Optics formula solvers

Categories — 73 of them, 1787 solvers in all

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A B C D F G I L M N P R S T

Apparent Depth

d' = \frac{d}{n}

OpticsPhysicsViewed from straight above, an object under water appears at depth d/n.

Beam Quality Factor M² (θ = M²λ/πw₀)

\theta = \frac{M^{2} \lambda}{\pi w_0}

OpticsSiegman's correction that turns the ideal-Gaussian relation into one a real laser obeys. M² is the factor by which a beam's waist-divergence product exceeds the diffraction limit, it is measured rather than chosen, and it is never below 1.

Brewster's Angle

\tan\theta_B = \frac{n_2}{n_1}

OpticsPhysicsThe incidence angle at which reflected light is completely polarized.

Critical Angle for Total Internal Reflection

\sin\theta_c = \frac{n_2}{n_1}

OpticsPhysicsBeyond this angle of incidence, light in the denser medium reflects totally instead of refracting.

Diffraction Grating Equation

m \lambda = d \sin\theta

OpticsWaves & OscillationsPhysicsBright beams leave a grating at angles where the path difference d sin θ is a whole number of wavelengths.

Diffraction-Limited Spot Diameter (Airy Disc)

d = \frac{2.44 \, \lambda f}{D}

OpticsThe diameter of the central bright disc — first dark ring to first dark ring — that a round aperture of diameter D produces at the focus of a lens of focal length f. Airy worked out the pattern in 1835 and it is still the floor under every imaging and focusing system: no lens, however perfect, focuses tighter than its own aperture allows.

Double-Slit Fringe Spacing

\Delta y = \frac{\lambda L}{d}

OpticsWaves & OscillationsPhysicsSpacing between adjacent bright fringes on a screen a distance L behind two slits d apart.

Fibre V-Number (Normalised Frequency)

V = \frac{2\pi a \, \mathrm{NA}}{\lambda}

OpticsOne dimensionless number that decides everything about how a step-index fibre guides. Below V = 2.405 — the first zero of the Bessel function J₀ — only the fundamental mode propagates and the fibre is single-mode; above it, more and more modes appear. Gloge published the weakly-guiding analysis this rests on in 1971.

Focal Length of a Spherical Mirror

f = \frac{R}{2}

OpticsPhysicsA spherical mirror focuses parallel rays at half its radius of curvature.

Focused Spot Diameter of a Gaussian Beam

d_f = \frac{4 M^{2} \lambda f}{\pi D}

OpticsThe 1/e² diameter a collimated Gaussian beam of diameter D is brought to by a lens of focal length f. It is the working spot-size equation of laser cutting, welding and marking, and it says the three ways to a smaller spot are a shorter lens, a bigger input beam, and a better beam.

Gaussian Beam Divergence (θ = λ/πw₀)

\theta = \frac{\lambda}{\pi w_0}

OpticsThe far-field half-angle a diffraction-limited Gaussian beam spreads at, set entirely by its wavelength and the radius of its own waist. It is the hardest constraint in beam delivery: a small waist and a small divergence are the same thing asked for twice, and you cannot have both.

Gaussian Beam Radius at Distance z

w(z) = w_0 \sqrt{1 + \left(\frac{z}{z_R}\right)^{2}}

OpticsThe whole propagation of a Gaussian beam in one line: a hyperbola that is flat near the waist and straightens into the far-field cone once z passes the Rayleigh range. Use it to find how wide the beam will be on the target, or to work backwards from two measured widths to where the waist actually sits.

Index of Refraction (n = c/v)

n = \frac{c}{v}

OpticsPhysicsHow much a medium slows light: the ratio of c to the speed of light in the medium.

Laser Fluence (Energy per Unit Area)

F = \frac{E_p}{A} = \frac{4 E_p}{\pi d^{2}}

OpticsPulse energy divided by the area it lands on. Fluence in J/cm² is the quantity every ablation threshold, every damage rating and every laser process window is written in, and it is the number that decides whether a pulse marks a surface, cuts it, or does nothing at all.

Lens Magnification (m = −d_i/d_o)

m = -\frac{d_i}{d_o}

OpticsPhysicsImage magnification from the ratio of image to object distance; the minus sign tracks inversion.

Lens Power in Diopters

P = \frac{1}{f}

OpticsPhysicsThe optician's unit: lens power in diopters is the reciprocal of focal length in meters.

Magnification from Heights (m = h_i/h_o)

m = \frac{h_i}{h_o}

OpticsPhysicsMagnification as the ratio of image height to object height.

Malus's Law

I = I_0 \cos^{2}\theta

OpticsPhysicsIntensity of polarised light after an analyser, from the angle between the light's polarisation and the analyser's transmission axis.

Number of Modes in a Step-Index Fibre

M \approx \frac{V^{2}}{2}

OpticsA count of how many modes a multimode step-index fibre supports, straight out of Gloge's weakly-guiding analysis: the modes fill a region of phase space of area proportional to V², and half of V² is the count. It is an approximation, it is only good when V is large, and it is close enough to be genuinely useful.

Numerical Aperture of a Step-Index Fibre

\mathrm{NA} = \sqrt{n_1^{\,2} - n_2^{\,2}} = \sin\theta_a

OpticsHow wide a cone of light a step-index fibre will actually accept and guide. It follows directly from total internal reflection at the core-cladding boundary, and the sine of the acceptance half-angle in air is the numerical aperture itself.

Pulse Peak Power, Average Power and Duty Cycle

P_{avg} = E_p f_{rep} = P_{pk} \, \tau \, f_{rep}

OpticsThe bookkeeping that connects the two power numbers on every pulsed-laser datasheet. Average power is what the wall socket and the cooling loop see; peak power is what the workpiece and the optics see, and on a Q-switched laser it is thousands of times larger. The duty cycle τ·f_rep is the whole of the ratio between them.

Rayleigh Range (z_R = πw₀²/λ)

z_R = \frac{\pi w_0^{\,2}}{\lambda}

OpticsThe distance from the waist at which a Gaussian beam's radius has grown by √2 — that is, at which its area has exactly doubled. It is the honest working definition of how far a beam stays collimated, and because it goes as the SQUARE of the waist radius, a beam twice as wide stays collimated four times as far.

Single-Slit Diffraction Minima

w \sin\theta = m \lambda

OpticsWaves & OscillationsPhysicsAngles of the dark fringes either side of a single slit's central bright band, from the slit width and the wavelength.

Snell's Law of Refraction

n_1 \sin\theta_1 = n_2 \sin\theta_2

OpticsPhysicsLight bends at an interface so that n sin θ stays the same on both sides.

Thin Lens Equation

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

OpticsPhysicsRelates a thin lens's focal length to its object and image distances.

Thin-Film Constructive Interference (Bright Reflection)

2 n t = \left(m + \tfrac{1}{2}\right)\lambda

OpticsWaves & OscillationsPhysicsBright-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a half-odd number of wavelengths.

Thin-Film Destructive Interference (Dark Reflection)

2 n t = m \lambda

OpticsWaves & OscillationsPhysicsDark-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a whole number of wavelengths.