Thick Lenses/Multiple Lens Systems

What we’ll do today ► Thick lens theory  Concepts  Cardinal points explained ► Schematic eyes  Exact  Reduced  Construction of retinal images ► Catoptric images ► Angle Kappa ► Accommodation and AC/A ratio

Thick Lens Theory ► A lens is not thin if the CT is sufficient to change the power ► The eye is a moderately complex thick lens system  Focusing power depends on curved surfaces, each separated by media of different indices of refraction

Thick Lens Theory Element by Element Imaging ► Use previously developed thin lens imaging techniques for each refracting surface.  Use the image of one lens as the object of the next lens  Each refracting surface is an element of the system.  The medium between the elements is the index of the lens system

Imaging of multiple lens systems- element by element RO RI n1n2 n3 VO RO VI E1 E2 E3

Thick Lens Power ► F=F 1 +F 2 - (t/n)F 1 F 2 ► If the n1=n3, then f=f’  In English, this means that the focal lengths on either side of the lens will be the same if the index of refraction is the same on both sides of the lens.

The Effect of Thickness on Power ► When thickness is 0, the F=F1+F2 ► If both surfaces are the same sign, increasing the thickness makes the net equivalent power more negative ► If one is negative and one is positive, then increasing CT makes the lens more negative.

Problem ► Lens system has 2 thin lenses, +15D and -3D with CT of 17cm. An object is placed 1m in front of the first lens. Where is the final image? ► Knowns F 1 =+15D, F 2 =-3D, t=17cm, obj dist l 1 =-1m, ► Unknown image dist l’ ► Equations L’=F+L, L=n/l and L’=n’/l’ RO n1 n2 n3 100cm=l1 17mm

RO n1 n2 n3100cm=l 1 17cm Incident vergence L 1 =n 1 /l 1 =1.00/-1.00=-1.0D Emergent vergence L’=L 1 +F 1 = =14D Image position l’=n 2/ L 1 =1/14=.0714m=7.14cm Incident vergence L 2 =n 2 /l 2 =1.00/-.0986=-10.14D Emergent vergence L 2 ’=L 2 +F 2 = (-3)=-13.14D Image position l’ 2 =n 3 /L 2 =1/-13.14=.0761= -7.61cm 7.61

Lens systems: Size and Orientation ► Product of Lateral Magnification of each object/image ► Note that you cannot predict orientation of final images ► LM system= (LM1)(LM2)(LM3)…. ► Recall LM=h’/h ► LM=nl’/n’l

Lateral Mag of Lens Systems ► So, if you have the following lenses 8x7x2x Total mag is (8)(7)(2) =112X

Problem

What we’ll do today ► Thick lens theory  Concepts  Cardinal points explained ► Schematic eyes  Exact  Reduced  Construction of retinal images ► Practical applications of thick lenses and schematic eyes ► Accommodation and AC/A ratio ► Catoptric images

Cardinal Points ► In thick lenses, not valid to assume that focal lengths are measured from the center of lens ► Convenient reference positions for all optical systems  Principal planes  Principal points  Nodal points ► They exist in thin lenses, but all coincide with the axial position of the lens.

Some defintions ► Neutralizing (front vertex) power: Incident vergence on front of lens that yields image at infinity ► Back Vertex Power: Emergent vergence from back surface of lens for object at infinity. Used in refraction and lens prescriptions. ► Effective Power: shows what the power is from the other surface

Cardinal Points: Locating H/H’ Principal Planes H’= emergent ray extended backward H=incident ray extended forward f H F’ f’ H’ Front vertex power Back vertex power

Bending a Thick Lens ► Changing the form of the lens does not change the separation between the two planes, but it does change the location of H/H’ ► In the concave or convex lenses, the H usually falls within the lens ► A meniscus lens shifts H/H’ towards the more curved surface.

Movement of H with lens shape H (principal plane) moves toward the most curved surface

Cardinal Points: Principal Planes ► Rays enter and leave H/H’ at the same height, a property called unitary linear magnification ► H/H’ are conjugate, meaning the optical image of each other These planes can replace all other optical elements HH’

Cardinal Points: Principal Planes ► CP are the reference planes- all object and image distances are measured relative to them ► Primary and secondary focal points also measured relative to the principal planes  We usually measure from the back of a lens (effective power/ vertex power)

HH’ Cardinal Points: Principal Planes F F’ FH and H’F’ will be equal IF the media composing the object and image spaces Is the same refractive index. If it is not, the focal length will be longer on the side with the higher index n n’

Cardinal Points: Principal Points ► Where the principal plane intersects the optic axis. ► They are a conjugate pair (object and image of each other) ► In the eye, P and P’ are separated by 0.3mm PP’

Cardinal Points: Nodal Points ► The place where the chief ray passes undeviated through the lens (also true for thin lenses) NN’

Cardinal Points: Nodal Points ► As long as n=n’, nodal point is at the same location as principal point and N’ is in the same place as P’ ► If n does not = n’ (like the eye) then both N and N’ are NOT coincident with P and P’  They will shift in the direction of the greater index

Nodal Points Any ray striking N will leave N’ with an identical inclination to the axis (Unitary Angular Magnification). NN’

Optical Center ► The place where an undeviated ray crosses the optic axis. ► In reality the nodal points represent the apparent position of the optical center o N N’

Thick Minus Lenses AVFL PVFL f’f F’ F H‘H The lensometer uses Back vertex power because It is relevant. Back vertex Power IS NOT vertex distance Note that the AVFL and The PVFL may not be equal

Thick Plus Lenses HH’ f f’ F F’ AVFL PVFL Convex Meniscus Lens Equivalent power (true) Back or front surface power

Topics ► Thick lens theory  Cardinal points explained ► Schematic eyes  Exact  Reduced  Construction of retinal images ► Angle Kappa ► Catoptric images

Schematic Eyes ► Refractive components  Corneal power  Anterior chamber depth (n, aqueous)  Lens power  Axial length of eye  N, vitreous  Powers determined by radii of curvature and n

Schematic Eyes- Gullstrand’s Exact ► Unique in that it:  specifies n and radii for both the nucleus and cortex of the lens  represents the cornea with both front and back surfaces  Provides values for the accommodative and relaxed state FF’ Principal planes Nodal points

Simplified Gullstrand Model ► Lens has just one pair of refracting surfaces and a single index ► Cornea is a single refracting surface Principal planes Nodal points FF’ n=1.336 for aqueous, vitreous n=1.413 lens r, cornea= 7.80 (43.25D) r, ant/post lens=10.0 (33.50),8.0 (42.12) AC depth, lens thickness 3.6mm

Reduced Eye- simplest F F’ n=4/3 17mm 24mm N All refraction takes place At the front surface of the cornea 7mm

Schematic Eye ► The cornea is so powerful because of the change in index of refraction between air and the tear film n (air) =1.0 n=1.376 n=1.336 n=1.406 n (water) =1.336 F=n’-n/r

Problem ► Recall that F=n’-n/r ► So, what is the power of the cornea of 7.5mm radius in air? ► F= /7.5 = 50.D ► What is the power in water? ► F= /7.5 = 5.3D

Schematic Eyes- Construction of Retinal Images Fa Fr N h Because the light subtends the same angle at the nodal point and Fa, we can say that retinal image size is related to the angle of incidence! Thus, h=tan  (fa) So, as an object approaches, it appears larger b/c the angle is greater. fa VISUAL ANGLE

Finding the Retinal Image Size ► Use similar triangles to solve these problems 17mm H’ object H’retinal image x

Problem

What we’ll do today ► Thick lens theory  Concepts  Cardinal points explained ► Schematic eyes  Exact  Reduced  Construction of retinal images ► Catoptric images ► Angle Kappa

Catoptric (Purkinje) Images ► Each refracting element of the eye is really also a mirror, as some of the light is reflected back at you  Front corneal surface  Back corneal surface  Front lens surface  Back lens surface

Catoptric (Purkinje) Images These are the apparent positions These are the actual positions They are different b/c the light is refracted upon exiting the eye

So what??? ► These images have been used clinically:  Hirschberg reflex, keratometry use image 1  Changes in the shape of the lens during accommodation can be gleaned by comparing the positions of images 3 and 4  Eye tracking systems use the 4 th purkinje image  Refractive procedures- do you center on the reflex or the pupil center? What about angle kappa? Where does best result occur?

What we’ll do today ► Thick lens theory  Concepts  Cardinal points explained ► Schematic eyes  Exact  Reduced  Construction of retinal images ► Catoptric images ► Angle Kappa (lambda)

Angle Kappa ► The optical components of the eye are not coincident with the line of sight, but are along an optical axis temporal to it ► The pupil is not usually centered on the optical axis of the eye  Pupillary axis: imaginary line normal to the cornea and containing the center of the pupil  Line of sight- not anatomical. Noted relative to the pupillary axis

Angle Kappa ► Route of LOS through pupil located by observing the corneal reflex  Usually.4mm nasal to center of pupil  1.0mm of displacement = 22 prism diopters/12.5 degrees of rotation  OD= OS normally ► Angle kappa is the difference between the pupillary axis and the LOS.  Usually 5 degrees (range 3-7) temporal (+)  If negative then kappa is nasal to pupillary axis

Angle Kappa ► Useful to determine strabismus ► Important in refractive surgery.  Do you center on the pupillary axis or the line of sight?  What would give you a better outcome? ► Lasik- LOS ► CK- center on pupil ► Custom with iris registration- center on pupil

Stiles- Crawford Effect ► It refers to the directional sensitivity of the cone photoreceptors; specifically to the phenomenon that light passing near the edge of the pupil is less efficient at evoking sensation than light passing through the center of the pupil. pupil ► A photoreceptor acts like a retinal optic fibre, it captures light that hits it at a narrow angle from its normal. The acceptance angle of a cone is narrow, approximately 5°, rods have larger acceptance angles. ► The "Stiles-Crawford" effect reduces the detrimental effects of light scatter on the retina at photopic levels photopic

Stiles Crawford Effect Effect of Position

Accommodation ► Anterior curvature of lens changes as the CM contracts, allowing zonular relaxation  Far point- object position that allows image to fall on the retina w/o accommodation  Near point- closest point at which object is seen clearly using maximum accommodation  Range is the difference between the two  Amplitude is the range in diopters

Accommodation ► Amplitude = near point (D)- far point (D) ► Accommodation needed = where you want to see (D) – far point (D)

Problem

AC/A ratio ► Neural linkage of accommodative triad ► How many prism diopters of convergence occurs for each diopter of accommodation  Normal is 3:1 to 5:1

AC/A ratio  Sometimes there is too much convergence for any given amount of accommodation (eso)  Sometimes there is not enough (exo) ► Either can cause problems with accommodative amplitudes ► If you converge too much, you will accommodate less ► If you converge not enough, you will accommodate more