Skip to main content

NBEO Part I formula sheet

The optics and vision science formulas for NBEO Part I, grouped by the areas of the NBEO content outline with the most scored items first. Every formula names the textbook chapter it comes from.

Free NBEO Part I formula sheet

The optics and vision science formulas on one printable sheet, front and back. Every symbol is defined with its units, and every formula names its textbook chapter.

Download PDFMore handouts

Vergence and refraction

Ametropia · 29–39 of 350 scored items

Vergence and the vergence equation

GVO 3
L=\displaystyle L ={}nl\displaystyle \frac{n}{l}{}
L′=\displaystyle L' ={}L+\displaystyle L +{}F\displaystyle F{}
L′=\displaystyle L' ={}n′l′\displaystyle \frac{n'}{l'}{}

L,L′L, L' = object and image vergence (D); l,l′l, l' = object and image distance (m); n,n′n, n' = index of the object and image media; FF = power of the lens or surface (D)

Index of refraction

GVO 1
n=\displaystyle n ={}cv\displaystyle \frac{c}{v}{}

nn = index of refraction, which varies with wavelength; cc = speed of light in a vacuum; vv = speed of light in the medium

Snell's law

GVO 1
nsin⁡θ=\displaystyle n \sin\theta ={}n′sin⁡θ′\displaystyle n' \sin\theta'{}

n,n′n, n' = index of the first and second medium; θ,θ′\theta, \theta' = angles of incidence and refraction, from the normal

Critical angle

GPVO 11
sin⁡θc=\displaystyle \sin\theta_c ={}n′n\displaystyle \frac{n'}{n}{}
(n>n′)\displaystyle (n > n'){}

θc\theta_c = critical angle of incidence; nn = index of the denser medium the light starts in; n′n' = index of the second medium

Beyond θc\theta_c the light totally internally reflects.

Power of a spherical surface

GVO 2
F=\displaystyle F ={}n′−nr\displaystyle \frac{n' - n}{r}{}

FF = surface power (D); n,n′n, n' = index before and after the surface; rr = radius of curvature (m), positive when the center of curvature lies to the right

Thin lens in air

GVO 4 · MOO 4
F=\displaystyle F ={}F1+\displaystyle F_1 +{}F2=\displaystyle F_2 ={}(n−1)(1r1−1r2)\displaystyle (n - 1)\left(\frac{1}{r_1} - \frac{1}{r_2}\right){}

F1,F2F_1, F_2 = front and back surface powers (D); nn = lens index; r1,r2r_1, r_2 = front and back radii (m), signed, so a convex back surface has r2<0r_2 < 0

This nominal power ignores thickness.

Apparent depth

GPVO 8
apparent distance=\displaystyle \text{apparent distance} ={}real distance×\displaystyle \text{real distance} \times{}n2n1\displaystyle \frac{n_2}{n_1}{}

n1n_1 = index of the medium holding the object; n2n_2 = index of the observer's medium

From air, apparent depth is real depth divided by nn.

Lateral magnification

GVO 4
m=\displaystyle m ={}LL′\displaystyle \frac{L}{L'}{}
m=\displaystyle m ={}l′l\displaystyle \frac{l'}{l}{}

mm = image size over object size; L,L′L, L' = object and image vergence (D); l,l′l, l' = object and image distance

The l′/ll'/l form needs one medium on both sides. For a mirror, m=−l′/lm = -l'/l.

Mirror power

GVO 16
F=\displaystyle F ={}−2nr\displaystyle \frac{-2n}{r}{}
F=\displaystyle F ={}−1f′\displaystyle \frac{-1}{f'}{}
r=\displaystyle r ={}2f′\displaystyle 2f'{}

FF = mirror power (D); nn = index of the medium in front of the mirror, 1 in air; rr = radius of curvature (m); f′f' = focal length (m)

A concave mirror has r<0r < 0, so F>0F > 0. Reflected light travels right to left, so L′=−1/l′L' = -1/l'.

Thick lenses and effective power

Equivalent power

GVO 6 · GPVO 16
Fe=\displaystyle F_e ={}F1+\displaystyle F_1 +{}F2−\displaystyle F_2 -{}tnF1F2\displaystyle \frac{t}{n} F_1 F_2{}

FeF_e = equivalent power (D); F1,F2F_1, F_2 = front and back surface powers (D); tt = center thickness (m); nn = lens index

Two thin lenses dd meters apart in air give Fe=F1+F2−dF1F2F_e = F_1 + F_2 - dF_1F_2.

Back vertex power

GVO 6
Fv=\displaystyle F_v ={}F11−tnF1+\displaystyle \frac{F_1}{1 - \frac{t}{n} F_1} +{}F2\displaystyle F_2{}

FvF_v = back vertex power (D); F1,F2F_1, F_2 = front and back surface powers (D); tt = center thickness (m); nn = lens index

Front vertex (neutralizing) power swaps the surfaces, Fn=F2/(1−(t/n)F2)+F1F_n = F_2 / (1 - (t/n)F_2) + F_1.

Effective power at a new position

GPVO 6
Fx=\displaystyle F_x ={}F1−dF\displaystyle \frac{F}{1 - dF}{}

FF = lens power at its original position (D); FxF_x = power that does the same job at the new position (D); dd = distance moved (m), positive toward the eye

Astigmatism

Transposition

GPVO 10
S  /  C×\displaystyle S \;/\; C \times{}θ  ⟹\displaystyle \theta \;\Longrightarrow{}  (S+C)  /  −C×\displaystyle \; (S + C) \;/\; {-C} \times{}(θ±90)\displaystyle (\theta \pm 90){}

SS = sphere (D); CC = cylinder (D); θ\theta = cylinder axis (degrees)

Add 90 to an axis of 90 or less, and subtract 90 from an axis above 90.

Spherical equivalent

GPVO 10
SE=\displaystyle SE ={}S+\displaystyle S +{}C2\displaystyle \frac{C}{2}{}

SESE = spherical equivalent, the dioptric position of the circle of least confusion (D); SS = sphere (D); CC = cylinder (D)

Power in an oblique meridian

IOO 5
Fθ=\displaystyle F_\theta ={}S+\displaystyle S +{}Csin⁡2θ\displaystyle C \sin^2\theta{}

FθF_\theta = approximate power in the meridian θ\theta (D); SS = sphere (D); CC = cylinder (D); θ\theta = angle from the cylinder axis

Javal's rule

GVO app. C
ocular astig.=\displaystyle \text{ocular astig.} ={}1.25×\displaystyle 1.25 \times{}corneal astig.+\displaystyle \text{corneal astig.} +{}(−0.50×090)\displaystyle (-0.50 \times 090){}

corneal astig. = keratometric astigmatism, written as minus cylinder; ocular astig. = predicted refractive astigmatism; −0.50×090-0.50 \times 090 = allowance for against-the-rule lenticular astigmatism

The eye

Far point

GPVO 6
far point (m)=\displaystyle \text{far point (m)} ={}1refractive error at the cornea (D)\displaystyle \frac{1}{\text{refractive error at the cornea (D)}}{}

refractive error = correcting power at the cornea, signed (D)

A myope's far point lies in front of the eye. A hyperope's is virtual and lies behind it.

Blur and defocus

GPVO 19
δ=\displaystyle \delta ={}bdmV\displaystyle \frac{b}{d_m} V{}
angular blur (rad)=\displaystyle \text{angular blur (rad)} ={}dm δ\displaystyle d_m \, \delta{}

δ\delta = defocus (D); bb = blur circle diameter on the retina (m); dmd_m = pupil diameter (m); VV = power of the eye, about 60 D

Retinal image size

GPVO 6
∣I∣=\displaystyle |I| ={}16.67 mm×\displaystyle 16.67 \text{ mm} \times{}∣O∣u\displaystyle \frac{|O|}{u}{}

II = retinal image size; OO = object size; uu = object distance, in the units of OO; 16.67 mm16.67 \text{ mm} = nodal point to retina in a +60.00+60.00 D reduced eye

Physical optics

Waves in a medium

GPVO 21
v=\displaystyle v ={}fλ=\displaystyle f\lambda ={}cn\displaystyle \frac{c}{n}{}
λm=\displaystyle \lambda_m ={}λn\displaystyle \frac{\lambda}{n}{}

vv = speed in the medium; ff = frequency (Hz); λ\lambda = wavelength in a vacuum; λm\lambda_m = wavelength in the medium; cc = speed of light in a vacuum; nn = index of the medium

Frequency does not change between media.

Photon energy

VP 2 · GPVO 24
E=\displaystyle E ={}hν=\displaystyle h\nu ={}hcλ\displaystyle \frac{hc}{\lambda}{}
E (eV)=\displaystyle E \text{ (eV)} ={}1239λ (nm)\displaystyle \frac{1239}{\lambda \text{ (nm)}}{}

EE = photon energy; hh = Planck's constant, 6.63×10−346.63 \times 10^{-34} J·s; ν\nu = frequency (Hz); cc = 3.00×1083.00 \times 10^{8} m/s; λ\lambda = wavelength

Fresnel reflection

GPVO 14
R=\displaystyle R ={}(n′−nn′+n)2\displaystyle \left(\frac{n' - n}{n' + n}\right)^2{}

RR = fraction reflected at one surface, at normal incidence; n,n′n, n' = index on each side of the surface

Quarter-wave antireflection coating

GPVO 21
nfilm=\displaystyle n_{\text{film}} ={}nsurround nlens\displaystyle \sqrt{n_{\text{surround}} \, n_{\text{lens}}}{}
t=\displaystyle t ={}λ4 nfilm\displaystyle \frac{\lambda}{4 \, n_{\text{film}}}{}

nfilmn_{\text{film}} = coating index; nsurroundn_{\text{surround}} = index outside the coating, 1 in air; nlensn_{\text{lens}} = lens index; tt = coating thickness for light at normal incidence; λ\lambda = design wavelength in a vacuum

Rayleigh criterion

GPVO 22
θmin⁡=\displaystyle \theta_{\min} ={}1.22 λd\displaystyle \frac{1.22 \, \lambda}{d}{}

θmin⁡\theta_{\min} = smallest resolvable angle (rad); λ\lambda = wavelength; dd = diameter of the circular aperture, in the units of λ\lambda

Double slit and grating

GPVO 21 · GPVO 22
x=\displaystyle x ={}mλth\displaystyle \frac{m \lambda t}{h}{}
hsin⁡θ=\displaystyle h \sin\theta ={}mλ\displaystyle m\lambda{}

xx = distance of bright fringe mm from the center; mm = fringe order (0, 1, 2, …); λ\lambda = wavelength; tt = slit to screen distance; hh = slit or grating line spacing; θ\theta = angle to fringe mm

The xx form needs small angles.

Rayleigh scattering

GPVO 23
Iscat∝\displaystyle I_{\text{scat}} \propto{}1λ4\displaystyle \frac{1}{\lambda^4}{}

IscatI_{\text{scat}} = light scattered by particles much smaller than the wavelength; λ\lambda = wavelength

Photometry

Illuminance from a point source

VP 4
E=\displaystyle E ={}Icos⁡θd2\displaystyle \frac{I \cos\theta}{d^2}{}

EE = illuminance (lux); II = luminous intensity (cd); dd = distance from the source (m); θ\theta = angle between the surface normal and the line to the source

Luminous flux

VP 4
lumens=\displaystyle \text{lumens} ={}683×\displaystyle 683 \times{}V(λ)×\displaystyle V(\lambda) \times{}watts\displaystyle \text{watts}{}

V(λ)V(\lambda) = photopic luminous efficiency, 1 at 555 nm; watts = radiant power at that wavelength

For mixed light, sum wavelength by wavelength.

Retinal illuminance

VP 4 · EYE 5
trolands=\displaystyle \text{trolands} ={}L×\displaystyle L \times{}A\displaystyle A{}

LL = luminance (cd/m²); AA = pupil area (mm²)

Lens form and thickness

Ophthalmic Optics / Spectacles · 17–25 items

Sagitta

GVO 11 · SOD 13
s=\displaystyle s ={}r−\displaystyle r -{}r2−y2  ≈\displaystyle \sqrt{r^2 - y^2} \;\approx{}  y22r\displaystyle \; \frac{y^2}{2r}{}

ss = sag; rr = radius of curvature; yy = half the chord

Use one unit throughout. The approximation needs yy much smaller than rr.

Center and edge thickness

GVO 11
ct=\displaystyle ct ={}et+\displaystyle et +{}s1−\displaystyle s_1 -{}s2\displaystyle s_2{}

ct,etct, et = center and edge thickness; s1,s2s_1, s_2 = front and back sags over the same chord, unsigned

This applies to a meniscus lens.

Lens clock reading

SOD 13
Fsurface=\displaystyle F_{\text{surface}} ={}Fclock×\displaystyle F_{\text{clock}} \times{}n−10.53\displaystyle \frac{n - 1}{0.53}{}

FsurfaceF_{\text{surface}} = true surface power (D); FclockF_{\text{clock}} = lens clock reading (D); nn = actual lens index; 0.530.53 = n−1n - 1 for the index 1.53 the clock assumes

Spectacle magnification

GPVO 18
SM=\displaystyle SM ={}11−tnF1×\displaystyle \frac{1}{1 - \frac{t}{n} F_1} \times{}11−hFv\displaystyle \frac{1}{1 - hF_v}{}

SMSM = spectacle magnification; F1F_1 = front surface power (D); tt = center thickness (m); nn = lens index; FvF_v = back vertex power (D); hh = vertex distance plus 3 mm, lens to entrance pupil (m)

The first factor is the shape factor and the second is the power factor.

Prism

Prism diopter

GVO 10
P=\displaystyle P ={}100tan⁡d=\displaystyle 100 \tan d ={}displacement (cm)distance (m)\displaystyle \frac{\text{displacement (cm)}}{\text{distance (m)}}{}

PP = prism power (Δ\Delta); dd = angle of deviation

Deviation by a thin prism

GVO 10 · SOD 15
d=\displaystyle d ={}a(n−1)\displaystyle a(n - 1){}

dd = minimum angle of deviation; aa = apex angle, under about 10°, in the units of dd; nn = prism index

Prentice's rule

GVO 10
Δ=\displaystyle \Delta ={}cF\displaystyle cF{}

Δ\Delta = induced prism (prism diopters); cc = distance from the optical center (cm); FF = lens power in that meridian (D)

Decentration for prism

SOD 16
c=\displaystyle c ={}ΔF\displaystyle \frac{\Delta}{F}{}

cc = decentration (cm); Δ\Delta = prism wanted (prism diopters); FF = lens power in that meridian (D)

Decenter a plus lens toward the base you want and a minus lens away from it.

Resolving and combining prism

SOD 15 · IOO 8
H=\displaystyle H ={}Pcos⁡θ\displaystyle P\cos\theta{}
V=\displaystyle V ={}Psin⁡θ\displaystyle P\sin\theta{}
R=\displaystyle R ={}H2+V2\displaystyle \sqrt{H^2 + V^2}{}
tan⁡θ=\displaystyle \tan\theta ={}VH\displaystyle \frac{V}{H}{}

PP = oblique prism to resolve (Δ\Delta); θ\theta = base direction, 360° notation; H,VH, V = horizontal and vertical components (Δ\Delta); RR = prism that combines HH and VV

Find tan⁡−1(V/H)\tan^{-1}(V/H) without signs, then set the quadrant: I θ\theta, II 180−θ180 - \theta, III 180+θ180 + \theta, IV 360−θ360 - \theta.

Prism thickness difference

IOO 8
g=\displaystyle g ={}Δ⋅d100 (n−1)\displaystyle \frac{\Delta \cdot d}{100 \, (n - 1)}{}

gg = thickness difference, base to apex; Δ\Delta = prism power (prism diopters); dd = lens size along the base-apex line, and gg takes its units; nn = lens index

Multifocals and materials

Vertical imbalance at the reading level

SOD 21
VI=\displaystyle VI ={}c (FR,90−FL,90)\displaystyle c \, (F_{R,90} - F_{L,90}){}
F90≈\displaystyle F_{90} \approx{}S+\displaystyle S +{}Ccos⁡2(axis)\displaystyle C \cos^2(\text{axis}){}

VIVI = vertical imbalance (Δ\Delta); cc = drop from the distance optical centers to the reading level (cm); F90F_{90} = power in the 90° meridian, signed (D); FR,90,FL,90F_{R,90}, F_{L,90} = that power for the right and left lens; S,C,axisS, C, \text{axis} = each lens's prescription

Image jump

SOD 19
jump (Δ)=\displaystyle \text{jump } (\Delta) ={}c×\displaystyle c \times{}add\displaystyle \text{add}{}

cc = seg top to seg optical center (cm), half the diameter for a round seg; add = add power (D)

Distance power does not change jump.

Abbe value and lateral color

MOO 1
νd=\displaystyle \nu_d ={}nd−1nF−nC\displaystyle \frac{n_d - 1}{n_F - n_C}{}
TCA (Δ)=\displaystyle \text{TCA } (\Delta) ={}cFνd\displaystyle \frac{cF}{\nu_d}{}

νd\nu_d = Abbe value; nd,nF,nCn_d, n_F, n_C = index at 587.6, 486.1 and 656.3 nm; TCA = transverse chromatic aberration (Δ\Delta); cc = distance from the optical center (cm); FF = lens power (D)

Malus's law

SOD 22 · GPVO 23
I=\displaystyle I ={}I0cos⁡2θ\displaystyle I_0 \cos^2\theta{}

II = transmitted intensity; I0I_0 = intensity of the already polarized incident light; θ\theta = angle between its polarization and the transmission axis

An ideal polarizer passes 50% of unpolarized light.

Brewster's angle

SOD 22
tan⁡θB=\displaystyle \tan\theta_B ={}n′n\displaystyle \frac{n'}{n}{}

θB\theta_B = angle of incidence at which the reflected light is fully polarized; nn = index of the medium the light comes from; n′n' = index of the reflecting medium

Accommodation and vergence

Accommodation / Vergence / Oculomotor Function · 10–18 items

Amplitude of accommodation

GPVO 6
A=\displaystyle A ={}Ufp−\displaystyle U_{fp} -{}Unp\displaystyle U_{np}{}

AA = amplitude of accommodation (D); Ufp,UnpU_{fp}, U_{np} = far point and near point vergences at the cornea (D)

Hofstetter's formulas

CPOE 2
min⁡=\displaystyle \min ={}15−\displaystyle 15 -{}0.25 age\displaystyle 0.25\,\text{age}{}
avg=\displaystyle \text{avg} ={}18.5−\displaystyle 18.5 -{}0.30 age\displaystyle 0.30\,\text{age}{}
max⁡=\displaystyle \max ={}25−\displaystyle 25 -{}0.40 age\displaystyle 0.40\,\text{age}{}

age = age in years; min⁡,avg,max⁡\min, \text{avg}, \max = expected minimum, average and maximum amplitude (D)

Tentative add

CPOE 3
add=\displaystyle \text{add} ={}1working distance (m)−\displaystyle \frac{1}{\text{working distance (m)}} -{}amplitude2\displaystyle \frac{\text{amplitude}}{2}{}

working distance = near working distance (m), unsigned; amplitude = push-up amplitude (D)

Refine the add by balancing NRA and PRA.

Meter angle

CMBV 23
MA=\displaystyle \text{MA} ={}1d (m)\displaystyle \frac{1}{d \text{ (m)}}{}
Δ=\displaystyle \Delta ={}MA×\displaystyle \text{MA} \times{}PD (cm)\displaystyle \text{PD (cm)}{}

MA = meter angles; dd = fixation distance (m); PD = interpupillary distance (cm); Δ\Delta = convergence demand (prism diopters)

Calculated AC/A

CMBV 1
AC/A=\displaystyle \text{AC/A} ={}PD (cm)+\displaystyle \text{PD (cm)} +{}NFD (m)×\displaystyle \text{NFD (m)} \times{}(Hn−Hf)\displaystyle (H_n - H_f){}

AC/A = accommodative convergence per diopter of accommodation (Δ\Delta/D); PD = interpupillary distance (cm); NFD = near fixation distance (m); Hn,HfH_n, H_f = near and far phorias (Δ\Delta), eso plus and exo minus

Gradient AC/A

CMBV 1
AC/A=\displaystyle \text{AC/A} ={}phoria without lens−phoria through lenslens power\displaystyle \frac{\text{phoria without lens} - \text{phoria through lens}}{\text{lens power}}{}

AC/A = accommodative convergence per diopter of accommodation (Δ\Delta/D); phoria = in Δ\Delta, eso plus and exo minus; lens power = the added lens, signed (D)

Visual perception

Perceptual Function / Color Vision · 10–18 items

Acuity

VP 7
MAR (arcmin)=\displaystyle \text{MAR (arcmin)} ={}Snellen denominatornumerator\displaystyle \frac{\text{Snellen denominator}}{\text{numerator}}{}
logMAR=\displaystyle \text{logMAR} ={}log⁡10MAR\displaystyle \log_{10} \text{MAR}{}

MAR = minimum angle of resolution (arcmin)

A 20/20 letter subtends 5 arcmin, and its detail subtends 1 arcmin.

Contrast

EYE 5 · VP 7
CMichelson=\displaystyle C_{\text{Michelson}} ={}Lmax⁡−Lmin⁡Lmax⁡+Lmin⁡\displaystyle \frac{L_{\max} - L_{\min}}{L_{\max} + L_{\min}}{}
CWeber=\displaystyle C_{\text{Weber}} ={}ΔLLb\displaystyle \frac{\Delta L}{L_b}{}

Lmax⁡,Lmin⁡L_{\max}, L_{\min} = peak and trough luminance; ΔL\Delta L = target luminance minus background luminance; LbL_b = background luminance

Use Michelson contrast for gratings and Weber contrast for a target on a uniform background.

Weber and De Vries–Rose laws

VP 3
ΔII=\displaystyle \frac{\Delta I}{I} ={}K\displaystyle K{}
ΔI∝\displaystyle \Delta I \propto{}I\displaystyle \sqrt{I}{}

ΔI\Delta I = threshold increment; II = background intensity; KK = Weber fraction, about 0.015 photopic and 0.14 scotopic

De Vries–Rose holds on very dim backgrounds and Weber on brighter ones.

Ricco's law

VP 3
I×\displaystyle I \times{}A=\displaystyle A ={}K\displaystyle K{}

II = threshold intensity; AA = stimulus area; KK = a constant

It holds inside the critical diameter, about 10 arcmin scotopic and smaller photopic.

Bloch's law

VP 3
I×\displaystyle I \times{}t=\displaystyle t ={}K\displaystyle K{}

II = threshold intensity; tt = stimulus duration; KK = a constant

It holds inside the critical duration, about 100 ms scotopic and 10 to 50 ms photopic.

Ferry–Porter law

ADLER 37
CFF=\displaystyle \text{CFF} ={}klog⁡L+\displaystyle k \log L +{}b\displaystyle b{}

CFF = critical flicker fusion frequency (Hz); LL = luminance; k,bk, b = constants

It holds only under photopic conditions, and rod flicker fusion stays near 20 Hz.

CIE chromaticity coordinates

VP 5
x=\displaystyle x ={}XX+Y+Z\displaystyle \frac{X}{X + Y + Z}{}
y=\displaystyle y ={}YX+Y+Z\displaystyle \frac{Y}{X + Y + Z}{}
x+\displaystyle x +{}y+\displaystyle y +{}z=\displaystyle z ={}1\displaystyle 1{}

X,Y,ZX, Y, Z = tristimulus values; x,y,zx, y, z = chromaticity coordinates

The chromaticity diagram plots xx against yy.

Stereopsis

Amblyopia / Strabismus · 9–17 items

Stereoacuity

ADLER 36
η=\displaystyle \eta ={}a Δbd2\displaystyle \frac{a \, \Delta b}{d^2}{}

η\eta = stereo threshold (rad), times 206,265 for arcsec; aa = interpupillary distance; Δb\Delta b = depth interval; dd = viewing distance, in the units of aa and Δb\Delta b

Contact lenses

Contact Lenses · 8–14 items

Keratometer conversion

GPVO 14
K (D)=\displaystyle K \text{ (D)} ={}337.5r (mm)\displaystyle \frac{337.5}{r \text{ (mm)}}{}

KK = corneal power (D); rr = anterior corneal radius (mm)

337.5 comes from the keratometric index, (1.3375−1)×1000(1.3375 - 1) \times 1000.

Tear lens

CLOP 7 · MOO 7
tear lens (D)=\displaystyle \text{tear lens (D)} ={}base curve (D)−\displaystyle \text{base curve (D)} -{}flat K (D)\displaystyle \text{flat K (D)}{}

base curve = back optic zone radius, as power (D); flat K = flattest keratometry reading (D)

Steeper than K adds minus and flatter adds plus (SAM–FAP), about 0.25 D per 0.05 mm.

Residual astigmatism

CLOP 11
RA=\displaystyle \text{RA} ={}refractive astigmatism−\displaystyle \text{refractive astigmatism} -{}corneal astigmatism\displaystyle \text{corneal astigmatism}{}

RA = residual astigmatism, what a spherical rigid lens leaves uncorrected; refractive astigmatism = cylinder in the refraction; corneal astigmatism = keratometric cylinder

Plain subtraction works when the axes match.

IOL power

Lens / Cataract / IOL · 7–13 items

SRK formula

AOOT 6
P=\displaystyle P ={}A−\displaystyle A -{}2.5L−\displaystyle 2.5L -{}0.9K\displaystyle 0.9K{}

PP = IOL power for emmetropia (D); AA = lens constant; LL = axial length (mm); KK = average keratometry (D)

Low vision

Low Vision · 5–9 items

Magnifier

GVO 13
M=\displaystyle M ={}F4\displaystyle \frac{F}{4}{}

MM = effective magnification (×); FF = magnifier power (D)

It assumes a 25 cm reference distance, with the object at the focal point.

Relative distance magnification

GPVO 17
M=\displaystyle M ={}u1u2\displaystyle \frac{u_1}{u_2}{}

MM = relative distance magnification; u1u_1 = original viewing distance; u2u_2 = new viewing distance

Telescope

GVO 14
M=\displaystyle M ={}−FeFo\displaystyle -\frac{F_e}{F_o}{}
d=\displaystyle d ={}1Fo+\displaystyle \frac{1}{F_o} +{}1Fe\displaystyle \frac{1}{F_e}{}

MM = angular magnification; Fo,FeF_o, F_e = objective and eyepiece powers (D); dd = tube length (m)

A Galilean (Fe<0F_e < 0) gives M>0M > 0 and an erect image.

Kestenbaum add

AOA table 6
starting add (D)=\displaystyle \text{starting add (D)} ={}Snellen denominatornumerator\displaystyle \frac{\text{Snellen denominator}}{\text{numerator}}{}

numerator, denominator = of the best-corrected distance Snellen acuity

For standard print, 20/200 gives +10.00+10.00 D.

Sources

GVO Geometrical and Visual Optics: A Clinical Introduction, 3rd ed.
ch. 1, Basic Terms and Concepts
ch. 2, Refraction at Spherical Surfaces
ch. 3, The Vergence Relationship
ch. 4, Thin Lenses
ch. 6, Optical Properties of Thick Lenses
ch. 10, Prisms
ch. 11, Lens Thickness
ch. 13, Magnifying Lenses and Electronic Magnification
ch. 14, Telescopes
ch. 16, Reflection
Appendix C, Javal's Rule
GPVO Geometric, Physical, and Visual Optics
ch. 6, Thin Lens Eye Models
ch. 8, Plane Refracting Interfaces and Reduced Systems
ch. 10, Astigmatism: On Axis
ch. 11, Prisms
ch. 14, Reflection
ch. 16, System Matrices
ch. 17, Angular Magnification
ch. 18, Spectacle Magnification and Relative Spectacle Magnification
ch. 19, Stops and Related Effects
ch. 21, Waves and Superposition
ch. 22, Diffraction
ch. 23, Scattering, Absorption, Dispersion, and Polarization
ch. 24, Emission, Absorption, Photons, and Lasers
MOO Modern Ophthalmic Optics
ch. 1, Ophthalmic Materials
ch. 4, Single Vision Lenses
ch. 7, Optics of Contact and Intraocular Lenses
IOO Introduction to Ophthalmic Optics
ch. 5, Sphero-cylindrical Lenses
ch. 8, Ophthalmic Prisms
VP Visual Perception: A Clinical Orientation
ch. 2, Introductory Concepts
ch. 3, The Duplex Retina
ch. 4, Photometry
ch. 5, Color Vision
ch. 7, Spatial Vision
EYE The Eye: Basic Sciences in Practice, 5th ed.
ch. 5, Physiology of Vision and the Visual System
SOD System for Ophthalmic Dispensing
ch. 13, Lens Curvature and Thickness
ch. 15, Optical Prism: Power and Base Direction
ch. 16, Optical Prism: Decentration and Thickness
ch. 19, Segmented Multifocal Lenses
ch. 21, Anisometropia
ch. 22, Absorptive Lenses
CPOE Clinical Procedures for Ocular Examination
ch. 2, Entrance Tests
ch. 3, Refraction
CMBV Clinical Management of Binocular Vision, 5th ed.
ch. 1, Diagnostic Testing
ch. 23, Development and Management of Refractive Error
ADLER Adler's Physiology of the Eye
ch. 36, Binocular Vision
ch. 37, Temporal Properties of Vision
CLOP Contact Lenses in Ophthalmic Practice
ch. 7, How to Fit Rigid Spherical Contact Lenses
ch. 11, Astigmatism and Toric Contact Lenses
AOOT Advances in Ophthalmic Optics Technology
ch. 6, IOL Power Calculation
AOA Optometric Clinical Practice Guideline: Care of the Patient with Visual Impairment (Low Vision Rehabilitation)
Table 6, Determining Magnification for Near Needs

Item ranges are from the NBEO’s Part I ABS discipline-based content outline. For practice with these formulas, the first half of Exam 1 is free.