Optics Study Guide

Formula Finder

This page is designed as a practical optics reference for opticians, optometrists, ophthalmologists, opticianry students, and other refracting practitioners. On mobile devices, it works best as a quick formula finder: tap a topic, review the formula and explanation, and then jump to the matching calculator when one is available.

Using This Page

Many sections now include a direct link to the corresponding calculator so you can move from the formula to a working tool without hunting through the calculator library.

Focal Length

Formula

\[ F.L. = \frac{1}{D} \]

Rearranged

\[ D = \frac{1}{F.L.} \]

Focal length in meters.

Why it matters: Focal length and dioptric power are inverse ways of describing the same optical relationship. This is basic but foundational in optics.

Prism

\[ \Delta = \frac{D}{m} \]
\[ D = \Delta \cdot m \]
\[ m = \frac{D}{\Delta} \]

Why it matters: Prism power describes light deviation. In dispensing, prism is often discussed in prism diopters and then estimated with Prentice's Rule when decentration is involved.

Prentice's Rule

\[ \Delta = \frac{D \cdot c}{10} \]

where \(c\) is decentration in mm.

\[ D = \frac{\Delta \cdot 10}{c} \]
\[ c = \frac{\Delta \cdot 10}{D} \]

Why it matters: Prentice's Rule is used constantly in dispensing optics to estimate induced prism from decentration. For horizontal induced prism, use the power in the 180 meridian. For vertical induced prism, use the power in the 90 meridian.

Related calculators: Prentice's Rule Calculator, Meridional Power Calculator, Lens Analyzer

Radius of Curvature (Contact Lens)

\[ R_{mm} = \frac{337.5}{K} \]
\[ K = \frac{337.5}{R_{mm}} \]

Radius of Curvature

\[ R = \frac{n - 1}{D} \]
\[ D = \frac{n - 1}{R} \]

Prismatic Power

\[ \Delta = \frac{\text{deviation (cm)}}{\text{distance away (m)}} \]
\[ \Delta = a(n - 1) \]

where \(a\) is the apical angle.

Back and Front Vertex Power

Back Vertex Power

\[ D_{eff} = D_f + D_b + \frac{t(D_f)^2}{n} \]

Front Vertex Power

\[ D_{eff} = D_f + D_b + \frac{t(D_b)^2}{n} \]

Oblique Total Power

\[ P_{total} = S + C\sin^2(\theta) \]

Why it matters: Oblique total power is another way of thinking about power in a chosen meridian. It helps explain why cylinder contributes differently as the meridian changes.

Related calculator: Meridional Power Calculator

Snell's Law

\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]
\[ n = \frac{\sin(\theta_i)}{\sin(\theta_r)} \]
\[ \sin(\theta_c) = \frac{1}{n_1} \]

Vertical Imbalance

\[ \text{Vertical Imbalance} = \frac{(\text{difference at } 90^\circ)\cdot(\text{reading level})}{10} \]

Why it matters: Vertical imbalance at near can make multifocal wear uncomfortable and may require slab-off or another compensation strategy.

Related calculator: Vertical Imbalance Calculator

Image Jump

\[ \text{Image Jump} = \frac{(\text{distance from segment top to segment O.C. in mm})\cdot(\text{Add Power})}{10} \]

Why it matters: Image jump is the sudden prism effect encountered when the line of sight crosses into a bifocal or multifocal segment.

Related calculator: Image Jump Calculator

True and Marked Power

\[ \frac{\text{True Power}}{\text{Marked Power}} = \frac{n - 1}{0.530} \]

Bifocal BTS

\[ BTS = \frac{w}{2} + h \]

Circle of Least Confusion

\[ \text{Circle of Least Confusion} = \frac{1}{\text{Spherical Equivalent}} \]

It is the spherical equivalent expressed in metric.

Thin Film Coatings

\[ n_{ideal\ film} = \sqrt{n_{lens}} \]

Decentration Caused by Vertical Imbalance

\[ c_{mm} = \frac{\Delta \cdot 10}{\text{Add}} \]

Use average add if dissimilar.

Why it matters: This relationship helps show how much decentration or compensating prism may be needed when vertical imbalance is present.

Related calculators: Vertical Imbalance Calculator, Compound Prism Calculator

Pantoscopic Tilt and Face Form Tilt

Pantoscopic Tilt

\[ S_{new} = D_{90}\left(1 + \frac{\sin^2(\theta)}{3}\right) \] \[ C_{new} = D_{90}\tan^2(\theta) \]

Axis is always \(180^\circ\).

Face Form Tilt

\[ S_{new} = D_{180}\left(1 + \frac{\sin^2(\theta)}{3}\right) \] \[ C_{new} = D_{180}\tan^2(\theta) \]

Axis is always \(90^\circ\).

Why it matters: Pantoscopic and face-form tilt can change the effective prescription the wearer actually receives. In stronger prescriptions and larger tilt values, the differences can become clinically meaningful.

Related calculators: Oblique Tilt Calculator, Lens Analyzer

Vergence

\[ V = U + F \] \[ V = \frac{100}{v} \qquad U = \frac{100}{u} \qquad F = \frac{100}{f} \]

Image distance to the right for lenses, left for mirrors.

\[ \text{Linear Magnification (mirrors)} = \frac{v}{u} \] \[ \text{Linear Magnification (lenses)} = -\frac{v}{u} \] \[ \text{Angular Magnification} = \frac{\theta_{image}}{\theta_{object}} \]
\[ F = -\frac{2}{R} = -\frac{1}{f} \]

Mirrors.

\[ \text{Telescope Power} = \frac{f_1}{f_2} \] \[ \text{Telescope Length} = f_1 + f_2 \]

Reading Field Size

\[ \frac{MQ}{AB} = \frac{MC}{AC} \] \[ MQ = \frac{MC \cdot AB}{AC} \]

MQ = width or height through segment

MC = fixation point to center of rotation

AB = segment diameter minus pupil size

AC = segment to center of rotation

Inset

\[ \text{Gerstman Inset} = 0.75 \cdot (\text{dioptric demand}) \]
\[ \text{Lebensohn Inset} = \frac{\text{Binocular PD (mm)}}{\text{Reading Distance (in)} + 1} \]
\[ \text{Approx. Inset} = \frac{\text{Monocular PD (mm)}}{\text{Working Distance (in)}} + \frac{\text{Distance Rx in 180th meridian}}{20} \]

Obliquely Crossed Cylinders

Put Rx into plus cylinder form with strongest cylinder on top.

\[ C^2 = A^2 + B^2 + 2AB\cos(2a) \]
\[ \sin(2a') = \left(\frac{B}{C}\right)\sin(2a) \]

New axis = strongest cylinder axis + \(a'\).

\[ S_{new} = \frac{A_{cyl} + B_{cyl} - C}{2} + S_1 + S_2 \]

Why it matters: Obliquely crossed cylinder calculations help combine cylinder effects at different axes into a single resulting spherocylindrical form.

Related calculator: Transposition Calculator

Spectacle Magnification

\[ M = (\text{Shape Factor})(\text{Power Factor}) \] \[ \text{Shape Factor} = \frac{1}{1 - \frac{cD_1}{n}} \] \[ \text{Power Factor} = \frac{1}{1 - zD_v} \]
\[ \%M = (M - 1)\cdot 100 \] \[ \Delta M = \%M_1 - \%M_2 \]
\[ \text{Approx. Shape Factor Change} = \frac{X \cdot D_1}{15} \] \[ \text{Approx. Power Factor Change} = \frac{z \cdot D_v}{10} \]
  • Flatter base curves decrease magnification.
  • Steeper base curves increase magnification.
  • Thinner lenses decrease magnification.
  • Thicker lenses increase magnification.
  • Higher vertex distances create more magnification.
  • Shorter vertex distances create less magnification.
  • Higher index yields less magnification.
  • Lower index yields more magnification.

Why it matters: Spectacle magnification affects binocular image size and can matter in anisometropia, aniseikonia, and iseikonic lens design.

Related calculators: Spectacle Magnification Calculator, Lens Analyzer

Illumination, Apparent Depth, Luminous Flux, Velocity of a Wave

\[ \text{Illumination} = \frac{\text{Candle Power}}{\text{Distance}^2} \]
\[ \text{Apparent Depth} = \frac{\text{Actual Depth}}{n_{medium}} \]
\[ \text{Luminous Flux} = 4\pi(\text{Candle Power}) \]
\[ v = f\lambda \]

Electromagnetic Spectrum

Type of Light Wavelength Sources Effects on Eyes
Short UV 14 nm - 310 nm Sunlight at high altitude, snow, sand, water reflections, mercury arc lamps, UV lamps Absorbed by the cornea and conjunctiva. Causes conjunctivitis and keratitis.
Long UV 310 nm - 380 nm Intense sunlight, fluorescent bulbs, UV and mercury arc lamps Solar retinitis, may cause cataracts, and may play a role in age-related macular degeneration.
Visible Spectrum 380 nm - 780 nm Sunlight and artificial light sources 570 nm is the most sensitive wavelength to the eye, yellow-green.
Short Infrared 780 nm - 1500 nm Direct sunlight, molten glass and metal, mercury and infrared lamps Short exposure can be harmful to the retina.
Long Infrared 1500 nm and longer Direct sunlight, molten glass and metal, mercury and infrared lamps May cause keratitis or conjunctivitis.

Shadows: Umbra and Penumbra

Point Source

\[ \frac{D_2}{L_1} = \frac{U}{L_1 + L_2} \]

Extended Source

\[ \frac{D_2}{L_1} = \frac{P + U}{L_1 + L_2} \] \[ \frac{D_1}{L_1} = \frac{P}{L_2} \] \[ \text{Total Shadow} = 2P + U \]

Boxing System and Frame Measurements

\[ \text{Frame PD} = A + DBL \] \[ \text{Set Minus} = \text{Pattern A} - \text{Machine Standard} \] \[ MBS = 2(\text{decentration}) + ED + 2 \] \[ \text{Edger Setting} = \text{Frame A} - \text{Set Minus} \]

A = horizontal boxing width

B = vertical boxing length

ED = effective diameter

DBL = distance between lenses

MBS = minimum blank size

Machine standards: 36.5 Shuron, 37.1 AO.

Why it matters: Frame PD, boxing measurements, and decentration are central to practical dispensing and to understanding how unwanted prism can be introduced.

Related calculators: Prentice's Rule Calculator, Lens Analyzer

Convergence and Accommodation

\[ AC = (\text{Accommodation in use}) \cdot (\text{distance PD in cm}) \]
\[ \frac{AC}{A} \]

Ratio of accommodative convergence for every diopter of accommodation.

Adding plus increases convergence. Adding minus decreases convergence.

Adding BI decreases convergence. Adding BO increases convergence.

Sagitta, Sag of a Lens, Thickness Difference Prism

\[ \text{Sagitta} = R - \sqrt{R^2 - r^2} \]
\[ C.T. = S_1 + E.T. - S_2 \]

For plus lenses.

\[ E.T. = S_2 + C.T. - S_1 \]

For minus lenses.

\[ \text{Sag of a Lens} = \frac{r^2 \cdot D}{2000(n - 1)} \]

Approximate formula.

\[ \text{Thickness Difference} = \frac{(\text{diameter of lens})(\text{prism})}{100(n - 1)} \]

Why it matters: Sagitta and thickness relationships matter in surfacing, lens shape, and thickness estimates.

Related calculator: Sagitta Calculator

Notes

  • Slab off, bi-centric grinding, is always base up. Reverse slab off is always base down.
  • With vertical imbalance, the problem is always with the strongest lens in the 90th meridian.
  • Shortest BTS goes on the most minus or least plus lens.
  • Average vertex distance is 13.75 mm.
  • Average center of rotation is 13.25 mm.
  • 1 meter = 100 centimeters = 1000 millimeters.
  • 1 inch = 2.54 cm = 25.4 mm.
  • 1 cm = 0.3937 inches.
  • The Snellen E has 5 minutes of arc and is 8.87 mm on the 20/20 line.
  • 2 degrees of pantoscopic tilt = 1 mm decentered down.
  • 1 diopter converges light at 1 meter.
  • 1 prism diopter deviates light 1 cm at 1 meter.

Note: This information may contain mistakes and should be independently verified. The original page included a similar disclaimer, and that caution should remain in force.

Bibliography

Thomas, Brian A., A.B.O.M. In-house publications from Raritan Valley Community College.

Velardi, Thomas, O.D. In-house publication from Raritan Valley Community College.

Stimson, Russell L. Ophthalmic Dispensing. 3rd ed., 1979.

Rubin, Melvin L. Optics for Clinicians. 25th Anniversary ed., 1993.

Milder, Benjamin, and Rubin, Melvin L. The Fine Art of Prescribing Glasses: Without Making a Spectacle of Yourself. 3rd ed., 2004.

Brooks, Clifford W., and Borish, Irvin M. System for Ophthalmic Dispensing. 2nd ed., 1996.

Stoner, Ellen, Perkins, Patricia, and Ferguson, Roy. Optical Formulas: Tutorial. 2nd ed., 2005.

Brooks, Clifford W. Essentials of Ophthalmic Lens Finishing. 2nd ed., 2003.

Appler, V. Thomas, Dennis, Raymond P., Muth, Eric P., and White, Debra R. Management for Opticians. 2nd ed., 1999.