The Fresnel Lens
Centuries ago, it was recognized that the contour of the refracting surface of a conventional lens defines its focusing properties. The bulk of material between the refracting sur-faces has no effect (other than increasing absorption loss) on the optical properties of the lens. In a F resnel (point focus) lens the bulk of material has been reduced by the extraction of a t of coaxial annular cylinders of material, as shown in Figure 1. (Positive focal length Fresnel lens are almost universally plano-convex.) The contour of the curved surface is thus approximated by right circular cylindrical portions, which do not contribute to the lens’ optical proper-ties, intercted by conical portions called “grooves.” Near the center of the lens, the inclined surfaces or “grooves”are nearly parallel to the plane face; toward the outer edge, the inclined surfaces become extremely steep, especially for lens of low f–number. The inclined surface of each groove is the corresponding portion of the original aspheric surface, translated toward the plano surface of the lens; the angle of each groove is modified slightly from that of the original aspheric profile to compensate for this translation.
The earliest stepped-surface lens was suggested in 1748
by Count Buffon, who propod to grind out material from the plano side of the lens until he was left with thin ctions of material following the original spherical surface of the lens, as shown schematically in F igure 2a). Buffon’s work was followed by that of Condorcet and Sir D. Brewster, both of whom designed built-up lens made of stepped annuli. The aspheric Fresnel lens was invented in 1822 by Augustin Jean F resnel (1788–1827), a F rench mathematician and physicist also credited with resolving the dispute between the classical corpuscular and wave theories of light through his careful experiments on diffraction. Fresnel’s original lens was ud in a lighthou on the river Gironde; the main innovation embodied in Fresnel’s design was that the center of curvature of each ring receded along the axis according to its distance from the center, so as practically to eliminate spherical aberration. Fresnel’s original design, including the spherical-surfaced central ction, is shown schematically in Figure 2b). The early Fresnel lens were cut and polished in glass – an expensive process, and one limited to a few large grooves. Figure 3 shows a Fresnel lens, constructed in this way, which is ud in the lighthou at St Augustine, Florida, USA. The large aperture and low absorption of F resnel lens were especially important for u with the weak lamps found in lighthous before the invention of high-brightness light sources in the 1900s. The illustrated system is catadioptric: the glass rings above and below the Fresnel lens band in the center of the light are totally-internally-reflecting prisms, which rve to collect an additional frac-tion of the light from the source. The u of catadioptric sys-tems in lighthous was also due to Fresnel.
Until the 1950’s, quality Fresnel lens were made from glass by the same grinding and polishing techniques ud in 1822. Cheap Fresnel lens were made by pressing hot glass into metal molds; becau of the high surface tension of glass, Fresnel lens made in this way lacked the necessary detail, and were poor indeed.
In the last forty years or so, the advent of optical-quality plastics, compression and injection molding techniques,Figure 1 Construction of a Fresnel lens from its correspond-ing asphere. Each groove of the Fresnel lens is a
small piece of the aspheric surface, translated to-
ward the plano side of the lens. The tilt of each sur-
face must be modified slightly from that of the
original portion of aspheric surface, in order to
compensate for the translation.
Figure 2 Early stepped–surface lens. In both illustrations the black area is material, and the dashed curves
reprent the original contours of the lens. a)
shows the lens suggested by Count Buffon (1748),
where material was removed from the plano side
of the lens in order to reduce the thickness. b)
跆拳道基本动作
shows the original lens of Fresnel (1822), the cen-
tral ring of which had a spherical surface. In
Fresnel’s lens, the center of curvature of each ring
was displaced according to the distance of that
ring from the center, so as to eliminate spherical
可积条件
aberration.
a)
b)
© Copyright Fresnel Technologies, Inc. 20032
© Copyright Fresnel Technologies, Inc. 2003
3
and computer-controlled machining have made possible the manufacture and wide application of F resnel lens of higher optical quality than the finest glass F resnel lens.Modern computer-controlled machining methods can be ud to cut the surface of each cone precily so as to bring all paraxial rays into focus at exactly the same point, avoid-ing spherical aberration. Better still, newe
r methods can be ud to cut each refracting surface in the correct aspheric contour (rather than as a conical approximation to this con-tour), thus avoiding even the width of the groove (typically 0.1 to 1 mm) as a limit to the sharpness of the focus. Even though each groove or facet brings light precily to a focus,the breaking up of the wavefront by the discontinuous sur-face of a F resnel lens degrades the visible image quality.Except in certain situations discusd later, Fresnel lens are usually not recommended for imaging applications in the visible light region of the spectrum.
妞妞The characteristics of the aspheric “correction”
The grinding and polishing techniques ud in the manufac-ture of conventional optics lead to spherical surfaces. Spher-ical surfaces produce optics with longitudinal spherical aberration, which occurs when different annular ctions of the optic bring light rays to a focus at different points along the optical axis. This phenomenon is illustrated for a positive focal length, plano-convex conventional lens in Figure 4 (in all optical illustrations in this brochure, light is taken to propagate from left to right). The lens illustrated is a ction of a sphere with 1" (25 mm) radius of curvature, 1.6"(36 mm) in diameter; the index of refraction of the material is 1.5, typical both for optical glass and for our plastics materials. The focal length of the illustrated lens is thus 2"(50 mm), and the aperture is /1.3. As is evident from the figure, the longitudinal spherical aberration is very strong.Single-element sphe
rical lens are typically restricted to much smaller apertures (higher –numbers) than this,becau longitudinal spherical aberration of the magnitude shown in Figure 4 is generally unacceptable. Figure 5 shows an aspheric lens of the same focal length and –number;note that the surface contour is modified from the spherical profile in such a way as to bring rays passing through all points on the lens to a focus at the same position on the opti-cal axis. A lens made with the aspheric profile illustrated in Figure 5, therefore, exhibits no longitudinal spherical aber-ration for rays parallel to the optical axis.
Since Fresnel lens are made from the beginning to the correct aspheric profile, the notion of “correcting for spheri-cal aberration” is not meaningful for F resnel lens. The lens are more accurately characterized as “free from spherical aberration.” The combination of the aspheric sur-face (which eliminates longitudinal spherical aberration)and the thinness of the lens (which substantially reduces both absorption loss in the material and the change of tho loss across the lens profile) allows F resnel lens with acceptable performance to be made with very large apertures. In fact, F resnel lens typically have far larger apertures (smaller –numbers) than the /1.3 illustrated in Figure 4.
Figure 6 compares an aspheric plano-convex lens with an aspheric F resnel lens (the F resnel lens’
groove structure is
f f f f f Figure 3 The light from the St Augustine, Florida (USA) light-hou, showin
g the glass Fresnel optical system ud in the lighthou. The optical system is about 12 feet (3.5 m) tall and 7 feet (2 m) in diameter.
Figure 4
Illustration of longitudinal spherical aberration.
The rays shown were traced through an /1.3 spherical-surface lens; the focus is evidently忙碌的一天作文
翁德隆spread out over a considerable distance along the
optical axis.
f
© Copyright Fresnel Technologies, Inc. 2003
4
tive focal length (EFL), quential, so that the Fresnel lens.
focus. (This type of F application and reverd.
for a given focal length tion (where object distances, i.e. the conjugates), and are found to be and for the conjugate ratio 3:1. Even though a lens may be designed for conjugates in some particular ratio, it can be ud at other finite conjugate ratios as well. The error introduced is usually reasonably small.
Fresnel lens are normally fabricated so that they are correct for the ca of grooves toward the collimated beam,plano side toward the focus (grooves “out”). They can, how-ever, be fabricated so that they are correct for the ca of grooves toward the focus, plano side toward the collimated beam (grooves “in”). In this ca, there is no refraction at all on the plano side for a collimated beam traveling parallel to the optical axis. In the grooves “out” ca, both surfaces refract the light more or less equally. The ca of grooves toward the collimated beam (“out”) is the optically preferred ca. The main difference is that in the grooves “in” ca, the grooves at the outer periphery of the lens are canted at much
f f f 1f ⁄1i ⁄1o ⁄+=i 4f 4f 3⁄ Figure 6 Comparison between an aspheric conventional
lens and an aspheric Fresnel lens, illustrating the optical quantities discusd in the text.smaller angles to the plano surface than they would be in spherical or grooves “out” lens. Becau the angles made with the plano surface are relatively small toward the periphery of the lens, any small warpage or tilt of the lens surface, or any small deviation of a light ray from parallelism with the optical axis, leads to a very large deviation from the ideal in the angle between the light ray and the lens surface.The errors lead directly to a decrea in the collection effi-ciency of a grooves “in” lens relative to a grooves “out” lens of the same focal length and –number.
A third ca which is sometimes encountered is that of a Fresnel lens which is correct for grooves “out,” ud with its grooves toward the focus (grooves “out” turned grooves
赛车总动员3f
© Copyright Fresnel Technologies, Inc. 2003
5
for angles of interction between a light ray and the normal
to a surface larger than the critical angle = ,where the ray is traveling from a medium of index of refr
ac-tion into a medium of index of refraction . It is evident that total internal reflection only occurs for , since in the ca is greater than π /2 and therefore not physically meaningful.) This phenomenon makes the portion of a grooves “out” lens turned grooves “in” lens past about /1 uless. The phenomenon is easily obrved as an appar-ent “silvering” of the outer portion of a grooves “out” lens when its grooves are turned to face the shorter conjugate.Total internal reflection does not occur for grooves “out”lens ud in their correct orientation becau the only large-angle interction between the light and the lens sur-face occurs at a transition from low to high refractive index.
Materials
Our standard materials for visible light applications are acrylic, polycarbonate and rigid vinyl. The materials are suitable for some near infrared applications as well, as dis-cusd later in this brochure. Figure 9 shows uful transmis-sion ranges for a variety of plastics materials. Materials suitable for infrared applications are described in detail in our POLY IR® brochure.
法超The first step in choosing a material is to match the mate-rial to the spectral domain of the application. Other consid-erations include thickness, rigidity, rvice temperature,weatherability, and other physical properties listed in the table of properties on the next page.
Acrylic
Optical quality acrylic is the most widely applicable mate-rial, and is a good general-purpo material in the visible. Its transmittance is nearly flat and almost 92% from the ultravi-olet to the near infrared; acrylic may additionally be speci-fied to be UV transmitting (UVT acrylic) or UV filtering (UVF acrylic). The transmittance of our standard acrylic materials between 0.2 µm and 2.2 µm is shown in F igure 10 for a thickness of 1/8" (3.2 mm). Standard acrylic thickness are 0.060" (1.5 mm), 0.090" (2.3 mm), and 0.125" (3.2 mm). Rigid vinyl
Rigid vinyl has a number of characteristics which make it both affordable and very suitable for certain applications. It has a high index of refraction; it is reasonably inexpensive;and it can be die-cut. However, polycarbonate has very sim-ilar properties, without the problems associated with rigid vinyl, and its u is encouraged over that of rigid vinyl in new applications. Rigid vinyl has about the same tempera-ture range as acrylic and is naturally fire-retardant. The trans-mittance of rigid vinyl between 0.2 µm and 2.5 µm is shown in F igure 11 for a nominal thickness of 0.030" (0.76 mm).Standard thickness for rigid vinyl are 0.010" (0.25 mm),0.015" (0.38 mm), 0.020" (0.51 mm), and 0.030" (0.76 mm). Polycarbonate
Polycarbonate is spectrally similar to acrylic, but is uful at higher temperatures and has a very high impact resistance.The transmittance of polycarbonate between 0.2 µm and 2.2 µm is shown in Figure 12 for a nominal thickness of 1/8"
θc sin –1
n n '⁄()n n 'n 'n >n 'n <θc f Figure 7 Illustration of the strong asymmetry of the aspheric芒鱼
Fresnel lens. The illustrated lens is correct for the grooves facing the longer conjugate (grooves “out”). When it is turned around so that the
grooves face the shorter conjugate (grooves “out” turned grooves “in”), on-axis performance suffers. As discusd in the text, however, in the ca where the grooves must face the shorter conjugate, a grooves “out” lens turned grooves “in” has some advantages over a lens correct for grooves “in.”
Figure 8 Aspheric Fresnel lens correct for the grooves facing
the shorter conjugate (grooves “in”).
