Optics Part 1 Vern Lindberg March 5, 2013 This section of the course deals with geometrical optics refraction, reflection, mirrors, lenses, and aberrations. Physical optics, interference, diffraction, and polarization, will be the subject of a different course. The third branch of optics, sometimes called modern optics, deals with quantum optics. If you are currently in Optics, or took it earlier you have an optics text. Either Pedrotti, Pedrotti, and Pedrotti (P 3 ) or Hecht are good references. I will put some optics books in the PAC. 1 Electromagnetic Spectrum and Radiometry P 2 Chapter 2 Sections 1-3, P 3 Chapter 1 Sections 3-4 When considered as a wave, light consists of coupled electric and magnetic fields. vacuum the light travels at the speed of light In c = 3.00 10 8 m/s (1) Monochromatic (single wavelength) light has a frequency, Greek letter nu ν, or f, and a wavelength λ c = ν λ f λ (2) Different names are given to different wavelength ranges as shown in Figure P 2 1.1 or Figure P 3 1.1 of text. The limits overlap somewhat, but here is a rough guide: Radio waves λ > 1 m Microwaves 1 m > λ > 1 mm Infrared 1 mm > λ > 770 nm Visible 770 nm > λ > 380 nm 1
Ultraviolet 380 nm > λ > 1 nm X-Ray 1 nm > λ > 0.001 nm Gamma Rays 0.01 nm > λ While all of these are electromagnetic waves, they differ in means of production. Identical frequencies may be called X-rays or gamma rays depending on their source. Likewise the different types of e-m radiation interact with materials in different ways. Radio Waves and Microwaves can be produced in a controlled fashion (AC voltage sources, antennae) or a natural fashion. Infrared is most often related to the thermal motion of electrons in an object, but occasionally from molecular transitions. The same is true for visible with the range of colors ROYGBV, with the eye most sensitive to yellow-green, 560 nm. The response of the eye is shown in Figure P 2 2-7 or Figure P 3 19-2. (Figure 1) Figure 1: CIE spectral luminous efficiency of human eye. From http://physics.nist. gov/divisions/div844/facilities/photo/candela/photo.html Ultraviolet light also arises from thermal motion of electrons or atomic transitions. X- rays arise from Bremstrahlung or from transitions in the inner core electrons of an atom. Gamma Rays arise from transitions in the nucleus. Radiometry is the measure of energy content of light, while Photometry applies to the standard human eye response. Two light sources with identical radiometric measurement may appear different to the human eye yellow will appear brighter to our eye than red or blue. Radiometric terms may include a subscript e for electromagnetic (or it may be omitted) and use radiant as an adjective as in radiant flux. Photometric terms will use the same symbols but have a subscript v for visible and use the adjective luminous as in luminous flux. 2
Table 1 defines various terms. Radiometric units are watts for flux, watts/m 2 for irradiance, and watts/sterradian for intensity. Luminous flux has units of lumens, lm. Irradiance has units of lux, lx, and luminous intensity has units of candela, cd. In this course we will not deal with photometry. Radiometric Symbol Defining Equivalent Term Radiometric Units Equation Photometric Units Radiant Energy Q e in J lm s, talbot Radiant Energy Density w e in J/m 3 w e = dq e /dv lm s/m 3 Radiant flux (power) Φ e in W Φ e = dq e /dt lm Exitance/Irradiance E e, M e in W/m 2 dφ e /da lm/m 2, lux Radiant Intensity I e in W/sr I e = dφ e /dω lm/sr, candela Table 1: Radiometric and Photometric Terms: V is volume, A is area, sr is steradian, the unit of solid angle, Ω. Lumen abbreviation is lm. Candela abbreviation is cd. 2 Huygen s, Fermat, Reflection and Refraction P 2 Chapter 3 P 3 Chapter 2 We will deal with smooth surfaces but what defines smooth? A smooth surface will have an rms-roughness much smaller than the wavelength of the light. For visible light a λ/4 smoothness means that the rms roughness is less than 150 nm, while for radio waves used at a radio telescope (450 MHz) the rms roughness for λ/4 smoothness is 7 cm. A flatness of λ/4 is the minimum level required for optical elements, research quality optics are flat to λ/10. Once we have a smooth surface we can characterize any small portion of the surface in terms of the normal to the surface at that point. Consider a flat surface separating two media, neither of which is vacuum. In a medium other than vacuum, the speed of light v will be less than the speed in vacuum, v c. We define the index of refraction, n for a medium so that v = c n (3) The index of refraction is related to the dielectric constant of a material, and is frequency dependent as shown in Figure 2. 3
Figure 2: Dispersion of some glasses. The sharp rise at short wavelengths is because we are approaching a resonant frequency of the glass. Send an incident ray of light toward the interface (boundary) between the media, with the direction given by the angle θ i between the ray and the normal. Some light will be reflected. For a smooth surface, the reflection is called specular (rather than diffuse) and Law of Reflection: The reflected ray remains in the plane defined by incident ray and normal, and makes an equal angle of reflection, θ r. Some light will be transmitted and Snell s Law, The Law of Refraction: The transmitted ray will be in the plane defined by incident ray and normal, and transmit at an angle satisfying n i sin θ i = n t sin θ t (4) 2.1 Huygen s Principle One way to prove these laws is Huygen s Principle. This starts with a wavefront, and says that to find the wavefront after some time t, pick many points on the original wave and draw circles of radius vt. The new wavefront is drawn as the envelope of the circles. This is shown in P 3 and suggested below. 4
Figure 3: Wavefronts emitted by points separated by one wavelength apart. The envelope of the wavefronts is the next wavefront. 2.2 Fermat s Principle Version 1 Hero of Alexandria second century B.C.E. Light travels the shortest path between two points. This can be used to get the Law of Reflection. Figure 4: Minimize the total distance travelled to determine that x = L/2. Version 2 Fermat Light travels in the path between two points that requires the least time. This allows us to get the Law of Refraction. Version 3 Generalized Fermat: The path taken by a ray is such that its optical path is to first order, the same as paths infinitesimally different than the ideal path specifically this leads to the path of maximum or minimum time. This is an application of Variational Calculus. 2.3 Reversability If a ray leaves point A (source) and arrives at point B, then if we make B the source and start the ray in the reverse direction, it must pass through A. 5
3 Plane Surfaces: Reflection, Refraction 3.1 Reflection We can describe a ray by a direction unit vector, r 1 = (x, y, z). If this ray reflects from a mirror lying in the x y plane, i.e. having a normal in the z, the reflected ray will have r 2 = (x, y, z). Application: Corner Reflector. Consider three mirrors arranged perpendicularly as in the corner of a cube, with surface normals in x, y, and z directions. Applying the reflection rule for the three surfaces, we see that for a ray reflecting from all three surfaces, r r = ( x, y, z), and therefore reflects back in exactly the opposite direction as the incoming ray. Retroreflectors are found in tail light covers, and bike reflectors. The Apollo missions to the moon brought reflectors left on the surface. By using laser light reflected from them, the distance to the moon can be measured to a precision of 1 mm! We will discuss image formation by a plane mirror: a virtual image is formed with magnification of 1 and at the same distance behind the mirror as the object is in front. All reflected rays, regardless of angle, when projected back appear to come from the virtual image. 3.2 Refraction, Paraxial Rays In dealing with image formation with refraction, we will go well beyond the idea of thin lenses, but will stay in the paraxial approximation, that is the small angle approximation Snell s Law in the paraxial approximation is sin θ tan θ θ (in rad) (5) n 1 θ 1 = n 2 θ 2 (6) Using the paraxial approximation, we can determine the apparent depth of an object below a flat surface separating media of two different indices of refraction. Suppose an object is located a depth s below the boundary in a medium of index n 1. For an observer in a medium of index n 2 a virtual image appears at a depth of ( ) s n2 = s (7) 6 n 1
Figure 5: Construction to find apparent depth of an object. E. g. A fish appears to be at a depth of 1.0 m below the surface of a lake. Where is it actually? Invert Equation 7 to get s = ( n1 n 2 ) s = ( 1.33 1 ) 1.0 = 1.33 m (8) E. g. An archer fish catches insects by spitting at them. If the insect is 120 cm above the water, what is the apparent location of the insect? The words internal and external have specific meaning in optics. Suppose n 2 > n 1, then events happening in the region of higher index, n 2, are called internal, while events happening in the lower index, n 1, are called external. An unpolarized external ray reaching the boundary will always have some portion reflected, and some portion refracted 1. An unpolarized internal ray reaching the boundary will always have some portion reflected, but refraction occurs only when the incident angle is less than the critical angle. ( ) θ critical = sin 1 n2 (9) n 1 If the angle is larger than the critical angle we will have total internal reflection. 1 In order to determine the relative amplitudes of the reflected and transmitted light you need the Fresnel coefficients that will be developed in E&M 2. For cases of total internal reflection, there exists an exponentially decaying evanescent wave in the lower index material. This is the same mathematics that you saw for a barrier of finite height in Modern Physics. 7
4 Imaging by an Optical System We start with a real object existing in real object space. Rays diverge from the object and some of the rays enter an optical system. The optical system changes the direction of the rays, and for a useful optical system the rays define an image. Two questions arise: What shapes of optical elements will take rays from a single object and form a single image? and How do we find the location of the image, the size of the image, and the nature of the image (real, virtual, erect, inverted, magnification)? For paraxial rays, spherical surfaces are a sufficiently accurate shape. I will not try to justify this. If we go larger other shapes are required, paraboloids of revolution for mirrors. Aspheric lenses are possible, especially if they can be molded. Aspheric lenses can simplify the optical design, but their treatment is well beyond what we are going to cover. Some definitions real object Rays from a point on a real object will diverge, i.e. spread apart as they progress virtual object Rays are converging to a point that is the virtual object. This is useful in systems of more than one lens or mirror. real image Rays converge to a point on the real image. If a piece of paper or film is put at this location, a real image will appear. virtual image Rays are diverging, but appear to come from a single point. optical axis The symmetry axis of a lens or mirror. vertex, V The point on the optical axis where the surface of the mirror or lens is located (one vertex for a mirror, two for a lens.) 5 Spherical Mirror The important terms for a spherical mirror are the optical axis, the vertex, V, the center of curvature, C, and the radius of curvature R. Before we discuss the mirror, we explain the sign convention, the same one as in P 3. 5.1 Sign Convention for Mirrors A real object is located to the left of the optical system so that light from the real object is moving from left to right 8
The radius of curvature, R, is positive when C is to the right of V, a convex surface. Object distance, s o, is positive if the object, O, is to the left of V. This corresponds to a real object. Image distance, s i, is positive when the image produced by the mirror, I, is to the left of V, corresponding to a real image. If the image is to the right of the vertex, no actual rays reach the point so it is a virtual image with a negative distance. Figure 6: Mirror: Object, O, Vertex, V, Image, I, Center of curvature, C, and optical axis. Object and image distances are s o and s i. For this configuration the P3 sign convention is s o, R positive, s i negative. This is a convex mirror. 5.2 Image Formation by a Spherical Mirror With a little geometry, and using the sign convention, we determine 1 s o + 1 s i = 2 R (10) Figure 7: Construction to find the equation for image formation for a mirror. Remember the sign convention. Here R and s o are positive, s i is negative. 9
It is convenient, and common, to define a focal point, F, and a focal length f. If we have rays parallel to the axis, for which s =, upon reflection they will form an image at the focal point. The distance from the vertex to the focal point is f = R 2 and is positive for a concave mirror, negative for a convex mirror. Then 1 + 1 s o s = 1 i f (11) (12) The magnification of the system will be defined so that its magnitude equals the ratio of image height to object height, and so that it is positive if the image and object point the same direction (erect), and negative if the image is inverted relative to the object. Simple geometry shows that m = s i s o (13) A convenient applet to play with is http://pdukes.phys.utb.edu/physapplets/thinlens/ lens&mirror/lensdemo.html. 5.3 Principal Rays for a Mirror Given a mirror of known radius of curvature R and a center of curvature C, we can draw the focal point F at the focal length f. Suppose an object extends from the axis to some distance above (or below) the axis. Many rays leave the tip of the object, but three can be easily drawn, and can be used to verify the computation. 1. Ray parallel to the axis reflects so that it passes through the focal point, or its virtual extension passes through the focal point. 2. Ray passing through the focal point, or heading toward the focal point, reflects parallel to the axis. 3. Ray passing through the center of curvature, or heading towards it is reflected back on itself. 6 Refraction at a Spherical Surface 6.1 Sign Convention We use almost the same sign convention as for mirrors. If the center of curvature is to the right of the vertex, it is positive. Light is assumed to move from left to right. Real objects 10
have positive distances. If the image forms on the right hand side of the surface where light is refracted, it is a real image with a positive image distance. 6.2 Images for a Spherical Refracting Surface For a single spherical surface we again have the vertex, the center of curvature, and the radius of curvature. Assume that the indices are n 1 for material on the left and n 2 for material on the right. The geometry is a little trickier here, and Snell s law must be used, but the result is with magnification n 1 s o + n 2 s i = n 2 n 1 R (14) m = n 1 s i n 2 s o (15) Figure 8: Concave refracting surface separating indices n 1, n 2. We assume a paraxial condition. Sign convention means that s o is positive but s i and R are negative. E.g. Suppose I make a rod of ice with one end being convex with a radius of curvature of 5.0 cm. Assume the index of ice is n = 4/3 = 1.333. (a) For an object located at 30.0 cm in front of the ice (in air) where is the image located? Is the image real or virtual, erect or inverted, and what is its magnification? (b) Repeat for an object located 6.0 cm from the vertex. (c) Repeat for a concave end of same radius and object located at 6.0 cm 11