## Abstract

Antireflective properties of one-dimensional periodically microstructured lens surfaces (refractive index 1.5) are studied with the Green’s function surface integral equation method, and design guidelines are obtained. Special attention is given to the requirement of having practically all incident light transmitted in the fundamental transmission diffraction order. The effect of the presence of higher transmission diffraction orders is studied to determine if such more easily fabricated structures will be useful. The decrease of optimum fill factor of a periodic array of subwavelength ridges with structure period is explained as a waveguiding effect. Near-fields are calculated illustrating standing-wave interference and waveguiding effects for ridge structures, and adiabatic field transformation for tapered structures, including evanescent near-fields in in- and out-coupling regions. The antireflective properties of tapered geometries are considered for a wide range of angles of light incidence. It is found that while the reflection can be very small this rarely implies high transmission into the fundamental transmission diffraction order when higher-order transmission diffraction orders exist. This leads to the guideline that for visible and normally incident light the surface structure period should not be larger than ~300 nm, and a smaller period is needed in the case of oblique light incidence.

©2010 Optical Society of America

## 1. Introduction

Advanced plastic lens geometries and even entire plastic optical systems can be fabricated by injection moulding and are inexpensive, light-weight, and well-suited for mass production [1]. Plastic optics can in many cases straightforwardly replace glass optics since the refractive indices can be very similar (refractive index ~1.5 for the visible). In optical systems with many refracting surfaces the reflection loss can become appreciable for both glass and plastic optics even though the loss per surface is only ~4%, and it is desirable to coat or microstructure the refracting surfaces to minimize these losses [1–23]. For glass surfaces the antireflection can be reduced by coating with one or more thin films. Such films should have both an optimum refractive index and thickness, and especially the possibility to choose the optimum refractive index is limited to the available materials suited for coating. Furthermore, in order to get a smooth enough film it is desirable to apply temperatures in the fabrication process that may have a destructive effect if applied to polymer optics. Thus for polymer surfaces antireflection by surface microstructuring is an interesting alternative method that does not require introducing another material, and the effective refractive index can be engineered to whatever is needed by choosing an appropriate surface geometry [2–7]. The fabrication steps can ultimately be fewer if the microstructured surfaces in the optics are fabricated with a mould having a microstructured surface. Another inexpensive fabrication possibility is hot embossing where a surface microstructure is imprinted or stamped into the surface [8–10].

Recently, tapered periodic triangular surface geometries have been fabricated for the THz range [11], and antireflection from silicon surfaces has been obtained by fabricating nanowire, nanocone and pyramid arrays [12–15]. Interestingly, similar periodic antireflection surface geometries exist on the eyes of some types of nocturnal butterflies, where the reduced reflection or glare has been a useful evolutionary survival strategy [16]. Other interesting possibilities to achieve antireflecting refracting surfaces are to apply plasma etching to obtain a rough randomly structured surface layer with an on-average lower refractive index [17–19], or to coat with a mixture of two materials and etch one of them away [20,21], resulting in a similar effect, or to apply a layer of spherical SiO_{2} or latex particles [22,23].

The aim of the present work is to obtain guidelines and basic physical understanding of how surfaces of polymer lenses with refractive index ~1.5 can be microstructured with *one-dimensional* periodic surface geometries to obtain the antireflection effect. Achieving low reflection is, however, not sufficient, since if the transmitted light is transmitted in an undesirable direction this will not be useful for a lens. We have noticed from the literature that papers usually consider the reflection but not the transmission into the fundamental transmission diffraction order [1–23]. For solar cell and photodetector applications the direction of transmitted light is not critical, and light transmission under an oblique angle due to higher transmission diffraction orders might even enhance the chance of photon absorption. For the case of applying microstructures to lens surfaces in order to achieve antireflection we cannot allow any significant amount of transmission into higher transmission diffraction orders, and thus requirements for microstructured lens surfaces are more stringent. In this paper we will quantify the effect of the presence of higher transmission diffraction orders in order to determine if such more-easily-fabricated structures (having a larger period) will be useful. We will also be able to add some basic physical understanding to the effective refractive index of some microstructured surface geometries in the case when the period is not extremely small compared with the wavelength.

The geometries that we will consider for achieving antireflection in this paper are shown in Figs. 1b and 1c.

The paper is organized in the following way. In Sec. 2 we will outline the numerical method that has been used to obtain practically all results of this paper. The optical properties and design guidelines for the structure in Fig. 1b will be presented in Sec. 3. The tapered antireflective surface of Fig. 1c will be considered in Sec. 4. Another tapered geometry that serves as a 1D analog of a 2D array of pyramids is considered in Sec. 5, and finally we offer our conclusions in Sec. 6.

## 2. Numerical method

The numerical results presented throughout the paper have all been obtained with the Green’s function surface integral equation method (GFSIEM) [24–29] for periodic structures. We consider a 2D geometry such as the ones in Figs. 1b and 1c. Along the third direction (*z*) the structure and the fields are assumed invariant. If we consider e.g. s-polarized light the electric field **E** only has a *z*-component, i.e. $\mathbf{E}(\mathbf{r})=\widehat{z}E(\mathbf{r}),$ where $\widehat{z}$ is a unit vector along the z-axis and **r** is a position in the *xy*-plane. We shall consider an incident field being a plane wave incident with some angle of incidence *θ* on the form ${E}_{0}(\mathbf{r})=A{e}^{-i{k}_{0}\mathrm{sin}\theta \text{\hspace{0.05em}}x}{e}^{i{k}_{0}\mathrm{cos}\theta \text{\hspace{0.05em}}y}$
$\left({e}^{i\omega t}\right)$, where *k*
_{0} is the free-space wave number.

The electric field at any position **r** in the region with refractive index *n*
_{1} can be expressed as the sum of the incident field and an integral involving the total field and its normal derivative at the interface between the two media via the following integral equation

*E*refers to the field at the side of the interface with refractive index

*n*

_{1}, and $\widehat{n}$ is the surface unit normal vector. Similarly, the electric field at a position in the region with refractive index

*n*

_{2}can be calculated from the same fields at the interface via

*n*

_{1,2}. The Green’s functions are given by

## 3. Periodic-ridge surface microstructure

In this section we will consider the geometry in Fig. 1b. If we initially consider a period Λ, which is small compared with the wavelength, we may treat the microstructured layer of height *h* as an effective medium with a refractive index lying between the refractive index of air (n_{1} = 1) and that of the polymer (n_{2} = 1.5), where the exact value of the effective refractive index will depend on the filling factor *f* and the light polarization [2,3]. In this approximate picture, and referring to Fig. 1a, it is well-known that for normally incident light of wavelength λ we can achieve 100% transmission if we use the optimum effective refractive index ${n}_{3}=\sqrt{{n}_{1}{n}_{2}}$ and e.g. height**
$h=\lambda /4{n}_{3}$ [30]. For other wavelengths or heights the reflection is given by**

**$$r={\left|\frac{{r}_{13}+{r}_{32}{e}^{2i{k}_{0}{n}_{3}h}}{1+{r}_{13}{r}_{32}{e}^{2i{k}_{0}{n}_{3}h}}\right|}^{2},$$**

**where ${r}_{ik}=({n}_{i}-{n}_{k})/({n}_{i}+{n}_{k})$ and ${k}_{0}=2\pi /\lambda .$ However, the effective refractive index for the microstructured layer in Fig. 1b will depend on wether the electric field is oriented parallel with or perpendicular to interfaces [2,3,31], and thus it will depend on the polarization. Thus, the effective refractive indices for s- and p-polarized light in the long-wavelength limit are given by [2,3,31] For s-polarized light the optimum filling factor**

*f*resulting in**${n}_{eff,\mathbf{s}}=\sqrt{{n}_{1}{n}_{2}}$is thus***f*= 0.4, whereas for p-polarized light it is instead*f*= 0.6. In order to illustrate how well this effective-index approach actually works for small periods we present in Fig. 2 a rigorous calculation for the structure in Fig. 1b with a period of Λ = 200 nm and for the applied wavelength λ = 633 nm. This data is shown together with a calculation based on Eq. (4) and the effective indices given in Eqs. (5) and (6) equivalent to the rigorous calculation in the limit Λ = 0. We notice that the deviation from the effective medium theory is small, and that a filling factor of 40% and 60%, respectively, for s- and p-polarized light are close to being optimum.If the period is very small, e.g. 200 nm, we may aim at a structure height of app. 130 nm and filling factor of 0.5, in which case the reflection minimum is obtained at only very slightly different structure heights for s- and p-polarized light. For the reflection minima at larger structure heights the optimum height for s- and p-polarized light will differ more, and thus it is more difficult to achieve good simultaneous antireflection for s- and p-polarized light.

It would be preferable from a fabrication point of view to use a period which is larger than 200 nm but in that case Eqs. (5) and (6) become inaccurate. A higher-order effective medium theory has been presented in Refs [6,7]. Here we will use an approach where we consider the modes in the microstructured layer and set the effective refractive index to the mode-index of the fundamental space-filling mode. A similar approach has been used in the field of photonic crystal fibers to obtain an effective refractive index for a microstructured fiber cladding [32]. Within this model we can obtain the effective refractive index ${n}_{eff}$ by solving the dispersion equation [33]

In Fig. 3(a) we show the calculated mode-index for the microstructured layer of the structure in Fig. 1b versus filling factor for both polarizations and a range of structure periods for a fixed wavelength of 633 nm. We also show (Fig. 3b) what filling factor will result in the effective refractive index of ${n}_{3}=\sqrt{{n}_{1}{n}_{2}}.$ Interestingly, the optimum filling factor decreases with increasing period. This can be explained as a waveguiding effect where we can view the microstructured layer as a periodic array of coupled waveguides. As the width of waveguides becomes larger compared with the wavelength it also becomes possible for the light to become more concentrated in the high-refractive-index regions due to the waveguiding or index-guiding effect. For photonic-crystal-fiber claddings the same phenomenon is seen where the electric field tends to move out of the low-index regions with decreasing wavelength or increasing period [32], and thus the effective index increases. Here, this means that if we use a larger period we will need to use a smaller filling factor in order to have the same optimum effective refractive index. From Fig. 3 we can see that if we use a period of 400 nm the optimum filling factor for p-polarized light is very close to 50%, whereas it is 32% for s-polarized light.

For s- and p-polarized light the difference in mode-index for a given *f* will not be larger than ~0.1, which means that the optimum height will not differ by more than ~15 nm, where the optimum height for a given mode index *n* is λ/4*n*.

We note that a curve similar to Fig. 3(a) can be found in Ref [5] for two-dimensional periodic surface geometries in silicon in the long-wavelength limit, and a similar result can also be found in Refs [2,34] for one-dimensional periodic surface microstructures in silicon or GaAs. Also, our Fig. 3(a) is similar to the effective-index depth profile for tapered geometries as in Fig. 1(c), and such a curve is available in Ref [7] for ZnSe. However, neither of these previous results [2,5,7,34] show the trend of how the effective refractive index increases with the structure period. Instead of solving the exact dispersion equation (Eq. (7)) an alternative method is to use the second-order effective medium theory, in which case an approximate expression for the effective refractive index for s- and p-polarized light exists in closed form (see e.g. Refs [6,7,34]). The approximate optimum fill factor that would be obtained with 2nd order EMT is shown in Fig. 3(b) for comparison with the exact result.

To illustrate that the approach of using the mode-index of the fundamental space-filling mode in Eq. (4) works very well as an approximation we present in Fig. 4 a calculation of the reflection in the case of a structure period of 400 nm. Here, we notice e.g. for s-polarized light that while the filling factor f = 0.4 works reasonably it is no longer optimum but now the filling factor of 32% results in practically zero reflection at optimum structure height. Similarly, for p-polarized light it is now the filling factor of 50% that results in zero reflection. A filling factor of 40% would work reasonably well for both polarizations.

From Fig. 4 we notice that for the optimum filling factor we have reflection minima at wavelengths close to $(2j+1)\lambda /4\sqrt{{n}_{1}{n}_{2}}$ corresponding to *h* = 129 nm, 387 nm, 646 nm etc., as would be predicted from Eq. (4). The magnitude of the electric field at optimum structure heights (Fig. 5
) clearly show that the field in the region of the surface microstructure resembles a standing wave interference pattern due to counterpropagating waves.

For s-polarization, and for the case of a very small period, the magnitude of the electric field would be the same in both high- and low-refractive index regions. Here we notice that the field is significantly larger in the dielectric region, which is a consequence of the period being comparable to the wavelength, in which case the field can become somewhat concentrated in the high-refractive-index regions due to the waveguiding effect. In the case of p-polarization the field magnitude appears to be larger in the air-region and not in the dielectric region. This is a consequence of the boundary condition at the air-dielectric interface that makes the field increase across the air-dielectric interface by a factor ${n}_{2}^{2}/{n}_{1}^{2}$ = 2.25. One should also notice that within a few hundred nanometers right below the surface microstructuring the field contains near-field components that decay evanescently into the material. We should mention that a waveguiding effect and standing-wave interference patterns have previously been discussed for 1D blazed-binary diffractive elements [35].

If we use a longer period or shorter wavelength, and consider normally incident light, we have to consider that only for periods satisfying $\Lambda <\lambda /{n}_{2}$ will there be only a fundamental transmission diffraction order, and for larger periods light can be coupled into higher transmission diffraction orders which may be undesirable if the antireflection surface is applied to e.g. lens surfaces. We shall consider to what extent this will impair the antireflecting properties of the surface. In Fig. 6 we consider the same geometry as before with period Λ = 400 nm but for three different wavelengths 633 nm (red light), 532 nm (green light) and 473 nm (blue light). Only the fundamental transmission diffraction order exists for the red light but higher orders exist for green and blue light. We notice that for green light the reflection can be practically zero for certain heights even though higher transmission diffraction orders exist. However, transmission into the fundamental order does not reach 100% due to light coupled into higher transmission diffraction orders. For the even shorter wavelength (blue light) transmission into the fundamental order can be seriously impaired. Thus, we may still achieve low reflection but a significant part of the transmitted light is propagating in an undesirable direction. If we had considered solar cell or photodetector applications it would be fine to just have light transmitted no matter what would be the transmission diffraction order. However, for a lens the requirements are more stringent. In Fig. 6 we have also shown an example of the electric field magnitude for p-polarized green light (λ = 532 nm) and a height of h = 530 nm where reflection is practically zero. In that case we notice an interference pattern in the transmitted field due to the presence of both fundamental and higher transmission diffraction orders.

We have also made similar calculations for s-polarized light (not shown) from which similar conclusions can be drawn. If the period is large or the wavelength small a very low reflection is still achievable but a significant fraction of the transmitted light is not in the fundamental transmission diffraction order. This means that for a lens with microstructured antireflective surfaces to work well for the whole visible spectrum, including blue light (~λ = 450 nm), then, for normally incident light, we should use a period which is not larger than ~Λ = 300 nm.

## 4. Linearly tapered geometry

A major concern with the geometry of the previous section is that the optimum structure height is rather sensitive to the wavelength of incident light, and furthermore the optimum fill factor is also sensitive to both wavelength and polarization. Thus, while it is possibly to obtain smaller reflection than in the case of not having a microstructure, it is not possible to obtain extremely low reflection with such one geometry for the whole visible range and both polarizations, not even when we consider just normally incident light. In this section we will therefore consider a tapered geometry (see Fig. 1c), in which case good antireflection can be achieved for all visible wavelengths and both polarizations simultaneously (Figs. 7 and 8 ). Tapered antireflection surface microstructures have previously been considered in e.g. Refs [4–7,11,12,14,15].

To begin with we will consider geometries such as Fig. 1c with period Λ = 300 nm in order to avoid that light is lost to higher transmission diffraction orders. We consider reflection and transmission for red, green and blue light versus structure height for p-polarized light in Fig. 7. We notice that except for some interference-related reflection minima and corresponding transmission maxima the general trend is that reflection decreases with increasing structure height. Thus, if the structure is high enough, e.g. *h* = 250 nm, then the reflection will be very low for the whole visible range. The exact structure height is not critical for a certain wavelength contrary to the previous case of non-tapered geometries. We have also shown in Fig. 7 the electric field magnitude for the wavelength λ = 473 nm and three different heights h where the reflection has a minimum. Similar to Fig. 5 for the non-tapered geometry we also here notice an interference pattern within the first few hundred nanometers in the polymer below the surface microstructure due to the presence of evanescently decaying higher-order diffraction field components. However, there is no pronounced standing wave-interference visible in the field profile within the tapered region.

A similar calculation for s-polarized light is presented in Fig. 8, in which case we notice the same trend, namely that if the structure height is large enough transmission will be very high no matter which of the three wavelengths we consider. This means that this geometry can work well for the whole visible range and both polarizations simultaneously.

We may ask ourselves if it works better to use a period being larger than $\lambda /{n}_{2}$ for the tapered geometry than it did for the non-tapered ridge geometry, so that we perhaps in this case with a gradual transition from one medium to the other could be allowed to use a larger structure period in a lens surface. One result from that investigation is shown in Fig. 9 where we consider again the tapered geometry but with period Λ = 400 nm. In this case only the wavelength 633 nm results in Λ being smaller than $\lambda /{n}_{2}$, in which case we notice as before that 0-order transmission becomes very high with increasing structure height. However, for the shorter wavelengths of 532 nm and 473 nm we notice that while reflection can be very small for large structure heights the 0-order transmission is seriously impaired by the presence of higher transmission diffraction orders.

A similar calculation for s-polarized light leads to the same conclusion, namely, that 0-order transmission will be seriously impaired if the period of the surface microstructuring is larger than $\lambda /{n}_{2}$, and that reflection can nevertheless still be very small for the large structure heights.

We notice in Figs. 7-9 that reflection minima are located at structure heights of *h*
_{min}
*≈An*, where *n* is an integer and *A* is a constant that depends on the wavelength. This is different from the ridge geometry where the optimum heights are on the form *h*
_{min}
*≈A(n-*1/2*)*. This has previously been observed for graded refractive index films in Refs [36,37].

While the period of 300 nm might be sufficient for normally incident light a smaller period is needed for larger angles of light incidence θ, since in that case the requirement for having only the fundamental transmission diffraction order becomes $\Lambda <\lambda /({n}_{2}+\left|{n}_{1}\mathrm{sin}\theta \right|).$ In fact for the previously considered geometry (Λ = 300 nm) and wavelength 473 nm this requirement is only fulfilled for $\theta <{4.4}^{\circ}.$ For the wavelength 532 nm we avoid higher-order transmission if $\theta <{15.9}^{\circ},$ while for the wavelength 633 nm the requirement is $\theta <{37.6}^{\circ}.$ Total transmission and 0-order transmission are considered for all three wavelengths and various angles of light incidence in Figs. 10
-12
. We notice that for the angle of incidence of ${10}^{\circ}$ and wavelength 473 nm the 0-order transmission is already slightly reduced and is seriously impaired for the larger angles of incidence. If we do not have a surface microstructure we will get large transmission for p-polarized light as we approach the Brewster angle as seen in e.g. Figure 10. Thus a height of *h* = 0 and angle of incidence ${50}^{\circ}$ results in very low reflection which the surface microstructure can almost only make worse.

If we make the same calculation but for the wavelength 532 nm (Fig. 11 ) we notice good 0-order transmission for the angles of incidence ${0}^{\circ}$ and ${10}^{\circ},$ but for larger angles the 0-order transmission is impaired. Similarly we notice that for the wavelength 633 nm (Fig. 12) the 0-order transmission is very good up to angles of light incidence of ${30}^{\circ}$ but already for ${40}^{\circ},$ which is only slightly too large, the 0-order transmission is significantly reduced. As a general trend we notice that the total transmission into all diffraction orders can be very high in some cases even when the requirement $\Lambda <\lambda /({n}_{2}+\left|{n}_{1}\mathrm{sin}\theta \right|)$ is not satisfied.

Finally, if we want our geometry to have only 0-order transmission up to angles of light incidence of ${50}^{\circ}$ it is necessary to use a period as small as ~200 nm. Structures with such dimensions can be made with embossing fabrication methods [10]. The calculation for this case is shown for both red and blue light in Fig. 13 for both polarizations. Here we just refer to transmission since 0-order transmission and total transmission is the same. Very good transmission is possible for the whole visible range but we notice that we need to use a larger structure height in the case of the red light compared with the blue light.

## 5. Quadratically tapered geometry as a 1D model for a 2D pyramid array

The tapered geometry considered in the previous section had a filling factor that varied linearly with height *y* from 0 to the total height of the structure *h* as $f(y)=(h-y)/h.$ In the case of a 2D periodic array of a similar structure, such as a 2D array of pyramids or cones, the corresponding filling factor of high-refractive-index material has a quadratic variation, namely $f(y)={(h-y)}^{2}/{h}^{2}.$ A periodic surface geometry (1D) with this quadratically varying filling factor is shown in the insets of Fig. 14
for a period of 300 nm and heights *h* of 300 and 700 nm. This structure thus can be thought of as the 1D analog of the 2D pyramid array. Compared with the case of the linearly tapered geometry (Figs. 7 and 8) the reflection and transmission is similar but oscillates less with the height. A calculation with structure period 400 nm (not shown), where higher-order transmission is possible for blue and green light, results in that the 0-order transmission is seriously impaired similar to Fig. 9. Thus also in this case should we require that the period is not larger than 300 nm for normally incident light, and that it must be smaller for larger angles of light incidence similar to Figs. 10-13.

## 6. Conclusion

In conclusion, we have obtained guidelines for the design of one-dimensional periodic surface microstructures for antireflective polymer lenses with refractive index 1.5. In the ultra-short-period limit, and for normally incident light, the optimum filling factor of high-refractive-index material in ridge surface structures should be ~40% for s-polarized light and ~60% for p-polarized light. For larger periods the optimum filling factor decreases due to the waveguiding effect and can be deduced from Fig. 3 or Eq. (7). The physical explanation is that light concentrates in the high-refractive-index material with increasing period, a phenomenon also found for e.g. photonic-crystal-fiber claddings [32], thus increasing the effective refractive index.

We found that if the period is large enough that higher transmission diffraction orders exist the reflection can still be very small. However, in that case low reflection rarely implies high transmission into the fundamental transmission diffraction order, which is unacceptable for applications where the microstructure should not change the direction of transmitted light such as would normally be the case for an ordinary lens surface. Thus, for normally incident light, and wavelengths in the visible, including blue light with wavelength ~450 nm, the period should not be larger than 300 nm. For a linearly and quadratically tapered microstructure surface geometry we found that essentially if the period is 300 nm and the structures are high enough the reflection from one single surface geometry will become small for the whole visible range, and for both s- and p-polarized light. If the period is larger we again found that the transmission into the fundamental diffraction order would be seriously reduced even for normally incident light. For light which is incident under an oblique angle in the range from ${0}^{\circ}$ to ${50}^{\circ}$ the reflection will also become very small if the structures are made high enough but now as a guideline high fundamental-diffraction-order transmission for the largest angle of ${50}^{\circ}$requires even shorter periods on the order of 200 nm.

## Acknowledgement

The authors acknowledge financial support for this work from the advanced technology foundation, project nr. 053-2007-3.

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