Essential Optics Review for the Boards

Clinical Optics—Ophthalmic Technical Skills Series, Appleton
Optics, Refraction, and Contacts Lenses, American Academy of Ophthalmology: Basic and Clinical Science Course, 1998-1999
Optics and Refraction: Ophthalmology Board Review, MacInnis
Clinical Ophthalmology, Volume 1, Chapters 31-52, 58, 60, Duane
Optics, Refraction, and Contacts Lenses, Basic and Clinical Science Course, Section 3 (2004-2005), American Academy of Ophthalmology

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Formulas at a Glance

Simple Lens Formula

U + D = V or 100/u (cm) + D = 100/v (cm)

Where: U = vergence of object at the lens u = object position = 100/U (cm)

D = lens power

V = vergence of image rays v = image position = 100/V (cm)

Lens Effectivity

The change in vergence of light that occurs at different points along its path. This is related to vertex distance.

Formula: F _new = F _current/(1-dF_current)

Where F is in Diopters and d is in meters.

Optical Media and Indices of Refraction

Object vergence V = n/u

Image vergence V’ =n’/u’

Where: n = index of refraction for where the light is coming from

n’ = index of refraction for where the light is going to

u = object distance

u’ = image distance

Snell’s Law of Refraction

n sin i = n’ sin r

Where: i = angle of incidence as measured from the normal

r = angle refracted as measured from the normal

n = index of refraction for where the light is coming from

n’ = index of refraction for where the light is going to

Critical Angle

sin i_c = n’/n x 1

Where: i_c = the critical angle and the refracted angle is 90°

n = index of refraction for where the light is coming from

n’ = index of refraction for where the light is going to

Apparent Thickness Formula

n/u = n’/u’

Where: n = index of refraction for where the light is coming from

n’ = index of refraction for where the light is going to

u = object distance

u’ = image distance

Mirrors

The focal length of a curved mirror is always ½ its radius of curvature (f= r/2)

The reflecting power of a mirror in diopters D_M = 1/f_(m)

For mirrors or reflecting surfaces: U + 2/r_m = V, (r_m is in meters) or U + 1/f = V

Where: f = focal length of the mirror in meters

r = radius of curvature of the mirror in meters

Prism Diopters

A Prism Diopter (∆) is defined as a deviation of 1 cm at 1 meter.

Approximation Formula

For angles under 45° (or 100 ∆), each degree (°) of angular deviation equals approximately 2 ∆

Prentice’s Rule

Deviation in prism diopters (PD) = h (cm) x F

Where: F = power of the lens

h = distance from the optical center of the lens

Convergence

Convergence ( ∆ ) = 100/working distance (cm) x Pupillary Distance (cm)

Convergence (in prism diopters) required for an ametrope to bi-fixate a near object is equal to the dioptric distance from the object to the center of rotation of the eyes, multiplied by the subject’s intra-pupillary distance in centimeters.

Spherical Equivalent

Spherical equivalent = ½ cylinder power + sphere power

Relative Distance Magnification

Relative Distance Magnification = r/d

Where: r = reference or original working distance

d = new working distance

Relative Size Magnification

Relative Size Magnification = S2/S1

Where: S1 = original size

S2 = the new size

Transverse/Linear Magnification

M _T= I/O=U/V = v/u

Where: I = Image size

O = Object size

U = Object vergence

V = Image vergence

u = object distance

v = image distance

Axial Magnification

M_A= M₁ X M₂

M_A = (M)² (Approximation formula for Axial Magnification of objects with relatively small axial dimensions)

Rated Magnification

M _r = F/4

Assumes that the individual can accommodate up to 4.00 diopters when doing close work which gives d = 25cm (25cm is the standard reference distance used when talking about magnification).

Effective Magnification

M_e = dF

Where: d = reference distance in meters to the object (image is formed at infinity)

F = the lens power

Conventional Magnification

M_c = dF + 1

Where: d = reference distance in meters to the object (image is formed at infinity)

F = the lens power

The underlying assumption in this equation is that the patient is “supplying” one unit (1X) of magnification

Angular Magnification of a Telescope

M_A Telescope = (-) F_E/F_O

Where: F_E = eyepiece lens power

F_O = objective lens power

Telescopic Approximation Formula ( for accommodation required to view a near object through an afocal telescope)

A_oc = M²U

Where: A oc = vergence at the eyepiece = accommodation

U = object vergence at the objective = 1/u

M = the magnification of the telescope

Aniseikonia

Total Magnification of a Lens: M_T = M_P + M_S.

Where: M_P is the magnification from the lens power

M_S is the magnification from the lens shape

Magnification from Power (M P): M_P = D_VH

Where: D_V is the dioptric power of the lens

H is the vertex distance measured in cm

Magnification from Shape (M S): M_S = D₁ (t_cm/1.5)

Where: D₁ is the curvature of the front surface of the lens

t_cm = the center thickness of the lens

The 1.5 in the following equation is the index of refraction (approximately) of glass or plastic

IOL Power (SRK Formula)

D_IOL = A – 2.5L – 0.9K

Where: D_IOL = recommended power for emmetropia

A = a constant (provided by manufacturers for their lenses)

L = axial length in mm

K = average keratometry reading in diopters for desired ametropia

Lens Clock

To calculate true power of a single refracting surface (SRS) using a lens clock

F_true = F_{lens clock} (n’_true – n)/(n’_{lens clock} – n)

Where: n’_true = the true index of refraction of the lens being measured

n’_{lens clock} = 1.53 (crown glass)

n = 1.00 (air)

Ophthalmoscopic Magnification

Direct: M = F/4

Where: F = the total refractive power of the eye. The image is upright.

Indirect: M A = (-)D Eye/C ondensing lens

The image of the fundus becomes the object of the condensing lens, which then forms an aerial image that is larger and inverted.

Astigmatism Estimation from Keratometry

Take the amount of with the rule astigmatism noted by keratometry readings, multiply that by 1.25, and then subtract that number from 0.75 diopters (lenticular astigmatism) to arrive at the estimated amount of refractive astigmatism.

When against the rule astigmatism is noted by keratometry, add 0.75 diopters to the full amount of corneal astigmatism to arrive at the estimated amount of refractive astigmatism.

Reflecting Power of the cornea to determine corneal curvature

D = (n-1)/r

Where: D is the reflecting power of the cornea

n is the standardize refractive index of the cornea (1.3375)

Lens Tilt

The change in power of the sphere through tilting is determined by the formula:

F (1 + 1/3 sin² a)

The created cylinder power is determined by the formula: F (tan² a)

Where: a = the angle of tilt

A simplified formula to determine the change in sphere power is to take (1/10 the amount of tilt)² = the percentage of power added to the original sphere. The increase in the cylindrical correct is approximately equal to 3x the induced sphere increase.

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Miscellaneous Information

a. LensTtilt: The position of the optical center will vary with the tilt of the lens before the eyes. The ideal tilt of standard lenses is 8 degrees in on the bottom of the lens. Such a tilt places the optical center 4mm below the center of the pupil when the line of sight passes normally through the lens surface.

When the lens is tilted, the incident light strikes the lens obliquely, inducing marginal or radial astigmatism even though the light passes through the center of the lens.

The change in power of the sphere through tilting is determined by the formula:
F (1 + 1/3 sin²a). The created cylinder power is determined by the formula: F (tan²a), where a = the angle of tilt.

If a cylinder lens is tilted on its axis, no actual sphere power is induced however the total new cylinder power is increased by the formula previously noted.

The effect of tilting a minus spherical lens is the production of minus cylinder at the axis of rotation - 180 degrees. The cylinder power increases with both the degree of the tilt and the power of the lens.

A simplified formula to determine the change in sphere power is to take (1/10 the amount of tilt)² = the percentage of power added to the original sphere. The increase in the cylindrical correct is approximately equal to 3x the induced sphere increase.

Examples of simplified formula:
A +3.00D sphere tilted 20 degrees will result in what spherical power increase? - (20/10)² = 4%, .04 x 3.00D = 0.12D

A +3.00D sphere tilted 20 degrees will result in a compound effect of +3.12 combined with +0.40 cylinder. Simplified formula - (20/10)² = 4%, .04 x 3.00D = 0.12D

A +1.00D sphere tilted 45 degrees will result in a compound effect of +1.16, combined with +1.00 cylinder.

An under corrected myope will therefore be able to obtain better distance acuity by tilting his glasses. For example, the effect of tilting a –10.00 diopter lens 10 degrees along the horizontal axis results in an optical correction of –10.10 –0.31 x 180 which gives a spherical equivalent of –10.25D. If the same lens is tilted 30 degrees, the resultant effective optical correction is –10.83 –3.33 x 180 with a spherical equivalent of –12.50 diopters. This is why an under-corrected myope tilts their spectacles to attain better distance vision.

Question: A point source is placed 50 cm from a cylindrical lens of +5.00 diopters, axis 90 degrees. Find the position and direction of the line foci formed by this lens.

Answer: Do this yourself to understand how this works.

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Instruments

a. Lens Clock = Lens Gauge = Geneva Lens Measure (Figure 41)

Click on image to enlarge.

The lens measure, lens clock, or lens gauge has two fixed pins on the outside and in the center, a spring-loaded, movable pin. This device physically measures the sagital depth of a refracting surface and calculates the refracting power of the surface. A pointer that is activated by a system of gears indicates the position of the movable pin in relation to the fixed pins. If the instrument is placed on a flat surface, the protrusion of the central pin is equal to that of the fixed pins, with the result that the scale reading is zero. If placed on a convex surface, the protrusion of the central pin is less than that of the fixed pins, but if placed on a concave surface, the protrusion of the central pin is greater. Because the chord length (the distance between the two outer pins) has a constant value for the instrument, the position of the central pin, indicates the sagitta of the surface, which provides a direct reading of diopters of refracting power of a surface of the lens.

The lens clock physically measures the sagital height/depth.
The reading is in power (diopters)
The lens clock assumes that n is in air and n’ = 1.53 (crown glass)

To calculate for the lens radius (assumes that s is very small) r = y²/2s (see diagram)

To calculate true power of a single refracting surface (SRS)

F true = F lens clock (n’ true – n)/(n’ lens clock – n)

Question: What is a Geneva lens clock?

Answer: A device used to determine the base curve of the back surface of a spectacle lens. It is often used clinically to detect plus cylinder spectacle lenses in an individual who is use to minus cylinder lenses. It is specifically calibrated for the refractive index of crown glass (n = 1.53). Special lens clocks are available for plastic lenses.

Question: A lens clock measures the power of a high index plastic lens (n=-1.66) to be –5.00 diopters. Has the lens clock overestimated, underestimated or accurately determined the power of the lens?

Answer: The lens clock has underestimated the power of the surface.

Question: A lens clock is used to measure the power of a SRS where n = 1.00 and n’ = 1.60.

What is the true power of the SRS if the lens clock reads –10.00D?

F true = F lens clock (n’ true – n)/(n’ lens clock – n) = -10.00D (1.6-1)/(1.53-1) = -10.00 (.6/.53) = -10.00D (1.132) = -11.32D

How much error was induced by the lens clock? -11.32 – (-10.00) = -1.32D

Question: A lens clock is used to measure the power of a SRS where n = 1.00 and n’ = 1.498.

What is the true power of the SRS if the lens clock reads –10.00D?

F true = F lens clock (n’ true – n)/(n’ lens clock – n) = -10.00D (1.498-1)/(1.53-1) = -10.00 (.498/.53) = -10.00D (0.939) = -9.39D

How much error was induced by the lens clock? -9.39 – (-10.00) = +0.61D

From these examples, you see that for lenses made with indexes of refraction greater than crown glass, the lens clock will underestimate the true lens power and for those lenses with indexes of refraction less than crown glass, the lens clock will overestimate the true lens power.

b. Lensometer

The lensometer measures the vertex power of the lens. The vertex power is the reciprocal of the distance between the back surface of the lens and its secondary focal point. This is also known as the back focal length. For this reason, a lensometer does not really measure the focal length of a lens. The true focal lengths are measured from the principal planes, not from the lens surface. The lensometer works on the Badel principle with the addition of an astronomical telescope for precise detection of parallel rays at neutralization. The Badel principal is Knapp’s law applied to lensometers.

A lensometer is really an optical bench consisting of an illuminated moveable target, a powerful fixed lens, and a telescopic eyepiece focused at infinity. The key element is the field lens that is fixed in place so that its focal point is on the back surface of the lens being analyzed. A lensometer measures the back vertex power of the spectacle lens. However, when measuring a bifocal addition, the spectacles must be turned around in the lensometer so that the front vertex power is measured. This is because the distance portion of the spectacle lenses is designed to deal with essentially parallel light. However, the bifocal addition is designed to work on diverging light, originating from a standard working distance of 40 centimeters. This diverging light from the near object is made parallel by the bifocal lens. The parallel light then enters the distance lens where it is refracted with the expected optical affect to give the patient clear vision. In this way, the bifocal exerts its effect on the light from the object before it passes through the rest of the lens. For strong bifocal corrections, there would be a significant difference in the bifocal strength measurement when using the front versus back vertex measurement.

Question: To measure the power of spectacles in a lensometer, when do you want the temples towards you and when do you want them away from you?

Answer: The distance correction is measured with the temples facing away from you (back/posterior vertex power). The bifocal power is measured with the temples pointing towards you (front/anterior with the vertex power). This is particularly important when checking the prescription of an individual with corrections > +4.00. Between 4.00D and 8.00D, there is approximately 0.25D difference between the fabricated add and the effective add. For lenses between 8.00D and 12.00D, the disparity is approximately 0.50D. For plus lenses, the effective add is always greater than the fabricated add, and for minus lenses it is just the opposite. In this case, you must measure the top of the lens with the temples away from you, then the bifocal segment with the temples towards you, in the opposite direction), to get an accurate reading of the bifocal power.

c. Ophthalmoscopes

Direct – image is upright. Magnification is based on the total refractive power of the eye. Using the basic magnification formula of M = F/4, an emmetropic eye of +60.00D would provide +60/4 = +15X. An aphakic eye of +40.00D would provide +40/4 = +10X.

Indirect – the image of the fundus becomes the object of the condensing lens, which then forms an aerial image that is larger and inverted. The two plus lenses (the eye and the condensing lens) determine the magnification of the aerial image. For the emmetropic eye, using the formula M_A = (-)D_Eye/C_{ondensing lens}= (-)60/D_{(condensing lens)}, we find that a 20D condensing lens results in (-)60/20 = -3X.

As the power of the condensing lens decreases, the magnification increases. Axial magnification increases exponentially, based on the formula Axial magnification: M_A = (M)².

d. Keratometer

Instrument used to measure the curvature/refractive power of the cornea. It accomplishes this by measuring the radius of curvature of the central cornea. The central cornea can be thought of as a high powered (~-250D) convex spherical mirror.

Question: How do you compute the anticipated astigmatic correction based on K-readings?

Answer: Take the amount of with the rule astigmatism noted by keratometry readings, multiply that by 1.25, and then subtract that number from 0.75 diopters (lenticular astigmatism).

Example: 1.00 diopter of with the rule corneal astigmatism would result in an expected refractive astigmatism of 0.50 with the rule. (1.00D x 1.25 = 1.25D - 0.75D = 0.50D)

Question: What instrument uses the reflecting power of the cornea to determine its readings?

Answer: The keratometer uses the reflecting power of the cornea to determine the corneal curvatures. The formula is: D = (n-1)/r. Where D is the reflecting power of the cornea and n is the standardize refractive index of the cornea (1.3375).

Question: How much of the cornea is measured with a keratometer?

Answer: Only the central 3-mm. For this reason, using a keratometer instead of a corneal mapping device may miss peripheral corneal scar defect.

e. Gonioscope

Total Internal Reflection (TIR) makes it impossible to view the anterior chamber angle without the use of a gonioscopic contact lens. Normally light from the angle undergoes TIR at the air-tear film interface. As result of this, the light from the angle is not able to escape from the eye making the angle impossible to visualize. This problem is overcome by the gonioscopic contact lens which sits on the cornea. In this way, the air at the surface of the cornea is eliminated. Total internal reflection occurs when light is trapped in the incident medium. Because TIR never occurs when light travels from a lower to a higher index, light is able to enter the gonioscopic contact lens where it is reflected by the gonioscopic mirror. This allows the angle of the anterior chamber to be visualized by the examiner.

Gonioscopic tilt angle should be approximately 7.5 degrees to the visual axis. This minimizes reflections and image distortion.

f. Retinoscopy

A retinoscope allows the clinician to objectively determine the spherocylindrical refractive error; irregular astigmatism, and also evaluate opacities and irregularities of the cornea and lens.

Most retinoscope today use a streak projection system. This streak of light is reflected from a mirror. Additionally, the streak can be moved in relation to a convex lens in the device by way of the sleeve. This allows the light to leave the device as if it were coming from a point behind the retinoscope ( plano mirror setting) or as if it were coming from a point between the examiner and the patient (concave mirror setting). For Copeland retinoscopes, the plano position is with the sleeve up, while the Welch Allyn retinoscope is in the plano position with the sleeve down.

Normally, the examiner will use their right eye to perform retinoscopy on the patient's right eye and their left eye for the patient's left eye. The examiner should align themselves just off-center to minimize lens reflections and to allow the patient to visualize the distance target to relax their accommodation. The patient should be instructed to look at a distance target such as a large Snellen letter (20/200-20/400).

When doing retinoscopy, the examiner is attempting to put the far point of the patient’s eye at the plane of the examiner’s pupil.

When the reflex shows “against” motion, the far point plane lies between the patient’s eye and the examiner’s eye, indicating myopia.
When the reflex shows “with” motion, the far point lies outside the interval between the patient’s eye and the observer’s eye, indicating hyperopia, emmetropia or mild myopia.

Question: If you obtain “with motion” during retinoscopy, is the far point of the patient in front of the peep hole, at the peep hole, or beyond the peep hole?

Answer: Beyond the peephole. The goal of neutralization is to have the light reflex of the patient’s far point at the peephole. The light at the patient’s pupil fills the entire space at once when neutrality is reached. “With” motion requires more plus to be added to the prescription to move the far point to neutralization. “Against” motion means that the far point is in front of the peephole. Therefore, more minus must be added to move the far point to neutralization.

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IOL Power (SRK Formula)

D_IOL = A – 2.5L – 0.9K. Where D_IOL = recommended power for emmetropia, A = a constant (provided by manufacturers for their lenses), L = axial length in mm, K = average keratometry reading in diopters for desired ametropia.

Change IOL power by 1.25 to 1.5D for each diopter of desired ametropia. Alternate formulas are needed for shorter or longer eyes.

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Lensmaker equation

The surface power of a lens = Ds = (n’- n)/r, where r is in meters, n = the index of the object space (air or fluid the lens is in) and n’ = the index of the lens. This is also called the refractive power or simply the power of a spherical refracting surface.

The power of a thin lens (IOL) immersed in fluid
D_air/D_fluid = (n_IOL – n_air)/(n_IOL – n_fluid)

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Ring Scotoma

Produced by the aphakic or high plus spectacle lens resulting from the prism effect induced by the peripheral edge of the lens which possesses the maximum prismatic power and creates the greatest deviation of rays.

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Night Myopia

Light rays at the edge of the human lens are refracted more than those at the center of the lens. Because our pupils are larger at night, more spherical aberration is present under lower light conditions. A refractive shift towards more myopia is needed to compensate for this increase in spherical aberration. Additionally, accommodation does not go to a neutral state under low light conditions. The visual system actually accommodates approximately 0.75D under low light conditions. These changes result in a need for more minus or less plus correction for those individuals, such as over the road truck drivers, who need to function with their highest visual clarity at night.

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Duochrome Test

Chromatic aberration of the eye results in green light being focused in front of the retina, yellow light at the retinal plane, and red light behind the retina. Therefore, when red is brighter, it indicates that more minus is needed to move the focus of red light further behind the retina? When green is brighter, more plus is needed to move the green light to focus further into the posterior chamber.

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Jackson Cross Cylinder (JCC)

Cross cylinders are combinations of two cylinders whose powers are numerically equal and of opposite sign and whose axes are perpendicular to each other. The Jackson Cross Cylinder is usually mounted in a ring with a handle at 45 degrees from the axis so that a twirl of the handle changes the cross cylinder to a second position.

Example: +0.25x90/–0.25x180 to –0.25x90/+0.25x180

When the JCC is placed in contact with a spherocylinder, it displaces both focal lines simultaneously in opposite directions, expanding the initial Interval of Sturm in the first position and contracting it in the second. However, there will be no displacement of the Circle of Least Confusion, only the diameter of the circle will increase in the first and decrease in the second position of the JCC.

When a +/-0.50 JCC is placed on a lensometer, with the red axis at 0 and 180 degrees, the lensometer will read the power as -0.50 +1.00 x 090. But, remembering that the JCC has no spherical power, only cylindrical power. For this reason, we can more accurately write the power of the JCC as –0.50 x 180 combined with +0.50 x 90

Question: When the Jackson cross cylinder is used to define the astigmatic axis, is the handle of the lens parallel to the axis or 45 degrees from it?

Answer: Parallel. To define the astigmatic power, the handle is rotated 45 degrees to the axis. Normally, you should define the axis before the power.

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Contrast Sensitivity

Contrast indicates the variation in brightness of an object. When an eye chart uses perfectly black ink on perfectly white paper, 100% contrast is achieved. Acuity charts approximate 100% contrast. Acuity charts are helpful for characterizing central visual acuity. However, they are less helpful for examining visual function away from fixation.

Contrast sensitivity is tested using alternating light and dark bars at varying intensity. The number of light bands per-unit length or per-unit angle is called the spatial frequency. During clinical testing of contrast sensitivity, patients are presented with targets of various spatial frequencies and peak contrasts. The minimum resolvable contrast is the contrast threshold. The reciprocal of the contrast threshold is defined as the contrast sensitivity, and the manner in which contrast sensitivity changes as a function of the spatial frequencies of the target is called the contrast sensitivity function.

Contrast sensitivity can be tested with sine wave gratings presented using either charts or video gratings. Because standard Snellen acuity charts test only the higher spatial frequency (30 cycles per degree), they do not provide an accurate picture of an individual's visual functioning, particularly when the individual has an ocular disease.

Acuity charts provide us with a quantitative assessment of visual functioning while contrast sensitivity charts provide us with a qualitative assessment of visual functioning. Contrast sensitivity testing is similar to current audiological testing which assesses an individual's ability to hear various tones and frequencies.

Contrast sensitivity testing can detect changes in visual function at times when Snellen visual acuity is normal. This can occur when corneal pathology, cataracts, glaucoma and various other ocular diseases are present.

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Types of Visual Acuity Testing

a. Minimum Visible Acuity : measures brightness discrimination; the person's ability to detect small differences in the brightness of two light sources. Minimum visible acuity is determined by the brightness of the object relative to its background illumination as opposed to the visual angle subtend by the object.

b. Minimum Perceptible Acuity : measures detection discrimination. Minimum perceptible acuity is concerned with simple detection of objects, not their identification or naming. An example of this type of acuity testing is determining if a child can see and grasp a small candy bead held in the examiner's hand.

c. Minimum Separable Acuity : measures the resolution threshold, or smallest visual angle at which two separate objects can be discriminated. Landolt C, and grating acuity are examples of minimum separable tasks.

d. Vernier Acuity (hyper acuity ): a precise form of visual discrimination still under study. Hyper acuity has been coined to classify the high precision (within a few seconds of arc) with which vernier alignment task can be performed. This level of precision is well above resolution or recognition acuity thresholds.

e. Minimum Legible Acuity: measures the individual's ability to recognize progressively smaller objects (letters, numbers or objects) called optotypes. The angle that the smallest recognized letter or symbol subtends on the retina is a measure of visual acuity. This type of acuity testing is used most often clinically.

f. Snellen Acuity uses a notation in which the numerator is the testing distance (in feet or meters) and the denominator is the distance at which a letter subtends the standard visual angle of 5 minutes. A 20/20 letter (6/6 in meters) subtends an angle of 5 minutes when viewed at 20 feet (6 meters).

Question: What are the dimensions of a 20/20 size “E” from an eye chart that is meant to be viewed at 20 feet? What is the visual angle?

Answer: A 20/20 “E” from a chart meant to be viewed at 20 feet is about 9 mm tall. Each leg and space between the legs is about 1.7 mm tall. The 20/20 “E” subtends 5 minutes of visual angle; each leg and space is 1 minute (1/60 degrees) = 0.017 degrees of visual angle.

For the Landolt “C”, the opening in the “C” is about 1.75 mm (1 minute of arc).

Question: How many minutes does the “E” on the 20/20 line of the Snellen eye chart subtend?

Answer: 5 minutes at 20 feet. Snellen eye chart measures the minimum legible acuity.

Question: What is the optimum size of pinhole used to measure “pinhole acuity”?

Answer: The optimum size is 1.2 mm. Larger pinholes do not effectively neutralize refractive error and smaller pinholes markedly increase diffraction and decrease the amount of light entering the eye.

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Multifocal Design

Bifocals are made in two different ways. (Figure 40) One piece and fused lenses.

The one-piece type is made from one piece of glass or plastic. The lens surface is ground with two different curvatures. The shorter radius of curvature in the bifocal area creates the additional power. Fusing two different types of glass together makes fused bifocals. Each type of glass has a different index of refraction, which is not possible with plastic lenses. The segment button has a higher refractive index (flint n = 1.7) than the basic lens (crown n = 1.523).

Most flat top fused segments are designed with the optical center 4 mm below the segment top and produce only a very small amount of image jump (see below). Larger, flat top segments are similar to an executive bifocal, in that they will have little to no image jump.

Click on image to enlarge.

a. Image Jump is produced by the sudden introduction of the prismatic power at the top of a bifocal segment. The object the individual sees in the inferior field suddenly jumps upward when the eye turns down to look at it. If the optical center of the segment is at the top of the segment, there is no image jump. Image jump is worse in glasses with a round top bifocal, because the optical center of the bifocal is farther from the distance lens optical center. A flat top bifocal is better because the optical center of the bifocal is close to the distance optical center.

b. Image Displacement is the prismatic effect induced by the combination of the bifocal type and the power of the distance lens prescription in the reading position. Image displacement is more bothersome than image jump for most people. Most bifocal corrected presbyopes read through a point about 10 mm below the optical center of their distance lenses. If that position is also at the bifocal segment’s optical center, as in most fused flat top bifocals, the bifocal segment produces no prismatic effect at all. The prismatic effect that is there is induced by the distance lens correction, not the segment. However, if the optical center of the bifocal segment is located below or above the reading position, the bifocal will contribute to image displacement at the reading position. The total prismatic displacement will be the sum of that produced by the bifocal and that induced by the distance lens.

A flat top lens is essentially a base up prism, whereas a round top lens is a base down prism at the normal reading spot, 10 mm below the optical center of the lens. A myopic distance lens has base down prismatic power in the reading position; thus, image displacement is worsened with a round top lenses. The prism effects are additive. Similarly, a hyperopic spectacle lens is a base up prism in the reading position; thus a flat top lens makes image displacement an issue.

Most individuals will physiologically adapt, or learn to fuse small vertical deviations. If they cannot, there are several ways to compensate for the problem including, using contact lenses instead of spectacles, prescribing dissimilar segments, or providing "slab-off" prism.

In the past, slab-off prism (base up) was added to the spectacle lens that had the most minus or least plus correction for distance. Now, with modern plastic lenses, slab-off prism is taken off the mold, effectively adding base down prism to the most plus or less minus lens. This is called "reverse slab". Slab off is fabricated by way of bicentric grinding which creates two optical centers in the lens. One optical center is for the distance correction and the other is for the reading correction of the lens.

Question: If a patient comes in wearing glasses: OD +2.00, OS -2.00, and complains of vertical diplopia when reading. Both eyes are reading 5 mm down from the optical center. How much slab-off do you prescribe?

Answer: 2.00 prism diopters base up OS. A slab-off prism is always put in front of the more minus eye because slab off provides base up prism. In this example, the right eye will have induced 1.00 prism diopter base up and the left eye will have induced 1.00 prism diopter base down.

Question: Should a hyperope use a round top or a flat top bifocal?

Answer: A plus lens will have significant image displacement with a flat top lens. Image displacement is lessened with a round top lens. Although image jump will be present, it is less disturbing than image displacement.

Question: Should a myope use a flat top or a round top bifocal?

Answer: A round top lens has significant image displacement with a minus lens. A flat top lens minimizes image displacement and image jump for a myope.

Question: What is the induced prism for an individual wearing +5.00D OU, when reading at the usual reading position of 2mm in and 8mm down from the optical center of his lenses?

Answer: PD = hF, therefore, vertically +5.00D x 0.8 = 4PD BU per eye horizontally +5.00D x 0.2 = 1PD BO per eye

Question: What vertically compensating prism is needed for an individual wearing +5.00D OD and +2.00D OS when they are viewed in the normal reading position of 8mm down from the optical center of the lens?

Answer: +5.00D x 0.8 = 4PD BU and +2.00 x 0.8 = 1.6PD BU, 4-1.6 = 2.4PD BU needed for the left eye.

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Aniseikonia

Aniseikonia…. “may be due to differences in the size of the optical images on the retina or may be anatomically determined by a different distribution in spacing of the retinal elements”. (Duke-Elder, 1963)

Aniseikonia is a term coined by Dr. Walter Lancaster in 1932. It means literally “not equal images (either size, shape, or both)” from the two eyes, as perceived by the patient and is one of the problems most frequently associated with the correction of anisometropia with spectacles. It is an anomaly of the binocular visual process that affects the patient’s perceptual judgment. The most common cause is the differential magnification inherent in the spectacle correction of Anisometropia. This difference in magnification produces different sized retinal images. Approximately 1/3 of the cases of aniseikonia are predicted from anisometropia. Aniseikonia is more commonly caused by unequal refractive errors common in conditions such as monocular aphakia or pseudophakic surprises. However, it is also found with retinal problems and occipital lobe lesions. Aniseikonia occurs in 5-10% of the population with only 1-3% having symptoms.

The perception of an image size disparity between the two eyes is due to the image on the retina not falling on corresponding retinal points. The ocular image is the final impression received in the higher cortical centers, involving the retinal image with modifications imposed by anatomical, physiologic, and perhaps psychological properties of the entire binocular visual apparatus. This is why there are cases of aniseikonia in individuals with emmetropia and isometropia (equal refractive errors).

In general aniseikonia is associated with a false stereoscopic localization and an apparent distortion of objects in space. Aniseikonia can be the cause of asthenopia, diplopia, suppression, poor fusion, headaches, vertigo, photophobia, amblyopia, and strabismus. The differences in size may be overall, that is, the same in all meridians, or meridional, in which the difference is greatest in one meridian and least in the meridian 90° away.

Clinically, aniseikonia usually occurs when the difference in image size between the two eyes approaches 0.75%. Individuals with greater than 4-5% image size difference, have such a large disparity in image size, that they generally do not have binocularity. It is usually assumed that patients can comfortably tolerate up to 1% of aniseikonia in non-astigmatic cases.

A change in refractive correction is always accompanied by some change in the retinal image size and in the conditions under which the patient sees. The magnitude of these changes and the patient’s tolerance determines whether these changes will produce symptoms of discomfort or inefficiency. Persons with normal binocular vision can readily discriminate differences in image size as low as 0.25 to 0.5 percent. For persons with normal binocular vision, a deviation of 4-5x the threshold of discrimination is usually considered significant.

Aniseikonia can be noted when a patient, for the sake of comfort, prefers to use one eye for reading or watching moving objects. If an individual can learn to rely on non-stereoscopic, rather than stereoscopic clues, they may be able to avoid irritation from aniseikonia, even when it is present.

Aniseikonic patients may see an apparent slant of level surfaces, such as tabletops and floors. The effect is more pronounced with objects on the surfaces, for instance, with an irregular pattern carpet on the floor. For high levels of cylinder correction, spherical equivalents may help reduce the aniseikonia.

The magnification for flat trial lens case cylinders is approximately 1.5% per diopter.

The uncorrected refractive myopic eye will have a larger image by 1.5% per diopter and the uncorrected refractive hyperopic eye will have a smaller image by 1.5% per diopter. This holds for anisometropia primarily of refractive origin.

The corrected refractive myopic eye will have a smaller image by 1.5% per diopter and the corrected refractive hyperopic eye will have a larger image by 1.5% per diopter. This holds for anisometropia primarily of refractive origin.

However, since anisometropia may be partially axial, an estimate of 1% per diopter is more clinically useful.

When considering axial versus refractive anisometropia:

If the amount of anisometropia is > 2D – assume it to be axial.

If the amount of anisometropia is < 2D or is in cylinder only – assume it to be refractive.

Spectacle correction of astigmatism produces meridional aniseikonia with accompanying distortion of the binocular spatial sense. Anisometropia is commonly stated to be present if the difference in the refractive correction is 2.00D or more either spherical or astigmatic. However, smaller differences than 2.00D may be significant.

When prescribing aniseikonic lenses, it is important to realize that the size and shape of the final image does not matter, it is only important that the images of each eye match each other. For this reason, instead of magnifying the image of one eye, it may be easier to minify the image of another. This may allow for a more cosmetically acceptable spectacles, or at least lenses that are easier to manufacture, and therefore, less costly.

a. Cylindrical Corrections

Cylindrical corrections in spectacle lenses produce distortion. This is a problem of aniseikonia, which may be solved by prescribing iseikonic spectacle corrections. Iseikonia is when perceived images are the same size. Iseikonic spectacle corrections may be complicated and expensive and the vast majority of practitioners prefer to prescribe cylinders according to cylinder judgment using guidelines that have evolved over the years. Remember the reason for intolerance of an astigmatic spectacle correction is distortion caused by meridional magnification which is more poorly tolerated. Unequal magnification of the retinal image in the various meridians produced monocular distortion manifested by tilted lines or altered shapes of objects. The monocular distortion by itself is rarely a problem. The effect is too small.

Oblique meridional aniseikonia causes a rotary deviation between fused images of vertical lines in the two eyes. The maximum tilting of vertical lines is called the Declination Error. The maximum declination error occurs when the corrected cylinder axis is at 45 or 135°, but even under these conditions, each diopter of correcting cylinder power produces only about 0.4° of tilt. This problem occurs more often with plus cylinder lenses which is why most spectacle lenses are now made in the minus cylinder form. Clinically significant problem begin to occur when the declination approaches 0.3%. Minor degrees of monocular distortion can produce major alterations in binocular spatial perception.

The Total Magnification of a Lens (M T) is found by adding the magnification from its power (M P) and the magnification from its shape (M S). Therefore, total magnification M T = M P + M S.

Magnification from Power (M P) is dependent on the dioptric power of the lens (D V) and its vertex distance (H). If H is measured in cm, the relationship is M P = D VH. From this formula, we see that moving a lens away from the eye increases the magnification of a plus lens and the minification of a minus lens. Moving a lens toward the eye (decreasing the vertex distance) decreases the magnification of a plus lens and the minification of a minus lens. These effects are especially notable with higher powered lenses.

Examples

+10.00D lens @ 10mm and 15mm vertex distance

@ 10mm, M P = D VH = +10.00 x 1.0 = +10

@ 15mm, M P = D VH = +10.00 x 1.5 = +15

-10.00D lens @ 10 and 15 mm vertex distance

@ 10mm, M P = D VH = -10.00 x 1.0 = -10

@ 15mm, M P = D VH = -10.00 x 1.5 = -15

For spectacle lenses remember, as you move a lens closer to the eye, you must add plus power to the lens. Therefore remember CAP = Closer Add Plus.

Magnification from Shape (M S) is dependent on the curvature of the front surface of the lens D 1 and the center thickness of the lens t. The 1.5 in the following equation is the index of refraction (approximately) of glass or plastic. M S = D 1 (t cm/1.5). Therefore, the more curved the lens, the larger the D 1 and the more magnification from shape the lens have. Also, the thicker the lens (t), the more magnification from shape.

Examples

Front curve of a +2.00D lens is +2.00D and +6.00D, Thickness is 2mm.

M S = D 1 (t cm/1.5) = +2.00(0.2/1.5) = +0.27

M S = D 1 (t cm/1.5). = +6.00(0.2/1.5) = +0.80

Front curve of a +2.00D lens is +2.00D and +6.00D, Thickness is 4mm.

M S = D 1 (t cm/1.5) = +2.00(0.4/1.5) = +0.53

M S = D 1 (t cm/1.5). = +6.00(0.4/1.5) = +1.60

Magnification may be reduced by making the front surface power of a lens less positive.

Decreasing center thickness also decreases magnification.

However, a change in either the front curve or the thickness of the lens will also cause the vertex distance (h) to be changed so that the magnification from the power factor (M P) is also affected. If the front curve is changed to give the magnification or minification needed, the back curve must also be changed to maintain the same power of the lens.

If, for example, the front curve is increased, the back curve must also be increased, which increases the vertex distance. If the front curve is flattened, the back curve must be flattened, which causes the vertex distance to decrease. If center thickness is increased to increase the magnification of the lens, but the front curve is left the same, the increase moves the back surface closer to the eye by the amount of the increase, therefore decreasing the vertex distance. On the other hand, it the center thickness is decreased, but the front curve is left the same, the decrease causes the vertex distance to be increased by that amount.

For further review on this subject, go to the THILL Aniseikonia Worksheet in Duane’s Clinical Ophthalmology (Lippincott Williams & Wilkins).

Contact lenses may provide a better solution than spectacles in most patients with anisometropia, particularly children, where fusion may be possible, because it gives the least change in image size from the uncorrected state in refractive ametropia.

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Telescopes

Telescopes are afocal optical systems consisting of two lenses, separated in space, in air. There are two types of telescopic systems, Keplerian and Galilean.

a. Keplerian telescopes (Figure 38) have a weak (+) objective lens and a strong (+) eyepiece lens.

The lenses are separated by the sum of their focal lengths. Keplerian (astronomical) telescopes form an inverted image so they require an erecting lens or prisms to make it a Terrestrial telescope.

Click on image to enlarge.

b. Galilean telescopes (Figure 39) has a weak (+) objective lens and a strong (-) eyepiece lens. The lenses are separated by the difference of their focal lengths. Galilean telescopes form an erect/upright image.

Click on image to enlarge.

The angular magnification of a telescope is equal to the power of the eyepiece divided by the power of the objective.

M_A Telescope = (-) F_E/F_O

The eyepiece in the Galilean telescope has a negative power. Therefore, the magnification given by the equation above is positive, indicating an upright image.
Keplerian telescopes have both positive objective and eyepiece lenses; the magnification is negative, indicating an inverted image.
With any telescope, the secondary focal point of the first lens must coincide with the primary focal point of the second lens. With Galilean telescopes, the second lens is minus and so the primary focal point is virtual.
Galilean telescopes have several practical advantages for low vision work. The image is upright, without the need for image erecting prisms and the device is shorter. Galilean telescopes typically are 2, 3 or 4x in strength, inexpensive, light, and have a large exit pupil, which makes centering less difficult.
4x telescopes and stronger are usually Keplerian in design which gives an optically superior image, but are more expensive with a smaller exit pupil requiring better centering and aiming. Keplerian binoculars, contain prisms to erect the otherwise inverted image.
Galilean telescopes used as surgical loupes, require an add to be combined with the objective lens. The field size is far smaller than that obtained with bifocal spectacles.
Telescopic loupes can produce asthenopia with any type of refractive error. If binocular loupes are not aligned properly, vertical or horizontal phorias can be induced. Adopting a working distance too far inside the focal distance of the “add” can require excessive accommodation, even for a myope.
When viewing a near object through an afocal telescope, the telescope acts as a vergence multiplier. The approximate accommodation required is given by A_oc= M²U, where A_oc = vergence at the eyepiece = accommodation, U = object vergence at the objective = 1/u, M = the magnification of the telescope.

Question: How far apart must a +5D lens and a –10D lens be placed to form a Galilean (afocal) telescope?

Answer: With any telescope, the secondary focal point of the first lens must coincide with the primary focal point of the second lens. With Galilean telescopes, the second lens is minus and so the primary focal point is virtual. To make the secondary focal point (20cm) of the plus lens coincide with the virtual primary focal point of the minus lens (10cm), the lenses must be separated by 20 – 10 = 10cm.

Question: You are a –5.00D spectacle corrected myope stranded on a small island with your significant other. Unfortunately, your companion has broken your glasses (which had an 11mm vertex distance). The only lens available to you is a –55D Hruby lens, which your companion had.

How many cm from the eye should you hold the lens to fully correct your refractive error?

Answer: a) first, locate the far point of your eye. The far point of the lens is 0.211m in front of the eye (F = 1/-5 which equals 0.20m + 0.011m vertex distance = 0.211m = 211mm). The Hruby lens has a power of –55D which means, its focal point is 1/55 which equals 0.018m or 18mm away from the lens. To correct the refractive error, the focal point of the lens should coincide with the far point of the eye. Therefore, it should be 18mm away from the far point or 211 - 18 = 193mm in front of the eye.

Question: Why would you not be able to read the 20/20 line with this correction?

Answer: b) The problem is magnification. This configuration turns the combination of the eye and its corrective lens into a reverse Galilean telescope, where the eyepiece is approximately +5D (the extra power of the myopic eye) and the objective lens is –55D. The resulting magnification is (-) 5/-55, which equals 0.1x. Thus, the 20/20 line, while in focus, subtends 1/10 of the angle it would in the eye of an emmetrope. Therefore, the best distance acuity obtainable is only about 20/200, assuming an otherwise normal eye.

It should be noted that properly corrected patients with high myopia might not be able to read 20/20 through their spectacle lenses even in the absence of other pathology. This is because the longer axial length commonly found in higher amounts of myopia, results in greater separation of the photoreceptors, which decrease the visual potential of the eye.

Question: You and a stowaway are ship wrecked on a lost island with your trial lens set, but only a few lenses survive the shipwreck. Your are left with a –20D, +4D, +5D, and a +20D. You build a viewing device to search the horizon for ships using the –20D and the +4D lens. The stowaway, Dr. Smith, uses the +20D and the +5D lens. Dr. Smith complains that his viewing device is inferior.

What did each of you build?
How did you position the lenses?
Why is Dr. Smith plotting to steal your telescope?

Answer: You use the –20D lens as the eyepiece and the +4D lens as the objective lens of a Galilean telescope. The secondary focal point of the plus lens should coincide with the primary focal point of the minus lens, thus the lenses are 25cm – 5cm = 20cm apart. Dr. Smith built a second telescope (astronomical) using the +20.00 diopter lens as the eyepiece and the +5D lens as the objective. The secondary focal point of the objective lens needs to coincide with the primary focal point of the eye piece lens, so he positions them 5 cm + 20 cm = 25 cm apart. Dr. Smith does not like having to stretch his arms the additional 5cm.

Question: Which telescope above will provide more magnification?

Answer: The angular magnification of a telescope is equal to the power of the eyepiece divided by the power of the objective. Magnification of the Galilean telescope is (-)-20/4 = 5x. The magnification of the astronomical telescope is (-) 20/5 = -4x. Therefore, Dr. Smith’s telescope will provide less magnification.

Question: Will the telescopes have an erect or inverted image?

Answer: The Galilean telescope will produce an upright image of the, hopefully approaching ships, while Dr. Smith’s astronomical telescope will produce an inverted image.

Question: How is the Galilean telescope modified when used as a surgical loupe?

Answer: The binocular surgical loupe is just a short Galilean telescope with an add to bring the working distance in from infinity. Powerful lenses are used so that the tube length of the telescope is kept to a minimum. A +25D object, combined with a –50D eyepiece, would provide 2x magnification. The additional add needed to focus the telescope at near is the reciprocal of the working distance in meters. Example: for a 25cm working distance, the add would be 100/25 = 4D.

Question:

How long is the 2x Galilean telescope described above?
What if it were made using a +5D objective lens and a –10D eyepiece lens?

Answer:

a) The focal length of the –50D lens is 1/50 =2cm. The +25D lens has a 100/25 = 4cm focal length. Thus, the telescope is 4 – 2 = 2cm long.

b) The +5/-10 telescope is 20 – 10 = 10cm long.

Question: You are working with a 2x afocal Galilean telescope that is fabricated with a +8D objective lens. We know that the ocular lens must be –16D and the 2 lenses are separated by 6.25cm (objective lens 1/8 = 12.5cm, ocular lens 1/16 = 6.25cm, tube length = 12.5 – 6.25 = 6.25cm).

When viewing at infinity by an uncorrected 4D hyperope, the ocular has an effective power of?

Answer: +4 is needed to correct for the hyperopic refractive error. This power must be taken from the ocular lens of the telescope and so the effective power of the ocular lens becomes -16 - 4 = - 20D. (The -20D effective ocular lens combined with the +4D correction lens gives us the -16D the ocular lens of the telescope actually has).

Question: For the telescope to remain afocal, the tube length must be?

Answer: The objective lens focal length is still 12.5cm, ocular lens is now 1/20 = 5cm. Therefore 12.5 - 5 = 7.5cm

Question: What is the telescopic power now?

Answer: M_A Telescope = (-)F_E/F_O = (-)-20/8 = 2.5x

Question: When viewed by an uncorrected 4D myope, the ocular has an effective power of?

Answer: The uncorrected –4D of the eye must act as a correcting lens so the ocular now has an effected power of -16 + 4 = -12D. (The -12D effective ocular lens combined with the -4D correction lens gives us the -16D the ocular lens of the telescope actually has).

Question: To make the telescope afocal, the tube length must be?

Answer: The objective lens focal length is still 12.5cm, ocular lens is now 1/12 = 8.33cm. Therefore 12.5 – 8.33 = 4.17cm.

Question: What is the telescopic power now?

Answer: M_A Telescope = (-)F_E/F_O = (-)-12/8 = 1.5x.

Question: An afocal Keplerian telescope has an objective lens that is +7D and an eyepiece lens that is +17.50D. What is the separation between the lenses?

Answer: The focal length of the objective lens is 1/7 = 14.3cm. The focal length of the eyepiece lens is 1/17.5 = 5.7cm. Therefore, the lens separation is 14.3 + 5.7 = 20cm

Question: What is the power of the lenses?

Answer: M_A Telescope = (-)F_E/F_O = (-)17.5/7 = -2.5x

Question: A patient uses a focusable 2x Keplerian telescope that has a +8D objective lens. What is the power and tube length of the afocal telescope when used by an emmetropic patient and focused for distance viewing?

Answer: The power is 2x because it is being used by an emmetrope.

The eyepiece lens power would be +16D (M_A Telescope = (-)F_E/F_O = (-)X/8 = -2x)

To find the tube length, the focal length of would be 1/8 = 12.5mm for the objective lens and 1/16 = 6.25mm for the eyepiece lens. Therefore, the tube length would be 12.5 + 6.25 = 18.75mm.

Question: When used by a 4D hyperope in a similar fashion?

Answer: For the uncorrected 4D hyperope, the eyepiece lens now has an effective power of 16 – 4 = 12D. (The +12D effective ocular lens combined with the +4D correction lens gives us the +16D the ocular lens of the telescope actually has). The power of the telescope would become (-) 12/8 = -1.5x. The tube length would be 20.83mm. (1/12 = 8.33mm + 12.5 = 20.83mm)

Question: When used by a 4D myope in a similar fashion?

Answer: For the uncorrected 4D myope, the eyepiece lens now has an effective power of 16 + 4 = 20D. (The +20D effective ocular lens combined with the -4D correction lens gives us the +16D the ocular lens of the telescope actually has). The power of the telescope would be (-)20/8 = -2.5x. The tube length would be 17.5mm (1/20 = 5mm + 12.5mm = 17.5)

Question: A focusable Galilean telescope with a +20D objective lens with a –40D ocular lens is dispensed to a patient for a variety of tasks.

a) What is the magnification of the telescope at distance?

Answer: M = (-) -40/20 = 2x

b) What tube length is required for viewing distance objects?

Answer: 1/20 = 5cm, 1/40 = 2.5cm, 5 - 2.5 = 2.5cm

c) What is the tube length required for viewing numbers that are 50cm away in an elevator?

Answer: The objective power would now be +20 + (-2) = +18D. 100/+18 = +5.55, +5.55 – 2.5 = 3.05cm.

Important to remember – 20 inches = 50 cm. To find the vergences when working in inches, use the formula V = 40/distance (inches) = 100/distance (cm)

Question: A 3x afocal Galilean telescope has a separation between the objective and ocular lens of 2cm. When viewing an object 25cm in front of the objective lens, what power reading cap would eliminate the need to accommodate for this target distance?

Answer: 100/25 = +4D

Question: A low vision patient needs a 10D add to read the text on a computer monitor but the 10cm working distance is too close. He wants to work at a 25cm distance. What theoretical telescope and reading cap combination would be needed?

Answer: 25/10 = 2.5x, 100/25 = +4D, therefore you would need a 2.5x telescope with a +4D reading cap.

Question: What is the equivalent lens that should be prescribed to replace a 4x telescope with a +2.50D reading cap (F_RC) so the patient has the same resolution ability through the lens that he has through the telemicroscopic system?

Answer: F_e = (F_RC) (power of telescope) = 2.5D (4) = 10D

Question: If a patient is able to read enlarged sheet music with a 3x telescope and a cap focus for 16 inches, what telescope and cap are needed to read the same sheet music set at 32 inches?

Answer: 32/16 = 2x additional magnification, 40/32 = 1.25D, Therefore you would need a 6x telescope with 1.25D cap