Retroreflective panels

Ernst Karl Abbe (1840-1905) was a German mathematician and physicist. In 1866 he met Carl Zeiss and together they founded what was known as the ‘Zeiss Optical Works’, now known as Zeiss. In addition, he also co-founded Schott Glassworks in 1884. Abbe was also the first person to define the term numerical aperture. In 1873, Abbe published his theory and formula which explained the diffraction limits of the microscope [2]. Abbe recognized that specimen images are composed of a multitude of overlapping, multi-intensity, diffraction-limited points (or Airy discs).

[12] “Wolfram Demonstrations Project,” Corner Reflector. [Online]. Available: https://demonstrations.wolfram.com/CornerReflector/.

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In order to increase the resolution, d = λ/(2NA), the specimen must be viewed using either a shorter wavelength (λ) of light or through an imaging medium with a relatively high refractive index or with optical components which have a high NA (or, indeed, a combination of all of these factors).

Retro reflectorvs retroreflector

Retroreflection itself stays unchanged, but retroreflection technology is far from stagnant. Companies have been developing increasingly brighter retroreflective beads, sheetings, and tapes. Sonar, which involves retroreflection, is improving in accuracy and effectiveness. Retroreflection has and still is being utilized in space to learn more about our moon. In the future, retroreflection will continue to be indispensable for research and safety technology.

As stated above, the shorter the wavelength of light used to image a specimen, then the more the fine details are resolved. So, if using the shortest wavelength of visible light, 400 nm, with an oil-immersion objective having an NA of 1.45 and a condenser with an NA of 0.95, then R would equal 203 nm.

[2] R. A. Serway and J. W. Jewett, “The Nature of Light and the Principles of Ray Optics,” in Physics for Scientists and Engineers with Modern Physics, 10th ed. Cengage, 2017, pp.902–908.

Abbe’s diffraction formula for axial (Z) resolution is:  d = 2λ/(NA)2 and again, if we assume a wavelength of 514 nm to observe a specimen with an objective having an NA value of 1.45, then the axial resolution will be 488 nm.

[9] O. Sanchez Jr. et al., CHARACTERIZATION OF THE INTENSITY DISTRIBUTION IN RETROREFLECTIVE ADHESIVES WITH NEAR FIELD GONIOPHOTOMETRY, 29th CIE Session, 2019.

These theoretical resolution values, derived from physical and mathematical assumptions, are estimates. They assume perfect imaging systems and a point light source in a vacuum or a completely homogeneous material as the sample or specimen. Of course, this assumption is almost never the case in real life, as many samples or specimens are heterogeneous. Because there is only a finite amount of light transmitting through the sample or reflecting from its surface, the measurable resolution depends significantly on the signal-to-noise ratio (SNR).

Taking all of the above theories into consideration, it is clear that there are a number of factors to consider when calculating the theoretical limits of resolution. Resolution is also dependent on the nature of the sample. Let’s look at calculating resolution using the Abbe diffraction limit, Rayleigh Criterion, and also FWHM.

Retro reflectorsensor

This light, which is caused by retroreflection, helps drivers see just about everything that is important to see. What if there was no retroreflection at night? You would be in serious danger of collisions unless you slowed down considerably. And forget about the freeway; your life would be at risk when making sharp turns or merging lanes! According to a study conducted by the Federal Highway Administration, crash frequency decreases as retroreflectivity increases. The study also found that drivers can see an average distance of 300 feet down a road at night at the minimum retroreflectivity level in California [1]. Driving at night is possible — and is safer — thanks to retroreflection.

Firstly, it should be remembered that: NA = n(sinα) where n is the refractive index of the imaging medium and α is half of the angular aperture of the objective. The maximum angular aperture of an objective is around 144º. The sine of half of this angle is 0.95. If using an immersion objective with oil which has a refractive index of 1.52, the maximum NA of the objective will be 1.45. If using a ‘dry’ (non-immersion) objective the maximum NA of the objective will be 0.95 (as air has a refractive index of 1.0).

The retroreflectivity of road markings and beaded retroreflective sheeting can be attributed to the transparent glass beads embedded into them. A combination of regular reflections and a phenomenon called refraction occur inside these beads to create retroreflection.

Retro ReflectorTape

[6] “Reflective Glass Beads for Road & Runway Markings – Potters Beads,” Potters, Dec 2020. [Online]. Available: https://www.pottersindustries.com/reflective-beads/.

In microscopy, the term “resolution” is used to describe the ability of a microscope to distinguish details of a specimen or sample. In other words, the minimum distance between 2 distinct points of a specimen where they can still be seen by the observer or microscope camera as separate entities. Resolution is intrinsically linked to the numerical aperture (NA) of a microscope’s optical components, like the objective lens, as well as the wavelength of light used. This article covers some of the history behind resolution concepts and explains each one using relatively simple terminology.

Figure 9. Three-dimensional rendering of retroreflection inside a microprism [12].A microprism, or corner cube prism, depends on regular reflections to work properly. There is no refraction involved, which distinguishes microprisms from microbeads. An incident light beam from headlights that enters a corner cube prism will strike one face. If that beam strikes at an appropriate angle it can be led towards a second face following the law of reflection. When the beam reflects off of the second face, it may be led towards the third face. Upon reflecting off of the third face, the light exits the prism along a path that is parallel to its entry path. Not all light will retroreflect. Beams may enter a prism, strike one or two faces, and exit the prism in a different direction. If a light ray strikes all three faces of a corner cube prism, as it does in Figure 9, it is guaranteed to retroreflect [13].

When a road is painted, transparent glass spheres (commonly called beads) are scattered onto the fresh paint. These beads act as retroreflectors. Roads are painted with melted thermoplastic, so the plastic-like paint will harden with the beads embedded on the surface [4]. For long-lasting retroreflection, glass beads can also be mixed into the paint prior to application. Beads covered by paint cannot act as retroreflectors, but they work properly once the paint wears down and they are exposed to the surface. An observer would be unable to discern beads even if they are on the paint surface. These beads can range from 100 to 1500 microns in diameter [5]. For reference, they range from the thickness of a sheet of paper to the size of a grain of sand. The presence of millions of these incredibly tiny beads helps road markings shine brightly as a car with headlights approaches.

Helen Situ is an undergraduate student studying mechanical engineering at the University of Southern California. She has stared at many miles of road lines at night.

[8] G. Burgess, M. R. Shortis, P. Scott, Photographic assessment of retroreflective film properties, ISPRS Journal of Photogrammetry and Remote Sensing, July 2011.

A more practical approach for resolution is the full width at half maximum (FWHM) intensity of an optically unresolved structure [4,5]. This value is relatively easy to measure with a microscope and has become a generally accepted parameter for comparison purposes. The theoretical value for the FWHM is RFWHM = 0.51λ/(NA) which is approximately λ/(2NA). So the FWHM as a resolution parameter is very close to Abbe’s diffraction limit, but also can be measured from microscope image data. For calibration or resolution-limit measurements, often beads or colloids of various diameters are imaged and measured.

Traffic signs made from repeated microscopic prisms are as prevalent as signs made from microbeads. A microprism is an extremely small corner cube prism. The basic construction of a corner cube prism, shown in Figure 8, is true to its name. It is one corner of a cube that is composed of three faces. However, a corner cube prism doesn’t need to literally be cubic. It typically isn’t cubic when used as a retroreflector. So long as the retroreflector has three regularly reflective faces at 90° angles to one another, it is a corner “cube” prism [11]. For example, Figure 9 is a corner cube prism constructed from 3 triangular faces. This triangular form is the most common shape for microprisms in retroreflective sheeting.

[3] “Physics Tutorial: The Role of Light to Sight,” The Physics Classroom. [Online]. Available: https://www.physicsclassroom.com/class/refln/Lesson-1/The-Role-of-Light-to-Sight.

[11] “Corner Cube Prisms,” RP Photonics Encyclopedia, June 2020. [Online]. Available: https://www.rp-photonics.com/corner_cube_prisms.html.

Figure 7 follows the path of a beam of light that meets the right conditions to be retroreflected in a glass bead. Light from headlights shines on nearby road markings, and the incident light beams strike the glass bead surface, as in Figure 7a. The incident beam will split into a reflected beam—  which is irrelevant to retroreflection—  and a refracted beam—  which is important to retroreflection. The refracted beam makes a smaller angle to the normal compared to the incident beam. This allows the refracted beam to strike the paint at the back of the bead, as in Figure 7b.  It is crucial that the beam reaches the paint-glass boundary. A majority of observed retroreflection stems from reflection off this part of the bead [10]. The beam reflects off of the paint-glass boundary, as in Figure 7c, but no simultaneous refraction occurs because the paint is not transparent. After returning to the boundary, this beam splits into two. The beam that reflects and remains in the glass is irrelevant to retroreflection. The beam that refracts as it exits the glass and enters the air, as in Figure 7d, travels parallel to the original incident beam. In traffic signs made with microbead retroreflective sheeting, this same process occurs to return light to its source. The only difference is that, instead of reflecting off paint as in Figure 7c, light must reflect off the sheeting that the microbeads are embedded in.

Retroreflection occurs when a light beam returns in a path parallel from which it came, as demonstrated in Figure 1. Beyond road use, retroreflection is used in aerial and maritime radar technology.

We encounter retroreflective materials every single day: at street corners, on crosswalks, and almost anywhere that cars can go. They mostly go unnoticed during the day, but under the cover of darkness, the roads become an optical playground. Light on the road is designed to go back towards the direction from which it came during dark conditions. This optical phenomenon is called retroreflection, and it makes nighttime driving safer by increasing the visibility of road markings and traffic signs.

Figure 2. The law of reflection states that θ1 = θ1’ [2].We are only able to see objects if light emits from or reflects off the objects and into our eyes, such as what occurs in Figure 1 [3]. If road paint and road signs utilized regular reflection without retroreflection, we would have no problem seeing them in the daylight. The chances of sunlight reflecting off our surroundings and reaching our eyes are high. But regularly reflecting signs would be problematic at night. Illumination coming from headlights would reflect away from the vehicle and the motorist’s eyes following a path similar to that of Figure 2. Retroreflection in road paint and road signs is needed to send vehicle lights back towards the driver and to increase road visibility.

There are 3 mathematical concepts which need to be taken into consideration when dealing with resolution: Abbe’s diffraction limit, Airy discs, and the Rayleigh criterion. Each of these are covered below in chronological order.

Retroreflector mirror

The diffraction pattern is determined by the wavelength of light and the size of the aperture through which the light passes. The central point of the Airy disc contains approximately 84% of the luminous intensity with the remaining 16% in the diffraction pattern around this point. There are of course many points of light in a specimen as viewed with a microscope, and it is more appropriate to think in terms of numerous Airy patterns as opposed to a single point of light as described by the term ‘Airy disc’.

Refraction occurs when an incident light ray travelling in a transparent medium, such as air, strikes the surface of another transparent medium, such as glass. A light beam will split into two light beams at the boundary of transparent surfaces. One beam behaves per the law of reflection from Figure 2 and reflects off the transparent surface. The other beam crosses the boundary and into the new medium instead of reflecting. More often than not, this light beam will bend upon entry. The incident ray and this new ray (the refracted ray) form different angles with the normal of the surface, seen in Figure 6 [2]. This bending phenomenon is called refraction.

Figure 6. An incident ray experiences refraction as it enters another medium [2].Snell’s law of refraction in Equation 1 describes the geometry behind refraction. In Snell’s law of refraction, n1 and n2 are indices of refraction. A refractive index is the ratio of the speed of light in vacuum to the speed of light within a medium [2]. Air has a refractive index of 1, and glass beads typically have a refractive index of 1.5. Therefore, for light travelling between glass and air, a light ray inside glass will always form a smaller angle with the normal compared to a light ray in the air.

The three-dimensional (3D) representation of the Airy pattern, as illustrated in the right half of Figure 1, is also known as the ‘point-spread function’ (PSF) of an optical instrument which has no appreciable aberration.

Retro reflectorthorlabs

Abbe’s diffraction formula for lateral (XY) resolution is:  d = λ/(2NA) where λ is the wavelength of light used to image a specimen. If using a green light of 514 nm and an oil-immersion objective with an NA of 1.45, then the (theoretical) limit of resolution will be 177 nm.

Figure 3. A road line illuminated under dark conditions. The glow texture indicates the presence of embedded beads [6].Traffic signs are made by applying retroreflective sheeting to a metal frame. Retroreflective sheeting is similar to a sticker: it is flexible, has an adhesive backing, and has a variety of color options readily available. The retroreflective sheetings that are widely used today fall under two categories. One type is a sheet embedded with microscopic glass beads. It uses the same retroreflecting mechanism as beads in road markings. The second type is made from repeated microscopic prisms molded into a sheet [7].

Taking the NA of the condenser into consideration, air (with a refractive index of 1.0) is generally the imaging medium between the condenser and the slide. Assuming the condenser has an angular aperture of 144º then the NAcond value will equal 0.95.

An Airy disc is the optimally focused point of light which can be determined by a circular aperture in a perfectly aligned system limited by diffraction. Viewed from above (Figure 1), this appears as a bright point of light around which are concentric rings or ripples (more correctly known as an Airy Pattern).

[10] K. Vedam and M. D. Stoudt, “Retroreflection from spherical glass beads in highway pavement markings 2: Diffuse reflection (a first approximation calculation),” Applied Optics, vol. 17, no. 12, p. 1859, 1978.

If using a green light of 514 nm, an oil-immersion objective with an NA of 1.45, condenser with an NA of 0.95, then the (theoretical) limit of resolution will be 261 nm.

Using the theory of Airy discs, if the diffraction patterns from two single Airy discs do not overlap, then they are easily distinguishable, ‘well resolved’ and are said to meet the Rayleigh criterion. When the center of one Airy disc is directly overlapped by the first minimum of the diffraction pattern of another, they can be considered to be ‘just resolved’ and still distinguishable as two separate points of light (Figure 2, mid). If the Airy discs are closer than this, then they do not meet the Rayleigh criterion and are ‘not resolved’ as two distinct points of light.

George Biddell Airy (1801-1892) was an English mathematician and astronomer. By the 1826 (aged 25) he was appointed professor of mathematics at Trinity College and two years later, he was appointed professor of astronomy at the new Cambridge Observatory. From 1835 to 1881 he was the ‘Astronomer Royal’ and even has a lunar and Martian crater named in his honor.

The Rayleigh Criterion is a slightly refined formula based on Abbe’s diffraction limits:  R = 1.22λ/(NAobj + NAcond) where λ is the wavelength of light used to image a specimen. NAobj is the NA of the objective. NAcond is the NA of the condenser. The value ‘1.22’ is a constant. This is derived from Rayleigh’s work on Bessel Functions. These are used for calculating problems in systems such as wave propagation.

However, even taking all of these factors into consideration, the possibilities with a real microscope are still somewhat limited due to the complexity of the whole system, transmission characteristics of glass at wavelengths below 400 nm, and the challenge to achieve a high NA in the complete microscope system. Lateral resolution in an ideal optical microscope is limited to around 200 nm, whereas axial resolution is around 500 nm (examples of resolution limits are given below).

retro-reflector on moon

[7] “Retroreflective Sheeting Identification Guide,” [Online]. Available: http://www.dot.state.mn.us/trafficeng/signing/doc/retroreflectivesheetingIDguide.pdf.

Not all light will bounce back parallel to the initial beam. Some beams never enter the bead, and some don’t strike the bead at a favorable angle. But enough beams will retro-reflect for the driver to perceive the road markings and drive on.

[5] “Using Glass Bead Composite Paints for Reflective Roads,” AZoM.com, Jul 2018. [Online]. Available: https://www.azom.com/article.aspx?ArticleID=16292.

John William Strutt, 3rd Baron Rayleigh (1842-1919) was an English physicist and a prolific author. During his lifetime, he wrote an astonishing 466 publications including 430 scientific papers. He wrote on a huge range of topics as diverse as bird flight, psychical research, acoustics and in 1895, he discovered argon (Ar) for which he was later awarded the Nobel prize for physics in 1904.

[13] J. H. Hannay and T. M. Haeusser, “Retroreflection by Refraction,” Journal of Modern Optics, vol. 40, no. 8, p. 1437, 1993. [Online]. Available: https://doi.org/10.1080/09500349314551501.

You are a passenger in a vehicle, cruising down the road at night. Take a look ahead of you. What you see could never match up to the brilliant sparkle of Guy Diamond from the 2016 Dreamworks hit, Trolls, but I would argue that it comes pretty close. The traffic sign that your vehicle approaches has a strong, eye-catching shine, while the road paint within your line of sight glows with almost otherworldly radiance.

Retro reflectorcar

Also in the year 1835, he published a paper in the Transactions of the Cambridge Philosophical Society entitled ‘On the Diffraction of an Object-Glass with Circular Aperture’ [1]. Airy wrote this paper very much from the view of an astronomer and in it he describes “the form and brightness of the rings or rays surrounding the image of a star as seen in a good telescope”. Despite writing in a different scientific field, these observations are relevant to other optical systems including microscopes.

[1] Synthesis of Pavement Marking Research, Federal Highway Administration, June 2015, pp.8-10. [Online]. Available: https://safety.fhwa.dot.gov/roadway_dept/night_visib/pavement_marking/pvmnt_mrkg_synth.pdf.

As already mentioned, the FWHM can be measured directly from the PSF or calculated using: RFWHM = 0.51λ/(NA). Again using a light wavelength of 514 nm and an objective with an NA of 1.45, then theoretical resolution will be 181 nm. This value is very close to the lateral resolution calculated just above from the Abbe diffraction limit.

The numerical aperture (NA) is related to the refractive index (n) of a medium through which light passes as well as the angular aperture (α) of a given objective (NA = n sinα). The resolution of an optical microscope is not solely dependent on the NA of an objective, but the NA of the whole system, taking into account the NA of the microscope condenser. More image detail will be resolved in a microscope system in which all of the optical components are correctly aligned, have a relatively high NA value and are working harmoniously with each other. Resolution is also related to the wavelength of light which is used to image a specimen; light of shorter wavelengths are capable of resolving greater detail than longer wavelengths.

Traffic signs, bike reflectors, safety vest tape, and other materials that glow when light is cast on them are popularly called “reflective,” but these items are actually retroreflective. In regular reflection, the reflected beam does not go back towards the direction from which it originated. In Figure 2, an incoming light ray that strikes a surface (the incident ray) and the regularly reflected ray must form equivalent angles to the normal of the smooth surface [2]. A regularly reflected ray travels far away from its source according to the law of reflection. Retroreflection is not regular reflection. Instead, retroreflection involves multiple regular reflections to manipulate a light beam to return to its source.

To achieve the maximum theoretical resolution of a microscope system, each of the optical components should be of the highest NA available (taking into consideration the angular aperture). In addition, using a shorter wavelength of light to view the specimen will increase the resolution. Finally, the whole microscope system should be correctly aligned.

[4] “Thermoplastic Road Marking Paint.” [Online]. Available: https://www.tenroadsglass.com/products/thermoplastic-traffic-road-line-marking-paint.html#:~:text=Thermoplastic%20paint%20is%20a%20road,reflection%2C%20and%20long%20service%20life.

[14] “Why do pavement markings disappear in the rain?,” 3M. [Online]. Available: https://www.3m.com/3M/en_US/road-safety-us/resources/road-transportation-safety-center-blog/full-story/~/why-do-pavement-markings-disappear-in-the-rain/?storyid=90b6c8e2-1c04-4491-adc6-9df3e09f022d.

Rayleigh built upon and expanded the work of George Airy and invented the theory of the ‘Rayleigh criterion’ in 1896 [3]. The Rayleigh criterion defines the limit of resolution in a diffraction-limited system, in other words, when two points of light are distinguishable or resolved from each other.