amount of phase difference between these two orthogonal components, in order to change the polarization of the wave. To do so, we propose a proper coordinate transformation that compresses or stretches the space. Based on the method of transformation electromagnetics, these perturbation in space will later be realized by calculated constitutive media parameters. Finally these constitutive parameters are modified in such a way that only one component of the wave is affected and
hence the desired phase difference between orthogonal components is achieved. We prove later that affecting only TM polarization can lead to a non-magnetic media that is a great advantage when it comes to fabrication. The proposed coordinate transformation is expressed as Eq. (4). (4)
{ Fig. 1 sketches the above transformation.
Virtual space in Fig. (1) is a square with side d. Dash-dotted line represents the physical space right boundary for expansion while dotted line indicates the right boundary of compressed physical space. Eq. (4) and Fig. (1) indicate that this linear transformation compresses the propagation medium in x direction when and expands it when . If the proposed transformation affects only one component of the wave, then the amount of phase difference between orthogonal components after propagation distance of d' in xˆ direction is obtained from the following equation: (5) ⁄ For example, if the phase difference of π/2 is desired over propagation distance of d'=25λ/4, m=25/24 is required according to Eq. (5). It is useful to note that the proposed transformation is linear and hence the transformation media will be homogeneous. Also, according to the reflectionless criteria in [24] and [25], proposed transformation is continuous at the boundaries and hence the media is reflectionless. (
)
→
Applying Eqs. (2)-(4), the relative permittivity and permeability tensors are calculated as: [
(6)
]
If we investigate Maxwell equations for 2D propagation scenario ( ⁄ ), we conclude that distinct constitutive parameters affect wave propagation for basic orthogonal polarization namely TE and TM. Parameters involved in Maxwell equations for TE and TM polarization are
and
respectively. It is good to recall that TE and TM polarization include respectively ( ) and ( ) components. Therefore the relative permittivity and permeability of medium for affecting only TE polarization are: [
]
[
]
(7)
]
(8)
And for affecting only TM polarization are as follows: [
]
[
Further investigation of Maxwell equations proves that dispersion relation for TM polarization is unchanged if multiplication of and remains constant. Similar rule applies to multiplication of and for TE polarization. So, alternative media parameters for Eq. (8) can be: [
]
[
]
(9)
which result in a non-magnetic medium. The original design procedure results in reflectionless media. However the parameters of Eq. (9) will not guaranty the reflectionless characteristic of the device; since a mismatch in impedance occurs. TM wave Impedance equals for media proposed by Eq. (9). The half space reflection √ ⁄ ⁄ coefficient equals . The total amount of reflection is easily calculated by turning the main problem into the calculation of reflection from a dielectric slab of width d' and impedance m [26]. Total reflection coefficient of the slab is as follows: ( (
⁄
) ⁄
(10) )
It is seen that a proper choice of d' can result in a reflectionless slab. We can choose d' so that the numerator of vanishes and the desired phase difference is obtained, i.e. →
(
( )
⁄
)
(11)
It is worthy to note that without using Eq. (11) for d', the reflection from media is generally negligible since values of m are very close to unity. In next section, a classical explanation of Eq. (9) media is given. 2.3. Classical explanation of derived reduced media The proposed media in Eq. (9) is non-magnetic and hence the principal refractive index tensor is as follows: √
(
)
(12)
Normal surfaces of this media at the boundary is shown in Fig. (2). It consists of a circle and an ellipse connected on y axis. In Fig. (2), circle represents normal surface for TE wave while ellipse shows normal surface for TM mode. As shown in Fig. (2), when incident wave vector is normal to the boundary, transmitted wave vector will also be normal to the boundary and the intersection of vector with wave normal surfaces, determines the refractive indices of polarizations. It is clear that refractive index of media for TE wave is unity while TM wave refractive index equals ⁄ . So phase difference between polarizations after propagating distance of d in media will be ( ⁄ ) that agrees with Eq. (5). 3. Simulation In this section we introduce several applications of the proposed polarization transformer. The operation frequency is set to 12 THz. Simulation domain is a square with side length of . All of our proposed designs are simulated and verified by commercial finite element software COMSOL Multiphysics. Fig (3) represents the geometry of proposed polarization transformer. PEC and PMC boundary conditions are used for different polarizations. Input and output ports are used to launch and absorb plane waves. If Gaussian wave is needed, the Gaussian wave is launched at the input and absorbed by scattering boundary condition at the output port. Effective width of incident Gaussian wave is set to . In all simulations, the fields for different polarizations is plotted along a direct line which connects the middle points of simulation domain sides. In all simulation results, the area covered with transformation media is highlighted with gray color. 3.1. linear to circular polarization transformer In order to create a circular polarization from an oblique linear polarization with both Ey and Ez components, we expand the space for TM polarization only, to achieve a phase difference of π/2 radians between Ey and Ez components. We choose d'=25λ/4 and m=25/24. Eq. (8) is used to calculate the parameters of media for slab. Fig. (4) illustrates the results. It can be seen that after propagating the distance d' inside transformer, TM component gains extra π/2 phase which creates right-hand circular polarization. It is clear that we can also compress the space for TM polarization and change the handedness of polarization. Since d'=25λ/4 meets the condition of Eq. (11), hence exactly the same functionality is expected from non-magnetic material realized by Eq. (9). 3.2. left hand to right hand circular polarization transformer The handedness of circular polarization is altered by applying an extra π radians phase difference between two orthogonal polarizations. Here, parameters m=13/12 and d'=13λ/2 are chosen and Eq. (8) is used in order to expand
the space for TM polarization and create this phase difference. Fig. (5) illustrates the results for right hand to left hand circular polarization transformation. Here d' meets the condition of Eq. (11) too and hence using the non-magnetic material of Eq. (9) results in the same functionality as shown by Fig. (6). 3.3. Vertical to Horizontal Linear polarization transformer In order to apply such a change in polarization of the wave, we rotate the device π/4 radians around x axis and utilize the previous transformation that results in π radians phase shift. It is seen in Fig. (7) that Ez component at input transforms into Ey component at the output. This method can be generalized to transform an arbitrary linear polarization into another arbitrary one. The only important parameter is the amount of rotation of the device and the fact that a π radians phase shift between TE and TM polarization is needed. 4. Conclusion In this paper, we designed and simulated a polarization transformer using transformation electromagnetics. By compressing or stretching the space, this design results in a media that can create arbitrary phase shift between different components of the wave which propagate in it. The proposed media affects only TM component. A nonmagnetic device is then achieved after modification of the media parameters which functions perfectly and is reflectionless at the boundaries. Using the proposed device, a linear to circular polarization, right hand to left hand circular and a linear to linear polarization transformers were designed and simulated using COMSOL Multiphysics.
References [1]
J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," science, vol. 312, pp. 1780-1782, 2006.
[2]
D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. B. Pendry, A. Starr, et al., "Metamaterial electromagnetic cloak at microwave frequencies," Science, vol. 314, pp. 977-980, 2006.
[3]
R. Liu, C. Ji, J. Mock, J. Chin, T. Cui, and D. Smith, "Broadband ground-plane cloak," Science, vol. 323, pp. 366-369, 2009.
[4]
T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, "Three-dimensional invisibility cloak at optical wavelengths," Science, vol. 328, pp. 337-339, 2010.
[5]
W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature photonics, vol. 1, pp. 224-227, 2007.
[6]
L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructure cloak operating at optical frequencies," Nature Photonics, vol. 3, pp. 461-463, 2009.
[7]
Q. Wu, J. P. Turpin, and D. H. Werner, "Integrated photonic systems based on transformation optics enabled gradient index devices," Light: Science & Applications, vol. 1, p. e38, 2012.
[8]
M. Rahm, D. Roberts, J. Pendry, and D. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Optics Express, vol. 16, pp. 11555-11567, 2008.
[9]
D.-H. Kwon and D. H. Werner, "Polarization splitter and polarization rotator designs based on transformation optics," Optics Express, vol. 16, pp. 18731-18738, 2008/11/10 2008.
[10]
T. Zhai, Y. Zhou, J. Zhou, and D. Liu, "Polarization controller based on embedded optical transformation," Optics express, vol. 17, pp. 17206-17213, 2009.
[11]
W. Wang, L. Lin, J. Ma, C. Wang, J. Cui, C. Du, et al., "Electromagnetic concentrators with reduced material parameters based on coordinate transformation," Optics express, vol. 16, pp. 11431-11437, 2008.
[12]
J. Yang, M. Huang, C. Yang, Z. Xiao, and J. Peng, "Metamaterial electromagnetic concentrators with arbitrary geometries," Optics Express, vol. 17, pp. 19656-19661, 2009.
[13]
D.-H. Kwon and D. H. Werner, "Flat focusing lens designs having minimized reflection based on coordinate transformation techniques," Optics Express, vol. 17, pp. 7807-7817, 2009/05/11 2009.
[14]
D.-H. Kwon and D. H. Werner, "Transformation optical designs for wave collimators, flat lenses and rightangle bends," New Journal of Physics, vol. 10, p. 115023, 2008.
[15]
E. Hecht, "Optics," Addison Weley, 2002.
[16]
I. V. Lindell and A. H. Sihvola, "Plane-wave reflection from uniaxial chiral interface and its application to polarization transformation," IEEE Transactions on Antennas and Propagation, vol. 43, pp. 1397-1404, 1995.
[17]
A. Viitanen and I. Lindell, "Uniaxial chiral quarter-wave polarisation transformer," Electronics letters, vol. 29, pp. 1074-1075, 1993.
[18]
F. Heismann, "Integrated-optic polarization transformer for reset-free endless polarization control," IEEE Journal of Quantum Electronics, vol. 25, pp. 1898-1906, 1989.
[19]
R. Alferness and L. Buhl, "Waveguide electro‐optic polarization transformer," Applied Physics Letters, vol. 38, pp. 655-657, 1981.
[20]
Z. Z. Cheng, Y. Z. Cheng, and C. Fang, "Compact asymmetric metamaterial circular polarization transformer based on twisted double split-ring resonator structure," Journal of Electromagnetic Waves and Applications, vol. 28, pp. 485-493, 2014.
[21]
Y. Cheng, Y. Nie, Z. Cheng, and R. Z. Gong, "Dual-band circular polarizer and linear polarization transformer based on twisted split-ring structure asymmetric chiral metamaterial," Progress In Electromagnetics Research, vol. 145, pp. 263-272, 2014.
[22]
Y. Z. Cheng, Y. Nie, Z. Z. Cheng, L. Wu, X. Wang, and R. Z. Gong, "Broadband transparent metamaterial linear polarization transformer based on triple-split-ring resonators," Journal of Electromagnetic Waves and Applications, vol. 27, pp. 1850-1858, 2013.
[23]
M. G. Silveirinha, "Design of linear-to-circular polarization transformers made of long densely packed metallic helices," IEEE Transactions on Antennas and Propagation, vol. 56, pp. 390-401, 2008.
[24]
Y. Wei, Y. Min, R. Zhichao, and Q. Min, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New Journal of Physics, vol. 10, p. 043040, 2008.
[25]
Y. Wei, Y. Min, and Q. Min, "Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background," arXiv:0806.3231, 2008.
[26]
P. Yeh and M. Hendry, "Optical Waves in Layered Media," Phys. Today, vol. 43, pp. 77-78, 2008.
Figure Captions
Figure 1. Geometry of the proposed transformation. (a) Virtual space, (b) Physical space.
Figure 2: normal surfaces for material results from Eq. (9) for m<1
Figure 3: Geometry of polarization transformer.
Figure 4. Oblique-linear to circular polarization transformation
Figure 5. left hand to right hand circular polarization transformation
Figure 6. left hand to right hand circular polarization transformation using non-magnetic material
Figure 7.Vertical to horizontal linear polarization transformation