Efficient and uniform illumination with microlens-based band-limited diffusers

Tasso R.M. Sales, RPC Photonics, Inc.

Uniform illumination over a certain target area is a common requirement in many applications, and oftentimes schemes must be devised to ensure that this requirement is achieved. In display systems, for example, light that illuminates a viewing interface such as a screen must appear uniform and devoid of any distracting image artifacts. Depending on the particularities of the system, different approaches must be developed to provide uniform illumination without sacrificing efficiency.

Side-illuminated backlit displays use painted dots and microstructures to extract guided light in a controlled fashion so that the display is visually uniform; lithographic illumination systems often use fly’s eye microlens arrays to generate uniform illumination, sometimes in combination with additional diffusers and/or motion to minimize the diffraction artifacts induced by the periodic lens arrays; laser displays make use of optical diffusers for shaping and uniform light distribution as well as speckle management; general lighting has routinely relied on simple diffusers and prismatic films that, although not the most efficient or aesthetically pleasing, are inexpensive and commonly found in light fixtures.

A frequent approach to uniformly illuminating a target uses a diffusing element in the optical path to spread and homogenize the incident illumination and provide the desired degree of uniformity. The requirements for the diffuser, or diffusers, depend upon a number of factors, such as the source properties (primarily spectrum, divergence and coherence), as well as on the performance goals, which depend strongly upon the application. We are particularly concerned with applications that require some sort of light shaping, either in intensity or distribution. For this type of application, optical performance is critical, and only a handful of diffuser technologies can be brought to bear.

Shown here is a microlens array in fused silica.

Among the available options, band-limited diffusers are unique because of their highly desirable qualities and good performance. Band-limited diffusers are defined as diffusers that scatter light within a well-defined angular range, usually – but not necessarily – with uniform intensity. An ideal band-limited diffuser attains 100 percent of the diffuse light within its band limit; in practice, however, it has been difficult to come close to this goal for a variety of reasons. Here we will review issues related to common diffuser technologies, typical performance associated with the technology and its precision in comparison with the band-limited ideal. We will also discuss recent advancements in microlens diffusers engineered to attain nearly band-limited performance.

Band-limited diffusers

The concept of a band-limited function is best known in the framework of linear systems theory, particularly with regard to signal recovery from sampled data¹ where band-limited functions have interesting properties. A band-limited function has nonzero values only within a finite domain. Outside of this domain, the function is identically zero. In the context of diffusers, the concept applies directly so that in the far field, or at the focal plane of a lens, the diffuse light is confined to a certain angular range with little or, ideally, no energy outside of that range.

Figure 1 illustrates an ideal band-limited diffuser in the geometrical optics limit where all of the incident illumination is contained within a cone angle Θ. Both blue and red curves represent band-limited diffusers. The case of uniform illumination (blue curve), however, plays a particularly important role in many applications, and for this reason, we will focus on uniform band-limited diffusers.

Figure 1. In an ideal band-limited diffuser, all light is scattered within a certain angular range, Θ.

Evidently, the geometrical limit is only an approximation, and, because of diffraction, the intensity fall-off at the edges of the intensity profile always has a finite extent. This depends on the nature of the surface features that define the diffuser, assuming that the incident illumination is collimated. In any case, the concept of a band-limited diffuser is simply extended to include the intensity fall-off so that, in the best possible scenario, the fall-off is limited only by diffraction.

Given the importance of uniform illumination, it is no surprise that band-limited diffusers have been the focus of continual, if sporadic, investigation. Significant advances in the understanding of the fundamental properties of band-limited diffusers seem to have started in the early 1970s, and research continues to this day. During this period, however, the goal of producing a band-limited diffuser has remained elusive.

Existing solutions

One of the earliest attempts at implementing a band-limited diffuser was the diffractive diffuser, which can create very uniform diffuse patterns with general distributions. However, it is intrinsic to diffractive elements that a certain fraction of the incident illumination spills into diffraction orders beyond the target region of interest. In the case of binary elements, at least 20 percent of the light is lost to higher diffraction orders. If the diffuser is produced with a continuous-phase function, the losses become much lower, but even in this case, typically at least 5 to 10 percent is lost to higher diffraction orders (Note that these figures do not include Fresnel losses). We can therefore say that, although a diffractive diffuser can provide uniform illumination, it is not band-limited, with the continuous-phase element coming very close.

A limiting issue with diffractive diffusers, however, is that they are often restricted to monochromatic illumination and relatively small spread angles, unless one can deal with a strong zero order, a bright center spot collinear with the incident illumination. Because of deviation from the design wavelength or fabrication errors, diffractive elements commonly display the presence of a zero diffraction order that is much stronger than any other order in the diffraction pattern, and may or may not be eliminated in practice.

Refractive diffusers, on the other hand, are fundamentally free of the intrinsic losses associated with high-order diffraction and zero order, and are thus the best candidates for band-limited behavior. Simple examples of refractive diffusers include ground glass and holographic diffusers created by exposing photosensitive materials to laser speckle. These diffusers have been around for quite some time but generate Gaussian scatter and therefore do not have a uniform scatter region or well-defined cutoff for the diffuse light and are thus not band-limited. Just a few decades ago, researchers at Kodak would comment² that “no known random phase diffuser is band-limited.” This includes diffractive diffusers, which, strictly speaking, are not random, and holographic diffusers, which are obviously not band-limited.

Since the early work at Kodak and elsewhere, not much had happened until recently, when microlens-based diffusers became commercially available.³ Alternative concepts also have been proposed⁴ such as a distribution of linear facets that are randomly combined to spread light over a specific angular range. Each facet scatters into a specific direction; by imposing an upper limit to the slope angles in the ensemble, one can theoretically guarantee band-limited behavior. By further ensuring the appropriate distribution of facets over the angular range of interest, one can produce uniform illumination.

The approach does provide, at least in principle, band-limited behavior, and some experimental demonstration can be found in the literature.⁵ However, because each linear facet corresponds to a specific angular direction, this type of diffuser would seem to require a large input beam size to sample a sufficiently large number of facets so that uniform illumination could result over some specified angular range. Making the linear facets small minimizes the requirements for a large beam but runs into manufacturing issues.

Microlens-based engineered diffusers

Microlens arrays are used in a great number of applications, and in certain instances, they are also used for beam shaping and diffusion. However, because microlens arrays are periodic, there is a limitation in the achievable shaping capabilities as well as in the diffraction effects that result from the regular arrangement of the microlenses. The microlens element itself, however, provides a basic component from which general band-limited diffusers can be built.

In recent years, we have introduced a class of microlens-based diffusers³ – also referred to as engineered diffusers – that provide not only homogenization but also general shaping capabilities. Each microlens unit in an engineered diffuser is defined by a certain number of parameters, such as a radius of curvature and a conic constant that are randomized to create the diffuser. Unlike common diffusers, however, engineered diffusers are created deterministically, such that each microlens element is individually designed and fabricated to produce a certain scatter pattern with a controlled intensity profile.

The design concept behind engineered diffusers can be described in simple terms. The prescription for the set of lenses that comprise the diffuser is defined, including feature sizes and slope angles based on the scatter requirements. These parameters are typically defined in terms of probability distribution functions that specify the likelihood that a certain lens will assume a specific prescription. The next step is the spatial distribution of the microlenses to create the diffuser surface structure according to probability distribution functions. An example of a commercially available engineered diffuser that shapes an incident beam into a 20° square pattern is shown in Figure 2.

Figure 2. Shown is the surface pattern of a diffuser that generates a square pattern (A) and its measured intensity profile (B).

The intensity profile (Figure 2B,) was measured in the far field with a detector large enough to average enough speckles and reveal the envelope of the intensity distribution. The intensity profile can be described by a Lorentzian profile – the red curve in Figure 2B – given by

where I₀ is a constant, 2Θ₀ is the full width half maximum, and p is a number directly related to the flatness of the intensity profile. For the plot shown in Figure 2, p = 30 and Θ₀ = 11.5°. For p = 2, one finds the usual Lorentzian function. For p >2, the function is sometimes referred to as a “super-Lorentzian” with power p. The parameter p provides an indication of the flatness, or uniformity, of the super-Lorentzian intensity profile, Figure 2. The larger the parameter p, the flatter will be the intensity profile and the narrower the fall-off region. For uniform diffusers, the steepness of the fall-off depends directly on the microlens feature size. For the diffuser shown in Figure 2, the lens feature sizes are ≤160 μm.

Figure 3. Shown are super-Lorentzian curves for Θ₀ = 15° and various power values.

Two items are readily noticeable from Figure 2. First, although the surface pattern on Figure 2B is clearly not regular, an underlying grid pattern seems apparent. Secondly, the super-Lorentzian fit matches nicely the flat portion of the intensity profile but not the bottom portion, toward wider angles. The effect of an underlying grid in the diffuser surface reveals itself in the diffuse pattern as diffraction artifacts that are particularly obvious with coherent sources such as lasers. A small illuminating beam size further emphasizes the nonuniformity. Examples of these artifacts are seen in Figure 4. As the beam size increases, the visibility of these artifacts is reduced but not completely eliminated.

The mismatch in the wide-angle fit, Figure 2B, is caused by scatter losses due to light falling outside the target. These losses originate from steep slopes between adjoining lenses. Each lens has a unique surface prescription that is generally distinct from another lens.

Figure 4. The scatter pattern from a diffuser that shapes a laser beam into a square pattern is pictured. The diffuser surface has some underlying periodicity that shows up as diffracting artifacts such as lines (A) and, on a finer scale, dots (B) with apparent local periodicity.

Consequently, the boundary between lenses generally contains discontinuities that spread light into wide angles outside the target and lead to the observed deviation from the super-Lorentzian fit. It clearly also reduces the system efficiency. Estimates place the amount of lost light to wide-angle scatter at approximately 10 to 15 percent, depending on the diffuser design, feature sizes and conditions of fabrication.

Band-limited engineered diffusers

Recent improvements in design and fabrication techniques have led us to develop a diffuser concept that considerably advances the state of the art in the production of microlens-based diffusers. At the root of these improvements is the effort to eliminate any underlying periodicity in the spatial distribution of the microlenses and to eliminate lens mismatches that lead to wide-angle scatter, thus maximizing the use of available light.

Figure 5. Shown is an example of a band-limited engineered diffuser profile (A) and its measured intensity profile (B).

The measured performance of this new class of diffusers comes much closer to the band-limited ideal without the presence of diffraction or image artifacts, which should be particularly significant not only for coherent illumination but also for incoherent illumination such as LEDs, because of the higher efficiency. An illustration of the surface structure of this band-limited engineered diffuser is shown in Figure 5.

Figure 6. A typical engineered diffuser produces a square pattern (A), whereas a band-limited engineered diffuser shows considerably less light loss outside the square pattern (B).

The intensity profile shown in Figure 5B indicates much improved super-Lorentzian behavior over the entire range of angles with p = 40 and lens feature sizes ≤160 μm. To further illustrate the higher efficiency of the band-limited engineered diffuser, Figure 6 shows a visual comparison of the light scattered outside of the target, where it seems clear that the new generation of diffusers significantly increases the utilization of light.

Finally, the careful randomization of the diffuser surface structure also creates a scatter distribution that is devoid of artifacts. In the case of coherent illumination, it gives rise to the typical uniformly random speckle pattern expected from random diffusers, as illustrated in Figure 7. For reference and side-by-side comparison, 4B is repeated in 7B.

It is now nearly half a century since the first significant efforts started in the understanding and fabrication of band-limited diffusers. Since then, we have developed a better understanding of the theoretical requirements for band-limited diffusers. The relatively recent introduction of microlens-based diffusers, in the form of engineered diffusers, came a step closer to the band-limited ideal, although they are not perfect.

Figure 7. Shown is a speckle comparison of a standard (A) and a band-limited (B) engineered diffuser.

The new generation of microlens diffusers that is now being unveiled, as described here, brings the band-limited diffuser idea from a purely conceptual realm into the real world.

Meet the author

Tasso R.M. Sales is chief technology officer at RPC Photonics Inc. in Rochester, N.Y. He can be reached at tasso@rpcphotonics.com.

References

1. J.W. Goodman (1996). Introduction to Fourier Optics, 2nd ed. McGraw-Hill, New York.

2. C.N. Kurtz, H.O. Hoadley and J.J. DePalma (1973). Design and synthesis of random phase diffusers. J Opt Soc Am, pp. 1080-1092.

3. www.rpcphotonics.com and www.thorlabs.com.

4. E.R. Méndez et al (2004). Design of two-dimensional random surfaces with specified scattering properties. Opt Lett, pp. 2917-2919.

5. E.R. Méndez et al (2001). Photofabrication of random achromatic optical diffusers for uniform illumination. Appl Opt, pp. 1098-1108.