Optics Calculator: Snell's Law, Critical Angle, Brewster Angle, Thin Lens and Mirror Equation

Work out how light bends at a surface and where a lens or mirror makes its image.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

For refraction, enter the refractive index of each medium and the angle of the ray from the normal. The calculator applies n1 sin theta1 = n2 sin theta2, and gives the critical angle, the Brewster angle and the share reflected.

If the ray is above the critical angle it says total internal reflection occurs.

For a lens or mirror, enter the focal length and the object distance. It gives the image distance, the magnification, the image height and the power in dioptres.

Formulas used

  • n_1\sin\theta_1 = n_2\sin\theta_2
  • \theta_c = \sin^{-1}\frac{n_2}{n_1}
  • \theta_B = \tan^{-1}\frac{n_2}{n_1}
  • \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
  • m = -\frac{d_i}{d_o}

Worked example

With the default numbers in the calculator above:

  1. Angle of the refracted ray (Snell's law): 19.47 degrees
  2. How much the ray is bent: 10.53 degrees
  3. Brewster angle: reflected light is fully polarised at this angle: 56.31 degrees
  4. Share of the light reflected at normal incidence: 4.00 percent
  5. Speed of light in the second medium: 2.00 x 10^8 m/s

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want? Thin lens or mirror: image position and size.

  1. Image distance (thin lens equation; for a mirror the same equation is used): 150.00 mm (a real image, on the far side of a lens)
  2. Magnification: -0.50 (the image is upside down) and smaller than the object
  3. Height of the image: -10.00 mm
  4. Power of the lens: 10.00 dioptres (focal length in metres, inverted)

Common questions

What is total internal reflection?

Light going from a denser to a less dense medium at an angle above the critical angle is all reflected, which keeps the light inside an optical fibre.

What is the Brewster angle?

The angle at which the reflected light is completely polarised. Polarising sunglasses cut the glare from water for this reason.

What is a virtual image?

One that rays only appear to come from. It cannot be projected onto a screen. A magnifier or a mirror makes a virtual image.

What does a negative magnification mean?

The image is upside down compared with the object.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Hecht, Optics; any university physics text (geometrical optics).

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