Lenses & Prism Consider light entering a prism At the plane surface perpendicular light is unrefracted Moving from the glass to the slope side light

Transcription

1 Lenses & Prism Consider light entering a prism At the plane surace perpendicular light is unreracted Moving rom the glass to the slope side light is bent away rom the normal o the slope Using Snell's law n sin( ϕ ) = n sin( ϕ ) o sin( ϕ ) =. 75sin( 30 ) = ϕ = arcsin( ) = 6 o

2 Prisms & Index o Reraction with Wavelength Dierent wavelengths have dierent index o reraction Index change is what makes prism colour spectrum Generally higher index at shorter wavelengths Most eect i use both sides to get max deviation & long distance Angle change is ~ only ratio o index change -2% Eg BSC glass red.5, violet.5, assume light leaves at 30 o Red φ R = arcsin [.5 sin(60)] = o Violet φ v = arcsin [.5 sin(60)] = o This 0.43 o dierence spreads spectrum 7.6 mm at m distance

3 Lens Lens is like a series o prisms Straight through at the centre Sharper wedge angles urther out More ocusing urther out Snell s law applied to get the lens operation

4 Why is Light Focus by a Lens Why does all the light ocus by a lens Consider a curved glass surace with index n on right side Radius o curvature r is centered at C Let parallel light ray P at height h rom axis hit the curvature at T Normal at T is through C orming angle φ to parallel beam Beam is reracted by Snell s law to angle φ to the normal n sin( ϕ ) = n sin( ϕ ) Assuming small angles then sin(φ)~φ and n n sin( ϕ ) = sin( ϕ ) or ϕ ϕ n n From geometry or small angles h h sin( φ ) = or φ r r Angle θ the beam makes to the axis is by geometry n n n h n n θ = φ φ = φ φ = φ n n r n Thus the ocus point is located at = h sin h h r h nr ( θ ) θ n n n n Thus all light is ocused at same point independent o h position n

5 Focal Points Two ocal points depending on surace & where light comes rom Primary Focal Points are Convex (a) where diverge beam orms parallel light Concave surace (b) where light appears to converge when it is converted into a parallel beam Secondary Focal Points Convex (c) where parallel beam is ocused Concave surace (d) where parallel light coming in appears to diverge rom.

6 Types o Lenses Convex (a) Biconvex or equiconvex (b) Planoconvex (c) positive meniscus Concave (d) biconcave or equiconve (e) Planoconcave () negative meniscus Primary and secondary ocal points very dependent on type Planoconvex/Panloconcave easiest to make Two surace lenses about twice the price

7 Fresnel Lens Lens with thickness remove Cheaper, but can be lower quality Reason: diraction eects at step boundries

8 Lens Conventions From Jenkins & White: Fundamentals o Optics, pg 50 Incident rays travel let to right Object distance s + i let to vertex, - i right to vertex Image distance s' + i right to vertex, - i let to vertex Focal length measured rom ocal point to vertex positive or converging, negative or diverging lens I center o curvature C is to right o vertex r positive C to let o vertex r negative Thus or curved surace on object side o vertex r positive or convex suraces r negative or concave Note radius sign reverses or surace on right side o vertex Object and Image dimension + i up, - i down rom axis

9 Gaussian Formula or a Spherical Surace The radius o curvature r controls the ocus Gaussian Lens ormula n s n + s n n = r where n index on medium o light origin n index on medium entered r = radius o curvature o surace Clearly or s' ininite (parallel light output) then s = (primary ocal length) n s n + = nr = n n Same as the previous calculations n n n = r

10 Thin Lens Assume that thickness is very small compared to s, s' distances This is oten true or large ocal length lenses Primary ocus on let convex lens, right concave Secondary ocus on right convex, let concave I same medium on both sides then thin lens approximation is =

11 Basic Thin Lens ormula Basic Thin Lens ormula Lens Maker's ormula s + = s = r r ( n ) 2

12 Magniication and Thin Lenses positive or convex, negative or concave Magniication o a lens is given by s m = = s = s s Magniication is negative or convex, positive or concave

13 Simple Lens Example Consider a glass (n=.5) plano-convex lens radius r = 0 cm By the Lens Maker's ormula 0.5 = ( n ) = (.5 ) = = r r2 0 0 = = cm Now consider a cm candle at s = 60 cm rom the vertex Where is the image + = s s = s = s = s' = = 30 cm M ' s 30 Magniication m = = = = 0. 5 M s 60 Image at 30 cm other side o lens inverted and hal object size What i candle is at 40 cm (twice ) s = = = 0.05 s' = = 40 cm m = = s s s = Image is at 40 cm other side o lens inverted and same size ( cm)

14 Lens with Object Closer than Focus Now place candle at 0 cm (s < condition) = = = 0.05 s s 20 0 s 20 m = = = 2 s 0 s' = 0.05 = 20 cm Now image is on same side o lens at 20 cm (ocal point) Image is virtual, erect and 2x object size Virtual image means light appears to come rom it

15 Graphic Method o Solving Lens Optics Graphic method is why this is called Geometric Optics Use some scale (graph paper good) Place lens on axis line and mark radius C & ocal F points Draw line rom object top Q to mirror parallel to axis (ray 4) Hits vertex line at T Then direct ray rom T through ocus point F and beyond Because parallel light rom object is ocused at Now direct ray rom object top Q through lens center (ray 5) This intersects ray 4 at image Q (point 7) This correctly shows both position and magniication o object This really shows how the light rays are travelling Eg Ray through the ocal point F (ray 6) becomes parallel Intersects ray 5 again at image Q

16 Thin Lens Principal Points Object and image distances are measured rom the Principal Points Principal point H Location depends on the lens shape H also depends on a thin lens orientation Note i you reverse a lens it oten does not ocus at the same point Need to look at lens speciications or principal points Thick lenses have separate Principal points

17 Thick Lens Formula As lens gets thicker optical suraces may be not meet Lens thickness t c (between vertex at the optical axis i.e. centre) Now lens ormula much more complicated Distances measured relative to the principal points H or light coming rom the ront H or light coming rom the back o lens ( n ) tc = ( n ) + r r2 n r r2 Note simple lens ormula assumes t c = 0 which is never true But i is large then r s large and t c is small so good approximation Note plano-convex r 2 = and thin = thick but principal point changes 2

18 Very Thick Lenses Now primary and secondary principal points very dierent A = ront vertex (optical axis intercept o ront surace) H = primary (ront) principal point A 2 = back vertex (optical axis intercept o back surace) H = secondary (back) principal point t c = centre thickness: separation between vertex at optic axis Relative to the ront surace the primary principal point is n A H = tc r2 Relative to the back surace the secondary principal point is n A 2 H = tc r el eective ocal length (EFL): usually dierent or ront and back

19 Numerical Aperture (NA) NA is the sine o the angle the largest ray a parallel beam makes when ocused NA = ( ) sin θ = where θ = angle o the ocused beam φ = diameter o the lens NA < are common High NA lenses are aster lenses NA is related to the F# F# = 2NA φ 2

20 Combining Lenses Can combine lenses to give Combination Eective Focal Length e I many thin lenses in contact then = + + e Two lenses and 2 separated by distance d To completely replace two lens or all calculations New image distance or object at ininity (eg laser beam) e = + 2 d 2 2 or 3 e L Distance rom irst lens primary principal point to combined lens primary principal point d D = e 2 = + Distance rom second lens secondary principal point to combined lens secondary principal point d D = Combined "thick lens" extends rom D to D' e 2 2 d

21 Combining Two Lens Elements Combined object distance s e se = s D Combined image distance s' e s e = s 2 D NOTE: Combined object/image distance may change sign The thick lens ollows the standard ormula s e + s Combined magniication s e me = se Secondary ocus distance relative to 2 nd lens vertex is: e == = + e Note some devices (e.g. telescopes) cannot use these ormulas D e