Lens Maker Equation

9 Key excerpts on "Lens Maker Equation"

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  Microlenses

  Properties, Fabrication and Liquid Lenses

  • Hongrui Jiang, Xuefeng Zeng(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  16 Microlenses: Properties, Fabrication and Liquid Lenses © 2008 Taylor & Francis Group, LLC Equations (2.7) and (2.8) are then substituted into Snell’s law, sin sin α β = n n 2 1 (2.9) to obtain n o n i n n R n R 1 2 2 1 + = -= ∆ (2.10) which is Gauss’ formula for refraction at a single refracting surface. Here Δ n is the difference between the refractive index of the material of the lens and that of the outside medium, and R is the radius of curvature of the surface. The refractive power of a surface is defined as the ratio of the refractive index of the medium over the focal length, as expressed in Equation (2.11). A surface or lens of short focal length has high refractive power, and vice versa. P n f n R = = ∆ (2.11) 2.2.3 Thin Lenses A lens is defined as thin if a ray entering on one surface emerges at approxi-mately the same position on the opposite surface; that is, negligible transla-tion occurs in the lens. However, there is no numerical limit distinguishing thin and thick lenses. The issue is decided entirely by the degree of precision required for solving a given problem. One lens may be considered thin for a preliminary and thick for a rigorous solution. A thin lens can be considered the sum of two single refractive surfaces: FIGURE 2.3 Refraction at single spherical surface. 17 Basic Physics of Liquid Microlenses © 2008 Taylor & Francis Group, LLC P P P n R n R = + = + 1 2 1 2 ∆ ∆ (2.12) P f n R R = = +       1 1 1 1 2 ∆ (2.13) Their powers added together represent the lens maker’s formula. Note that in Equation (2.13), n is assumed to be 1. 2.2.4 Aberrations 2.2.4.1 Spherical Aberration Spherical aberration arises when rays passing through different zones of a lens come to different focal points. In general, rays close to the optic axis are refracted less and come to a focus further away from the lens than the marginal rays, as shown in Figure 2.4.

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  Physics, Student Study Guide

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  It produces a final image which is inverted and virtual. Spherical aberration An effect in which rays from the outer edge of a spherical lens are not focused at the same point as rays that pass through the center of the lens, thus causing images to be blurred. Chromatic aberration An undesirable lens effect arising when a lens focuses different colors at different points. Equations The index of refraction of a material is _ Speed oflight in a vacuum _ c n- -- Speed of light in the material v (26.1) Snell's law of refraction (26.2) 338 THE REFRACTION OF LIGHT: LENSES AND OPTICAL INSTRUMENTS The apparent depth of an object with the observer directly above the object is (26.3) The critical angle for total internal reflection is given by (26.4) Brewster's Law (26.5) The thin-lens equation (26.6) The magnification equation _ Image height _ hi _ di m- ----- Object height ho d 0 (26. 7) The refractive power of a lens is given by: Refractive power (in diopters) = 1 f(in meters) (26.8) The angular magnification of an optical instrument is: M = Angular size produced by instrument = .!t_ Reference angular size 9 (26.9) The angular magnification of a magnifying glass is: (26.10) The angular magnification of a compound microscope is given by: (26.11) Chapter 26 339 The angular magnification of an astronomical telescope is: (26.12) DISCUSSION OF SELECTED SECTIONS 26.2 Snell"s Law and the Refraction of Light When light travels across an interface from one medium to another, it undergoes a change in its speed as indicated by the different indices of refraction of the materials. This speed change results in a bending (refraction) of the path of the light unless it strikes the interface at normal incidence (See figure 26.2 in the text.). Snell's law (equation 26.2) can be used to calculate the amount of bending. Example 1 Light strikes an interface between two materials of refractive indices, n1 and n2, at an angle el to the normal to the surface.

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  Applied Optics and Optical Design, Part One

  • A. E. Conrady(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  The exact ray-tracing formulae with which we have dealt up to now find their chief application in the later stages of the evolution of new optical designs. A very large amount of practically useless labour would be required if we attempted to test every rough preliminary scheme by these formulae and thus to feel our way, largely in a more or less blindfold manner, to the finished instrument. Progress towards the ultimate perfection will be quickest if we begin with a rough first approximation and gradually apply more refined methods as the work proceeds. In the majority of cases we can arrive at such a simple scheme most quickly and with very little trouble by determining what combination of thin simple lenses would produce the intended effect. Therefore we must now deduce suitable formulae which enable us to calculate the properties of simple lenses.

  Equation (6p)** can be applied to the two surfaces of a simple lens so as to yield a single expression for its focal effects.

  Referring to Fig. 18 , we shall have by the transfer-equations (5p)* and , therefore for the first surface

  for the second surface

  If the first equation is multiplied throughout by the stated factor its left side becomes equal to the last term on the right of the second equation ; these two terms therefore cancel each other when the second equation is added to the extended first, with the result

  Taken together with the equation for the first surface, this is the most general, but decidedly inconvenient, solution of the problem of a thick lens. The equation becomes simpler if we assume that , that is, that the lens is employed immersed in a medium of index N 1 . We then have

  or on division throughout by N 1

  Here is the relative refractive index of the material of the lens referred to the surrounding medium and using the simple symbol N for this index, we find the solution

  (IIP)

  in which

  The second equation following on dividing the original equation of the first surface by N 1 and introducing

  FIG. 18 .

  The equation will be almost exclusively applied to lenses used in air, and N will then be the ordinary refractive index of the glass. But it should be remembered that the equation can be applied to a lens which, together with its object and image, is immersed in a medium other than air, provided that N is then taken as the relative index of the glass to the medium.

  (IIp) is an exact paraxial equation : it will be modified on a subsequent occasion so as to render it a useful solution of the problem of a thick lens. For the present we merely use it as a means to deduce the simplest possible equations for use in rough and rapid preliminary studies of lenses and lens systems by restricting the equations to very thin lenses in which may be treated as negligible in comparison to . The inconvenient factor in (IIp) then becomes = 1, but as this would be strictly correct only for an infinitely thin lens, we must note once for all that the convenient equations obtained on this assumption are (with certain rare exceptions) only a more or less rough approximation when applied to actual lenses, and too much dependence must not be placed on the results obtained by them. As a constant reminder of the inaccuracy of these equations we will distinguish them by putting TL (= thin lens) in front of the numbers assigned to them. The fundamental equation is found by omitting the factor

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  The Science of Imaging

  • Graham Saxby(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  47 4 Chapter Lens Principles A Model for the Geometry of Camera Lenses Although there are a number of models describing the behavior of light, the one used by lens designers is the simplest, namely the ray model. Before there were computers, the designing of a new lens was a complicated business involving months of laborious ray tracing. Today, designers can optimize a lens design in a few minutes using an off-the-shelf computer program. The variables to juggle are the refractive indices of the components; the color dispersions of the glasses; the number of components and their curvature, thickness, and separation; and the position of the stop. This all represented a heavy meal for the old-time designer equipped with only a slide-rule and a set of tables of glass types. But to a mod-ern computer they signify no more than a light breakfast. The result still doesn’t predict everything about the performance of the final lens; this demands a more sophisticated model, as we will see in Chapter 6. However, a simple ray model is sufficient to describe how a lens forms an image, and to explain why it is necessary to have more than one element in a camera lens. To begin with we will look at the properties of a simple convex lens. The Simple Lens The most basic ray model for a lens makes a number of assumptions: 1. The thickness of the lens can be ignored. 2. The lens aperture is small compared with the focal length. 3. The refractive index of the material is the same for all wavelengths. The geometrical optics of a simple or “thin” lens was established by Sir Isaac Newton, and for this reason is usually known as Newtonian optics . The Lens Laws The Newtonian lens laws consist of three basic principles concerning a simple lens. The first gives the relationship between the distances of the object and its image from the lens. The positions of the object and the image (which is inverted) are together termed conjugate foci (Figure 4.1).

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Both options are based on the law of reflection. The mirror and magnification equations relate the distances d o and d i of the object and image from the mirror to the focal length f and magnification m. For an object placed in front of a lens, Snell’s law of refraction leads to the technique of ray tracing and to equations that are identical to the mirror and magnification equa- tions. Thus, mirrors work because of the reflection of light, whereas lenses work because of the refraction of light, a distinction between the two devices that is important to keep in mind. The equations that result from applying Snell’s law to lenses are referred to as the thin-lens equation and the magnification equation: Thin-lens equation 1 d o + 1 d i = 1 f (26.6) Math Skills The thin-lens equation ( 1 d o + 1 d i = 1 f ) is sometimes thought to imply that d o + d i = f . To emphasize that the focal length f does not equal the object distance d o plus the image distance d i , we can solve the thin lens equation for f. First, we multiply the left side of the thin-lens equation by 1 in the form of d o d i d o d i : ( d o d i d o d i )( 1 d o + 1 d i ) = 1 f or ( 1 d o d i )( d o d i d o + d o d i d i ) = 1 f Simplifying this result gives ( 1 d o d i )( d o d i d o + d o d i d i ) = 1 f or d i + d o d o d i = 1 f Taking the reciprocal of both sides of the simplified result shows that ( d i + d o d o d i ) −1 = ( 1 f ) −1 or d o d i d i + d o = f Clearly, it is not true that d o + d i = f . Do not make this mistake when solving problems. 1 26.8 The Thin-Lens Equation and the Magnification Equation 753 Magnification equation m = Image height Object height = h i h o = − d i d o (26.7) Figure 26.30 defines the symbols in these expressions with the aid of a thin converging lens, but the expressions also apply to a diverging lens, if it is thin. The derivations of these equations are presented at the end of this section.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  (c) The image is of the object’s upper half only, but its brightness is not reduced. 26.8 | The Thin-Lens Equation and the Magnification Equation When an object is placed in front of a spherical mirror, we can determine the location, size, and nature of its image by using the technique of ray tracing or the mirror and magnifica- tion equations. Both options are based on the law of reflection. The mirror and magnifi- cation equations relate the distances d o and d i of the object and image from the mirror to the focal length f and magnification m. For an object placed in front of a lens, Snell’s law of refraction leads to the technique of ray tracing and to equations that are identical to the mirror and magnification equations. Thus, mirrors work because of the reflection of light, whereas lenses work because of the refraction of light, a distinction between the two devices that is important to keep in mind. The equations that result from applying Snell’s law to lenses are referred to as the thin-lens equation and the magnification equation: Thin-lens equation 1 d o 1 1 d i 5 1 f (26.6) Magnification equation m 5 Image height Object height 5 h i h o 5 2 d i d o (26.7) Figure 26.30 defines the symbols in these expressions with the aid of a thin converging lens, but the expressions also apply to a diverging lens, if it is thin. The derivations of these equations are presented at the end of this section. Certain sign conventions accompany the use of the thin-lens and magnification equations, and the conventions are similar to those used with mirrors in Section 25.6. The issue of real versus virtual images, however, is slightly different with lenses than with mirrors. With a mirror, a real image is formed on the same side of the mirror as the object (see Figure 25.18), in which case the image distance d i is a positive number.

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  eBook - PDF

  Introduction to Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  (c) The image is of the object’s upper half only, but its brightness is not reduced. 26.8 | The Thin-Lens Equation and the Magnification Equation When an object is placed in front of a spherical mirror, we can determine the location, size, and nature of its image by using the technique of ray tracing or the mirror and magnifica- tion equations. Both options are based on the law of reflection. The mirror and magnifi- cation equations relate the distances d o and d i of the object and image from the mirror to the focal length f and magnification m. For an object placed in front of a lens, Snell’s law of refraction leads to the technique of ray tracing and to equations that are identical to the mirror and magnification equations. Thus, mirrors work because of the reflection of light, whereas lenses work because of the refraction of light, a distinction between the two de- vices that is important to keep in mind. The equations that result from applying Snell’s law to lenses are referred to as the thin- lens equation and the magnification equation: Thin-lens equation 1 d o 1 1 d i 5 1 f (26.6) Magnification equation m 5 Image height Object height 5 h i h o 5 2 d i d o (26.7) Figure 26.30 defines the symbols in these expressions with the aid of a thin converging lens, but the expressions also apply to a diverging lens, if it is thin. The derivations of these equations are presented at the end of this section. Certain sign conventions accompany the use of the thin-lens and magnification equations, and the conventions are similar to those used with mirrors in Section 25.6. The issue of real versus virtual images, however, is slightly different with lenses than with mirrors. With a mirror, a real image is formed on the same side of the mirror as the object (see Figure 25.18), in which case the image distance d i is a positive number. With a lens, a positive value for d i also means the image is real.

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  College Physics

  • Michael Tammaro(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  For incident angles greater than or equal to c θ , we say that there is total internal reflection and no light passes into the other medium. The critical angle c θ for total internal reflection is given by n n sin c θ = ′ (26.2.1) where n is the index of refraction of the medium in which the ray is traveling, and n′ is the medium on which the light ray is incident. 26.3 Dispersion The index of refraction of a material depends on the wavelength of light (Table 26.3.1). This phenomenon is called dispersion and allows a prism to separate white light into its colors and is responsible for rainbows. 26.4 Thin Lenses and Ray Diagrams A converging lens deflects rays that are parallel to the principal axis toward the principal axis (Animated Figure 26.4.2). A properly formed converging lens will deflect these rays so that they pass through a unique point on the axis called the focal point. A diverging lens deflects rays that are parallel to the principal axis away from the principal axis (Animated Figure 26.4.3). A properly formed diverging lens will deflect these rays so that they appear to originate from a unique focal point on the axis. Ray tracing can be used to determine the characteristics of images formed by lenses. If an object is placed beyond the focal point of a converging lens, then an inverted, real image is formed (Animated Figure 26.4.6). If an object is placed between the focal point of a con- verging lens and the lens, then an upright, virtual image is formed (Animated Figure 26.4.7). Summary | 747 If an object is placed in front of a diverging lens, then an upright, virtual image is formed (Animated Figure 26.4.8). 26.5 The Thin Lens Equation and Magnification Suppose that an object is placed beyond the focal point of a converging lens (Figure 26.5.1). The focal length f is the distance from the lens to either focal point. The symbols d o and d i denote the object and image distances, respectively.

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  Physics, Student Solutions Manual

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Since the focal length is also known, the object distance can be found from the thin-lens equation. SOLUTION The object distance d o is related to the focal length f and the image distance d i by the thin-lens equation: o o i 1 1 1 1 1 or 28.0 cm 35.1 cm 138 cm d d f d = − = − = − (26.6) 75. REASONING The far point is 5.0 m from the right eye, and 6.5 m from the left eye. For an object infinitely far away (d o = ∞), the image distances for the corrective lenses are then –5.0 m for the right eye and –6.5 m for the left eye, the negative sign indicating that the images are virtual images formed to the left of the lenses. The thin-lens equation [Equation 26.6: o i (1/ ) (1/ ) (1/ ) d d f + = ] can be used to find the focal length. Then, Equation 26.8 can be used to find the refractive power for the lens for each eye. SOLUTION Since the object distance o d is essentially infinite, o 1/ 1/ 0 d = ∞ = , and the thin-lens equation becomes i 1/ 1/ d f = , or i . d f = Therefore, for the right eye, 5.0 m f = − , and the refractive power is (see Equation 26.8) [Right eye] Refractive power 1 1 –0.20 diopters (in diopters) (–5.0 m) f = = = Chapter 26 Problems 347 Similarly, for the left eye, 6.5 m, f = − and the refractive power is [Left eye ] Refractive power 1 1 –0.15 diopters (in diopters) (–6.5 m) f = = = 79. REASONING The angular magnification M of a magnifying glass is given by i 1 1 M N f d   ≈ −     (26.10) where f is the focal length of the lens, d i is the image distance, and N is the near point of the eye. The focal length and the image distance are related to the object distance d o by the thin-lens equation: i o 1 1 1 f d d − = (26.6) These two relations will allow us to determine the angular magnification. SOLUTION Substituting Equation 26.6 into Equation 26.10 yields i o 1 1 72 cm 18 4.0 cm N M N f d d   ≈ − = = =     84. REASONING AND SOLUTION The information given allows us to determine the near point for this farsighted person.

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