Progress In Electromagnetics Research M, Vol. 9, 9 20, 2009

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1 Prgress In Electrmagnetics Research M, Vl. 9, 9 20, 2009 WIDE-ANGLE REFLECTION WAVE POLARIZERS USING INHOMOGENEOUS PLANAR LAYERS M. Khalaj-Amirhsseini and S. M. J. Razavi Cllege f Electrical Engineering Iran University f Science and Technlgy Narmak, Tehran, Iran Abstract In this paper, inhmgeneus planar layers are ptimally designed as reflectin wave plarizers in a desired incidence angles range. First, the electric permittivity functin f the structure is expanded in a truncated Furier series. Then, the ptimum values f the cefficients f the series are btained thrugh an ptimizatin apprach. The validatin and the perfrmance f the prpsed structure are verified using sme examples. 1. INTRODUCTION Wave plarizers have fund sme applicatins in micrwave and ptical systems such as free-space ptical switching netwrks, read-write magnet-ptic data strage systems and plarizatin-based imaging systems [1 14]. In a wave plarizer nly ne f tw pssible mdes (TE r TM) can be appeared in the transmissin r reflectin waves. One may classify plarizers int the fllwing types [9, 14]: dichric plarizers, anistrpic crystal plarizers, Brewster angle plarizers, wire-grid plarizers and gyrtrpic slab plarizers. The mst straightfrward idea t design a wave plarizer is using the Brewster s angle t eliminate the TM mde in the reflectin wave [8, 9]. Hwever, the Brewster s plarizers wrk nly at a narrw range f angles f incidence. On the ther hand, Inhmgeneus Planar Layers (IPLs) are widely used in micrwave and antenna engineering [15 19]. In this paper, we prpse utilizing IPLs as reflectin wave plarizers in a desired incidence angles range. T ptimally design IPLs, their electric permittivity functin is expanded in a truncated Furier series, first. Then, the ptimum values f the cefficients f the series are Crrespnding authr: M. Khalaj-Amirhsseini (khalaja@iust.ac.ir).

2 10 Khalaj-Amirhsseini and Razavi btained thrugh an ptimizatin apprach. The identical prcedure has been used t ptimally design IPLs as Radmes [18] and impedance matchers between tw different mediums [19] previusly. Finally, the usefulness f the prpsed structure is verified using sme examples. 2. ANALYSIS OF IPLS In this sectin, the frequency dmain equatins f the IPLs are reviewed. Fig. 1 shws a typical IPL with thickness d, whse left and right mediums are the free space and whse electric permittivity functin is ε r (z). One way t fabricate the IPLs is t place several thin hmgeneus dielectric layers beside each ther. It is assumed that the incidence plane wave prpagates bliquely twards psitive x and z directin with an angle f incidence θ i and electric filed strength E i. Als, tw different plarizatins are pssible, ne is the TM and the ther is the TE. We desire t design the intrduced plarizer s that nly TE mde is reflected t the input medium, i.e., z < d, and TM mde can t be reflected. The differential equatins describing IPLs have nn-cnstant cefficients and s except fr a few special cases n analytical slutin exists fr them. There are sme methds t analyze the IPLs such as cascading many shrt sectins [20], Finite Difference [21], Taylr s Series Expansin [22], Furier Series Expansin [23], the Equivalent Surces Methd [24], the Methd f Mments [25] and an apprximate clsed frm slutin [26]. Almst in all f these methds, the IPLs are mdeled as a nnunifrm transmissin line with the fllwing Figure 1. A typical IPL as a reflectin wave plarizer.

3 Prgress In Electrmagnetics Research M, Vl. 9, characteristic impedance and prpagatin cefficient η 0 εr (z) sin 2 (θ i ), TE Z c (z) = η 0 ε r (z) sin 2 (θ i ), TM ε r (z) β = 2π = ω λ t c (1) ε r (z) sin 2 (θ i ) (2) where c is the velcity f the light, λ t is the transverse wavelength and η 0 is the wave impedance in the free space. After finding the ABCD parameters f the equivalent nnunifrm transmissin lines [20 26], the input reflectin cefficient will be determined as fllws where Γ = (A η 0tC) η 0t + (B η 0t D) (A + η 0t C) η 0t + (B + η 0t D) η 0 η 0t = cs(θ i ), η 0 cs(θ i ), TE TM is the transverse impedance f the free space. It is well knwn that a high perfrmance reflectin plarizer has t perate ver a wide range f incidence angles, the extinctin rati f the unwanted t the desired plarized wave in reflectin is small and the reflectance fr the desired plarizatin is high. Hence t shw the perfrmance f a plarizer, we define the extinctin rati R Γ as the rati f reflectin cefficient f TM mde t that f TE mde, as fllws R Γ = Γ TM Γ TE (5) It is evident that as R appraches zer the perfrmance f the plarizer becmes better. Mrever, a gd plarizer shuld have TE reflectin cefficient Γ TE as higher as pssible. 3. SYNTHESIS OF POLARIZERS In this sectin, a general methd is prpsed t ptimally design the IPLs as plarizers. First, we cnsider the fllwing truncated Furier series expansin fr the electric permittivity functin N ln (ε r (z) 1) = C n cs(πnz/d) (6) n=0 (3) (4)

4 12 Khalaj-Amirhsseini and Razavi T enfrce the designed plarizers t be symmetric, we have t use the fllwing truncated Furier series instead f (6) fr the electric permittivity functin. N ln (ε r (z) 1) = C n cs(2πnz/d) (7) n=0 An ptimum designed plarizer has t have a parameter R Γ defined in (5) as small as pssible in a desired incidence angle range, while the amplitude f the reflected TE mde is being cnstant at that range. Therefre, the ptimum values f the cefficients C n in (6) r C n in (7) can be btained thrugh minimizing the fllwing errr functin crrespnding t J incidence angles between θ i,min and θ i,max. Errr = 1 J ( ) R Γ θ (j) 2 i J + 1 J + 1 J j=1 J ( R Γ j=1 J ( Γ TE j=1 θ (j) i θ (j) i ) 2 1 J ) 2 1 J J ( R Γ j=1 θ (j) i J ( Γ TE j=1 ) θ (j) i 2 ) The secnd and third terms in (8) act as variance functins t keep cnstant R Γ and Γ TE, respectively, in desired incidence angles range. Mrever, defined errr functin shuld be restricted by sme cnstraints such as easy fabricatin given by 2 (8) max {ε r (z)} (ε r ) max (9) where (ε r ) max is the maximum value f ε r (z), in the fabricatin step. 4. EXAMPLES AND RESULTS In this sectin we investigate and validate the capability f IPLs as wide-angle reflectin wave plarizer. First f all, cnsider a hmgeneus planar layer with permittivity ε r = 7.55 as a plarizer perating at Brewster s angle θ i, = 70. Figs. 2 and 3, illustrate the magnitude f TE and TM reflectin cefficients and R Γ versus the incidence angle fr sme thickness d. It is seen that the plarizers with thickness d equal t a quarter f transverse wavelength (λ t /4 = 5.81 mm fr f = 10 GHz) at Brewster s angle has the best perfrmance,

5 Prgress In Electrmagnetics Research M, Vl. 9, i.e., mre TE reflectin. Imprtantly, it is seen that Brewster s plarizers perate at a very narrw incidence angle range. Nw, we wuld like t design an IPL as a plarizer fr incidence angles between θ i,min = 60 and θ i,max = 80 at frequency 10.0 GHz, cnsidering (ε r ) max = 10. Using the prpsed ptimizatin apprach, cnsidering N = 10 spatial harmnics and J = 10 incidence angles fur plarizers were synthesized assuming their thickness t be d = 5, 10, 20 and Figure 2. θ i = 70. The Γ T E and Γ T M f hmgeneus planar layer fr Figure 3. The R Γ f hmgeneus planar layer fr θ i = 70.

6 14 Khalaj-Amirhsseini and Razavi 25 mm. The unknwn cefficients f the truncated Furier series related t the synthesized plarizers are written in Table 1. Fig. 4 illustrates the btained electric permittivity functin ε r (z) and Figs. 5 and 6 illustrate the magnitude f TE and TM reflectin cefficients and R Γ versus the incidence angle. It is bserved that the designed plarizers have a lw TM reflectin at the incidence angle range f Figure 4. The electric permittivity functin ε r (z) f plarizers Ns. 1 4 fr f = 10 GHz, (ε r ) max = 10 and θ i = [60, 80 ]. Figure 5. The Γ T E and Γ T M f plarizers Ns. 1 4 fr f = 10 GHz, (ε r ) max = 10 and θ i = [60, 80 ].

7 Prgress In Electrmagnetics Research M, Vl. 9, θ i = [60, 80 ]. Als, it is seen that as the thickness f plarizer is selected higher the magnitude f TE reflectin is increased. Fig. 7 illustrates the Γ TE, Γ TM and R Γ f the fifth plarizer versus the incidence angle cnsidering θ i,min = 65 and θ i,max = 75 and d = 25 mm. Als, Fig. 8 illustrates the Γ TE, Γ TM and R Γ f the sixth plarizer versus the incidence angle cnsidering θ i,min = 40 Figure 6. The R Γ f plarizers Ns. 1 4 fr f = 10 GHz, (ε r ) max = 10 and θ i = [60, 80 ]. Figure 7. The Γ T E, Γ T M and R Γ f plarizer N. 5 fr f = 10 GHz, (ε r ) max = 10 and θ i = [65, 75 ].

8 16 Khalaj-Amirhsseini and Razavi and θ i,max = 50 and d = 50 mm. The btained electric permittivity functin and the unknwn cefficients f the fifth and sixth plarizer are shwn in Fig. 9 and Table 1, respectively. Cmparing Figs. 5 8 with each ther gives us that as the desired relative incidence angle Figure 8. The Γ T E, Γ T M and R Γ f plarizer N. 6 fr f = 10 GHz, (ε r ) max = 10 and θ i = [40, 50 ]. Figure 9. The electric permittivity functin ε r (z) f plarizers Ns. 5 and 6 fr f = 10 GHz and (ε r ) max = 10.

9 Prgress In Electrmagnetics Research M, Vl. 9, range defined by w θ = 2 θ i,max θ i,min θ i,max + θ i,min (10) is decreased, the perfrmance f the ptimum designed plarizers is increased. It is an interesting result frm Fig. 8 that IPLs can be designed as plarizers fr the incidence angles smaller than 45 (a) TM, θ i = 60 (b) TE, θ i = 60 (c) TM, θ i = 70 (d) TE, θ i = 70 (e) TM, θ i = 80 (f) TE, θ i = 80 (g) TM, θ i = 20 (h) TE, θ i = 20 Figure 10. The abslute f electric and magnetic fields, respectively fr TE and TM plarized incident wave at f = 10 GHz simulated by CST sftware.

10 18 Khalaj-Amirhsseini and Razavi Table 1. The unknwn cefficients f the truncated Furier series f designed plarizers fr f = 10 GHz and (ε r ) max = 10. N. 1: θ i = [60, 80 ] d = 5 mm N. 2: θ i = [60, 80 ] d = 10 mm N. 3: θ i = [60, 80 ] d = 20 mm N. 4: θ i = [60, 80 ] d = 25 mm N. 5: θ i = [65, 75 ] d = 25 mm N. 6: θ i = [40, 50 ] d = 50 mm C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C (despite lw reflectivity f TE plarizatin), which it is nt pssible fr Brewster s plarizers. T validate the results, the perfrmance f the designed plarizer N. 3 is simulated by full wave simulatr CST sftware. Fig. 10 illustrate the abslute f electric and magnetic fields fr TE and TM plarized incident wave, respectively, cnsidering f = 10 GHz and θ i = 20, 60, 70 and 80. It is seen the reflectin f TM wave is nearly equal t that f TE wave at θ i = 20, while the reflectin f TM wave is very weaker than that f TE wave at θ i = 60, 70 and 80. Therefre, Fig. 10 is in agreement with Fig. 5 assuming d = 20 mm. 5. CONCLUSION Inhmgeneus Planar Layers (IPLs) were ptimally designed as reflectin wave plarizers in a desired incidence angles range. First, the electric permittivity functin f the structure is expanded in a truncated Furier series. Then, the ptimum values f the cefficients f the series are btained thrugh an ptimizatin apprach. The validatin and the perfrmance f the prpsed structure were verified using sme examples. It was bserved that the designed plarizers have a gd perfrmance in the desired incidence angle range. Als, as the thickness f the plarizer is chsen larger r its desired relative incidence angle range is decreased, whse perfrmance is increased. The prpsed methd can be extended fr IPLs, whse magnetic permeability is inhmgeneus slely r alng with their electric permittivity. Als, we can cnsider IPLs fr spherical wavefrnts instead f planar nes in the future.

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