= Optical microscope
An <optical microscope> uses <light> and <optical lens> assemblies to reveal small structures, such as the boundaries of <biological cells> or defects in a manufactured surface. In a compound <optical microscope>, the <microscope objective> supplies an enlarged <real image>; the <microscope eyepiece> makes that <intermediate microscope image> subtend a larger viewing <angle>. The <transverse magnification> of the first stage and the <angular magnification> of the second answer different questions.<University Physics Volume 3 by Samuel J. Ling and others/2.8 Microscopes and Telescopes>[University Physics Volume 3, §2.8 Microscopes, image formation and Eqs. 2.35–2.37]{ref}
Consider an original ideal <finite-conjugate microscope> in air, with both <optical lens> assemblies treated as converging <thin lens> elements in the <paraxial approximation>. Choose objective <focal length> $f_o=10\,\mathrm{mm}$ and object distance $u_o=12\,\mathrm{mm}$. The <thin lens> equation and signed <transverse magnification> give
$$
\frac1{v_o}=\frac1{10\,\mathrm{mm}}-\frac1{12\,\mathrm{mm}}=\frac1{60\,\mathrm{mm}},\qquad v_o=60\,\mathrm{mm},\qquad m_o=-\frac{v_o}{u_o}=-5.
$$
Place a <microscope eyepiece> of <focal length> $f_e=25\,\mathrm{mm}$ a further $25\,\mathrm{mm}$ beyond the <intermediate microscope image>. The objective-to-eyepiece separation is therefore $85\,\mathrm{mm}$. Each object point then supplies a parallel outgoing bundle of <light rays>, appropriate to a relaxed eye focused at infinity. Define the unaided reference as the same object viewed at distance $D=250\,\mathrm{mm}$; this is a rating convention, not a universal property of eyesight. The eyepiece <angular magnification> and total signed <angular magnification> are
$$
M_e=\frac D{f_e}=10,\qquad M=m_oM_e=-50.
$$
For instance, a hypothetical object height $h_o=0.020\,\mathrm{mm}$ becomes $h_i=-0.100\,\mathrm{mm}$ at the <intermediate microscope image>. The signed viewing <angles> are approximately $h_i/f_e=-0.004\,\mathrm{rad}$ with the instrument and $h_o/D=0.00008\,\mathrm{rad}$ unaided. Their ratio is $-50$: enlarged and inverted. These are stipulated model dimensions, not measurements or specifications of a modern commercial <optical microscope>.
\Image[/microscope-objective-eyepiece-conjugates.png]
{title=<Microscope objective> and <microscope eyepiece> in an ideal <finite-conjugate microscope>}
{description=Original <light ray> construction for $f_o=10\,\mathrm{mm}$, $u_o=12\,\mathrm{mm}$, $v_o=60\,\mathrm{mm}$ and $f_e=25\,\mathrm{mm}$. Horizontal distances are proportional; object and image heights are enlarged for visibility. The <real image> at $x=60\,\mathrm{mm}$ is the eyepiece's object, and the emerging bundle from its tip is parallel. A detector at that <real image> plane would record <transverse magnification> $-5$; the relaxed-eye viewing convention gives total <angular magnification> $-50$.}
The attraction of an <optical microscope> is that seeing can become measuring: a <stage micrometer> supplies a known object-space <length>, and <microscope scale calibration> converts an image readout into a specimen <length>. This also separates a large-looking image from a trustworthy size estimate. Meanwhile, <diffraction-limited resolution> depends on <numerical aperture> and <wavelength>; <empty magnification> enlarges an existing blur without revealing additional structure.
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