Engineering Optics Understanding light? Reflection and refraction Geometric optics ( << D): ray tracing, matrix methods Physical optics ( ~ D): wave equation, diffraction, interference Polarization Interaction with resonance transitions Optical devices? Lenses Mirrors Polarizing optics Microscopes and telescopes Lasers Fiber optics ……………and many more

Engineering Optics In mechanical engineering, a primary motivation for studying optics is to learn how to use optical techniques for making measurements. Optical techniques are widely used in many areas of the thermal sciences for measuring system temperatures, velocities, and species on a time- and space-resolved basis. In many cases these non-intrusive optical devices have significant advantages over physical probes that perturb the system that is being studied. Lasers are finding increasing use as machining and manufacturing devices. All types of lasers from continuous-wave lasers to lasers with femtosecond pulse lengths are being used to cut and process materials.

Huygen’s Wavelet Concept Collimated Plane Wave Spherical Wave At time 0, wavefront is defined by line (or curve) AB. Each point on the original wavefront emits a spherical wavelet which propagates at speed c away from the origin. At time t, the new wavefront is defined such that it is tangent to the wavelets from each of the time 0 source points. A ray of light in geometric optics is found by drawing a line from the source point to the tangent point for each wavelet.

Geometric Optics: The Refractive Index The refractive index n: fundamental property of all optical systems, a measure of the effective speed of propagation of light in a medium The optical path length in a medium is the integral of the refractive index and a differential geometric length: a b ds

Fermat’s Principle: Law of Reflection Fermat’s principle: Light rays will travel from point A to point B in a medium along a path that minimizes the time of propagation. Law of reflection: x y (x 1, y 1 ) (0, y 2 ) (x 3, y 3 ) rr ii

Fermat’s Principle: Law of Refraction Law of refraction: x y (x 1, y 1 ) (x 2, 0) (x 3, y 3 ) tt ii A nini ntnt

Imaging by an Optical System Light rays are emitted in all directions or reflected diffusely from an object point. Spherical wavefronts diverge from the object point. These light rays enter the (imaging) optical system and they all pass through the image point. The spherical wavefronts converge on a real image point. The optical path length for all rays between the real object and real image is the same. Later we will discuss scattering, aberrations (a geometric optics concept), and diffraction (a physical optics concept) which cause image degradation.

Imaging by Cartesian Surfaces Consider imaging of object point O by the Cartesian surface . The optical path length for any path from Point O to the image Point I must be the same by Fermat’s principle. The Cartesian or perfect imaging surface is a paraboloid in three dimensions. Usually, though, lenses have spherical surfaces because they are much easier to manufacture.

Reflection at Spherical Surfaces I Use paraxial or small-angle approximation for analysis of optical systems: Reflection from a spherical convex surface gives rise to a virtual image. Rays appear to emanate from point I behind the spherical reflector.

Reflection at Spherical Surfaces II Considering Triangle OPC and then Triangle OPI we obtain: Combining these relations we obtain: Again using the small angle approximation:

Reflection at Spherical Surfaces III Now find the image distance s' in terms of the object distance s and mirror radius R: At this point the sign convention in the book is changed and the imaging equation becomes: The following rules must be followed in using this equation: 1. Assume that light propagates from left to right. Image distance s is positive when point O is to the left of point V. 2. Image distance s' is positive when I is to the left of V (real image) and negative when to the right of V (virtual image). 3. Mirror radius of curvature R is positive for C to the right of V (convex), negative for C to left of V (concave).

Reflection at Spherical Surfaces IV The focal length f of the spherical mirror surface is defined as –R/2, where R is the radius of curvature of the mirror. In accordance with the sign convention of the previous page, f > 0 for a concave mirror and f < 0 for a convex mirror. The imaging equation for the spherical mirror can be rewritten as

Reflection at Spherical Surfaces V CFOI I' O' Ray 1: Enters from O' through C, leaves along same path Ray 2: Enters from O' through F, leaves parallel to optical axis Ray 3: Enters through O' parallel to optical axis, leaves along line through F and intersection of ray with mirror surface V

Reflection at Spherical Surfaces VI C FO I I' O' V

Reflection at Spherical Surfaces VII Real, Inverted Image Virtual Image, Not Inverted

Geometrical Optics Index of refraction for transparent optical materials Refraction by spherical surfaces The thin lens approximation Imaging by thin lenses Magnification factors for thin lenses Two-lens systems

Optical Materials Source: Catalog, CVI Laser Optics and Coatings.

Optical Materials Source: Catalog, CVI Laser Optics and Coatings.

Optical Materials Source: Catalog, CVI Laser Optics and Coatings.

Optical Materials Source: Catalog, CVI Laser Optics and Coatings.

Refractive Index of Optical Materials Source: Catalog, CVI Laser Optics and Coatings.

Refractive Index of Optical Materials Source: Catalog, CVI Laser Optics and Coatings.

Refraction by Spherical Surfaces n 2 > n 1 At point P we apply the law of refraction to obtain Using the small angle approximation we obtain Substituting for the angles  1 and  2 we obtain Neglecting the distance QV and writing tangents for the angles gives

Refraction by Spherical Surfaces II n 2 > n 1 Rearranging the equation we obtain Using the same sign convention as for mirrors we obtain

Refraction at Spherical Surfaces III CO I I' O' V 11 22

The Thin Lens Equation I O O' t C2C2 C1C1 n1n1 n1n1 n2n2 s1s1 s' 1 V1V1 V2V2 For surface 1:

The Thin Lens Equation II For surface 1: For surface 2: Object for surface 2 is virtual, with s 2 given by: For a thin lens: Substituting this expression we obtain:

The Thin Lens Equation III Simplifying this expression we obtain: For the thin lens: The focal length for the thin lens is found by setting s = ∞:

The Thin Lens Equation IV In terms of the focal length f the thin lens equation becomes: The focal length of a thin lens is >0 for a convex lens and <0 a concave lens.

Image Formation by Thin Lenses Convex Lens Concave Lens

Image Formation by Convex Lens Convex Lens, focal length = 5 cm: F F hoho hihi RI

Image Formation by Concave Lens Concave Lens, focal length = -5 cm: F F hoho hihi VI

Image Formation Summary Table

Image Formation Summary Figure

Image Formation: Two-Lens System I 60 cm

Image Formation: Two-Lens System II 7 cm

Matrix Methods Development of systematic methods of analyzing optical systems with numerous elements Matrices developed in the paraxial (small angle) approximation Matrices for analyzing the translation, refraction, and reflection of optical rays Matrices for thick and thin lenses Matrices for optical systems Meaning of the matrix elements for the optical system matrix Focal planes (points), principal planes (points), and nodal planes (points) for optical systems Matrix analysis of optical systems

Translation Matrix

Refraction Matrix

Reflection Matrix

Thick Lens Matrix I

Thick Lens Matrix II

Thin Lens Matrix The thin lens matrix is found by setting t = 0: nLnL

Summary of Matrix Methods

System Ray-Transfer Matrix Introduction to Matrix Methods in Optics, A. Gerrard and J. M. Burch

System Ray-Transfer Matrix Any paraxial optical system, no matter how complicated, can be represented by a 2x2 optical matrix. This matrix M is usually denoted A useful property of this matrix is that where n 0 and n f are the refractive indices of the initial and final media of the optical system. Usually, the medium will be air on both sides of the optical system and

Summary of Matrix Methods

System Ray-Transfer Matrix The matrix elements of the system matrix can be analyzed to determine the cardinal points and planes of an optical system. Let’s examine the implications when any of the four elements of the system matrix is equal to zero.

System Ray-Transfer Matrix Let’s see what happens when D = 0. When D = 0, the input plane for the optical system is the input focal plane.

Two-Lens System f 1 = +50 mm f 2 = +30 mm q = 100 mm r s Input Plane Output Plane F1F1 F2F2 F1F1 F2F2 T1T1 R1R1 R2R2 T3T3 T2T2

Two-Lens System f 1 = +50 mm f 2 = +30 mm q = 100 mm r s Input Plane Output Plane F1F1 F2F2 F1F1 F2F2 T1T1 R1R1 R2R2 T3T3 T2T2

Two-Lens System: Input Focal Plane f 1 = +50 mm f 2 = +30 mm q = 100 mm r s Input Plane Output Plane F1F1 F2F2 F1F1 F2F2 T1T1 R1R1 R2R2 T3T3 T2T2

System Ray-Transfer Matrix Let’s see what happens when A = 0. When A = 0, the output plane for the optical system is the output focal plane.

Two-Lens System: Output Focal Plane f 1 = +50 mm f 2 = +30 mm q = 100 mm r s Input Plane Output Plane F1F1 F2F2 F1F1 F2F2 T1T1 R1R1 R2R2 T3T3 T2T2

System Ray-Transfer Matrix Let’s see what happens when C = 0. When C = 0, collimated light at the input plane is collimated light at the exit plane but the angle with the optical axis is different. This is a telescopic arrangement, with a magnification of D =  f /  0.

f 1 = +50 mm f 2 = +30 mm q = 80 mm r s Input Plane Output Plane F1F1 F2F2 F1F1 F2F2 T1T1 R1R1 R2R2 T3T3 T2T2 Telescopic Two-Lens System

System Ray-Transfer Matrix Let’s see what happens when B = 0. When B = 0, the input and output planes are object and image planes, respectively, and the transverse magnification of the system m = A.

Two-Lens System: Imaging Planes f 1 = +50 mm f 2 = +30 mm q = 100 mm r s Object Plane Image Plane F1F1 F2F2 F1F1 F2F2 T1T1 R1R1 R2R2 T3T3 T2T2

Cardinal Points (Planes) of an Optical System Distances measured to the right of the respective reference plane are positive, distances measured to the left are negative. As shown: p 0 f 1 0 r > 0 s < 0 v > 0 w < 0

Cardinal Points (Planes) of an Optical System

Thick Lens Analysis R 1 = +30 mm R 2 = +45 mm RP 1 RP 2 V1V1 V2V2 t = 50 mm n L = 1.8 n 0 = 1.0 Find for the lens: (a)Principal Points (b)Focal Points (c)Focal Length (d)Nodal Points

Thick Lens Analysis In Lecture 4 we found the matrix for a thick lens with the same refractive index on either side of the lens

Thick Lens Analysis

R 1 = +30 mm R 2 = +45 mm RP 1 RP 2 V1V1 V2V2 t = 50 mm n L = 1.8 n 0 = 1.0 PP 1 F1F1 F2F2 PP 2 H1H1 H2H2

Thick Lens Analysis R 1 = +30 mm R 2 = +45 mm RP 1 RP 2 t = 50 mm PP 1 F1F1 F2F2 PP 2 H1H1 H2H2 s i = mm s o = -95 mm In general, for any optical system: