Incident-angle-insensitive and polarization independent ... fileIncident-angle-insensitive and...

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Incident-angle-insensitive and polarization independent polarization rotator Mingkai Liu, Yanbing Zhang, Xuehua Wang, and Chongjun Jin* State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China *[email protected] Abstract: This paper proposes a method to design an incident-angle-insensitive polarization-independent polarization rotator. This polarization rotator is composed of layers of impedance-matched anisotropic metamaterial (IMAM) with each layer’s optical axes gradually rotating an angle. Numerical simulation based on the generalized 4 × 4 transfer matrix method is applied, and the results reveal that the IMAM rotator is not only polarization-independent but also insensitive to the angle of incidence. A 90° polarization rotation with tiny ellipticity variation is still available at a wide range of incident angles from 0 to 40°, which is further confirmed with a microwave bi-split-ring resonator (bi-SRR) rotator. This may be valuable for the design of optoelectronic and microwave devices. ©2010 Optical Society of America OCIS codes: (160.3918) Metamaterials; (230.0230) Optical devices; (230.5440) Polarization-selective devices. References and links 1. D. Y. Yu, and H. Y. Tan, Engineering Optics(in chinese) (China Machine Press, Beijing, 2006). 2. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008). 3. S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68(2), 449–521 (2005). 4. A. Salandrino, and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). 5. W. Zhang, J. Liu, W. P. Huang, and W. Zhao, “Self-collimating photonic-crystal wave plates,” Opt. Lett. 34(17), 2676–2678 (2009). 6. J. Zhao, Y. Feng, B. Zhu, and T. Jiang, “Sub-wavelength image manipulating through compensated anisotropic metamaterial prisms,” Opt. Express 16(22), 18057–18066 (2008). 7. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). 8. J.-M. Lourtioz, “Photonic crystals and metamaterials,” C. R. Phys. 9(1), 4–15 (2008). 9. M. Beruete, M. Navarro-Cía, M. Sorolla, and I. Campillo, “Polarization selection with stacked hole array metamaterial,” J. Appl. Phys. 103(5), 053102 (2008). 10. J. Zhao, Y. Chen, and Y. Feng, “Polarization beam splitting through an anisotropic metamaterial slab realized by a layered metal-dielectric structure,” Appl. Phys. Lett. 92(7), 071114 (2008). 11. H. Luo, Z. Ren, W. Shu, and F. Li, “Construct a polarizing beam splitter by an anisotropic metamaterial slab,” Appl. Phys. B 87(2), 283–287 (2007). 12. V. Zabelin, L. A. Dunbar, N. Le Thomas, R. Houdré, M. V. Kotlyar, L. O’Faolain, and T. F. Krauss, “Self-collimating photonic crystal polarization beam splitter,” Opt. Lett. 32(5), 530–532 (2007). 13. J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007). 14. K. Bayat, S. K. Chaudhuri, and S. Safavi-Naeini, “Ultra-compact photonic crystal based polarization rotator,” Opt. Lett. 17, 7145–7158 (2009). 15. J. Y. Chin, J. N. Gollub, J. J. Mock, R. Liu, C. Harrison, D. R. Smith, and T. J. Cui, “An efficient broadband metamaterial wave retarder,” Opt. Express 17(9), 7640–7647 (2009). 16. T. Li, H. Liu, S. M. Wang, X. G. Yin, F. M. Wang, S. N. Zhu, and X. Zhang, “Manipulating optical rotation in extraordinary transmission by hybrid plasmonic excitations,” Appl. Phys. Lett. 93(2), 021110 (2008). 17. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95(22), 227401 (2005). 18. S. K. Awasthi, and S. P. Ojha, “Wide-angle, broadband plate polarizer with 1D photonic crystal,” Prog. Electromag. Res. 88, 321–335 (2008). #126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010 (C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11990

Transcript of Incident-angle-insensitive and polarization independent ... fileIncident-angle-insensitive and...

Page 1: Incident-angle-insensitive and polarization independent ... fileIncident-angle-insensitive and polarization independent polarization rotator Mingkai Liu, Yanbing Zhang, Xuehua Wang,

Incident-angle-insensitive and polarization

independent polarization rotator

Mingkai Liu, Yanbing Zhang, Xuehua Wang, and Chongjun Jin*

State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-sen

University, Guangzhou 510275, People’s Republic of China

*[email protected]

Abstract: This paper proposes a method to design an

incident-angle-insensitive polarization-independent polarization rotator. This

polarization rotator is composed of layers of impedance-matched anisotropic

metamaterial (IMAM) with each layer’s optical axes gradually rotating an

angle. Numerical simulation based on the generalized 4 × 4 transfer matrix

method is applied, and the results reveal that the IMAM rotator is not only

polarization-independent but also insensitive to the angle of incidence. A 90°

polarization rotation with tiny ellipticity variation is still available at a wide

range of incident angles from 0 to 40°, which is further confirmed with a

microwave bi-split-ring resonator (bi-SRR) rotator. This may be valuable for

the design of optoelectronic and microwave devices.

©2010 Optical Society of America

OCIS codes: (160.3918) Metamaterials; (230.0230) Optical devices; (230.5440)

Polarization-selective devices.

References and links

1. D. Y. Yu, and H. Y. Tan, Engineering Optics(in chinese) (China Machine Press, Beijing, 2006).

2. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk

metamaterials of nanowires,” Science 321(5891), 930 (2008).

3. S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68(2), 449–521 (2005).

4. A. Salandrino, and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory

and simulations,” Phys. Rev. B 74(7), 075103 (2006).

5. W. Zhang, J. Liu, W. P. Huang, and W. Zhao, “Self-collimating photonic-crystal wave plates,” Opt. Lett. 34(17),

2676–2678 (2009).

6. J. Zhao, Y. Feng, B. Zhu, and T. Jiang, “Sub-wavelength image manipulating through compensated anisotropic

metamaterial prisms,” Opt. Express 16(22), 18057–18066 (2008).

7. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional

optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).

8. J.-M. Lourtioz, “Photonic crystals and metamaterials,” C. R. Phys. 9(1), 4–15 (2008).

9. M. Beruete, M. Navarro-Cía, M. Sorolla, and I. Campillo, “Polarization selection with stacked hole array

metamaterial,” J. Appl. Phys. 103(5), 053102 (2008).

10. J. Zhao, Y. Chen, and Y. Feng, “Polarization beam splitting through an anisotropic metamaterial slab realized by a

layered metal-dielectric structure,” Appl. Phys. Lett. 92(7), 071114 (2008).

11. H. Luo, Z. Ren, W. Shu, and F. Li, “Construct a polarizing beam splitter by an anisotropic metamaterial slab,”

Appl. Phys. B 87(2), 283–287 (2007).

12. V. Zabelin, L. A. Dunbar, N. Le Thomas, R. Houdré, M. V. Kotlyar, L. O’Faolain, and T. F. Krauss,

“Self-collimating photonic crystal polarization beam splitter,” Opt. Lett. 32(5), 530–532 (2007).

13. J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave

polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007).

14. K. Bayat, S. K. Chaudhuri, and S. Safavi-Naeini, “Ultra-compact photonic crystal based polarization rotator,” Opt.

Lett. 17, 7145–7158 (2009).

15. J. Y. Chin, J. N. Gollub, J. J. Mock, R. Liu, C. Harrison, D. R. Smith, and T. J. Cui, “An efficient broadband

metamaterial wave retarder,” Opt. Express 17(9), 7640–7647 (2009).

16. T. Li, H. Liu, S. M. Wang, X. G. Yin, F. M. Wang, S. N. Zhu, and X. Zhang, “Manipulating optical rotation in

extraordinary transmission by hybrid plasmonic excitations,” Appl. Phys. Lett. 93(2), 021110 (2008).

17. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant

optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95(22), 227401 (2005).

18. S. K. Awasthi, and S. P. Ojha, “Wide-angle, broadband plate polarizer with 1D photonic crystal,” Prog.

Electromag. Res. 88, 321–335 (2008).

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11990

Page 2: Incident-angle-insensitive and polarization independent ... fileIncident-angle-insensitive and polarization independent polarization rotator Mingkai Liu, Yanbing Zhang, Xuehua Wang,

19. Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic

metamaterial,” Phys. Rev. B 79(4), 045131 (2009).

20. J. L. Tsalamengas, ““Interaction of electromagnetic waves with general bianisotropicslabs,” IEEE Trans. on

Micro,” Theo. and Tech 40(10), 1870–1878 (1992).

21. R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland Pub. Co., New York,

1977).

22. B. Bai, Y. Svirko, J. Turunen, and T. Vallius, “Optical activity in planar chiral metamaterials: Theoretical study,”

Phys. Rev. A 76(2), 023811 (2007).

23. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at

oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).

24. J. D. Baena, L. Jelinek, R. Marques, and J. Zehentner, “Electrically small isotropic three-dimensional magnetic

resonators for metamaterial design,” Appl. Phys. Lett. 88(13), 134108 (2006).

1. Introduction

As an important property of transverse EM waves, polarization has been widely applied in

engineering and scientific researches. Many approaches have been employed to manipulate the

polarization of light. Traditional anisotropic crystals and chiral liquid crystals [1] are commonly

used as wave retarders and polarization rotators. However, the polarization rotator made of

anisotropic crystal is generally polarization-dependent and incident-angle-sensitive, and the

polarization rotator composed of liquid crystals is also incident-angle-sensitive. These

shortcomings limit their applications in microwave and optoelectronic devices. To our

knowledge, how to design and fabricate an incident-angle-insensitive and

polarization-independent polarization rotator still remains a challenge.

Thanks to the advent of metamaterials, this gives a chance to overcome the challenge.

Metamaterials are artificial structures, which usually have periodic arrangements and exhibit

exotic electromagnetic properties [2–8]. These manmade structures provide completely new

mechanisms and novel methods to control light. Early and ongoing researches on metamaterials

have shown that it is possible to obtain strong anisotropy or chirality via deliberate design and

fabrication. This can be used in the design of polarization devices. One of them is polarization

beam splitter achieved by anomalous reflection and transmission [9–12]. Another application is

the polarization rotator based on modes coupling, extraordinary optical transmission (EOT),

chirality of the structure, or phase mutation at resonance frequency [5,13–17]. Up till now,

wide-angle polarizer [18], splitter [10,11] and absorber [19] have been realized with these

artificial structures. In this paper, we proposed a method to design an incident-angle-insensitive

and polarization-independent polarization rotator. The polarization rotator is composed of

layered impendence-matched anisotropic metamaterials (IMAM) with the crystal axes rotated,

as shown in Fig. 1. The cross polarization conversion becomes polarization-independent and

insensitive to the incident direction of the light beam. It can be proved that only two layers of

IMAMs are enough to construct a cross-polarization rotator, which greatly alleviates the

difficulty in fabrication. It might be valuable for the design of optoelectronic and microwave

devices.

2. General formalism of IMAM polarization rotator

To study the transmission and reflection of plane waves in an anisotropic slab, a generalized 4 ×

4 transfer-matrix method is applied [20]. By transforming the permittivity and permeability of

IMAM with a rotation matrix: 1

2 1,ε ε −= ℜ ℜ 1

2 1µ µ −= ℜ ℜ , where

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11991

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Fig. 1. Schematic structure of a multilayered metamaterial rotator. The plane of incidence is in

x-z plane. φ is the angle between the axes of the adjacent layers. ϕ and θ are the

polarization angle and incident angle of the incident wave, respectively.

cos sin 0

sin cos 0 ,

0 0 1

φ φφ φ

ℜ = −

(1)

one can easily obtain the transfer matrix of an anisotropic slab with its crystal axes rotated along

z axis by an angle .φ When many layers are aligned together, the total transfer matrix is given

by

1 11 1( ) ( ) ( ) ( ),total m mm m

P H P h P h P h− −= ⋅⋅ ⋅ (2)

where 1 i

,m

iH h

==∑ which is the total thickness of the stratified slabs. The 2 × 2 Jones

transmission coefficient matrix can be acquired after some tedious algebraic operation that is

not presented here [21]. To simplify the discussion, we restrict the incidence in x-z plane, i.e.

y0,k = and one of the crystal axes is chosen as z axis.

When the polarization of the transmitted wave is perpendicular to a linearly-polarized

incident wave 0

( , ) (cos ,sin ),iM iE

E E E ϕ ϕ= one can obtain

0

sin cos,

sincos

MM ME

t

EM EE

T TE E

T T

ϕ ϕϕϕ

= −

(3)

where ϕ is the polarization angle of the incident wave. MM

T and EE

T are the co-polarization

transmission coefficients of TM and TE waves, respectively; while the cross terms are the

cross-polarization transmission coefficients. By multiplying ( )cos , sinϕ ϕ on both sides of

the equation, we have

2 2

00 [ cos sin ( )sin cos ].

MM EE ME EME T T T Tϕ ϕ ϕ ϕ= + + + (4)

For a polarization-independent rotator, the following condition must be satisfied:

MM EE0,T T= →

EM ME0.T T+ → Generally, it cannot be fulfilled with a single anisotropic

medium but with a layered structure. To give an intuitionistic discussion, we choose an IMAM

slab with the following permittivity and permeability tensors:

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11992

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Fig. 2. Schematic view of the polarization vector projections on x-y plane when the light is

normally incident. The crystal axes of the anisotropic slabs are rotated along z direction by an

angleφ in sequence.

0 0

0 0 ,

0 0

xx

r yy

zz

εε ε

ε

=

(5)

0 0

0 0 ,

0 0

xx

r yy

zz

µµ µ

µ

=

(6)

where

/ / / 1 / .yy xx xx yy sur sur Zε µ ε µ ε µ= = = (7)

Z is the wave impedance, and , ,xx yy xx yy

ε ε µ µ≠ ≠sur

ε and sur

µ are permittivity and

permeability of the surrounding material. When the light strikes normally at the interface

between the lossless surroundings and an IMAM slab with thickness ,h the transmission

coefficients of TE and TM waves can be simplified as

0 0

exp( ' ) exp( '' ),E yy yy

T i k hZ k hZε ε= − (8)

0 0

exp( ' / )exp( '' / ),M yy yy

T i k h Z k h Zµ µ= − (9)

where0

k is the wave vector in vacuum, and 'ε ( 'µ ) and ''ε ( ''µ ) are the real and imaginary

parts of permittivity (permeability). When

2 2

'' '' ''/ ''/ ,yy xx xx yy

Z Zε ε µ µ= = = (10)

TE and TM waves experience the same attenuation, i.e. | | | | 1.E M

T T= ≤

Since the impendence-matched condition is satisfied, reflections at the interface either

between slab and surroundings or between slabs disappear. Then we can give an explicit

expression on the transmission coefficient of the stratified slabs. When the plane wave

( ),iM iE

E E is normally incident onto x-y plane, the projections of the transmitted wave

polarization on the crystal axes (denoted by suffix 1, 2) of the (n + 1)th slab can be written as

1 1

2 1

cos sin cos sin.

sin cos sin cos

n

n iME E E E

M M M Mn iE

E ET T T T

T T T TE E

φ φ δ δφ φ δ δ

< + >

< + >

− − = ×

(11)

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11993

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As shown in Fig. 2, δ is the angle between the x axis and the crystal axis 1 of the first

layer; φ is the angle of the crystal axes between adjacent layers. We once again project

1 1 2 1( , )

n nE E< + > < + > onto the coordinate axes and the Jones matrix can be obtained:

cos( ) sin(n )

sin( ) cos( )

cos sin cos sin.

sin cos sin cos

MM ME

EM EE

n

E E E E

M M M M

T T n

T T n n

T T T T

T T T T

φ δ φ δφ δ φ δ

φ φ δ δφ φ δ δ

+ + = − + +

− − ×

(12)

By carefully tuning the thickness of IMAM slab that makes it as a half-wave retarder, one

can obtain .E M

T T T= − = When n is odd, the Jones matrix can be simplified as follows:

1

cos[( 1) ] sin[( 1) ].

sin[( 1) ] cos[( 1) ]

MM ME n

EM EE

T T n nT

T T n n

φ φφ φ

+ + + = − + +

(13)

It is clear that, if T is not considered, the Jones matrix is a coordinate transformation matrix

by rotating an angle of ( 1)n φ+ anticlockwise. Hence the polarization rotation angle ( 1)n φ+

can be tuned dynamically and is independent of ϕ and .δ When ( 1) / 2,n φ π+ = ±

1

0 1.

1 0

MM ME n

EM EE

T TT

T T

+ ± =

∓ (14)

This is the condition for the cross polarization conversion. It is clear that only two (n = 1)

IMAM slabs are enough to construct a polarization-independent rotator.

Equation (14) indicates that the eigen polarization states of the rotator are left and right

circular polarizations. Although it is the same for planar chiral structures with four-fold

symmetry, it is very difficult to achieve linear polarization rotation in planar metallic chiral

structures due to dichroism; while in dielectric chiral structures, one can achieve linear

polarization rotation but this property strongly depends on incident direction, as has been

pointed out [22].

3. Incident-angle-insensitivity and polarization independency

It is more interesting to find that the IMAM rotator is not only polarization-independent, but

also incident-angle-insensitive. This is distinct from conventional anisotropic or chiral rotators.

Without loss of generality, we choose air as surroundings, and a lossless IMAM bilayered

rotator with parameters as 2,xx yy

ε µ= = 1,yy xx

ε µ= = 1.zz zz

ε µ= = The thickness h of each

layer is chosen as0

/ 1.5h λ = so that exp[ ( )]E M x

T T i kϑ= − = is satisfied at normal incidence

and each layer of slab acts as a half-wave retarder. To construct a polarization-independent

rotator, two half-wave plates are aligned together in the above mentioned way, i.e. 2 90 .φ = °

To study the effects of incident direction on the performance of the polarization rotator, incident

angle θ (denoted by0

/x

k k ), angleδ and incident polarization angle ϕ should be considered.

Figures 3(a) to 3(d) show the polarization changes of the transmitted light. The ellipticity ∆

and the polarization rotation anglet

ψ of the transmitted light are defined as

/ ,a b

E E∆ = (15)

,t t

ψ ϕ ϕ= − (16)

where the electric field amplitudesa

E and b

E are the minor and major polarization

components of the transmitted light, respectively; t

ϕ is the transmitted polarization angle.

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11994

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Fig. 3. The projections of (a) the ellipticity ∆ and (b) the polarization rotation angle t

ψ with

respect to 0

/x

k k and δ when ϕ = 45°. The projection views of (c) ∆ and (d) t

ψ with

respect to δ and ϕ at 40° incident angle, in which ϕ ranges from −90° to 90°. The insets

demonstrate the three-dimensional relations. The lossless IMAM bilayered polarization rotator is

characterized with 2,xx yy

ε µ= = 1,yy xx

ε µ= = 1,zz zz

ε µ= = and a total thickness

02 / 3.h λ =

Since we have restricted the incident beam in x-z plane, the variation of δ is actually

equivalent to the change of incidence in polar direction. Now we look about how the

performance of the polarization rotator depends on δ when ϕ = 45°. Figure 3(a) is the

projection of the three-dimensional (3D) curvature of ellipticity ∆ as a function of δ and

0/

xk k onto the

0~ /

xk k∆ plane; Fig. 3(b) is the projection of the 3D curvature of

polarization rotation angle t

ψ as a function of δ and 0

/x

k k onto the 0

~ /t x

k kψ plane. It

is clear that, at normal incidence (0

/ 0x

k k = ), 0∆ → and 90 ,t

ψ → − � the bilayered structure

acts as a perfect cross-polarization rotator. As the incident angle increases, the variation ranges

of ∆ and t

ψ become larger but still acceptable within a broad range of incident angles. It can be

seen in Fig. 3(a) and Fig. 3(b) that when δ ranges from 0° to 360°, ( 0.077, 0.077)∆∈ −

(indicating a value of 0.006 as an intensity contrast of two polarization components) and

tψ ∈ ( 90.69 , 89.05 )− ° − ° even at a 40° incident angle (

0/ 0.64

xk k ≈ ). As the incident angle

decreases to 30°, variations can drop to ( 0.024, 0.024)∆∈ − and t

ψ ∈ ( 90.4 , 89.6 ).− ° − °

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11995

Page 7: Incident-angle-insensitive and polarization independent ... fileIncident-angle-insensitive and polarization independent polarization rotator Mingkai Liu, Yanbing Zhang, Xuehua Wang,

Figures 3(c) and 3(d) depict the effect of incident polarization angle ϕ on the polarization

rotation at the oblique incidence of 40°. The insets plot the variations of ∆ and t

ψ with

respect to δ and ϕ at a 40° incident angle. The projections of these curvatures onto ~ δ∆

and ~t

ψ δ planes are shown in these figures. When ϕ changes from −90° to 90°, the

maximum variation range of ∆ is ( 0.077, 0.077),− while the maximum variation of t

ψ is

( 90.69 , 89.05 ).− ° − ° After studying the effects of incident direction under various0

/ ,x

k k ϕ

and ,δ we confirm that the IMAM rotator is more insensitive to incident direction when the

incident angle is smaller.

Moreover, we studied the effect of material loss. As is expected, the loss has a small impact

on the performance of the IMAM rotator that satisfies Eq. (10). We set

'' '' '' '' 0.2yy xx xx yy

ε ε µ µ= = = = while other parameters maintain the same. The results show

that for a lossy rotator, the corresponding curvatures of Figs. 3(a), 3(b) and 3(c) are similar to

the lossless ones; while for the one correspond to Fig. 3(d), the variation range of t

ψ is

enlarged to (−91.66°,-88.37°). Even so, a less than 2° deviation in polarization rotation at 40°

incidence is still quite acceptable.

4. Comparison and discussion

To make a comparison, we choose three lossless anisotropic slabs (surrounded by air) with

parameters as 2 ,xx

bε = 1,yy zz

ε ε= = 2 / ,yy

bµ = 1xx zz

µ µ= = , and the thickness h of each

layer is chosen as 0

/ 1.5h λ = , where 1, 2, 0.5b = respectively. Only the case b = 1 is

impedance-matched among the three cases. Since 1/2

y 0( )

xx yk hε µ = 6π for TM waves and

1/2

yy xx 0( ) k hε µ = 3π for TE waves, exp[ ( )]

E M xT T i kϑ= − = is satisfied for all three

half-wave plates at normal incidence; when two identical plates are stacked together in the

aforementioned way, i.e. 2 90 ,φ = ° they all act as polarization-independent rotators at normal

incidence. However, the impedance-matched conditions are not satisfied at oblique incidence,

and this will bring in negative influences on the performance of the rotators. Nevertheless, as is

shown above, the structure composed of IMAM slabs (b = 1) is more insensitive to incident

angle.

Figures 4(a) to 4(d) demonstrate the cross-polarization transmission coefficients and

polarization rotations with respect to 0

/x

k k when 45 ,δ = ° ϕ = 45°. It is clear that for the

impendence-matched polarization rotator, the incident angle region with | | | | 1EM ME

T T≈ ≈

( | |EE

T and | |MM

T tend to zero, which are not shown here) and EM ME

T T π∠ −∠ = ± is much

wider than the impendence-mismatched ones. Thus, the polarization rotation angle | | 90t

ψ → °

and the polarization ellipticity | | 0∆ → can be realized within a much wider incident angle as

well. For the IMAM polarization rotator, when the incident angle varies from 0° to 40°

(0

/ 0.64x

k k ≈ ), ( 0.077, 0.077)∆∈ − and t

ψ ∈ ( 90.69 , 89.05 )− ° − ° . While for the

impendence-mismatched rotators, the polarization rotation and ellipticity become very instable

as the incident angle increases. To take the nonmagnetic rotator with b = 2 as an example, the

polarization rotation becomes −105° and the ellipticity is around-0.6 at a 40° incidence.

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11996

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Fig. 4. (a) The amplitudes of transmission coefficients EM

| |T and | |ME

T , (b) the phase

difference EM ME

T T∠ −∠ , (c) the polarization rotation angle t

ψ and (d) the ellipticity ∆

with respect to the incident transversal vector 0

/x

k k for three different anisotropic bilayered

structures characterized as 2xx

bε = , 1yy zz

ε ε= = , 2 /yy

bµ = , 1xx zz

µ µ= = , and

the total thickness 0

2 / 3h λ = , when 45δ = ° , 45ϕ = ° . (e) Longitudinal wave vector

difference MZ EZ

( )k k− between TM and TE components and (f) p + 1/p as a function of

0/

xk k for different b values.

The basic mechanism of the incident-angle-insensitivity of the IMAM polarization rotator

can be understood by a simple argument. A lossless incident-angle-insensitive polarization

rotator actually requires that ( ) ( ) exp[ ( )]E x M x x

T k T k i kϑ≈ − ≈ can be satisfied for each single

slab even at oblique incidence. When 0,x

k ≠ the transmission coefficient of TM wave through

a homogeneous slab surrounded by air (the transmission coefficient of TE wave can be obtained

by duality) can be written as

1

,2 2cos( ) ( 1/ )sin( )M MZ MZ

M MT k d i p p k d

− = × − + (17)

where

2 2 1/ 2 2 2 1/2

yy 0 zz 0

[( / ) / ( )] / ( ) .M xx x xx x

p k k k kµ ε ε ε= − − (18)

The key issue is that for the impendence-mismatched slab, yy

( / ) 1xx

µ ε ≠ and

thus 1 2.M M

p p −+ > Calculations show that (not shown here), the reflection increases quickly

and become fluctuating when 0;x

k > meanwhile, ( )E x

T k and ( )M x

T k change periodically with

respect to kx, indicating the existence of high-order Fabry-Perot interference. While for the

IMAM slab, yy

( / ) 1,xx

µ ε = 1M

p ≈ (i.e. 1 / 2M M

p p+ ≈ ) can be maintained at a much larger

,x

k as depicted in Fig. 4(f), and thus ( )E x

T k and ( )M x

T k vary smoothly as a function of

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11997

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.x

k Moreover, it can be seen from the dispersion curves in Fig. 4(e) that 0MZ EZ

k k k− ≈ is

maintained better in the IMAM polarization rotator as 0

/x

k k increases, which indicates a more

stable phase difference. Hence, ( ) ( )E x M x

T k T k≈ − ≈ exp[ ( )]x

i kϑ can be obtained, and Eq. (14)

can be fulfilled even at a large-angle incidence. This is the reason that a bilayered polarization

rotator composed of such impedance-matched half-wave retarders will thus exhibit

incident-angle-insensitivity.

It should be noted that we mainly consider about the effects of transversal parameters

(xx

ε ,yy

ε ,xx

µ andyy

µ ) in the above discussion. Our calculations disclosed that the ratio between

zzε and

zzµ will also affect the property of angle-insensitivity. Numerical simulations show that

the optimized ratio generally locates around one when the surroundings is air, i.e.

zz zzε µ≈ .Since

2 2 1/ 2

yy 0( / )

EZ xx xx x zzk k kε µ µ µ= − and

2 2 1/2

yy 0 zz( / ) ,

MZ xx xx xk k kµ ε ε ε= − as the

longitudinal parameters zz

ε and zz

µ increase, the dispersion curve z

~x

k k of the metamaterial

will become flatter. Thus, the IMAM polarization rotator will become more insensitive to

incident angle and even exhibits a self-collimating property when /xx zz

ε ε and /xx zz

µ µ are

sufficiently small [5]. Then, the ratio deviation betweenzz

ε and zz

µ can be larger within the

same tolerable variation range of ∆ and t

ψ . To take the lossless IMAM polarization rotator

discussed in Fig. 3 as an example, within the same tolerable variation range of ∆ and t

ψ (i.e.

|∆| < 0.077 and t

ψ ∈ ( 90.69 , 89.05 )− ° − ° at 40° incidence), when 5,zz zz

ε µ≈ = the maximum

ratio deviation can be up to 20% (i.e. ,6.0 5.0zz zz

ε µ= = ).

5. Construct a microwave IMAM rotator with bi-SRR structure

To demonstrate an incident-angle-insensitive and polarization-independent polarization rotator,

we need to construct an IMAM half-wave retarder first, and then stack two retarders together

with their optical axes rotated 45° to form such a polarization rotator. Various metamaterial

structures can be employed to fabricate an IMAM retarder; however, metamaterials with both

electric and magnetic resonances are preferred, since the effective permeability of nonmagnetic

structures generally lies around one, which indicates that high refractive index and

impedance-matching condition can hardly be achieved simultaneously. Moreover, spatial

dispersion and anisotropy are inevitable because metamaterials are artificial mesostructures

[23]. In order to diminish such effects, electrically small non-bianisotropic microwave

bi-split-ring resonator (bi-SRR) is selected to construct an IMAM half-wave retarder [24].

Figures 5(a) and 5(b) are the schematic layouts of the IMAM retarder. The metallic bi-SRR

patterns are fabricated on one side of the FR-4 (lossless) substrate characterized as 04.9ε ε= ,

0µ µ= ,the substrate thickness t = 0.5 mm. The other dimensions of a unit cell are as follows:

lattice constant az = ay = 5 mm; the length of metal slices (perfect electric conductor) in Z and Y

directions z = y = 4 mm, the separation distance of the metal slices p = 0.12 mm, the gap g = 0.2

mm, the separation distance between adjacent unit cells s = 1.0 mm, and the width w and

thickness of metal are 0.2 mm and 0.08mm respectively. The design principle of the IMAM

half-wave retarder is that only TM component can excite magnetic resonances while TE wave

propagates “quietly” through the structure. Thus nearly full transmission and low effective

refractive index can be obtained for TE wave, while high effective index is available for TM

component at the impedance-matched frequency near the resonance. By tuning the dimensions

or the substrate material, one can change the impedance-matched frequency and the phase

difference between the two orthogonal components of the transmitted waves.

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11998

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Fig. 5. (a) and (b) are schematic layouts of the bi-SRR half-wave retarder. az = ay = 5 mm, z = y

= 4 mm, w = 0.2 mm, p = 0.12 mm, g = 0.2 mm, the substrate thickness t = 0.5 mm, the separation

distance between adjacent layers s = 1.0 mm, and the thickness of metal (perfect electric

conductor) is 0.08mm. (c) The structure of the polarization rotator, where yellow arrow indicates

the incident waves, pink arrows indicate the direction of electric field.

The simulation is performed using the software package CST Microwave Studio, in which

periodic boundary conditions are applied. Figure 6(a) depicts the amplitudes of co-polarization

terms of transmission coefficient 1T ( )x

k and reflection coefficient 1R ( )x

k for the first retarder,

where the cross-polarization terms are negligible. There are two dips in the TM wave reflection

coefficient curve corresponding to two possible impedance-matched frequencies at 3.12 GHz

and 3.25 GHz. Here the working frequency is chosen to be 3.25GHz. Because the ratio of unit

cell size to wavelength is less than 1/9, the effective medium description is valid, and the

retrieval effective parameters at normal incidence are: 1.156,TE

n = 3.457,TM

n = 0.854TE

Z =

and 1.001TM

Z = . Though the structure is not perfectly impedance-matched for TE wave at the

working frequency, the transmission is still very high (>99%) and it will have less influence on

the transmitted polarization state.

Note that in Fig. 6(b), the phase difference between two polarization states at 3.25GHz is

180°, which indicates that the structure acts as a half-wave retarder. Figure 6(c) reveals the

amplitudes of transmission coefficients for TE and TM waves with respect to 0

/x

k k at

3.25GHz, and Fig. 6(d) shows the transmitted phase difference between the two orthogonal

components, in which a 174° phase difference is still available at a 40° incidence

(0

/ 0.64x

k k ≈ ). It is clear that high transmission and a near 180° phase difference can be

maintained within a large incident angle.

Then, we construct a polarization rotator by aligning two such half-wave retarders in the

aforementioned way, i.e. 2 90φ = ° , as depicted in Fig. 5(c). In the simulation, the polar angle

of incident plane is rotated by 45° to perform an equivalent rotation of the IMAM retarder.

Thus, we can get the transmission and reflection Jones matrices of the two IMAM

retarders 1T ( )x

k , 1R ( )x

k and 2T ( )x

k , 2R ( )x

k directly from the simulation rather than from the

effective parameters. The cross-polarization terms of the transmission coefficient for the second

retarder 2T ( )x

k satisfy ( ) ( )EM x ME x

T k T k≈ ≈ exp[ ( )]x

i kϑ at 3.25GHz, as expected, and are not

shown here. The total transmission Jones matrix of the rotator can then be written

as 2 1T ( ) T ( )T ( ) ( )( ),R x x x xk k k O T k= + where 2 1 2 1( ) T ( )R ( )R ( )T ( ) ,

x x x xO T k k k k= +⋯ which is

the small quantity due to multiple reflections between the slabs. The first term of the expression

of O(T) represents the first order approximation. The ellipticity and polarization rotation angle

with respect to incident wave polarization under different incident angles can then be worked

out. For clarity but without loss of generality, we only give out the results for incident polar

angle o0δ = in Figs. 6(e) and 6(f). The results under zeroth order and first order

approximations are compared, it is clear that the variation ranges of ellipticity and polarization

rotation can be well described under zeroth order approximation. The variations of ellipticity

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11999

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Fig. 6. (a) The amplitudes of the transmission and reflection coefficients, (b) the phase of the

transmission coefficient of TE and TM waves at normal incidence for polarization rotator with

lossless substrate. (c) The transmission coefficients and (d) the phase difference between the two

orthogonal polarized waves with respect to 0

/x

k k at 3.25 GHz.(e) and (g) Polarization

rotation angle, (f) and (h) ellipticity variations with respect to incident wave polarization under

different angles of incidence, where (e) and (f) are for lossless substrate with 4.9,r

ε = (g) and

(h) are for lossy substrate with 4.9 0.01.r

ε = +

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 12000

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and polarization rotation are still quite acceptable even at the angle of incidence 40θ = ° ,

showing that the structure can act as an incident-angle-insensitive and polarization-independent

polarization rotator.

When a lossy substrate with a permittivity 4.9 ''r r

ε ε= + is considered in the polarization

rotator, it is found that the imaginary term may lead to a stronger absorption for TM wave near

resonant frequencies. However, we can still find the two dips in reflection curve of the

half-wave retarder. Although they are not as sharp as that in the lossless case, but they are small

enough to suppress the influence of multiple reflections between the two retarders in a

polarization rotator. For a commercial available low loss substrate with '' 0.01,r

ε ≤ our

simulations show that the phase difference has a tiny change (less than 2°), and there is a small

decrease in the transmission amplitude of TM wave due to absorption. The influences of the

substrate loss are illustrated in Figs. 6(g) and 6(h) when '' 0.01r

ε = and the polarization rotator

still works at 3.25GHz. Compared with the lossless rotator, the variations of ellipticity and

polarization rotation angle are slightly increased. But we have to emphasize that the depicted

results for the lossy rotator above are just for comparison, without any structural adjustment

according to the loss, so the performance can be further improved after a proper structural

optimization.

6. Conclusion

In conclusion, we proposed a design method to realize broad–angle and polarization-

independent polarization rotator based on impedance-matched anisotropic metamaterial

(IMAM). The mechanism and the influencing factors of the polarization rotator are discussed

analytically, in conjunction with the generalized 4 × 4 transfer matrix method. Also, we

illustrate how to construct a microwave IMAM rotator by using a bi-SRR structure. Compared

with traditional polarization rotators, the IMAM polarization rotator is more insensitive to

incident direction. This may offer the possibility to unyoke the limitation of the narrow working

angle and enhance the compactness and stability of microwave and optoelectronic systems.

Acknowledgement

The authors acknowledge the financial support from the National Natural Science Foundation

of China (NSFC) under the contracts 10774195, U0834001 and 10974263. The work is also

partially supported by Program for New Century Excellent Talents in University and the

Chinese National Key Basic Research Special Fund (2010CB923200).

#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 12001