instrument

lens

A lens is an optical instrument that focuses or disperses light beams through refraction to form images or manipulate light paths.
31 related articles · 3 formulas · 1 who worked here

Description

A lens is typically a transparent body with two spherical surfaces. Its curvature together with the refractive index (how much the material slows light) determines the focal length—the distance from the lens to the camera sensor when focused at infinity. A camera objective contains several lenses made of different glasses, because a single lens produces rainbow fringes and blurriness at the edges. The human eye is an amazing example: its crystalline lens changes thickness, focusing without a single moving part.

History

The earliest lenses made of ground rock crystal date back to 700 BCE, Assyria. The scientific approach to lens systems began with Newton's invention of the reflecting telescope (to avoid the rainbow fringe) and the subsequent development of achromatic doublets by Chester Moore Hall in the 1730s. In 1665, Robert Hooke, using a compound microscope assembled from several lenses, first revealed the cellular structure of living matter in his 'Micrographia'.

How it works

The law of refraction (Snell's law) n1 sinθ1 = n2 sinθ2, when light hits a curved boundary, gives different ray slopes at different points. For a thin lens, the lens formula works: 1/f = 1/u + 1/v, where f is the focal length, u is the object distance, v is the image distance. In the wave picture, the lens introduces a spatial phase delay proportional to the square of the distance from the axis, and thereby uses diffraction to turn a diverging wavefront into a converging one.

In practice

Lenses in phone cameras and in telescopes catching distant galaxies. Tiny lenses also live in the eye, focusing the world onto the retina.

Fun fact

The largest objective lens ever made—the Yerkes Observatory telescope (1897)—has a diameter of 102 centimeters. It is so heavy that the glass sags under its own weight, which is why modern telescopes use mirrors.

Concept graph →

Formulas

$$y = x - \alpha(x)$$
Relates source position to image position via deflection angle.
What is in the formula
yvariableSource position on the sky — angular position radian
xvariableImage position on the sky — angular position radian
αvariableDeflection angle of light — deflection angle radian
y=x(1-\bar{\kappa}(x)) · 2607.21834v1
$$\kappa(\boldsymbol{\theta}) = \frac{3H_0^2\Omega_m}{2c^2} \int_0^{\chi} d\chi' \, \frac{q(\chi')}{a(\chi')} f_K(\chi') \delta(f_K(\chi')\boldsymbol{\theta}, \chi')$$
Convergence is a weighted integral of density contrast along the line of sight.
What is in the formula
kappavariableweak lensing convergence — convergence
thetavariableangular position on the sky — angular position radian
chivariablecomoving distance to the source — comoving distance metre
chi'variablecomoving distance along the line of sight — comoving distance metre
qvariablelensing efficiency function — lensing efficiency
avariablescale factor — scale factor
f_Kvariablecomoving angular diameter distance — comoving angular diameter distance metre
deltavariablematter density contrast — density contrast
H_0constantcurrent expansion rate of the universe — hubble constant = 2.2e-18 per second
Omega_mconstantratio of matter density to critical density today — matter density parameter = 0.315
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
∫operatorsummation over infinitesimal elements — integral
\kappa(\boldsymbol{\vartheta})=\frac{3h_0^2ω_m}{2c^2}\int_0^{χ_{\lim}}dχ\frac{q(χ)}{a(χ)}f_k(χ)δ(f_k(χ)\boldsymbol{\vartheta},χ) · 2606.31988
$$d_{ls} = d_s \sqrt{1 + \Omega_K d_l^2} - d_l \sqrt{1 + \Omega_K d_s^2}$$
Lens-source distance depends on curvature and observer distances.
What is in the formula
d_{ls}variableangular diameter distance between lens and source — angular diameter distance metre
d_svariableangular diameter distance from observer to source — angular diameter distance metre
d_lvariableangular diameter distance from observer to lens — angular diameter distance metre
\Omega_Kvariablecurvature density parameter — curvature density parameter
\frac{d_{ls}}{d_s}=\sqrt{1+ω_kd_l^2}-\frac{d_l}{d_s}\sqrt{1+ω_kd_s^2} · 2606.03634v1
d_{δt}=(1+z_l)\frac{d_ld_s}{d_{ls}} · 2606.25205

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