2: Lens and Mirror Calculations
- Page ID
- 7079
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 2.1: Introduction to Lens and Mirror Calculations
- This page addresses the complexities of sign conventions in optics, emphasizing the challenges posed by differing conventions among educators. It suggests that clarity in describing image characteristics—such as position, type, orientation, and magnification—is essential for understanding. The author introduces a personal sign convention and the convergence method for problem-solving, advocating that mastering these approaches can enhance confidence in arriving at correct solutions.
- 2.2: Limitations
- In this chapter, I am going to ignore lens and mirror aberrations. I may possibly prepare a separate chapter on lens and mirror aberrations sometime, but that is not the topic of the present chapter. Thus, in this chapter, I am going to assume that all angles are small.
- 2.3: Real and Virtual
- Whereas you can project a real image on to a piece of card or a photographic film, you cannot do this with a virtual image. The reason that you can see a real image with your eye is that the additional optics of your eye bends the diverging light from the virtual image and makes it converge on to a real image on your retina.
- 2.4: Convergence
- Converging light has positive convergence; Diverging light has negative convergence.
- 2.5: Power
- The function of a lens is to change the convergence of a beam of light. Indeed the difference between the initial and final convergence is called the power P of the lens, or of a refracting interface, or of a reflecting mirror.
- 2.6: Magnification
- If the magnification is positive, the image is erect; If the magnification is negative, the image is inverted.
- 2.7: Examples
- This page discusses the behavior of light through converging lenses, focusing on image formation, magnification, and refractive indices. It provides examples and exercises on determining image positions, focal lengths, and magnifications. It also addresses the effects of various media on lens behavior and includes practical applications like projectors.
- 2.8: Derivation of the Powers
- Up to this point I have defined what is meant by convergence, and I have defined power as the difference between the final and initial convergences. I asserted without proof formulas for the powers of a lens, a refracting interface, and a mirror. It is now time to derive them.
- 2.9: Derivation of Magnification
- This page covers an optical element that serves as an interface between two media with varying refractive indices, detailing how an image of height \(h'\) is produced from an object of height \(h\) at designated distances \(p\) and \(q\). It presents the magnification relationship through angles as governed by Snell's law, ultimately deriving a formula that incorporates both refractive indices and distances, emphasizing the impact of optical properties on image formation.
- 2.10: Designing an Achromatic Doublet
- A combination of two lenses in contact, a converging lens made of crown glass and a weaker diverging lens made of flint glass, can be designed so that the combination is a converging lens that is almost achromatic.
- 2.11: Thick Lenses
- This page covers the principles of thick lenses, focusing on calculating image positions and understanding optical convergence through various surfaces and refractive indices. It provides examples and complex scenarios for deeper problem-solving.
- 2.12: Principal Planes
- This page outlines the properties of thick lenses and systems of two separated lenses, detailing focal points, principal planes, and calculations for distances based on focal lengths. It explains how to simplify these systems into a single effective lens, including formulas for lens power and chromatic aberration. The page emphasizes that to minimize color variation, the distance between lenses should be half the sum of their focal lengths.
- 2.13: The Lazy Way
- The convergence and power method has great advantages when you have a complex systems of many lenses, mirrors and interfaces in succession. You just add the powers one after the other. But I expect there are some readers who don’t want to be bothered with all of that, and just want to do simple single-lens calculations with a simple formula that they are accustomed to, which is appropriate for the “real is positive” sign convention.
- 2.14: Exercise
- This page presents complex lens exercises focused on calculating radii of curvature and refractive indices for thin and thick lenses in various media, highlighting real and virtual image formation. It covers algebraic manipulations for deriving focal lengths and provides exercises for practical application of concepts like lens configuration and magnification.
Thumbnail: Parallel rays coming into a parabolic mirror are focused at a point F. The vertex is V, and the axis of symmetry passes through V and F. For off-axis reflectors (with just the part of the paraboloid between the points P1 and P3), the receiver is still placed at the focus of the paraboloid, but it does not cast a shadow onto the reflector. (Public Domain).