4: sag, lens form and lens thickness.

sign conventions for lens calculations

when measuring radius , if compass point is placed to the right its positive if left to the surface then its negative.

signs of angles 

using sing angles for snells law

n normally air

n’ refractive index of material

angle of incidence is i

angle of refraction is i’

refractive index

refractive index of medium= velocity of light in vacuum/ velocity of light in medium

vacum is used in definition of asolute refractive index

air at standardised pressure 1 atm and temp 0 degrees is commonly used instead of velocity in vacuum.

velocity of light in vacuum= 299792458

refractive index of air - 1

so velocity of light in ait = 299702547 m/s

light travelling from left to right 

radius of the front surface would have a positive sign and back surface of radius would have a negative surface 

radius of front surface is on the right side so thats why it is positive 

miniscus lens showing light travelling from left to right

both surfaces will have a positive sign as theyre to the right of surface

biconcave lens showing light moving from left to right

front surface the compass points to the left so f1 i snegative

back surface points to the right to the sign will be positive

centre thickness, t 

thickness of the lens in the centre 

surface power (F) of a refracting surface made from a material of refractive index n’, in a medium of refractive index n is given by:

F= (n’ - n ) / r

if r is given in metres then F will be in dioptres

refractive index has no units

light going into lens, going through surface and its the new refractive index that youre moving into minus the refractive index the light has left from

surface power calculation

if it doesnt say, assume the lens is in air

n= air= 1.00

n’ = 1.498

r = +8cm

so

F= (1.498-1.00)/ 0.08

F= +6.23 D

back surface of a lens in air

going to be F= (n-n’) / r

rear surface power, refractive index of air is th new refractive index the light is moving into

n= 1.00

n’= 1.498

r= +6cm

so F= (1.00-1.498) / 0.06

F= -8.30 D 

using both front and rear surface equation

front surface power f1

(n’-n) / r

radius = +8cm

F1= (1.498-1.00)/0.08 = +6.23 D

back surface power F2

(n-n’) / r

radius = +6cm

F2= (1.00-1.498) / 0.06

F2= -8.30 D

overall power of the lens=

thin lens equation = f = f1 + f2

overall lens= 6.23 + - 8.30 = -2.07 D 

need to take into account the thickness of lens 

lens measure- device for measuring surface power

clock with needle that moves around

measures sag, determines radius of curvature from sag across a known chord , and then displays surface power using an assumed refractive index

if put on flat surface- with 0 power the pins will be in direct allignment

if use cinvex surface- pin will be pushed further upwards

if use concave surface pin will be pushed down

what is sag

refers to the height of a curved surface- like a lens or mirror

measured from the centre of the surface to the edge along line perpendicular to optical axis

sag formula 

derived from a² + b² = c²

semi diameter is Y / a 

r is radius / c is hypotenuese

b

s is- apex of curve to the line drawn 

s is the edge thickness worked out from centre thickness 

to find s need to find out b

r- b is going to be s 

rearranging s y and r

a² + b² = c²

Y= a

r= c

r- b = s 

to work out b : c² -a² then square root 

( radius² -  y² ) then square root this for b 

then to work out s = radius - b

rearrange sag for r

r = (y² + s²) / 2s : spherical surface

so the spacing of the outer pins on the lens measure will give 2y. the length of the centre pin will give s

lens measure- underlying maths using opthalmic crown glass

sag told us the radius of the surface, but lens measures are designed to tell us the surface power

most lenses are calibrated for opthalmic crown glas with a refractive index of 1.523

F= (n-1) / r

so if n is 1.523, surface power, F, = 0.523 / r (where radius of surface in metres)

what is we are measuring a lens made from a material other than crown glass

true surface power , F true = F measured x ( n true -1 ) / 0.523

0.523 is the surface power 

n true is the actual refractive index of the lens being measured 

so actual refractive index of lens, n true= 1+ ( ( F true / F measures ) x 0.523 )

what is the actual surface power of a lens made from flint glass (n= 1.654) when the lens measure displays a surface power of + 6.00 D

true surface power , F true = F measured x ((n true - 1) / 0.523

F true= +6.00 x ((1.654 -1.00 ) / 0.523

F treuu = + 6.00 x (0.654/0.523)

F true= + 7.50 D

Where n true is actual refractive index of lens being measured

so actual refractive index of the lens , n true = 1+ (( F true / F measured) x 0.523)

lens is made from higher refractive material than air there is more surface power

true surface power= ( n true - 1 / n calibrated -1 ) measured surface power

if not stated what the refractive index calibrated, it is always going to be 1.523

so it would be x-1 / 1.523-1

what is the refractive index of a lens when a lens measure calibrated for crown glass ( n+ 1.523) displays a surface power of +6.00 D, but actual surface power is +7.50 D

actual refractive index of the lens , n true= 1 + ((F true / F measured) x 0.523 )

n true = 1+ 7.50 / 6.00 × 0.523

n true = 1.654

altenative formula from henson, optometric instrumentation 

surface power = ((n-1) / ( n0 -1 )) x lens measure reading 

where n = actual refractive index of les being measured

n0 is refractive index to which lens measure is calibrated 

what is surface power if n = 1.523

n’ = refractive index of lens

n= refractive index of air = 1.00

radius- radius of curvature

surface power = 1.523 -1.00 / radius

0.523 / r is surface power

thick lens calculations: BVP 

the power of the lens measured with respect to the back vertex ; the reciprocal of the back vertex focal length

example shows a positive lens

correcting myope with a plus lens but will be upside down 

BVP= 1/ bacl vertex focal length in metres 

second principal focus behind the lens 

BVP with a negative lens

second principal focus is behind the lens

for a myopic person if you put the focus point coincident with far point then its corrected

shift the far point to infinity by placig concave lens in front of eye, so it diverges the rays making them appear to originate from the far point

FVP= 1/ front vertex focal length

thick lens calculation : back vertex power

F1= surface power of the front surface of the lens D

F2= surface power of the back surface of the lens D

t= centre thickness of the lens in meters

n’= refractive index of lens material

formula is reducing the front surface by placing F1 in the same plane as F2 , which turns it into the thin lens equation

thin lens calc

power of the surface = F1 + F2 

2 lenses F1= +5.00DS and F2 = +7.00DS are separated by 15cm. what is the equivalent power of this lens combination

Fequivalent= F1+F2 - d x F1 x F2

= ( 5+7) - 0.15 × 5 × 7

=

thick lens; front vertex power 

F1= surface power of the front surface of the lens D

F2= surface power of the back surface of the lens 

t= the centre thickness of the lens in metres 

n’ = the refractie index of the lens material 

BVP calculation

F1= +6.00 DS

F2= -8.00 DS

t= 2mm

BVP =-1.95 DS

if you give a lens more thickness it goes more plus

thin lens would be = + 6.00 + - 8.00 = -2.

thick is more plus than thin.

general effect of centre thickness, t, on BVP 

increasing centre thickness will make the bvp more positive or less negative, if other parameters remain the same 

effect on t on BVP when one surface is plano- negative lens eg

F1= plano

F2= -8.00

t= 2mm

bvp will be the rear surface power

bvp = F2= -8.00 DS

effect of t on the bvp when one surface is plano - positive lens eg

F1= + 8.00 DS 

F2= plano 

t= 2, 4, and 6mm 

positive lens with plano rear surface, theres no power at the back surface as f2 is 0 F1 is moved forwards so reduced surface changes as we go thicker

effect of t on back surface in biconcave lens form

F1 is negative in coconcave

as cntre thickness gets thicker, it gets more positive.

effect on t on the bvp in biconvex lens form

F2 is positive

lens form, sags and lens thickness- lecture week 4

sag of spherical lens surface: definitions

r= radius of surface

y= semi diameter of the chord

s= sag of surface at the chord diameter 2y

as you increase y ( as spectacle lense increases) the sag gets bigger

in small lenses- small field of view

approx sag formula

error increases as surface power increases larger radius- flatter lens srface so error goes down 

flat form conves lenses

plano- convex- one convex surface one plano surface

bi convex- two convex surfaces of differing radius

equi-convex- two convex surfaces of equal radius

flat form lenses seen in trial lenses, in refractor heads

flat form tend to be thinner and lighter, but not good for spectacles

fllat from concave lenses- minus powered lens

plano concave- one concave surface, one plano surface

bi concave- two cncave surfaces of differing radius

equi concave- two concave surfaces of equal radius

meniscus or curved form lenses

a lens that is not flat form is curved form 

one surface is convex one surface is concave 

the ralationship between the surface radii, lens thickness and refractive index determine the sign and power of the lens

LENS ON LEFT:

front surface is more positive so front is convex back is concave so negative power but more negative than front 

likely that front is negative but more positive than the rear 

LENS ON RIGHT:

front surface is positive, steeper curve so its more plus

rear surface is concave, lower minus power than front. Likely that front is positive, back is negative  

myopes- rear surface negative 

hyperopes- rear surface little bit minus or positive 

relationship between sag, centre thickness and edge thickness of opthalmic lenses

for this plano concave lens:

S1= 0 as F1 is flat/plano

S2= sag of the back surface

tc= centre thickness

te= edge thickness

for this form of lens: tc + S2= te

if the surface powers, centre thickness and lens diameter are known then the edge thickness can be predicted

this can help practitioner to infrom the patient of the likely cosmetic appearance of the finished spectacles

for a negative curved lens form

S1= sag of front surface

S2= Sag of back furface

tc= centre thickness

te= edge thickness

if surface powers, centre thickness and lens diameters are know, then edge can be predicted

minus powered lens as front surface flat and rear is curved

tc + S2= S1 +te

positive curved lens form

positive powered lens 

front surface more powerful as its very curved compared to the lower powered rear surface 

indicates that front surface is a high plus surface rear surface low minus lens. showing the lens is positive tc +S2 = S1 +te 

calculating sag from surface power - working out edge thickness from surface power

adding sag and surface power

curved surface so more needed

working out rear surface power 

rear surface power needed to work out S2 if front surface is +2.00 and and overall is -8 

so need the rear surface to be around -10 

calculating edge thickness when base curve ( front surface of lens) and its effect on edge thickness

as the base has changed to +4.00 more positive, more negative

sag is going to be bigger, radius smaller

power is needed in the rear surface

means edge surface is going to be thicker

a more positive base curve increases sag of both surfaces and increases edge thickness

edge thickness of lenses with different values of y

useful when calculating the temporal and nasal edges of spectacle lens

for each eye, temporal y and nasal y = horizontal lens size

distance between the centres of the two lenses is usually larger than the patients PD

temporal y is usually greater than nasal y

if frame was made smaller, tempral y value reduces as thickness reduces 

if made bigger, value increases 

box lens size

measured from temporal y to nasal y

will see in the case of box lens size being 50 : see 50 20 135

50 is horizontal box lens

20 is the distance between lenses

135 is total length of the side 

whats decetration of pupil 

pupil isnt centere correctly, so the geometric centre of the pupil no longer represents the true optical centre of vision 

impact of patient wearing a larger frame

larger box lens size

so larger temporal edge thickness

patient would not want thick lenses so important to pick th right frame

keeping lenses thin for a myope

consider a high refractive indes lens material

go for smallest frame possible, but keep field of view in ming, keeps pupils in middle, which keeps thickness even

small eye size- minimises thickness

avoid lens shapes with points a long way from required optical centre

try match distance between geometric centres with patients PD

choose a frame where the BVP can be as small as possible

choose frame where the rims and side configuration can hide some of the edge thickness

edge the lenses so that the bevel is at the front of the lens- this prevents a thick lens edge protruding form the front of the frame

keeping lenses thin for a hypermetrope

consider a high refractive index lens material

choose the finished lens blank size with care. many workshops will carry a stock of finished single vision lenses ; 65 m diameter for plus powers and 70 mm diameter for minus powers

glazing a 65mm lens of high plus power into a small frame can produce bad results - make eyes look small

should use smaller blank size