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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 8 — Apr. 18, 2005
  • pp: 3055–3067
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Design of polarization gratings for broadband illumination

Hanna Lajunen, Jari Turunen, and Jani Tervo  »View Author Affiliations


Optics Express, Vol. 13, Issue 8, pp. 3055-3067 (2005)
http://dx.doi.org/10.1364/OPEX.13.003055


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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

1. Introduction

Diffractive phase elements based on dielectric surface-relief structures can be used to realize a large variety of optical functions with high efficiencies [1

1. J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie-Verlag, Berlin, 1997).

]. However, one of the remaining fundamental problems in diffractive optics is the chromatic sensitivity of the phase elements which is caused by the wavelength-dependence of their optical functions. This means that the elements will usually give the desired results for a single wavelength or a narrow wavelength band, but the performance is not satisfactory if broadband illumination is used. Consequently, diffractive elements cannot be fully utilized, e.g., in connection with fields produced by thermal sources or femtosecond lasers. Therefore microstructured elements having the same response for a broad band of frequencies could improve the performance, or even allow novel applications of diffractive optics associated with broadband light.

The first steps towards achromatic diffractive phase elements are based on form birefringent subwavelength gratings. By optimizing the grating parameters, achromatic phase retarders can be designed, and the performance of such elements has also been experimentally demonstrated [2

2. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566–1572 (1997). [CrossRef] [PubMed]

4

4. D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003). [CrossRef]

]. In addition, some other types of subwavelength grating structures for different purposes, such as antireflection surfaces and polarizers as well as blazed binary gratings, optimized to work with broadband light have been recently reported [5

5. Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings fabricated upon silicon substrates,” Opt. Lett. 24, 1422–1424 (1999). [CrossRef]

9

9. C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. 29, 1593–1595 (2004). [CrossRef] [PubMed]

]. However, most of the introduced broadband elements are restricted to some specialized optical functions and they do not reach the versatility of general diffractive phase elements.

On the other hand, form birefringence of diffractive subwavelength structures can be utilized for realizing dielectric polarization gratings that are based on space-variant modulation of the polarization state of the incident field [10

10. F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]

13

13. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003). [CrossRef]

]. By that means paraxial-domain diffractive elements, such as beam-splitters, can be designed and fabricated [11

11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). [CrossRef]

17

17. C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. 26, 1651–1653 (2001). [CrossRef]

]. It has also been shown that by taking into account the polarization freedom of signals, the vectorial nature of light can be exploited in order to obtain paraxial-domain diffractive elements with 100% diffraction efficiency [12

12. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000). [CrossRef]

14

14. M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

], which is not possible in the realm of scalar optics [18

18. F. Wyrowski, “Upper bound of efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991). [CrossRef] [PubMed]

]. Moreover, we recently introduced the possibility to design broadband diffractive elements based on dielectric polarization grating structures [19

19. H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. 29, 803–805 (2004). [CrossRef] [PubMed]

].

In this paper we will discuss further the design of polarization gratings for broadband illumination. The theory presented in Ref. 19 is extended to the cases of arbitrarily polarized and partially polarized fields, and polarization-insensitive broadband designs are introduced. In addition, the broadband performance of polarization gratings is illustrated with numerical examples and some practical issues concerning the fabrication of the elements are discussed. Although only a few examples of the possible broadband elements are presented herein, the theory introduced in this paper embodies the general method for designing broadband elements that can be used for all the same various purposes as conventional paraxial-domain diffractive phase elements.

It is important to keep in mind that while, e.g., constant efficiencies over a wavelength band can be obtained with broadband gratings, the field propagation after the elements is still wavelength dependent. Thus, for instance, the diffraction angles are different for different wavelengths. However, while the wavelength-dependence of the field propagation is a separate problem, it will not prevent the usefulness of broadband diffractive elements in different situations. For example, the angular dispersion of gratings is, instead, utilized in many applications such as spectroscopic use or in hybrid diffractive-refractive optics, and the broadband performance of diffractive elements may improve their function also in such cases.

The paper is organized as follows. In Section 2 we discuss diffractive polarization elements and their achromatization in general terms. The signals attainable using periodic broadband polarization gratings are discussed in Section 3 assuming that the incident field has some well-defined polarization state. Special polarization-insensitive designs are presented in Section 4 for arbitrarily polarized and partially polarized fields. In Section 5 we show an example of replacing a conventional scalar phase element by a polarization grating working over a broad wavelength band and study the tolerance of elements made of real materials to fabrication errors by numerical simulations. Finally, the main results of the paper are summarized in Section 6.

2. Broadband diffractive polarization elements

In this Section we consider the transmission of polarized plane waves through polarization modulating diffractive elements. In general, the response of such elements may be wavelength-dependent in the same way as in the case of surface-relief phase elements. However, by optimizing the parameters of the subwavelength gratings that form the desired structure, the polarization gratings may be designed to perform the same function at a broad band of wavelengths [19

19. H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. 29, 803–805 (2004). [CrossRef] [PubMed]

], as we describe in the following.

The local effect of a polarization modulating element on the incident field can be described using the Jones matrix formalism. Let us consider a structure consisting of a subwavelength-period grating with spatially varying fringe orientation defined by a function θ(x,y) that denotes the local rotation angle of the direction of the grating grooves, measured in respect to the x-axis. The polarization modulating element may hence be represented by the Jones matrix [20

20. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.

]

T(x,y)=[tcos2θ(x,y)+tsin2θ(x,y)(tt)sinθ(x,y)cosθ(x,y)(tt)sinθ(x,y)cosθ(x,y)tsin2θ(x,y)+tcos2θ(x,y)],
(1)

where t and t are the complex-amplitude transmittances of the local electric field components parallel and perpendicular, respectively, to the optical axis. In general cases also t and t could be spatially varying but in this paper we will concentrate on binary structures for which only θ is a slowly varying function of the coordinates x and y.

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.

Let us first consider the properties of the form birefringent subwavelength grating forming the polarization modulating structure. The geometry and parameters characterizing such binary, y-invariant grating with period d, depth h, and fill factor f=g/d are illustrated in Fig. 1. The functions describing the transmittances of the field components parallel and perpendicular to the grating grooves can be written as

t=texp(iknh),
(2)

and

t=texp(iknh),
(3)

where h is the depth of the structure, k=2π/λ is the wave number and n and n are the effective refractive indices that depend on the grating parameters and the refractive indices of the used materials. If the period of the subwavelength grating is considerably smaller than the wavelength of incident light, d≪λ, the effective refractive indices can be determined using approximate formulas. However, if the period is close to the wavelength, rigorous diffraction theory [21

21. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]

] must be used to accurately evaluate the effective indices.

In an ideal situation all the light is transmitted through the structure so that |t |=|t |=1. In such a case the subwavelength grating modulates only the phase of the field introducing a phase difference

ϕ(λ)=argtargt=kh(nn)
(4)

between the parallel and perpendicular field components. Generally, this phase difference depends on the wavelength through the wave number k. However, the rigorously determined effective refractive indices are also wavelength-dependent, and it has been shown that by optimizing the grating parameters, i.e. the period, depth, and the fill factor, the dispersion of the effective indices can be employed to keep the phase difference approximately the same over a broad wavelength band [2

2. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566–1572 (1997). [CrossRef] [PubMed]

]. Thus, form birefringent subwavelength gratings can be used as achromatic wave plates [2

2. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566–1572 (1997). [CrossRef] [PubMed]

4

4. D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003). [CrossRef]

].

Let us now assume that we have a diffractive polarization modulating element made of an achromatic subwavelength grating, such that the phase difference between the field components is constant, ϕ(λ)=ϕ 0, over some wavelength band Δλ. In such a case the optical function of the element depends on the local phase difference between the electric field components that is controlled by the fringe orientation angle θ(x,y) only and thus it remains the same for all the wavelengths within Δλ. To illustrate this possibility more closely we consider in the following the transmission of an elliptically polarized plane wave through the polarization modulating element characterized by Eq. (1).

Any elliptical polarization state of the incident field can be represented by a superposition of the two orthogonal circular polarization states, i.e. as

Ji=αJR+βJL
(5)

where

JR=12(1i)
(6)

is the Jones vector associated with right-handed circularly polarized field and

JL=12(1i).
(7)

denotes correspondingly the left-handed circularly polarized state. The complex weighting coefficients α and β are defined by equations

α=(EixiEiy)2,
(8)
β=(Eix+iEiy)2,
(9)

where E ix and E iy are the x and y components of the incident plane wave, and they are normalized as |α|2+|β|2=|E ix|2+|E iy|2=1. Furthermore, we will now assume that α and β are wavelength-invariant, so that the polarization state of the incident field is the same at all the wavelengths in Δλ.

The field transmitted through the polarization modulating element can be determined by the equation

E(x,y)=T(x,y)Ji.
(10)

Inserting Eqs. (1) and (5) into (10) we obtain the expression

E(x,y)=αER(x,y)+βEL(x,y)
(11)

for the transmitted field with an arbitrary initial polarization state. The functions

ER(x,y)=12(t+t)JR+12(tt)exp[i2θ(x,y)]JL
(12)

and

EL(x,y)=12(t+t)JL+12(tt)exp[i2θ(x,y)]JR
(13)

characterize the transmitted forms of the initially right- and left-handed circularly polarized field components, respectively [11

11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). [CrossRef]

,15

15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express 12, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

]. Since the polarization state of the field is changed in the transmission, we emphasize that the subscripts of E R and E L imply only to the polarization state of the incident field components, and not to the properties of the transmitted fields themselves.

By examining Eqs. (11)(13) it can be easily seen that any polarized field transmitted through an arbitrary binary polarization grating depends only on the coefficients

t±t=exp(iargt)[1±exp(iϕ0)]
(14)

and the wavelength-independent rotation angle θ(x,y). We note that the phase difference ϕ 0 was assumed to be constant over the considered wavelength band Δλ. The phase term arg t in Eq. (14) may be wavelength-dependent, but usually such spatially invariant phase terms can be ignored since they will not affect the optical effect of the element. Thus, the polarization element will perform the same optical function for all the wavelengths of the certain band Δλ. This means that the transmitted field right behind the element has the same form given by Eqs. (11)(13) at all the different wavelengths. However, the further propagation of the field will still be wavelength-dependent as discussed in Section 1.

3. Designs for polarized fields

If the polarization modulating structure is periodic, the transmitted field can be represented by the Rayleigh expansion

E(x,y)=m=n=Jm,nexp[i2π(mxdx+nydy)],
(15)

where dx and dy denote the grating periods in the x- and y-directions, respectively, and

Jm,n=1dxdy0dx0dyE(x,y)exp[i2π(mxdx+nydy)]dxdy
(16)

are the Jones vectors associated with different diffraction orders (m,n). We assume that the grating period is large compared to the wavelengths of the incident field, and thus the diffraction orders are paraxial. The diffraction efficiencies can be simply defined as

ηm,n=Jm,n2
(17)

since the energy of the incident field per period is normalized to unity as ‖J i2=1.

Let us first consider the field transmitted through a polarization grating with some yet undefined complex-amplitude transmittances t and t and fringe rotation angle θ(x,y). The Jones vectors associated with the diffraction orders generated from an arbitrarily polarized incident field can be expressed in the form

Jm,n=Tm,n(0)Ji+αTm,n(1)JL+βTm,n(1)JR
(18)

which is obtained by inserting Eq. (11) into Eq. (16). Here we have defined scalar complex amplitudes Tm,n(0) and Tm,n(±1) characterizing the diffraction orders as

Tm,n(0)=12(t+t)δ0,0,
(19)

where δ m,n is the Dirac delta function, and

Tm,n(±1)=12(tt)tm,n(±1),
(20)

where

tm,n(±1)=1dxdy0dx0dyexp[±i2θ(x,y)]exp[i2π(mxdx+nydy)]dxdy.
(21)

It is noticed that Tm,n(0) vanishes for all but the zeroth diffraction order and its contribution in Eq. (18) is independent of the polarization state of the incident field. Furthermore, tm,n(±1) are seen to equal the complex-amplitude transmittances of scalar phase elements with transmission functions exp[±i2θ (x,y)]. The diffraction efficiencies are obtained from Eq. (18) by using definition (17). In general cases the result may be expressed as

ηm,n=Tm,n(0)2+αTm,n(1)2+βTm,n(1)2+2Re[α*Tm,n(0)*βTm,n(1)]+2Re[β*Tm,n(0)*αTm,n(1)]
(22)

but now T m,n=0 if (m,n)≠(0,0) and a simple form

ηm,n=αTm,n(1)2+βTm,n(1)2
(23)

is obtained for all but the zeroth diffraction order. This result is also known from studies concerning polarization gratings illuminated by a single wavelength [15

15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express 12, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

], but now these diffraction efficiencies are the same for all wavelengths of Δλ.

Assuming that the incident field is right-handed circularly polarized, such that α=1 and β=0, Eq. (22) reduces to the form

ηm,n=Tm,n(0)2+Tm,n(1)2=14t+t2δ0,0+14tt2tm,n(1)2.
(24)

The first term affects only the zeroth diffraction order, and the latter term depends on the complex amplitudes tm,n(1) given by Eq. (21). Thus the polarization grating produces an equivalent signal to that of a paraxial scalar phase element with phase transmission function 2θ(x,y) and an additional zeroth order contribution, in proportion to the coefficients determined by t and t . If we choose the parameters of the subwavelength grating so that it works as an ideal half-wave plate, i.e. ϕ 0=π and |t |=|t |=1, over the wavelength band Δλ, the additional zeroth order vanishes and the polarization grating corresponds directly to the phase element. In the same way as suggested for elements designed for a single wavelength [15

15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express 12, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

], the relation between the additional zeroth order and the rest of the signal can also be adjusted in a desired manner by varying the phase difference ϕ 0 introduced by the subwavelength grating.

Consequently, any scalar phase transmission function can be made to work over a broad wavelength band using a circularly polarized incident field and a polarization modulating element the fringes of which are rotated according to the desired phase function. This result is not restricted only to periodic elements as can be seen by considering the transmitted field given by Eq. (12), using the same assumptions as above [19

19. H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. 29, 803–805 (2004). [CrossRef] [PubMed]

]. Since the diffractive phase elements are usually strongly wavelength-dependent, the broadband performance offered by polarization gratings extends the functionality of their designs considerably. However, the total diffraction efficiencies are in this case restricted to those obtainable with scalar phase elements [18

18. F. Wyrowski, “Upper bound of efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991). [CrossRef] [PubMed]

].

As an example of polarization gratings with 100% efficiency, let us consider a one-dimensionally periodic structure based on binary subwavelength grating for which t and t are spatially invariant. The groove direction varies according to equation [10

10. F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]

, 12

12. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000). [CrossRef]

]

θ(x,y)=θ(x)=πxD,
(25)

i.e. the fringes are linearly rotated within a period D. This design produces only three nonvanishing diffraction orders the efficiencies of which are given by [12

12. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000). [CrossRef]

]

η0=14t+t2
(26)

and

η±1=18tt2(1EixEiysinϑ),
(27)

where ϑ=arg E ix-arg E iy. If we assume that all the light is transmitted completely and the phase difference introduced by the subwavelength grating is π, the associated complex amplitude transmittances satisfy t =-t and |t |=|t |=1. In this case the zeroth diffraction order disappears and energy distribution between the ±1 orders is determined by the polarization state of the incident field. For linearly polarized light η 1=η -1=1/2, and a diffractive 1→2 beam splitter with 100% total efficiency is obtained. On the other hand, assuming that arg t -arg t =cos-1(-1/3) and the incident light is still linearly polarized, η 0=η ±1=1/3 and the grating acts thus an ideal 1→3 beam splitter [12

12. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000). [CrossRef]

]. The performance of this kind of polarization-grating beam splitters has also been demonstrated experimentally for single wavelengths [11

11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). [CrossRef]

, 16

16. J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26, 587–589 (2001). [CrossRef]

, 17

17. C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. 26, 1651–1653 (2001). [CrossRef]

]. Using achromatic subwavelength structures we can obtain broadband diffractive elements with 100% efficiency based on the design described above [19

19. H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. 29, 803–805 (2004). [CrossRef] [PubMed]

].

Designs for polarization gratings producing larger arrays of diffraction orders with 100% efficiency has also been introduced [14

14. M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

]. Usually, in such designs the transmittances t and t are spatially varying besides the rotation angle θ, and the extension to the elements with broadband performance is not straightforward. However, designs for polarization-grating beam splitters can also be obtained by using iterative algorithms [13

13. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003). [CrossRef]

, 15

15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express 12, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

], and it has been found that the efficiencies of binary polarization gratings designed taking the vectorial nature of light into account are generally higher than those of any corresponding scalar designs [13

13. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003). [CrossRef]

]. All such binary structures can easily be converted to work over a broad wavelength band using achromatic subwavelength gratings.

4. Polarization-insensitive designs

Let us consider further Eqs. (19)(23) that define the diffraction efficiencies of any binary polarization grating for incident fields with any state of polarization. Concentrating on diffraction orders (m,n)≠(0,0) we see from Eq. (23) that the efficiencies equal the sum of the diffraction efficiencies associated with the complex amplitudes Tm,n(±1) weighed by the absolute values of the coefficients α and β. It is also noted from Eq. (20) that the quantities tm,n(±1) satisfy a symmetry condition tm,n(1)=tm,n(1)* and consequently |Tm,n(1)=|Tm,n(1)|. Hence, the far-field diffraction patterns produced by right- and left-handed circular polarization states are inversely symmetrical, and the diffraction pattern of an arbitrarily polarized consists of these signals in proportion of the circularly polarized field components. The zeroth diffraction order includes the contribution from these diffraction patterns and the additional terms defined in Eq. (19).

As a consequence of the inverse symmetry of the diffraction patterns of right- and left-handed circularly polarized incident fields, symmetric signals can be produced by polarization gratings independently of the polarization of the incident field. Namely, if we assume that |Tm,n(1)|=|Tm,n(1)|, it immediately follows that |Tm,n(1)|=|Tm,n(1)|. Thus the efficiencies of the diffraction orders of arbitrarily polarized illumination are, based on Eq. (23),

ηm,n=(α2+β2)Tm,n(1)2=Tm,n(1)2
(28)

when (m,n)≠(0,0), and the generated signal is the same for any incident polarization state. With the assumed symmetry Eq. (22) gives an expression

η0,0=T0,0(0)2+T0,0(1)2+(t2t2)Re[αβ*t0,0(1)]
(29)

The results above are valid for any polarized fields, but in real situations the direction of the electric field vector may often fluctuate in a more or less random manner as a function of time whereupon the field is partially polarized [22

22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

]. In that case the field can be described by the polarization matrix

J(r,ω)=[Jxx(r,ω)Jxy(r,ω)Jyx(r,ω)Jyy(r,ω)]=[Ex*(r,ω)Ex(r,ω)Ex*(r,ω)Ey(r,ω)Ey*(r,ω)Ex(r,ω)Ey*(r,ω)Ey(r,ω)],
(30)

where the angle brackets denote ensemble average, representing a space–frequency domain generalization of the often used representation in the space–time domain [22

22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

, 23

23. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004). [CrossRef]

]. The transmission of partially polarized field through a polarization modulating element can be studied by the matrix equation

Jt=TJiT,
(31)

where T is the Jones matrix of the element given by Eq. (1), J i is the polarization matrix of the incident field, and the dagger denotes the adjoint. Some studies considering the transmission of partially polarized and partially coherent fields through polarization gratings using this formalism can be found [24

24. G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18, 1399–1405 (2001). [CrossRef]

, 25

25. G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001). [CrossRef]

]. In the following, we will use a somewhat different approach and assume that the field components have enough spatial coherence to produce diffraction effects.

Since the polarization matrix is Hermitian and non-negative definite, it may be represented as [26

26. L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. 81, 1104–1114 (1963). [CrossRef]

, 27

27. J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astr. Soc. 34, 403–419 (1973). [CrossRef]

]

J(r,ω)=n=12λn(ω)ϕn(r,ω)ϕn(r,ω)
(32)

where λn and ϕn are the eigenvalues and the eigenvectors of the matrix. The eigenvalues are real and non-negative and the eigenvectors are orthonormal in the sense ϕn ϕ m =δmn . Furthermore, the sum of the eigenvalues represent the total energy of the field, and thus it may be normalized as λ 1+λ 2=1. The expansion (32) represents the polarization matrix as a sum of polarized fields that are mutually uncorrelated and, consequently, we can use it in order to examine arbitrary partially polarized planar fields by means of the fully polarized parts. Inserting (32) into (31) we obtain

Jt=T(λ1J1+λ2J2)T=λ1TJ1T+λ2TJ2T,
(33)

ηm,n=λ1η1m,n+λ2η2m,n,
(34)

where η 1m,n and η 2m,n are the diffraction efficiencies generated by the polarized field components.

Using this approach, we can study the signals generated by arbitrary partially polarized fields if the polarization matrix is known. However, let us now consider the special case of symmetric signals discussed above. In that case the diffraction efficiencies of any polarized field are given by Eqs. (28) and (29). Consequently η 1m,n=η 2m,n and Eq. (34) reduces to the form

ηm,n=(λ1+λ2)η1m,n=η1m,n.
(35)

Hence, the polarization gratings based on scalar designs producing symmetric signals will perform the same optical function over a broad wavelength band, independently of whether the incident field is completely or partially polarized or even unpolarized.

5. Numerical examples

As an example of the broadband implementation of scalar designs, let us first consider a simple 1→ 3 beam splitter. An optimal phase profile for such an element can be derived analytically [28

28. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998). [CrossRef]

], but we will now use a somewhat simpler design based on a sinusoidal transmission function

t(x,λ)=exp[iΦ(λ)sin(2πxD)],
(36)

where

Φ(λ)=2π(n2n1)h0λ
(37)

and h 0 is the modulation depth of the surface-relief grating. It can be shown that the diffraction efficiencies of this kind of element are given by

ηm(λ)=Jm2[Φ(λ)2],
(38)

where Jm (x) is the Bessel function of order m. Now η 0(λ 0)=η±1(λ 0) at some single wavelength if we choose the grating depth h 0 so that Φ(λ 0)≈0.91π, but at other wavelengths the efficiencies change. At that certain wavelength the total diffraction efficiency of this design is η≈90%, which is relatively good also when compared to the maximum obtainable efficiency η≈92.6% of the optimal scalar triplicator [28

28. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998). [CrossRef]

].

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).

The diffraction efficiencies of the triplicators based on the same scalar design described above but realized differently as a surface-relief grating and a polarization grating are illustrated in Fig. 2. In order to obtain comparable results, we have to take into account the Fresnel reflections also in the case of the phase element for which we use the material parameters n 1=n 0=1.8 and n 2=n 3=1. Thereby its diffraction efficiencies are multiplied by the Fresnel transmission coefficient that for the interface between the materials n 0 and n 3 at normal incidence angle is T≈0.918. The results clearly show how the diffraction efficiencies of the phase element change strongly as a function of the wavelength but the polarization grating produces uniformly the same efficiency for all three orders over the whole wavelength band considered.

Let us now use as an example the 1→2 beam splitter design with 100% efficiency presented in Section 3. On the other hand, it can be noted that the total diffraction efficiency of such a beam splitter equals the coefficient of Eq. (24) by which the efficiencies of the scalar designs are multiplied if the elements are realized as polarization gratings. Thus the results obtained for the beam splitter will reflect also the broadband performance of all polarization gratings based on scalar designs.

In order to achieve the phase retardation ϕ=π, relatively deep subwavelength-period structures should be used, which makes the fabrication process more difficult. The required depth is decreased if high-index materials are used in the modulated region (see e.g. Ref. 30). Therefore we have chosen to study a polarization grating fabricated in a thin TiO2 film on glass (SiO2) substrate. The refractive index data including material dispersion for these materials n 1 and n 0, respectively, is taken from Refs. [31

31. S.-C. Chiao, B. G. Bovard, and H. A. Macleod, “Optical-constant calculation over an extended spectral region: application to titanium dioxide film,” Appl. Opt. 34, 7355–7360 (1995). [CrossRef] [PubMed]

] and [32

32. H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Orlando, 1985), pp. 749–763.

], and the material behind the grating structure is assumed to be air (n 2=n 3=1).

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).

The subwavelength parameters in the basic example are chosen to be d=220 nm, f=0.6, and h=800 nm. Since the fabrication processes often introduce some errors causing deviations from the desired structure, we will also examine the sensitivity of the broadband beam splitter to such fabrication errors. Figure 3 illustrates the total diffraction efficiency of the beam splitter made of TiO2 with the basic parameters and additionally with different grating depths h=800±50 nm. It can be seen that 100% total diffraction efficiency is not fully reached owing to the losses at the interfaces, and the efficiency fluctuates slightly as a function of the wavelength. However, the diffraction efficiencies with all parameter combinations are still higher than the upper bound for scalar duplicators that is η u≈81% over nearly the whole a 350-nm-wide wavelength band. We also emphasize that the parameters used here represent just an example and not the most optimal solution for an achromatic subwavelength structure.

It can be seen that the efficiency curves at different grating depths clearly differ from each other, but the magnitude is not drastically reduced within the used depth error limits. Moreover, the results show that the positions of the highest efficiencies of the curves are shifted in a definite manner when h is changed. This feature of the broadband gratings is useful if one wants to design elements to be used with some fixed wavelengths. As another example of the parameter sensitivity of the element, the dependence of the results on the fill factor of the same grating is illustrated in Fig. 4 using deviated parameters f=0.6±0.05. The performance of the grating is seen to be slightly more sensitive to errors in the fill factor than in the depth, but still no drastic deterioration is noticed within these error bounds.

6. Conclusions

In this paper we have presented a method for designing diffractive broadband elements based on dielectric polarization gratings. The signals produced by the elements were studied theoretically and it was shown that any paraxial-domain scalar design can be converted to work over a broad wavelength band. In addition, broadband elements with efficiencies higher than obtainable with scalar phase elements can be designed taking the vectorial nature of light into account. It is also possible to design elements producing symmetric signals that work independently of the polarization state of the incident field.

The broadband performance of the elements was illustrated by numerical examples. It was shown that broadband elements can be obtained also if the dispersion of the materials used is taken into account. Although the fabrication of the structures required is challenging since they usually contain deep subwavelength features, the designs seem to be relatively tolerant to small fabrication errors according to the numerical simulations presented.

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).

Acknowledgments

The work of H. Lajunen is supported by the Graduate School of Modern Optics and Photonics, and that of J. Tervo by the Academy of Finland (project 203967) and the Alexander von Humboldt Foundation. The authors acknowledge the Network of Excellence in Micro-Optics (NEMO), http://www.micro-optics.org.

References and links

1.

J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie-Verlag, Berlin, 1997).

2.

H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566–1572 (1997). [CrossRef] [PubMed]

3.

G. P. Nordin and P. C. Deguzman, “Broadband form birefringent quarter-wave plate for the mid-infrared wavelength region,” Opt. Express 5, 163–168 (1999). [CrossRef] [PubMed]

4.

D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. 227, 49–55 (2003). [CrossRef]

5.

Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings fabricated upon silicon substrates,” Opt. Lett. 24, 1422–1424 (1999). [CrossRef]

6.

I. R. Hooper and J. R. Sambles, “Broadband polarization-converting mirror for the visible region of the spectrum,” Opt. Lett. 27, 2152–2154 (2002). [CrossRef]

7.

D. Kim and K. Burke, “Design of a grating-based thin-film filter for broadband spectropolarimetry,” Appl. Opt. 31, 6321–6326 (2003). [CrossRef]

8.

D. Yi, Y. Yan, H. Liu, S. Lu, and G. Jin, “Broadband polarizing beam splitter based on the form birefringence of a subwavelength grating in the quasi-static domain,” Opt. Lett. 29, 754–756 (2004). [CrossRef] [PubMed]

9.

C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. 29, 1593–1595 (2004). [CrossRef] [PubMed]

10.

F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]

11.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). [CrossRef]

12.

J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000). [CrossRef]

13.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003). [CrossRef]

14.

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).

15.

U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express 12, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

16.

J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26, 587–589 (2001). [CrossRef]

17.

C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. 26, 1651–1653 (2001). [CrossRef]

18.

F. Wyrowski, “Upper bound of efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991). [CrossRef] [PubMed]

19.

H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. 29, 803–805 (2004). [CrossRef] [PubMed]

20.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.

21.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]

22.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

23.

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004). [CrossRef]

24.

G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18, 1399–1405 (2001). [CrossRef]

25.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001). [CrossRef]

26.

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. 81, 1104–1114 (1963). [CrossRef]

27.

J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astr. Soc. 34, 403–419 (1973). [CrossRef]

28.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. 157, 13–17 (1998). [CrossRef]

29.

J. Turunen“Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems, and Applications, H.-P. Herzig, ed. (Taylor & Francis, London, 1997), Chapter 2.

30.

S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, and H. Launois, High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nmOpt. Lett. 23, 552–554 (1998). [CrossRef]

31.

S.-C. Chiao, B. G. Bovard, and H. A. Macleod, “Optical-constant calculation over an extended spectral region: application to titanium dioxide film,” Appl. Opt. 34, 7355–7360 (1995). [CrossRef] [PubMed]

32.

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Orlando, 1985), pp. 749–763.

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(050.1970) Diffraction and gratings : Diffractive optics
(230.1360) Optical devices : Beam splitters
(260.1440) Physical optics : Birefringence
(260.5430) Physical optics : Polarization

ToC Category:
Research Papers

History
Original Manuscript: March 8, 2005
Revised Manuscript: April 5, 2005
Published: April 18, 2005

Citation
Hanna Lajunen, Jari Turunen, and Jani Tervo, "Design of polarization gratings for broadband illumination," Opt. Express 13, 3055-3067 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3055


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References

  1. J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie�??Verlag, Berlin, 1997).
  2. H. Kikuta, Y. Ohira, and K. Iwata, �??Achromatic quarter-wave plates using the dispersion of form birefringence,�?? Appl. Opt. 36, 1566�??1572 (1997). [CrossRef] [PubMed]
  3. G. P. Nordin and P. C. Deguzman, �??Broadband form birefringent quarter-wave plate for the mid-infrared wave-length region,�?? Opt. Express 5, 163�??168 (1999). [CrossRef] [PubMed]
  4. D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si-Lu, and G.-F. Jin, �??Broadband achromatic phase retarder by subwavelength grating,�?? Opt. Commun. 227, 49�??55 (2003). [CrossRef]
  5. Y. Kanamori, M. Sasaki, and K. Hane, �??Broadband antireflection gratings fabricated upon silicon substrates,�?? Opt. Lett. 24, 1422�??1424 (1999). [CrossRef]
  6. I. R. Hooper and J. R. Sambles, �??Broadband polarization-converting mirror for the visible region of the spectrum,�?? Opt. Lett. 27, 2152�??2154 (2002). [CrossRef]
  7. D. Kim and K. Burke, �??Design of a grating-based thin-film filter for broadband spectropolarimetry,�?? Appl. Opt. 31, 6321�??6326 (2003). [CrossRef]
  8. D. Yi, Y. Yan, H. Liu, S. Lu, and G. Jin, �??Broadband polarizing beam splitter based on the form birefringence of a subwavelength grating in the quasi-static domain,�?? Opt. Lett. 29, 754�??756 (2004). [CrossRef] [PubMed]
  9. C. Sauvan, P. Lalanne, and M.-S. L. Lee, �??Broadband blazing with artificial dielectrics,�?? Opt. Lett. 29, 1593�??1595 (2004). [CrossRef] [PubMed]
  10. F. Gori, �??Measuring the Stokes parameters by means of a polarization grating,�?? Opt. Lett. 24, 584�??586 (1999). [CrossRef]
  11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, �??Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,�?? Opt. Lett. 27, 1141�??1143 (2002). [CrossRef]
  12. J. Tervo and J. Turunen, �??Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,�?? Opt. Lett. 25, 785�??786 (2000). [CrossRef]
  13. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, �??Design of space-variant diffractive polarization elements,�?? J. Opt. Soc. Am. A 20, 282�??289 (2003). [CrossRef]
  14. M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, �??Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,�?? J. Mod. Opt. 47, 2351�??2359 (2000).
  15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, �??Design, fabrication and characterization of subwave-length computer-generated holograms for spot array generation,�?? Opt. Express 12, 5345�??5355 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345.</a> [CrossRef] [PubMed]
  16. J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, �??Polarization beam splitters using polarization diffraction gratings,�?? Opt. Lett. 26, 587�??589 (2001). [CrossRef]
  17. C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, �??Polarizing diffraction-grating triplicators,�?? Opt. Lett. 26, 1651�??1653 (2001). [CrossRef]
  18. F. Wyrowski, �??Upper bound of efficiency of diffractive phase elements,�?? Opt. Lett. 16, 1915�??1917 (1991). [CrossRef] [PubMed]
  19. H. Lajunen, J. Tervo, and J. Turunen, �??High-efficiency broadband diffractive elements based on polarization gratings,�?? Opt. Lett. 29, 803�??805 (2004). [CrossRef] [PubMed]
  20. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.
  21. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]
  22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  23. J. Tervo, T. Setälä, and A. T. Friberg, �??Theory of partially coherent electromagnetic fields in the space�??frequency domain,�?? J. Opt. Soc. Am. A 21, 2205�??2215 (2004). [CrossRef]
  24. G. Piquero, R. Borghi, and M. Santarsiero, �??Gaussian Schell-model beams propagating through polarization gratings,�?? J. Opt. Soc. Am. A 18, 1399�??1405 (2001). [CrossRef]
  25. G. Piquero, R. Borghi, A. Mondello and M. Santarsiero, �??Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,�?? Opt. Commun. 195, 339�??350 (2001). [CrossRef]
  26. L. Mandel, �??Intensity fluctuations of partially polarized light,�?? Proc. Phys. Soc. 81, 1104�??1114 (1963). [CrossRef]
  27. J. C. Samson, �??Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,�?? Geophys. J. R. Astr. Soc. 34, 403�??419 (1973). [CrossRef]
  28. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, �??Analytical derivation of optimum triplicator,�?? Opt. Commun. 157, 13�??17 (1998). [CrossRef]
  29. J. Turunen, �??Diffraction theory of microrelief gratings,�?? in Micro-Optics: Elements, Systems, and Applications, H.-P. Herzig, ed. (Taylor & Francis, London, 1997), Chapter 2.
  30. S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, and H. Launois, "High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm", Opt. Lett. 23, 552�??554 (1998). [CrossRef]
  31. S.-C. Chiao, B. G. Bovard, and H. A. Macleod, �??Optical-constant calculation over an extended spectral region: application to titanium dioxide film,�?? Appl. Opt. 34, 7355�??7360 (1995). [CrossRef] [PubMed]
  32. H. R. Philipp, �??Silicon Dioxide (SiO2) (Glass),�?? in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Orlando, 1985), pp. 749�??763.

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