Optics

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1 Optics

Properties of Light

Light behaves both as waves and as particles (photons)

Its speed (velocity) (v) is directly proportional to wavelength (λ) and frequency (ν): v = λν

In any given medium, speed of light is constant (v_vacuum = c = 3.0 × 10¹⁰ cm/s); therefore, wavelength and frequency are inversely proportional

Light slows down in any substance other than air or vacuum, amount of slowing depends on medium; frequency of light remains unchanged, but wavelength changes (becomes shorter) (Figure 1-1)

Figure 1-1 The electromagnetic spectrum. The pictures of mountains, people, buttons, viruses, and so forth, are used to produce a real (i.e. visceral) feeling of the size of some of the wavelengths.

(From Miller D, Burns SK: Visible light. In: Yanoff M, Duker JS (eds) Ophthalmology, 2nd edn. St Louis, Mosby, 2004.)

Its energy is directly proportional to frequency and inversely proportional to wavelength: E = hν = h(c/λ)

Refraction

Light changes direction when it travels from one material to another of different refractive index (e.g. across an optical interface); direction of refraction is toward the normal when light passes from a medium with a lower index of refraction to a medium with a higher one, and away from the normal when light passes from a more dense to a less dense medium (higher refractive index materials are more difficult for light to travel through, so light takes a shorter path [closer to the normal]); light does not deviate if it is perpendicular to interface (parallel to the normal)

Prisms

Prisms displace and deviate light (because their surfaces are nonparallel); light rays are deviated toward the base; image is displaced toward the apex (Figure 1-3)

Figure 1-3 A, Displacement of image toward apex. B, Displacement and deviation of light by prism.

Prism diopter (PD, Δ)

displacement (in cm) of light ray passing through a prism, measured 100 cm (1 m) from prism

Example: 15 Δ = ray displaced 15 cm at a distance of 1 m (1 Δ = 1 cm displacement/1 m); 1° ≈ 2 Δ (this approximation is useful for angles smaller than 45°)

Angle of minimum deviation

total angle of deviation is least when there is equal bending at both surfaces of prism

Plastic prisms are calibrated by angle of minimum deviation: back surface parallel to frontal plane

Glass prisms are calibrated in Prentice position: back surface perpendicular to visual axis

Prism placed in front of the eye creates a phoria in the direction of the base

Example: Base-out (BO) prism induces exophoria; to correct, use prism with apex in the opposite direction

Apex is always pointed in direction of deviation: base-out for esotropia, base-in for exotropia, base-down (BD) for hypertropia

Stacking prisms is not additive; 1 prism in front of each eye is additive

Risley prism

2 right-angle prisms positioned back to back, which can be rotated to yield variable prism diopters from 0 to 30 Δ; used to measure prismatic correction for tropias

Fresnel prisms

composed of side-by-side strips of small prisms; prism power is related to apex angle, not the size of the prism; available as lightweight, thin press-on prism to reduce base thickness of the spectacle prism; disadvantage is reflection and scatter at prism interface, causing decreased visual acuity

Prismatic effect of lenses (Figure 1-4)

spectacles induce prism; all off-axis rays are bent toward or away from axis, depending on lens vergence

Figure 1-4 A, Plus lenses act like 2 prisms base to base. B, Minus lenses act like 2 prisms apex to apex.

Perceived movement of fixation target when lens moves in front of the eye:

Plus lenses: produce ‘against’ motion (target moves in opposite direction from lens)

Minus lenses: produce ‘with’ motion (target moves in same direction as lens)

Amount of motion is proportional to the power of the lens

Prismatic effect of glasses on strabismic deviations: 2.5 × D = % difference; minus lenses make deviation appear larger (‘minus measures more’); plus lenses decrease measured deviation

Prentice’s rule

prismatic power of lens = Δ = hD (h = distance from optical axis of lens [cm], D = power of lens [D])

Prismatic power of a lens increases as one moves farther away from optical center (vs power of prism, which is constant)

Example: Reading 1 cm below optical center: OD − 3.00; OS + 1.00 + 3.00 × 90

OD (Oculus Dexter – right eye): prism power = 1 cm × 3 D = 3 Δ BD

OS (Oculus Sinister – left eye): prism power vertical meridian: 1 cm + 1 D = 1 Δ BU (base-up) (Note: power of cylinder in 90° meridian is zero)

Net prismatic effect = 4 Δ (either BD over OD, or BU over OS)

Treatment of vertical prismatic effect of anisometropia:

1 Contact lenses (optical center moves with eyes)

2 Lower optical centers of lenses (reduce amount of induced prism)

3 Prescribe slab-off prism (technique of grinding lens [done to the more minus of the 2 lenses] to remove BD prism [to reduce amount of induced prism])

4 Single vision reading glasses

Prismatic effect of bifocal glasses

Image jump: produced by sudden prismatic power at top of bifocal segment; not influenced by type of underlying lens; as line of sight crosses from optical center of lens to bifocal segment, image position suddenly shifts up owing to base-down prismatic effect of bifocal segment (more bothersome than image displacement; therefore, choose segment type to minimize image jump)

Image displacement: displacement of image by total prismatic effect of lens and bifocal segment; minimized when prismatic effect of bifocal segment and distance lens are in opposite directions

Prismatic effect of underlying lens:

Hyperopic lenses induce BU prism, causing image to move progressively downward in downgaze

Myopic lenses induce BD prism, causing image to move progressively upward in downgaze

Prismatic effect of bifocal segment:

Round top (acts like BD prism): maximum image jump; image displacement less for hyperope than for myope

Flat top (acts like BU prism): minimum image jump; image displacement more for hyperope than for myope

Executive type or progressive lenses: no image jump (optical centers at top of segment)

Plus lens: choose round top

Minus lens: choose flat top or executive type (for myopes, image jump is very difficult to ignore because it is in same direction as image displacement, so avoid round tops in myopes)

Chromatic effects

prismatic effect varies with wavelength

Shorter wavelengths are bent farther, causing chromatic aberration

White light shines through prism: blue rays closer to base (bend farthest), red rays closer to apex

In the eye, blue rays come to focus closer to lens than do red rays; difference between blue and red is 1.5–3 D

Duochrome test

red and green filters create 0.5 D difference; use to check accuracy of refraction

If letters on red side are clearer, focal point is in front of retina (eye is ‘fogged’ or myopic)

If letters on green side are clearer, focal point is behind retina (eye is overminused or hyperopic)

Technique: start with red side clearer and add minus sphere in 0.25 D steps until red and green sides are equal (focal point on retina); works in colorblind patients because it is based on chromatic aberration rather than color discrimination

Vector addition of prisms

prismatic deviations in different directions are additive, based on pythagorean theorem (a² + b² = c²) (Figure 1-5)

Figure 1-5 Addition of BU and BO prisms.

Vergence

The amount of spreading of a bundle of light rays (wavefront) emerging from a point source

Direction of light travel must be specified (by convention, left to right)

Convergence (converging rays)

plus vergence; rare in nature; must be produced by an optical system

Divergence (diverging rays)

minus vergence

Parallel rays

zero vergence

Diopter

unit of vergence; reciprocal of distance (in meters) to point at which rays intersect; reciprocal of focal length of lens

Lens

adds vergence to light (amount of vergence = power of lens [in diopters])

Plus (convex) lens adds vergence; minus (concave) lens subtracts vergence

Basic lens formula

U + D = V (U = vergence of light entering lens; D = power of lens; V = vergence of light leaving lens)

Power of a spherical surface in a fluid: D_s = (n′ − n)/r

(n′ − n = difference in refractive indices; r = radius of curvature of surface [in meters])

Example: Power of corneal surface: back = −5.7 D; front = +52.9 D (front: n′ = 1.37, n = 1.00; back: n′ = 1.33, n = 1.37; r = 0.007)

Power of a thin lens immersed in fluid: D_air/D_fluid = (n_lens − n_air)/(n_lens − n_fluid)

Refracting power of a thin lens is proportional to difference in refractive indices between lens and medium

Objects and images

Object rays: rays that define the object; always on incoming (left) side of lens

Image rays: rays that define the image; always on outgoing (right) side of lens

Objects and images can be on either side of lens: real if on same side as respective rays; virtual if on opposite side from rays (locate by imaginary extension of rays through lens)

If object is moved, image moves in same direction relative to light

Adding power to system also moves image: plus power pulls image against light; minus power pushes image with light

Lenses

Real (thick) lenses (Figure 1-6):

6 CARDINAL POINTS: 2 principal planes (H and H’); 2 nodal points (N and N′); 2 focal points (F and F′)

Refraction occurs at principal planes (U is measured from H; V is measured from H′)

Focal lengths are also measured from principal planes

Nodal points coincide with principal planes (exception: if different refractive media are on opposite sides of the lens, then nodal points are both displaced toward the medium with the higher refractive index)

Central ray: passes through both nodal points (tip of object to N, across to N′; then, emerges parallel to original direction)

MENISCUS LENSES: difference between anterior and posterior curvature determines power

Steeper anterior curvature = convergent lens (+power); principal planes displaced anteriorly

Steeper posterior curvature = divergent lens (−power); principal planes displaced posteriorly

Power of lens is measured at posterior surface (posterior vertex power)

Conjugate points: each pair of object-image points in an optical system is conjugate; if direction of light is reversed, position of object and image is exactly reversed

Conjugate planes:

Figure 1-6 Conjugate points of real lens: each pair of object–image points in an optical system is conjugate; if direction of light is reversed, positions of object and image are exactly reversed.

Example: Viewing a slide presentation (image of slide is formed on retina of each person in audience, and very faint image of each person’s retina is projected on the screen), direct ophthalmoscope (patient’s retina and examiner’s retina are conjugate), indirect ophthalmoscope (3 conjugate planes [patient’s retina, aerial image plane, examiner’s retina]; any object placed at far point [aerial image] will be imaged sharply in focus on patient’s retina)

Ideal (thin) lenses: special case of thick lens; as lens gets thinner, principal planes move closer together; in an ideal lens, principal planes overlap at optical center

2 FOCAL POINTS:

PRIMARY (f): object point for image at infinity (point on optical axis at which object is placed so parallel rays emerge from lens)

SECONDARY (f′): image point for object at infinity (point on optical axis at which incident parallel rays are focused)

FOCAL LENGTH: distance between lens and focal points; reciprocal of lens power (f = 1/D)

Example: Focal length of +20 D lens is 1/20 = 0.05 m

NODAL POINT (N): point through which light ray passes undeviated; located at center of thin lens (optical center)

RAY TRACING: use to determine image size, orientation, and position

3 PRINCIPAL RAYS:

1 Central ray: undeviated ray passing from tip of object through nodal point of lens (or center of curvature of mirror) to tip of image; gives size and orientation of image (form similar triangles; thus, sizes of object and image are in same ratio as their distances from the lens)

2 Ray from tip of object through F emerges from lens parallel to optical axis

3 Ray from tip of object parallel to optical axis emerges from lens and passes through F′ (Figure 1-7)

Figure 1-7 Three principal rays of ideal lens.

Lens effectivity

function of lens power and distance from desired point of focus; depends on vertex distance and refractive index of media in which lens is placed (D_air = [n_IOL − n_air]/[n_IOL − n_aqueous] × D_aqueous); moving a lens forward away from eye increases effective plus power, so plus lens becomes stronger and minus lens becomes weaker; when vertex distance decreases, a more plus lens is required to maintain the same distance correction (i.e. as desired point of focus is approached, more plus power is needed); the more powerful the lens, the more significant is the change in position

Pure cylindrical lens

power only in 1 meridian (perpendicular to axis of lens); produces focal line parallel to axis

Spherocylindrical lens

power in 1 meridian greater than other

Spherical equivalent: average spherical power of a spherocylindrical lens; equal to sphere plus

the cylinder; places circle of least confusion on retina

Conoid of Sturm: 3-dimensional envelope of light rays refracted by a circular spherocylindrical lens; consists of: vertical line → vertical ellipse → circle (of least confusion) → horizontal ellipse → horizontal line (Figure 1-8)

Circle of least confusion: circular cross section of conoid of Sturm, which lies halfway (dioptrically) between the 2 focal lines at which image is least blurred; dioptrically calculated by spherical equivalent

Interval of Sturm: distance between anterior and posterior focal lines

Figure 1-8 Conoid of Sturm.

Cylinder transposition

converting cylinder notation (plus → minus; minus → plus)

New sphere = old sphere + old cylinder

New cylinder = magnitude of old cylinder but with opposite power

New axis = change old axis by 90°

Example: +3.00 +1.50 × 45 → +4.50 −1.50 × 135

Power cross diagram

depicts 2 principal meridians of lens with the power acting in each meridian (90° from axis), rather than according to axis (Figure 1-9)

Figure 1-9 Various lens notations.

Combining cylinders at oblique axis

complex calculation; therefore, measure with lensometer

Power of cylinder at oblique axis

power of the cylinder +1.00 × 180° is +1.00 @ 90°; +0.75 @ 60°; +0.50 @ 45°; +0.25 @ 30°; 0 @ 0°

Example:

JACKSON CROSS CYLINDER: special lens used for refraction to determine cylinder power and orientation; combination of plus cylinder in 1 axis and minus cylinder of equal magnitude in axis 90° away; spherical equivalent is zero (e.g.: −1.00 + 2.00 × 180)

Therefore, does not change the position of the circle of least confusion with respect to the retina; normally, have 0.25 D cross cylinder in phoropter; if patient’s acuity is 20/40 or worse, must use larger power cross cylinder so patient can discriminate difference

TIGHT SUTURE AFTER CATARACT OR CORNEAL SURGERY: steepens cornea in meridian of suture, inducing astigmatism

Aberrations

Lenses behave ideally only near optical axis; peripheral to this paraxial region, aberrations occur

Coma

comet-shaped image deformity from off-axis peripheral rays

Curvature of field

spherical lens produces curved image of flat object

Astigmatism of oblique incidence

tilting spherical lens induces astigmatism (oblique rays encounter different curvatures at front and back lens surfaces)

Example: pantoscopic tilt (amount of induced sphere and cylinder depends on power of lens and amount of tilt)

Distortion

differential magnification from optical axis to lens periphery alters straight edges of square objects; shape of distortion is opposite of shape of lens (plus lens produces pincushion distortion; minus lens produces barrel distortion); effect increases with power of lens

Chromatic

light of different wavelengths is refracted by different amounts (shorter wavelengths are bent farther; chromatic interval between blue and red is 1.5-3.0 D)

Example: at night with Purkinje shift, chromatic aberration moves focal point of eye anteriorly, producing myopia

Magnification

Transverse (linear or lateral):

magnification of image size (away from optical axis); must be able to measure object and image size; ratio of image height to object height (or image distance to object distance); if image is inverted, magnification is negative M_L = I/O

Axial

magnification of depth; magnification along optical axis; equal to square of transverse magnification M_Ax = M_L²

Angular

magnification of angle subtended by an image with respect to an object; useful when object or image size cannot be measured

Example: moon gazing with telescope

M_A = xD = D/4 (standardized to 25 cm [ m], the near point of the average eye)

Example: with direct ophthalmoscope, eye acts as simple magnifier of retina: M_A = 60/4 = 15× magnification

Size of image seen through glasses

Shape factors: for any corrective lens, an increase in either front surface curvature or lens thickness will increase the image size (therefore, equalize the front surface curvatures and lens thickness for the 2 lenses)

FRONT SURFACE CURVATURE:

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Prismatic effect of lenses (Figure 1-4)

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