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
Diffractive phase elements based on dielectric surface-relief structures can be used to realize a large variety of optical functions with high efficiencies . 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 [2–4]. 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 [5–9]. 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 [10–13]. By that means paraxial-domain diffractive elements, such as beam-splitters, can be designed and fabricated [11–17]. 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 [12–14], which is not possible in the realm of scalar optics . Moreover, we recently introduced the possibility to design broadband diffractive elements based on dielectric polarization grating structures .
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 , 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 
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.
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
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  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
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 . Thus, form birefringent subwavelength gratings can be used as achromatic wave plates [2–4].
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
is the Jones vector associated with right-handed circularly polarized field and
denotes correspondingly the left-handed circularly polarized state. The complex weighting coefficients α and β are defined by equations
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
for the transmitted field with an arbitrary initial polarization state. The functions
characterize the transmitted forms of the initially right- and left-handed circularly polarized field components, respectively [11,15]. 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.
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
The diffractive polarization elements can be designed to produce desired far-field signals in the same way as conventional diffractive elements [11–15]. In this Section we will discuss designs of such elements for incident fields with particular initial polarization states. The subwavelength structures forming the elements are assumed to be achromatic as described in the previous Section, and thus the designs to be presented work over the wavelength band Δλ. While the considerations above are valid for any polarization modulating element regardless of the periodicity of the structure, in the following we will restrict our considerations into periodic polarization gratings and examine the produced far-field signals by means of diffraction efficiencies.
If the polarization modulating structure is periodic, the transmitted field can be represented by the Rayleigh expansion
where dx and dy denote the grating periods in the x- and y-directions, respectively, and
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
since the energy of the incident field per period is normalized to unity as ‖J i‖2=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
where δ m,n is the Dirac delta function, and
It is noticed that 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, 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
but now T m,n=0 if (m,n)≠(0,0) and a simple form
is obtained for all but the zeroth diffraction order. This result is also known from studies concerning polarization gratings illuminated by a single wavelength , 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
The first term affects only the zeroth diffraction order, and the latter term depends on the complex amplitudes 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 , 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 . 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 .
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, 12]
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 
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 . The performance of this kind of polarization-grating beam splitters has also been demonstrated experimentally for single wavelengths [11, 16, 17]. Using achromatic subwavelength structures we can obtain broadband diffractive elements with 100% efficiency based on the design described above .
Designs for polarization gratings producing larger arrays of diffraction orders with 100% efficiency has also been introduced . 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, 15], 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 . All such binary structures can easily be converted to work over a broad wavelength band using achromatic subwavelength gratings.
4. Polarization-insensitive designs
In the previous Sections all the wavelengths of the incident field were expected to have the same given polarization state. Although such a situation may be obtained by using suitable polarizers, this requirement is still quite restricting, and, especially if the incident field is partially polarized, the elements discussed above may not work in the desired manner. However, it is also possible to obtain polarization-insensitive elements, as has been already shown in the case of single-wavelength polarization elements .
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 weighed by the absolute values of the coefficients α and β. It is also noted from Eq. (20) that the quantities satisfy a symmetry condition = and consequently |=||. 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 ||=||, it immediately follows that ||=||. Thus the efficiencies of the diffraction orders of arbitrarily polarized illumination are, based on Eq. (23),
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
for the diffraction efficiency of the zeroth order. However, it is usually assumed that |t ‖|=|t ⊥|=1 in which case the zeroth diffraction order is also polarization-insensitive. This shows that if the desired signal is symmetric, scalar phase elements may be replaced with polarization gratings with broadband performance in the same way as discussed in the previous Section for circularly polarized light, but now the elements will work similarly with arbitrarily polarized incident fields.
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 . In that case the field can be described by the polarization matrix
where the angle brackets denote ensemble average, representing a space–frequency domain generalization of the often used representation in the space–time domain [22, 23]. The transmission of partially polarized field through a polarization modulating element can be studied by the matrix equation
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, 25]. In the following, we will use a somewhat different approach and assume that the field components have enough spatial coherence to produce diffraction effects.
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 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
in which the transmission of J 1 and J 2 representing polarized fields can be studied alternatively using the Jones vector formalism as in the previous Sections. Assuming a periodic polarization grating, the produced far-field signal is characterized by diffraction orders with efficiencies
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
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
In practice, the ideal theoretical situation cannot be usually reached, but the efficiencies of the broadband gratings introduced in previous Sections are reduced by reflection losses at the interfaces or non-ideal phase retardations at different wavelengths. The quality of performance is determined mainly by the characteristics of the subwavelength gratings. In this Section we show some numerical examples of the broadband performance of polarization gratings in order to illustrate that highly satisfactory results may be obtained also when the subwavelength structures are considered more realistically.
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 , but we will now use a somewhat simpler design based on a sinusoidal transmission function
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
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 .
On the other hand, the same design can be realized by using a polarization grating with an achromatic subwavelength carrier grating that introduces the phase retardation ϕ=π. In this case the fringe rotation angle is chosen as θ(x)=t(x,λ 0)/2, and it naturally is wavelengthin-dependent. The maximum efficiency is the same as that of the phase element, but the polarization grating works as a uniform triplicator over a wider band of wavelengths. Since the signal is symmetric, the element will work in the same manner if illuminated by any polarized or partially polarized field as discussed in the previous Section.
For the numerical example the refractive indices of the materials used in the subwavelength grating structure are n 0=1.8, n 1=2 and n 2=n 3=1 as illustrated in Fig. 1. The period of the subwavelength grating is assumed to be d=0.4λ 0, where λ 0 is any fixed wavelength, the fill factor is f=0.8, and the desired phase retardation ϕ=π is obtained with modulation depth h=3λ 0. These parameters for the achromatic subwavelength structure are obtained from Ref. 2, but, instead of the sandwiched structure that would yield somewhat higher theoretical efficiencies , we have now chosen the material behind the element to be air as in the most realistic situations. The complex amplitude transmission functions t ‖ and t ⊥ of the subwavelength grating are computed rigorously using the Fourier modal method , and the efficiency of the diffraction orders of the polarization grating is then determined by Eq. (27).
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.
In the previous example we used the idealized assumption that the subwavelength grating structures are made of some unnamed materials with certain constant refractive indices, and the wavelength dependence of the refractive indices usually occurring with all real materials was ignored. However, this is not a defect since achromatic phase retarders can be designed also taking the material dispersion into account. In the following, we will present numerical examples of broadband performance of polarization gratings designed using the refractive index data of real materials and consider the tolerance of the broadband performance to possible fabrication errors.
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.  and , and the material behind the grating structure is assumed to be air (n 2=n 3=1).
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.
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.
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.
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