Geometrical Optics Revision

Overview §

• The Nature of Light and the Principles of Ray Optics
  • The Nature of Light
  • Measurements of the Speed of Light
  • The Ray Approximation in Ray Optics
  • Analysis Model: Wave Under Reflection
  • Huygens’ Principle
  • Dispersion
  • Total Internal Reflection
• Image Formation
  • Images Formed by Flat Mirrors
  • Images Formed by Spherical Mirrors
  • Images Formed by Refraction
  • Images Formed by Thin Lenses
  • Lens Aberrations
  • The Camera
  • The Eye
  • The Simple Magnifier
  • The Compound Microscope
  • The Telescope
• Wave Optics
  • Young’s Double-Slit Experiment
  • Analysis Model: Waves in Interference
  • Intensity Distribution of the Double-Slit Interference Pattern
  • Change of Phase Due to Reflection
  • Interference in Thin Films
  • The Michelson Interferometer
• Diffraction Patterns and Polarisation
  • Introduction to Diffraction Patterns
  • Diffraction Patterns from Narrow Slits
  • Resolution of Single-Slit and Circular Apertures
  • The Diffraction Grating
  • Diffraction of X-Rays by Crystals
  • Polarisation of Light Waves

Equations §

The Nature of Light and the Principles of Ray Optics §

Law of reflection: ${\theta }^{\prime }=\theta$. The angle of the incidence ray is equal to the angle of the reflected ray, on a perfectly flat surface.

Index of refraction: $n=\frac{c}{v}:c=3.00\times 10^8$ . Alternatively this can be expressed in terms of wavelength, as they’re proportional. The index of refraction is equal to the ratio between the speed of light in a vacuum, and its speed in a given medium.

Snell’s law: $n_1\sin {\theta }_1=n_2\sin {\theta }_2$. The ratio between refractive indices is equal to the reciprocal of the ratio between the sine value at their angles of incidence.

Critical angle: $\sin {\theta }_c=\frac{n_2}{n_1}(\text{for }{n}_1>n_2)$. The angle at which the refracted ray travels parallel to the boundary, perpendicular from the normal.

Image Formation §

Lateral magnification: $M=\frac{h^{\prime }}{h}$. The ratio between the height of the image and the height of the object. For a plane mirror, this value is always equal to 1.

Lateral magnification of a spherical mirror: $M=-\frac{q}{p}$. The ratio between the image distance and the object distance. A negative value indicates an inverted image.

Focal length of a spherical mirror: $f=\frac{R}{2}$. The distance from the vertex of the mirror to the focal point, F, located midway between the centre of curvature and the vertex of the mirror. For a concave mirror, the focal point is in front of the mirror and f is positive. For a convex mirror, the focal point is back of the mirror and f is negative. R represents the radius of curvature.

Thin lens equation: $\frac{1}{p}+\frac{1}{q}=\frac{1}{f}$. Used to locate the position of an image formed by reflection of paraxial rays.

Lens-Maker’s equation: $\frac{1}{f}=(n-1)(\frac{1}{R_1}-\frac{1}{R_2})$. An equation relating focal length to the physical properties of a lens. If the lens is surrounded by a medium other than air, the index of refraction given in the equation must be the ratio of the index of refraction of the lens to that of the surrounding medium. A hollow convex lens (“air lens”), if immersed in water, would have a negative focal length.

Thin lens combinations: $\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2},M=M_1{M}_2$. If two thin lenses are in contact, then their new focal point is the sum of the two, and the magnification is the product.

Power of a lens (in dioptres, D): $P=\frac{1}{f}$. The power of a lens is the reciprocal of the focal length, in meters, where the correct sign convention is used. This equation is usually only used for the eye.

Sign Conventions §

For spherical mirrors,

$p,q$: Positive if in front of of the mirror (real), negative if behind (virtual). $f,R$: Positive if the centre of curvature is in front (concave), negative if behind (convex). $M$: Positive if upright, negative if inverted.

For refracting surfaces and thin lenses, the same theoretical conventions are used: positive in the direction of the ray travel and areas of thickness/convergence, otherwise negative.

Wave Optics §

Geometrical Optics Revision 2024-01-09 15.32.08.excalidraw

Concepts §

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Review Checklist §

Describe the methods used by Roemer and Fizeau for the measurement of $c$ and make calculations using sets of typical values for the quantities involved. (Section 34.1)

Make calculations using the law of reflection and the law of refraction (Snell’s law). (Sections 34.3 and 34.4)

Calculate the angle of deviation and the angular dispersion in a prism. (Section 34.6)

Understand the conditions under which total internal reflection can occur in a medium and determine the critical angle for a given pair of adjacent media. (Section 34.7)

Correctly use the required equations and associated sign conventions to calculate the location of the image of a specified object as formed by a plane mirror, spherical mirror, plane refracting surface, spherical refracting surface, thin lens, or a combination of two or more of these devices. Determine the magnification and character of the image (real or virtual, upright or inverted, enlarged or diminished) in each case. (Sections 35.1, 35.2, 35.3, and 35.4)

Construct ray diagrams to determine the location and nature of the image of a given object when the geometrical characteristics of the optical device (lens or mirror) are known. (Sections 35.2 and 35.4)

Make calculations of magnification for a simple magnifier, compound microscope, and refracting telescope. (Section 35.6)

Describe Young’s double-slit experiment to demonstrate the wave nature of light. Account for the phase difference between light waves from the two sources as they arrive at a given point on the screen. State the conditions for constructive and destructive interference in terms of each of the following: path difference $\delta$, phase difference, distance from the centre of the screen $y$, and angle subtended by the observation point at the source mid-point $\theta$. (Sections 36.1 and 36.2)

Calculate the ratio of average intensities for different points in a double-slit interference pattern. (Section 36.3)

Recall that in general, an electromagnetic wave undergoes a phase change of upon reflection from a medium that has a higher index of refraction than the one in which the wave is traveling.

State the conditions, and write the corresponding equations, for constructive and destructive interference in thin films considering both path difference and any expected phase changes due to reflection. Calculate the minimum film thickness to produce constructive/destructive interference in a film between media of known indices of refraction. (Section 36.5)

Determine the positions of the maxima and minima in a single-slit diffraction pattern and calculate the intensities of the secondary maxima relative to the intensity of the central maximum. (Sections 37.1 and 37.2)

Calculate the intensities of interference maxima due to a double slit, expressed as a fraction of the intensity at the center of the pattern. (Sections 37.2)

Determine whether or not two sources under a given set of conditions are resolvable as defined by Rayleigh’s criterion. (Sections 37.3)

Determine the positions of the principal maxima in the diffraction pattern of a diffraction grating. (Sections 37.4)

Describe the technique of x-ray diffraction and make calculations of the lattice spacing using Bragg’s law. (Sections 37.5)

Describe how the state of polarization of a light beam can be determined by use of a polarizer-analyzer combination. Describe qualitatively the polarization of light by selective absorption, reflection, scattering, and double refraction. Make appropriate calculations using Malus’s law and Brewster’s law. (Sections 37.6)