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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.