Lens

Topics

• Thin Lensmaker Equation
• Full Lensmaker Equation

	$s>0$	$s^{\prime }>0$	$R>0$
If the X is on the same side as Y light rays	object; incoming	image; outgoing	center of curvature; outgoing

$m=\frac{y^{\prime }}{y}=-\frac{s^{\prime }}{s}=-\frac{n_a{s}^{\prime }}{n_{b}s}$
The lateral magnification of an image
$m<0$ corresponds to an inverted image
$y^{\prime }$ is the image height
$y$ is the object height
$s$ is the distance from the object to the mirror surface
$s^{\prime }$ is the distance from the object to the mirror surface
$s>0$ when the object’s on the same side as the incident ray
$s^{\prime }>0$ when the image’s on the same side as the reflected or refracted ray
- $m_{\text{total}}=m_1{m}_2\dots m_n$
The total magnification of a series of lens is the product of each lens’ magnifications
- $\frac{1}{s}+\frac{1}{s^{\prime }}=\frac{1}{f}=\frac{2}{R}$
Image position for a spherical mirror of radius $R$
$f$ is the focal length of the spherical mirror
Same sign conventions as in above equation
$R>0$ when the center of the spherical mirror is on the same side as the reflected light
Essentially assumes $R\ll s$
- $\frac{n_a}{s}+\frac{n_b}{s^{\prime }}=\frac{n_b-n_a}{R}$
Spherical lens equation for refraction between two media with different refractive indexes
$n_a$ is the refractive index of the medium that the incoming light first passes through
$n_b$ is the refractive index of the medium that the refracted light passes through
Same sign rules as above equation
Essentially assumes $R\ll s$
- $f\text{-number}=\frac{f}{D}$
The $f\text{-number}$ of a camera depends on the focal length $f$ and the diameter of the aperature $D$
$f\text{-numbers}$ are generally given in the format of $\frac{f}{f\text{-number}}$. For example, $\frac{f}{1.4}$ would correspond to an $f\text{-number}$ of $1.4$
$I\propto \frac{1}{(f\text{-number}{)}^2}$
- $P=\frac{1}{f}$
$P$ is the optical power of the lens
$f$ is the focal length of the lens and $=\frac{R}{2}$, where $R$ is the radius of the lens
The optical power of a lens determines how much bends light toward a point
$P>0$ means $f>0$ and that the lenses are converging for hyperopia (far-sighted, short eyeballs)
$P<0$ means $f<0$ and that the lenses are diverging for myopia (near-sighted, long eyeballs)
Increasing $|P|$ means lowering $f$ and $R$, which means more converging or divering of light
- $M=\frac{{\theta }^{\prime }}{\theta }=\frac{s_{\text{nearpoint}}}{f}$
$M$ is the angular magnification of
${\theta }^{\prime }$ is the angle subtended by the image height at the point of observation
$\theta$ is the angle subtended by the object height at the point of observation
$s_{\text{nearpoint}}$ is often taken as 25 cm for the human eye
$f$ is the focal length of the lens
Assumes that the object angle is small to allow for ${\theta }^{\prime }\approx \theta \approx \sin \theta =\frac{y}{f}$
- $M_{\text{microscopes}}=m_1{M}_2=\frac{s_{\text{nearpoint}}{s}_1^{\prime }}{f_1{f}_2}$
Microscopes use two converging lens: the objective lens and the eyepiece
$s_{\text{nearpoint}}$ is often taken as 25 cm for the human eye
$s_1^{\prime }$ is the length between the objective lens and the focal point of the eyepiece
$f_1$ and $f_2$ are the focal length of the lenses
- $M_{\text{telescopes}}=-\frac{f_1}{f_2}$
Telescopes use two converging lens like microscopes