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Design of polarization gratings for broadband illumination

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Abstract

Design of broadband diffractive elements is studied. It is shown that dielectric polarization gratings can be made to perform the same optical function over a broad band of wavelengths. Any design of paraxial-domain diffractive elements can be realized as such broadband elements that may, e.g., give constant diffraction efficiencies over the wavelength band while the field propagation after the elements remains wavelength-dependent. Furthermore, elements producing symmetric signals are shown to work with arbitrarily polarized or partially polarized incident planar broadband fields. The performance of the elements is illustrated by numerical examples and some practical issues related to their fabrication are discussed.

©2005 Optical Society of America

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Figures (4)

Fig. 1.
Fig. 1. Geometry of a form-birefringent, y-invariant, sub-wavelength-period grating with period d, modulation depth h, minimum transverse feature size g, and refractive indices n 0, n 1, n 2, and n 3.
Fig. 2.
Fig. 2. Diffraction efficiencies of the 1→3 beam splitter based on a scalar design, realized as a polarization grating, η 0=η ±1 (solid line), and as a surface-relief phase grating, η 0 (dashed line) and η ±1 (dash-dotted line).
Fig. 3.
Fig. 3. Diffraction efficiency of a 1→2 beam splitter made of TiO2 using subwavelength grating with parameters d=220 nm, f=0.6, and h=750 nm (dashed line), h=800 nm (solid line), and h=850 nm (dotted line).
Fig. 4.
Fig. 4. Diffraction efficiency of a 1→2 beam splitter made of TiO2 using subwavelength grating with parameters d=220 nm, h=800 nm, and f=0.55 (dashed line), f=0.60 (solid line), and f=0.65 (dotted line).

Equations (38)

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T ( x , y ) = [ t cos 2 θ ( x , y ) + t sin 2 θ ( x , y ) ( t t ) sin θ ( x , y ) cos θ ( x , y ) ( t t ) sin θ ( x , y ) cos θ ( x , y ) t sin 2 θ ( x , y ) + t cos 2 θ ( x , y ) ] ,
t = t exp ( i kn h ) ,
t = t exp ( i kn h ) ,
ϕ ( λ ) = arg t arg t = kh ( n n )
J i = α J R + β J L
J R = 1 2 ( 1 i )
J L = 1 2 ( 1 i ) .
α = ( E i x i E i y ) 2 ,
β = ( E i x + i E i y ) 2 ,
E ( x , y ) = T ( x , y ) J i .
E ( x , y ) = α E R ( x , y ) + β E L ( x , y )
E R ( x , y ) = 1 2 ( t + t ) J R + 1 2 ( t t ) exp [ i 2 θ ( x , y ) ] J L
E L ( x , y ) = 1 2 ( t + t ) J L + 1 2 ( t t ) exp [ i 2 θ ( x , y ) ] J R
t ± t = exp ( i arg t ) [ 1 ± exp ( i ϕ 0 ) ]
E ( x , y ) = m = n = J m , n exp [ i 2 π ( mx d x + ny d y ) ] ,
J m , n = 1 d x d y 0 d x 0 d y E ( x , y ) exp [ i 2 π ( mx d x + ny d y ) ] d x d y
η m , n = J m , n 2
J m , n = T m , n ( 0 ) J i + α T m , n ( 1 ) J L + β T m , n ( 1 ) J R
T m , n ( 0 ) = 1 2 ( t + t ) δ 0 , 0 ,
T m , n ( ± 1 ) = 1 2 ( t t ) t m , n ( ± 1 ) ,
t m , n ( ± 1 ) = 1 d x d y 0 d x 0 d y exp [ ± i 2 θ ( x , y ) ] exp [ i 2 π ( mx d x + ny d y ) ] d x d y .
η m , n = T m , n ( 0 ) 2 + α T m , n ( 1 ) 2 + β T m , n ( 1 ) 2 + 2 Re [ α * T m , n ( 0 ) * β T m , n ( 1 ) ] + 2 Re [ β * T m , n ( 0 ) * α T m , n ( 1 ) ]
η m , n = α T m , n ( 1 ) 2 + β T m , n ( 1 ) 2
η m , n = T m , n ( 0 ) 2 + T m , n ( 1 ) 2 = 1 4 t + t 2 δ 0 , 0 + 1 4 t t 2 t m , n ( 1 ) 2 .
θ ( x , y ) = θ ( x ) = π x D ,
η 0 = 1 4 t + t 2
η ± 1 = 1 8 t t 2 ( 1 E i x E i y sin ϑ ) ,
η m , n = ( α 2 + β 2 ) T m , n ( 1 ) 2 = T m , n ( 1 ) 2
η 0 , 0 = T 0 , 0 ( 0 ) 2 + T 0 , 0 ( 1 ) 2 + ( t 2 t 2 ) Re [ α β * t 0 , 0 ( 1 ) ]
J ( r , ω ) = [ J xx ( r , ω ) J xy ( r , ω ) J yx ( r , ω ) J yy ( r , ω ) ] = [ E x * ( r , ω ) E x ( r , ω ) E x * ( r , ω ) E y ( r , ω ) E y * ( r , ω ) E x ( r , ω ) E y * ( r , ω ) E y ( r , ω ) ] ,
J t = T J i T ,
J ( r , ω ) = n = 1 2 λ n ( ω ) ϕ n ( r , ω ) ϕ n ( r , ω )
J t = T ( λ 1 J 1 + λ 2 J 2 ) T = λ 1 T J 1 T + λ 2 T J 2 T ,
η m , n = λ 1 η 1 m , n + λ 2 η 2 m , n ,
η m , n = ( λ 1 + λ 2 ) η 1 m , n = η 1 m , n .
t ( x , λ ) = exp [ i Φ ( λ ) sin ( 2 π x D ) ] ,
Φ ( λ ) = 2 π ( n 2 n 1 ) h 0 λ
η m ( λ ) = J m 2 [ Φ ( λ ) 2 ] ,
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