Waves & Oscillations

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1 Physics Waves & Oscillations Lecture 27 Geometric Optics Spring 205 Semester Matthew Jones

2 Sign Conventions > + = Convex surface: is positive for objects on the incident-light side is positive for images on the refracted-light side is positive if is on the refracted-light side

3 Sign Conventions > + = (same formula) Concave surface: is positive for objects on the incident-light side is negative for images on the incident-light side is negative if is on the incident-light side

4 Magnification Using these sign conventions, the magnification is = Ratio of image height to object height Sign indicates whether the image is inverted

5 Thin Lenses The previous examples were for one spherical surface. Two spherical surfaces make a thin lens Thinner in the middle Thicker in the middle

6 Thin Lens Classification A flat surface corresponds to All possible combinations of two surfaces: (positive) (negative) R <0 R 2 = R <0 R 2 >0 R >0 R 2 >0

7 Thin Lens Equation First surface: Second surface: Add these equations and simplify using =and 0: (Thin lens equation)

8 Gaussian Lens Formula Recall that the focal point was the place to which parallel rays were made to converge Parallel rays from the object correspond to and : = This lens equation: + = =

9 Gaussian Lens Formula 0 Gaussian lens formula: + = Newtonian form: = (follows from the Gaussian formula after about 5 lines of algebra) All you need to know about a lens is its focal length

10 Example =.5 =50 mm Plano-convex spherical lens What is the focal length of this lens? Let, then = The flat surface has and we know that = 50mm =.5 50 = 00 =

11 Example f= 00 mm Objects are placed at =600 mm,200 mm,50 mm,00 mm,80 mm Where are their images? =20 mm,200 mm,300 mm,, 400 mm

12 Focal Plane Thin lens + paraxial approximation: All rays that pass through the center,, do not bend All rays converge to points in the focal plane (back focal plane) lies in the front focal plane

13 Imaging with a Thin Lens For each point on the object we can draw three rays:. A ray straight through the center of the lens 2. A ray parallel to the central axis, then through the image focal point 3. A ray through the object focal point, then parallel to the central axis.

14 Converging Lens: Principal Rays Object F o F i Optical axis Image Principal rays: )Rays parallelto principal axis pass through focal point F i. 2) Rays through center of lens are not refracted. 3)Rays through F o emerge parallel to principal axis. Assumptions: Monochromatic light Thin lens In this case image is real, inverted and enlarged Paraxial rays (near the optical axis) Since n is function of λ, in reality each color has different focal point: chromatic aberration.contrast to mirrors: angle of incidence/reflection not a function of λ

15 Diverging Lens: Forming Image F o O.A. Object F i Image Principal rays: Assumptions: paraxial monochromatic rays thin lens )Rays parallelto principal axis appear to come from focal point F i. 2) Rays through center of lens are not refracted. 3)Rays toward F o emerge parallel to principal axis. Image is virtual, upright and reduced.

16 Converging Lens: Examples s o > 2F This could be used in a camera. Big object on small film F < s o < 2F This could be used as a projector. Small slide(object) on big screen (image) This is a magnifying glass 0 < s i < F

17 Lens Magnification s o y o F i optical axis Image Object F o y i s o + s i = f s i Green and blue triangles are similar: Example: f=0 cm, s o =5 cm = 5cm si 0cm + s i = 30 cm Magnification equation: M T y y i o si = s T= transverse M 30cm = T 5 cm o = 2

18 Longitudinal Magnification s o + s i = f The 3D image of the horse is distorted: transverse magnification changes along optical axis longitudinal magnification is not linear Longitudinal magnification: M L dx dx i o = f x 2 2 o = M 2 T Negative: a horse looking towards the lens forms an image that looks away from the lens x x f 2 2 o i = x i f / xo dx dx = ( 2 ) ( 2 2 / ) i = f x = f x o d dx o / o o

19 Two Lens Systems Calculate using = + Ignore the first lens, treat as the object distance for the second lens. Calculate using = + Overall magnification: = =

20 Example: Two Lens System An object is placed in front of two thin symmetrical coaxial lenses (lens & lens 2) with focal lengths f =+24 cm & f 2 =+9.0 cm, with a lens separation of L=0.0 cm. The object is 6.0 cm from lens. Where is the image of the object?

21 Example: Two Lens System An object is placed in front of two thin symmetrical coaxial lenses (lens & lens 2) with focal lengths f =+24 cm & f 2 =+9.0 cm, with a lens separation of L=0.0 cm. The object is 6.0 cm from lens. Where is the image of the object? (not really to scale )

22 Example: Two Lens System An object is placed in front of two thin symmetrical coaxial lenses (lens & lens 2) with focal lengths f =+24 cm & f 2 =+9.0 cm, with a lens separation of L=0.0 cm. The object is 6.0 cm from lens. Where is the image of the object? Lens : = + = 8 Image is virtual. Lens 2: Treat image as O 2 for lens 2. O 2 is outside the focal point of lens 2. So, image 2 will be real & inverted on the other side of lens 2. = = + =8.0 Image 2 is real. Magnification: = =.33

Converging Lenses. Parallel rays are brought to a focus by a converging lens (one that is thicker in the center than it is at the edge).

Converging Lenses. Parallel rays are brought to a focus by a converging lens (one that is thicker in the center than it is at the edge). Chapter 30: Lenses Types of Lenses Piece of glass or transparent material that bends parallel rays of light so they cross and form an image Two types: Converging Diverging Converging Lenses Parallel rays