7. step- along calculations and ray tracing

vergence 

given in dioptres and calculated by dividing the refractive index of the medium in which the pensic s travelling by the distance to or from the focal point 

light travelling left to right through air 

rays passing through a medium other than air

no longer a reciporcal

refractive distance of material/ distance

at 1.0 metre, have more dioptres, and reduces as moves further from source

what do diverging rays look like 

light travels left to right 

light diverging away from a source of light or focal point has negative vergence 

what do converging rays look like 

light travels from left to right 

light converges towards a focal point has positive vergence 

converging & diverging in air

positive dioptric powers of increasig numbers until reaches focal point where it is infinity

past the point- light diverges, dioptres are negative

converging and diverging through opthalmic flint glass with a refractive index of 1.654

higher refractive index than air

value/ distance gives dioptric value

moving left to right, more dioptres until go through infinity

then goes to minus

calculating vergence at a transition between 2 media of different refarctive index

interface: boundary between 1 medium and the other 

light source in air 40cm away from interface 

light passes through air and vergence when hits the surface = 1/-0.4 = -2.50D 

eg of light converging towards focal point but meeting an interface before focal point is reached

calculating vergence when interface has dioptric power

can work out radius of curvature: ( n’-n)/F

incident vergence AT interface will be 1/-0.2 = -5.00D

vergence of light AFTER surface: -5.00 + 10D = +5D

add the surface power to the incident vergence at the interface

because refractive index is not in air anymore its 1.523

so focal point is formed at : 1.523 /+5 = 0.305m

what happens if something else gets in the way of light before it reaches focal point 

light hits front surface of lens- diverges 

converges as hits spectacle lens so positive

before focus, it hits the back surface of the lens 

need to work out vergeance at this point 

at the dashed point: light converging, wavefront has plus power , represents where the rear surface of the lens is 

1.523/ 0.120 = +12.69D 

if the interface was behind the focal point

dashed line is behind focal point

light at that point is diverging- negative power

1.523/-0.150 = -10.15D

thick lens in air - ray trace to determine back vertex focal length 1

input numbers into BVP equation

or ray trace and check using formula

flat form lens- biconvex lens

1. light travelling from a source at infinity so has 0D when hits front surface 

2. need to determine vergence of light immediately after front surface 

3. to do this, 0D + 8 = 8D 

4. light converges to F2, so dioptric power goes up 

5. imagine the light after converging carries on throughout - dashed lines , to work out how far it needs to travel to hit focal point 

6. 1.523/+8 = 0.1904m :rays would form a focus at this distance if the lens was really thick ( this is 16 lens thicknesses ), but it is not its 0.012

7. to work f2 vergence, subtract thickness of lens from the back focal length of F1 - 1.523/8 = 0.1904m 

8. so F2  incident vergence - 1.523 / (0.1904-0.012) = +8.537 D #

9. this shows the vergence as it just hits F2 

how to work out vergence immediatly after it hits the second surface 2

all you need to do is add the surface power of F2 to the vergence AT the lens

so at the lens it was +8.537D

surface power of F2 = 7D

= 8.537 +7 = 15.537D

Can now work out where light will form a focus 

back in AIR 

1/15.537 = 0.644m from the rear surface is where we form a focus BVP = +15.537 can check its rouphly right by thin lens equation = F1+F2 = +7+8 = 15 

Can use the BVP formula

consider the form of this lens : ray trace through to work out BVP 1

meniscus form

thin lens equation would indicate power of the lens would be -5 D

1. light hitting F1 at infinity is 0D

2. vergence leaving F1 would be 0+4 = +4D

3. As F2 is only 0.012m away from F1, the rays will be intercepted by F2 before it reaches focus

4. the rays are convergent so vergence at F2 will have greater positive power

5. F1= 1.523 / 4 = 0.3808m ; back focal length of F1

6. vergence at F2 calculated by subtracting thickness of lens from back focal length of F1

7. 1.523 /( 0.3808-0.012) = +4.130D

8. vergence leaving F2 diverging so negative number , add on -9 to the vergence at F2

9. +4.13-9 = -4.870 D after the rear surface

10. as its a minus lens, trace rays back to the focal point

11. 1/ -4.870 = -0.205m to the left of surface F2

can check using the BVP formula 2 

ray tracing through a focimeter

used to show why a low power lens can make the image of the target in a focimeter really blurred

this is because focimeter contains an astronomical telescope- from this we get vergence amplification

this means if you got a small error in vergence of light incoming telescope, when it passes through telescope, the error is made bigger so the blurr is made more prominent

standrad lens distance from lens rest is the focal length of standard lens 

target at the focal point of standard lens so 0 vergence light goes into telescope 

converges, then diverges once gone through the focal point 

rendered paralell when enters eye 

what is the astronomical telescope

it has 2 positive lenses that are separated by the sum of their focal lengths and that the lens closest to the eye is the highest power (hsortest focal length) of the 2 lenses

vergence of light as it moves through objective lens

objective lens +25 D and eyepiece lens +100D

target 40mm 1/-0.04 = -25D

as standrad lens is +25D vergence here is +25-25 = 0D

as it hits objective lens it converges at the +25D objective lens and forms a focus 0.04m from the objective lens

light now diverges from this point, but at 1cm from this point it hits the eyepiece lens

vergence at eyepiece - 1/-0.01 = -100D

as focal length of eyepeice lens is +100 D

vergence immediatly after the eyepiece les is -100 +100 = 0D

the observer will see a clear image of the target

ray tracing through focimeter when a lens is put in place 1 eg 

target hasnt moved

vergence from target to stabdard lens is 1/-0.04 = -25 D

vergence after standard lens is 0 as standard lens was +25D

as it hits the -1.00DS lens vergence is now 0+-1= -1D so diverges

0.05 m distance between objective lens and -1.00DS lens

vergence immediatly after the lens is -1.00DS , that is then going to pass 0.05m until it hits front surface of astronomical telescope

to work out vergence of light just before it hits the front surface of astronomical telescope:

1/-1 = -1 m

-1 -0.05 = -1.05 m

1/ -1.05m = -0.95 D

vergence after hitting objective lens: -0.95 +25 = +24.05 D

forms a focus at: 1/24.05 = 0.0416m

as distance between the plus lenses are 0.05 m apart, distance from focus to +100D lens = 0.05 - 0.0416 = 0.0084m

so vergence at the eyepiece lens is 1/-0.0084 = -119 D

what is vergence after the 100D lens 2

so vergence at the eyepiece was - 119 D

after lens = -119 +100 = -19 D

assuming a distance of 10mm between the 100D lens and the eye( 10 mm gap between your cornea and the lens , the vergence at the eye will be: 1/-19 = -0.0526 m

-0.0526 -0.01 = -0.0626m

1/-0.0626 = -15.97D

observer will see a very blurred image of the target

eg when plus 1.00 lens placed on focimeter

light converges from the lens until pbjective lens with a distance of 0.05m apart

vergence before hitting the objective lens

1/1 = 1m

1-0.05 = 0.95m

1/0.95 = +1.05 D

vergence after hitting objective lens = 0.95+25 = 26.05

forms a focus at 1/26.05 = 0.0384m 

vergence at eyepiece lens: distance from focus to +100 D len = 0.05 -0.0384 = 0.0116m 

so vergence at 100 D lens 1/- 0.0116  = -86.2 D 

vergence after 100 D lens : -86.2 +100 = +13.79 D 

1/13,79 = +0.0725 m 

0.0725 -0.01 ( countering in distance between cornea and lens)  = 0.0625 

1/0.0625 

+16D 

will see bluured image 

an object of sie 20mm is placed 10cm to the left of a +20.00DS lens. What is the size of the image?

object size :