Optical Microscopy

An optical microscope forms a magnified image of a small specimen with visible light and lenses, and in modern instruments that image falls on a camera, so the microscope is an imaging sensor whose output is an array of pixels. What sets it apart from an ordinary camera is the scale. Structures in the specimen are often comparable to the wavelength of light, so the image is shaped less by geometry than by diffraction: a point in the specimen becomes a spot a few hundred nanometres wide, two points closer than about 0.2 µm merge into one, and most cells, being transparent, are nearly invisible unless the microscope converts their phase shift into contrast. This article describes the light microscope from the point of view of someone who analyzes its images. Bioimage analysis surveys the modalities and the analysis that follows; fluorescence microscopy, which images light emitted by labelled molecules, is a variant of the same instrument with its own illumination, filters and noise.

How It Works

The optical path

A compound microscope uses at least two converging lenses. The objective, nearest the specimen, forms a real, magnified, inverted image; the eyepiece magnifies that intermediate image again into a virtual image for the eye, so the total magnification is the product of the two (Mualla et al., 2018). Because the objective’s image is real, a camera chip placed there records it directly as a digital image (Mualla et al., 2018). By the magnification relation $M = h'/h$, the image of a structure of size $h$ in the specimen has size $Mh$ on the sensor; it follows that a camera with pixel pitch $a$ samples the specimen at a pitch of $a/M$. A camera with 6.5 µm pixels behind a 100× objective (an illustrative pairing) therefore samples the specimen every 65 nm.

Below the specimen, the condenser concentrates light from the source onto the specimen (Mualla et al., 2018). Current microscopes arrange this as Köhler illumination: a spatially incoherent source, a set of lenses including the condenser, and two adjustable iris diaphragms. The field diaphragm sets the size of the illuminated area, which is lit very uniformly; the aperture diaphragm sets the angle, or numerical aperture, of the cone of light that reaches each point of the specimen, and every point receives the same cone (Sauer & Surrel, 2017). The illumination aperture matters again for resolution.

Objectives and numerical aperture

An objective is described by its magnification and its numerical aperture (NA), written for example as 50×/0.80 (Sauer & Surrel, 2017). The NA is

$$\mathrm{NA} = n \sin\theta ,$$

where $n$ is the refractive index of the medium between the objective and the specimen and $\theta$ is the half-angle of the largest cone of light the objective can collect (Mualla et al., 2018). Abbe coined the term (Cremer & Masters, 2013). The NA measures how much light the lens gathers and, as the next section shows, how fine a detail it can resolve. Since $\sin\theta < 1$, a dry objective has an NA below 1; immersion objectives place water ($n \approx 1.33$) or oil ($n \approx 1.51$) between lens and specimen to go higher (Mualla et al., 2018), and Cremer and Masters (2013) put the practically useful maximum at around 1.5. Bankhead (2022–2025) adds a practical caution: when the immersion medium’s refractive index does not match that of the medium around the specimen, the result is spherical aberration, which makes the point-spread function asymmetric and increasingly distorted with depth, so a matched objective of lower NA can give better data than a mismatched one of higher NA.

Diffraction and the point-spread function

Geometrical optics predicts that a point in focus is imaged as a point. Because light is a wave, the finite aperture of the objective diffracts it instead, and a point source is imaged as an Airy pattern: a central disc surrounded by faint rings (Mualla et al., 2018). The radius of the central disc, from its centre to the first dark ring, is

$$r_{\mathrm{Airy}} = \frac{0.61\,\lambda}{\mathrm{NA}}$$

in specimen coordinates, and $M\,r_{\mathrm{Airy}}$ on the sensor, where $\lambda$ is the wavelength (Mualla et al., 2018). For an ideal circular pupil the in-focus intensity has the standard form $I(v) = \left( 2 J_1(v)/v \right)^2$ with $v = 2\pi\,\mathrm{NA}\,r/\lambda$ and $J_1$ the Bessel function of the first kind; its first zero, at $v \approx 3.83$, gives the 0.61 factor. The image of a point is the microscope’s point-spread function (PSF). For mutually incoherent sources, such as self-luminous points, intensities add, and the image of an extended object is the convolution of the object’s intensity with the PSF (Cremer & Masters, 2013). The PSF of a real objective is never better than this ideal: aberrations, among them chromatic, defocus, spherical, coma, astigmatism and field curvature, degrade resolution, and Strehl showed that even small ones lower the central maximum and move the light into the outer pattern (Cremer & Masters, 2013).

In the frequency domain the same fact reads differently. The optical transfer function (OTF) is the Fourier transform of the PSF, the ratio of the image spectrum to the object spectrum, and its magnitude is the modulation transfer function; because the aperture is finite, the OTF is zero above a cutoff frequency, and that cutoff is one way to define the diffraction limit (Cremer & Masters, 2013). Detail finer than the cutoff is not blurred but absent from the image, which is why enlarging the image beyond a certain point shows nothing new; Mualla et al. (2018) call this empty magnification.

The PSF is three-dimensional. Along the optical axis, the distance from the centre to the first minimum is about $2\lambda n/\mathrm{NA}^2$, so axial blur depends on the square of the NA and is at least about three times the lateral radius even at the highest NA (Bankhead, 2022–2025). Seen from the side, a widefield PSF has an hourglass shape: a point above or below the focal plane still sends its light to the detector, spread over a wider and dimmer disc (Bankhead, 2022–2025). In a widefield image of a thick specimen, every plane therefore carries out-of-focus light from the planes around it.

Resolution criteria

Resolution is the smallest distance at which two structures can still be told apart, and there are several criteria for it (Cremer & Masters, 2013).

Abbe treated the object as a diffraction grating. In his theory, the objective must collect at least the undiffracted beam and one diffracted order for the grating to appear in the image, which gives the smallest resolvable period $d = \lambda/(n \sin\alpha)$ for direct, axial illumination and $d = \lambda/(2 n \sin\alpha)$ for oblique illumination, or illumination by a condenser of the same NA as the objective (Cremer & Masters, 2013). With the condenser’s aperture filled, the limit is therefore $\lambda/(2\,\mathrm{NA})$. From those two cases we infer that closing the condenser’s aperture diaphragm moves the limit from the second value toward the first, trading resolution for contrast.

Rayleigh’s criterion is defined for two incoherent point sources of equal brightness: they are just resolved when the central maximum of one Airy pattern falls on the first zero of the other, that is, at a separation of $r_{\mathrm{Airy}} = 0.61\lambda/\mathrm{NA}$ (Cremer & Masters, 2013; Mualla et al., 2018). The criterion rests on the assumption that a human observer needs a minimum contrast to see two spots, and it fails for patterns without well-defined zeros (Cremer & Masters, 2013).

Sparrow’s criterion drops the assumption about the observer: two points are just resolved at the smallest separation at which the combined intensity still has a minimum between them, the point at which the second derivative at the midpoint is zero; in angular terms it is $0.95\lambda/D$ against Rayleigh’s $1.22\lambda/D$ (Cremer & Masters, 2013). Rescaling it by the same ratio to the microscope’s form, our step rather than the source’s, gives about $0.47\lambda/\mathrm{NA}$, a value the example below finds numerically.

A fourth, practical criterion is the full width at half maximum (FWHM) of the measured PSF: two objects are taken to be resolvable when they are farther apart than the FWHM (Cremer & Masters, 2013). With $\lambda = 550$ nm and the oil-immersion bound $\mathrm{NA} = 1.51$, Mualla et al. (2018) obtain a Rayleigh resolution of 222 nm, the origin of the often-cited 0.2 µm limit. Cremer and Masters (2013) give about 0.2 µm laterally and about 0.6 µm axially for visible light, and stress that these limits assume a single objective, linear single-photon absorption and emission, and uniform illumination; the super-resolution methods that bioimage analysis describes work by breaking those assumptions.

Contrast Methods

In brightfield microscopy the condenser lights the specimen from below and the image records how much light each point absorbs, so empty background appears bright, which gives the method its name; it is the simplest and cheapest set-up (Mualla et al., 2018). Absorbing specimens, which Zernike called amplitude objects, show up directly. Most biological specimens are instead phase objects: transparent details that leave the amplitude almost unchanged but delay the light, and staining is the traditional way of turning them into amplitude objects (Zernike, 1953). A homogeneous object of thickness $q(x,y)$ and refractive index difference $\Delta n$ from its surroundings shifts the phase by

$$\varphi(x,y) = k\,\Delta n\, q(x,y), \qquad k = \frac{2\pi}{\lambda},$$

so the phase records the optical path difference between object and surroundings (Mualla et al., 2018). Cameras, like the eye, record intensity and are insensitive to phase (Mualla et al., 2018), so an unstained cell in focus is almost invisible. Microscopists traditionally made such details visible by defocusing slightly, at the cost of a blurred, watery image (Zernike, 1953). Hence the methods that follow.

Darkfield illumination keeps the direct light out of the objective, so that only light scattered by the specimen forms the image. The ultramicroscope of Siedentopf and Zsigmondy (1903) lit colloids perpendicular to the optical axis and detected particles far below the resolution limit as bright points of scattered light, although the size of each spot was set by the microscope, not by the particle; Siedentopf also built a darkfield condenser that blocked the incident illumination from entering the objective (Cremer & Masters, 2013).

Phase contrast, invented by Frits Zernike, separates the two parts of the light that leave a specimen. The direct, undiffracted light comes to a focus in the back focal plane of the objective, while the light diffracted by the specimen’s details fills the whole aperture; in the image the two interfere (Zernike, 1953). For a phase object, the diffracted wave is nearly a quarter period out of step with the direct wave, so their sum has the same intensity as the background. Zernike placed a phase strip in the back focal plane that advanced the direct light by 90°, which makes a transparent detail appear darker than the background, as if stained. Making the strip also absorb the direct light raises the contrast further: a strip transmitting 25% of the energy halves the direct amplitude and doubles the contrast. Annular strips replaced straight ones, and the residual effect of the strip on part of the diffracted light leaves a halo, bright around dark details and dark around bright ones (Zernike, 1953). For analysis, the halo and the method’s validity only for small phase shifts mean that phase-contrast intensities are not quantitative measures of phase (Mualla et al., 2018).

Differential interference contrast (DIC) is a form of lateral shearing interferometry: the wavefront is split into two copies, displaced sideways by less than the objective’s resolution, which then interfere. Each image point is modulated by a two-beam term of the form $\tfrac{1}{2}\left(1 + \cos(\Delta\varphi + \varphi_0)\right)$, where $\Delta\varphi$ is proportional to the local gradient of the wavefront in the direction of the shear and $\varphi_0$ is an offset the user adjusts (Sauer & Surrel, 2017). With $\varphi_0 = \pm\pi/2$ the image shows the sign of the phase gradient as brighter or darker than a mid-grey background, which gives the impression of relief under grazing side illumination. The split is made by birefringent prisms between a polarizer and an analyzer: a Wollaston prism belongs in the back focal plane of the objective, which is inaccessible inside modern objectives, and Georges Nomarski’s modified prism of 1952 works the same way when placed outside that plane; for transmitted light a second prism sits in the condenser’s front focal plane, at the aperture diaphragm (Sauer & Surrel, 2017). Expanding the cosine for small $\Delta\varphi$, it follows that with $\varphi_0 = \pi/2$ the intensity is approximately $\tfrac{1}{2}(1 - \Delta\varphi)$: a DIC image is close to a directional derivative of the optical path, so the same cell is bright on one side and dark on the other, and intensity thresholds that work on brightfield images of stained cells do not apply.

Quantitative phase methods recover the phase itself. One, the transport-of-intensity equation, computes it from a brightfield image in focus and one or more defocused images; the axial derivative is sensitive to the chosen defocus distance, and the solution carries a low-frequency bias that must be corrected (Mualla et al., 2018).

Fluorescence, finally, uses an excitation filter to illuminate the specimen at one wavelength band and an emission filter to keep only the light that labelled molecules emit at another wavelength, which gives high contrast for whatever has been labelled (Mualla et al., 2018).

Data It Produces

A camera on a transmitted-light microscope records 2D images, in colour for stained histology, and moving the focus produces a z-stack. What an analysis receives is the specimen convolved with a PSF of the sizes given above, sampled on a grid whose pitch must travel with the image, plus the camera’s noise. Bioimage analysis describes the multidimensional arrays, metadata and file formats; whole-slide scanners acquire entire slides as tiles that are stitched into one image (Peng et al., 2017).

Cameras and Sampling

Cameras

The camera converts the image into electrons and counts them; image sensor covers CCD and CMOS readout in general. Three properties matter most for microscopy. Quantum efficiency is the fraction of photons that produce an electron, typically high for a CCD (perhaps 90%) and much lower for the photomultiplier tubes that point-scanning instruments use (Bankhead, 2022–2025). Read noise, added each time a pixel is read, dominates when few photons arrive. Binning sums the charge of, for example, 2 × 2 pixels before readout, so the signal grows about fourfold relative to the read noise at the cost of resolution; an electron-multiplying CCD (EMCCD) instead amplifies the charge in a gain register of more than 500 stages before readout, which overcomes read noise but adds a randomness that Bankhead describes as roughly equivalent to detecting half as many photons, so EMCCDs suit low light and plain CCDs bright widefield images (Bankhead, 2022–2025). Scientific CMOS (sCMOS) cameras have gained ground for their higher frame rates, larger field of view and lower electrical noise, and the generation they describe in 2020 reaches a read noise of 1–2 electrons; because each pixel converts its own charge to voltage, pixels differ in offset, variance and gain, which adds a pixel-dependent pattern noise that matters at low light (Mandracchia et al., 2020). Image noise gives the Poisson–Gaussian model that combines photon and read noise.

Nyquist sampling

The image the objective delivers is band-limited, so the sampling theorem tells how fine the pixel grid must be. Cremer and Masters (2013) give the microscopy version: to transmit detail down to 0.2 µm, a spatial frequency of 5 lines per µm, the image must be sampled at twice that frequency, 10 points per µm. Applying the same rule to Abbe’s limit gives a general expression: the finest period that passes is $\lambda/(2\,\mathrm{NA})$, so the pixel pitch in the specimen must satisfy

$$\Delta x \le \frac{\lambda}{4\,\mathrm{NA}} ,$$

about 98 nm for $\lambda = 550$ nm and NA 1.4. This is our derivation from Abbe’s limit and the sampling theorem, not a formula quoted from the sources; the rule quoted in bioimage analysis, at least two pixels per resolvable element, is the same idea applied to a resolution distance rather than to Abbe’s period, and with the Rayleigh distance it gives a somewhat coarser pitch, about $0.31\lambda/\mathrm{NA}$. A coarser grid aliases detail that the optics deliver into false, coarser patterns, as sampling and aliasing explains. A much finer grid is not free either. Cremer and Masters (2013) note that the Nyquist criterion suits extended structures such as a folded filament, whereas locating two isolated points needs only that their diffraction patterns be distinguishable, and the example below shows the photon cost of oversampling.

Practical Example: Two Points at the Diffraction Limit

The example images two equal, mutually incoherent point sources through an ideal objective, first as a continuous intensity profile and then as a camera would record it. It uses NumPy 2.3.5 and SciPy 1.16.3, with $\lambda = 0.55$ µm and NA 1.4; all parameters were fixed before the first run. The PSF is the ideal in-focus Airy pattern of the formula above, with no aberrations, no background and no defocus. The first script computes the resolution criteria, then samples the pair at the Rayleigh distance with square pixels of several pitches, integrating the light over each pixel. Two placements are compared: the pair’s midpoint on a pixel centre and on a pixel boundary.

import numpy as np
from scipy.optimize import brentq
from scipy.special import j1

lam, NA = 0.55, 1.4            # wavelength (um) and numerical aperture (oil objective)
k = 2 * np.pi * NA / lam       # radial scale of the Airy pattern (1/um)

def airy(r):
    """In-focus intensity PSF of an ideal circular pupil, peak 1, r in um."""
    v = np.maximum(k * np.abs(r), 1e-12)
    return (2 * j1(v) / v) ** 2

def pair(x, s):
    """Profile through two equal, incoherent point sources at x = -s/2 and +s/2."""
    return airy(x - s / 2) + airy(x + s / 2)

x = np.linspace(-1, 1, 40001)                       # 0.05 nm steps
d_rayleigh = 0.61 * lam / NA
d_abbe = lam / (2 * NA)
fwhm = 2 * brentq(lambda r: airy(r) - 0.5, 0.01, 0.2)
# Sparrow: the dip vanishes where the curvature at the midpoint changes sign
curv = lambda s: pair(1e-4, s) + pair(-1e-4, s) - 2 * pair(0.0, s)
d_sparrow = brentq(curv, 0.1, 0.23)

print(f"Rayleigh 0.61 lam/NA = {d_rayleigh:.3f} um, Abbe lam/(2NA) = {d_abbe:.3f} um")
print(f"Sparrow (numerical)  = {d_sparrow:.3f} um = {d_sparrow * NA / lam:.3f} lam/NA")
print(f"FWHM of the PSF      = {fwhm:.3f} um = {fwhm * NA / lam:.3f} lam/NA")
print(f"Nyquist pixel lam/(4NA) = {lam / (4 * NA):.3f} um in the sample")
print("separation       s (um)   midpoint / maximum")
for name, s in [("1.5 x Rayleigh", 1.5 * d_rayleigh), ("Rayleigh", d_rayleigh),
                ("Abbe", d_abbe), ("Sparrow", d_sparrow), ("0.7 x Rayleigh", 0.7 * d_rayleigh)]:
    f = pair(x, s)
    print(f"{name:<15}  {s:.3f}    {f[len(x) // 2] / f.max():.3f}")

# The same pair at the Rayleigh distance, recorded by square pixels of pitch p
fine = 0.005                                        # integration step (um)

def pixel_row(s, p, phase):
    """Pixel values along the row through the pair; phase 0: pair centred on a
    pixel centre, phase 0.5: centred on a pixel boundary."""
    n = round(p / fine)
    sub = (np.arange(n) + 0.5) * fine - p / 2       # sample offsets inside a pixel
    centres = (np.arange(-15, 16) + phase) * p
    X = centres[:, None, None] + sub[None, :, None]
    Y = sub[None, None, :]
    val = (airy(np.hypot(X - s / 2, Y)) + airy(np.hypot(X + s / 2, Y))).sum(axis=(1, 2))
    return centres, val * fine**2

def dip(centres, val):
    left = np.where(centres < -1e-9)[0]
    right = np.where(centres > 1e-9)[0]
    iL, iR = left[np.argmax(val[left])], right[np.argmax(val[right])]
    if iR - iL < 2:
        return None
    ratio = val[iL + 1:iR].min() / min(val[iL], val[iR])
    return ratio if ratio < 1 else None

print("pixel (um)  dip, pixel-centred  dip, boundary-centred")
for p in (0.05, 0.10, 0.20, 0.30):
    out = [dip(*pixel_row(d_rayleigh, p, ph)) for ph in (0.0, 0.5)]
    print(f"  {p:.2f}      " + "        ".join(f"{r:.3f}" if r else " none" for r in out))

Rayleigh 0.61 lam/NA = 0.240 um, Abbe lam/(2NA) = 0.196 um
Sparrow (numerical)  = 0.186 um = 0.474 lam/NA
FWHM of the PSF      = 0.202 um = 0.514 lam/NA
Nyquist pixel lam/(4NA) = 0.098 um in the sample
separation       s (um)   midpoint / maximum
1.5 x Rayleigh   0.359    0.141
Rayleigh         0.240    0.735
Abbe             0.196    0.983
Sparrow          0.186    1.000
0.7 x Rayleigh   0.168    1.000
pixel (um)  dip, pixel-centred  dip, boundary-centred
  0.05      0.763        0.776
  0.10      0.803        0.920
  0.20       none         none
  0.30       none         none

At the Rayleigh distance the profile between the two points drops to about three quarters of the peaks, a clear dip. At Abbe’s $\lambda/(2\,\mathrm{NA})$ the dip is under 2%, and at the numerically found Sparrow distance it vanishes; the Sparrow value matches the $0.47\lambda/\mathrm{NA}$ obtained by rescaling Cremer and Masters’ angular formula. Below that, the two points form a single, broader peak. The FWHM, about $0.51\lambda/\mathrm{NA}$, lies between Abbe and Rayleigh. Which one is called “the resolution” is a convention; the example shows how much contrast each leaves.

Pixels then remove some of that contrast. Each pixel averages the profile over its width, so even at 50 nm the dip is shallower than in the continuous profile. At 0.10 µm, close to the Nyquist pitch, the dip survives but its depth depends on where the pair falls on the grid: it is more than twice as deep (20% against 8%) when a pixel sits at the midpoint as when the midpoint falls on a boundary between two pixels. At 0.20 and 0.30 µm no dip remains at either placement: the pair is optically resolved but not recorded as two objects.

The second part, which continues the first script, adds photon noise. Each source sends $N$ photons to the image plane, the expected pixel values follow from the PSF integrated over each pixel (its total over the plane is $4\pi/k^2$), and each of 2000 noisy images (random generator seed 0) is drawn by Poisson sampling. The dip counts as seen when the pixel at the midpoint is darker than both peak pixels of the noise-free image. The pair’s midpoint is on a pixel centre.

# Continues the first script: photon noise at the Rayleigh distance
rng = np.random.default_rng(0)
total = 4 * np.pi / k**2      # integral of the Airy PSF over the plane, so each source sends N photons
trials = 2000
print("photons per source:   " + "  ".join(f"{N:>6}" for N in (300, 1000, 3000, 10000)))
for p in (0.05, 0.10, 0.20):
    centres, val = pixel_row(d_rayleigh, p, 0.0)
    left, right = np.where(centres < -1e-9)[0], np.where(centres > 1e-9)[0]
    iL, iR = left[np.argmax(val[left])], right[np.argmax(val[right])]
    i0 = np.argmin(np.abs(centres))                 # pixel at the midpoint
    kept = []
    for N in (300, 1000, 3000, 10000):
        counts = rng.poisson(N * val / total, size=(trials, len(val)))
        ok = (counts[:, i0] < counts[:, iL]) & (counts[:, i0] < counts[:, iR])
        kept.append(ok.mean())
    print(f"pixel {p:.2f} um, dip seen:  " + "  ".join(f"{f:>6.2f}" for f in kept))

photons per source:      300    1000    3000   10000
pixel 0.05 um, dip seen:    0.57    0.81    0.97    1.00
pixel 0.10 um, dip seen:    0.74    0.96    1.00    1.00
pixel 0.20 um, dip seen:    0.00    0.00    0.00    0.00

With a few hundred photons per source, the Rayleigh dip is lost in a quarter to almost half of the images, so in practice two points at the textbook limit are often not seen as two. The finer 50 nm grid, which had the deeper dip without noise, does worse than the 0.10 µm grid at every photon count: each of its pixels collects about a quarter of the light, and since Poisson noise grows only as the square root of the count, its relative noise is about twice as large. Near-Nyquist sampling is the better compromise here, and the 0.20 µm grid fails regardless of light, because no noise level restores a dip that the sampling has removed. The test is deliberately simple, a single row of pixels and a fixed decision rule, and real data add background, out-of-focus light and aberrations; the general point, that the resolution actually achieved depends on sampling and photon count as well as on NA and wavelength, agrees with Bankhead’s (2022–2025) caution that brightness differences, finite pixel sizes and noise mean a greater distance may be needed in practice.

Characteristics

Property	What sets it	Typical consequence for analysis
Lateral resolution	$\lambda$, objective NA, condenser NA	About 0.2 µm at best with visible light; detail below it is absent, not blurred
Axial resolution	$\lambda$, $n$, NA²	Several times coarser than lateral; z-stacks are anisotropic
Pixel pitch in the specimen	Camera pixel ÷ magnification, binning	Must be read from metadata; near $\lambda/(4\,\mathrm{NA})$ for full resolution
Contrast	Brightfield, darkfield, phase contrast, DIC, fluorescence	Each method maps the specimen to intensity differently
Noise	Photon count, camera type	Read noise matters at low light; sCMOS adds pixel-dependent noise

Calibration

The pixel size in the specimen is a calibration: it depends on the objective, any intermediate optics and binning, and Bankhead (2022–2025) recommends always checking that it was written correctly into the file. The PSF can be measured by imaging a z-stack of a small fluorescent bead, which approximates a point source; a measured PSF differs from a theoretical one partly because the bead has finite size and partly because the objective is imperfect (Bankhead, 2022–2025). Illumination is calibrated through the shading model below: prospective methods estimate it from extra reference images, retrospective ones from the image data themselves (Peng et al., 2017). Köhler alignment, the setting of field and aperture diaphragms, is the optical calibration that makes the illumination uniform in the first place (Sauer & Surrel, 2017).

Artifacts

• Uneven illumination (shading). Peng et al. (2017) model a measured image as $I_{\mathrm{meas}}(x) = I_{\mathrm{true}}(x)\,S(x) + D(x)$, where the multiplicative flat-field $S(x)$ describes the variation of effective illumination across the field and the additive dark-field $D(x)$ is dominated by camera offset and thermal noise. Shading or vignetting typically dims the image from the optical axis toward the edges, biases measurements of tissue composition or single cells, and produces visible discontinuities between tiles of stitched whole-slide images; their method, BaSiC, estimates $S$ and $D$ from a set of images and corrects each one by inverting the model (Peng et al., 2017).
• Out-of-focus light. In widefield imaging of a thick specimen, light from other planes adds an unhelpful background, which hampers the detection of small structures (Bankhead, 2022–2025). Laser-scanning confocal microscopes reject it with a pinhole that admits light only from the focused point (Bankhead, 2022–2025).
• Aberrations. Refractive-index mismatch causes spherical aberration that grows with depth (Bankhead, 2022–2025); chromatic aberration shifts the focus with wavelength (Cremer & Masters, 2013), which in our reading misaligns colour channels relative to one another.
• Halos and shadow-casting. Phase-contrast halos surround objects with strong contrast (Zernike, 1953) and disturb phase-based measurements (Mualla et al., 2018); DIC’s relief is directional, set by the shear direction (Sauer & Surrel, 2017), so object boundaries look different on different sides.
• Dust and debris. Dust and fluorescent dye particles are common bioimage artifacts; Peng et al. (2017) point out that the robustness of earlier shading-correction methods to them had not been tested.
• Undersampling and pixelation. As the example shows, a grid coarser than about $\lambda/(4\,\mathrm{NA})$ loses detail that the optics deliver.

Processing the Data

The sensor’s properties translate into a few steps before or during analysis. Shading correction comes first when intensities are measured or tiles are stitched (Peng et al., 2017). Blur can be modelled as convolution with the PSF, which Bankhead (2022–2025) notes is well approximated in the focal plane by a Gaussian, so a Gaussian of matching width can stand in for it in simulation and filtering. Deconvolution inverts that model. Brightfield, phase-contrast and DIC images then go to cell segmentation and tracking, where the contrast mechanism matters, since the same cell looks different under each. Stained tissue imaged in brightfield is the material of digital pathology.

Strengths

• Images unstained, transparent cells by phase contrast, and works outside the vacuum that restricts electron microscopes to dead specimens and without their complicated preparation (Mualla et al., 2018); brightfield is the cheapest and simplest set-up (Mualla et al., 2018).
• A well-understood, nearly linear image formation model, so blur can be simulated and partly inverted.

Limitations

• The diffraction limit, made larger in practice by noise and sampling.
• Transparent specimens need a contrast method, each with its own nonlinear mapping and artifacts.
• Intensities depend on illumination, staining, focus and camera, so they are comparable across instruments only after correction and calibration.

Typical Applications

In clinical routine, brightfield microscopy serves mostly cytology, the study of cells and their inner structure, and histology, the study of cells within the surrounding tissue, both with stained specimens; hematoxylin and eosin, which stain nuclei blue and cytoplasm magenta, are the most common histology stain (Mualla et al., 2018). Whole-slide scanners digitize such slides for digital pathology. In cell biology, phase contrast is convenient for qualitative imaging of unstained, transparent specimens such as thin adherent cell cultures (Mualla et al., 2018), and fluorescence variants image specific labelled molecules. Microscope makers offer phase contrast and DIC for reflected as well as transmitted light, and DIC is used from materials science to biology (Sauer & Surrel, 2017).

History

Microscopes long predate their theory: until Abbe joined Carl Zeiss, microscope design proceeded largely by trial and error (Cremer & Masters, 2013). In 1873 Ernst Abbe published a 55-page paper on the theory of the microscope, containing not a single diagram or equation, which explained image formation by the diffraction of light at the object and set the resolution limit at about half the wavelength; Hermann von Helmholtz derived a similar limit in 1874 and acknowledged Abbe’s priority in a note, and Lord Rayleigh’s 1896 analysis reached similar conclusions (Cremer & Masters, 2013). Abbe demonstrated the theory with gratings and apertures before the Royal Microscopical Society in 1876, and Porter published a mathematical account of it in 1906 (Cremer & Masters, 2013). In 1903 Siedentopf and Zsigmondy built the ultramicroscope, which used darkfield illumination to detect particles below the resolution limit (Cremer & Masters, 2013).

Frits Zernike came to the microscope from diffraction gratings. Around 1930 he found that the spurious lines of a grating differed in phase from the main line, made phase strips to change this, and then tried them in a microscope; Zeiss, unenthusiastic at his 1932 demonstration, brought out phase-contrast objectives in 1941 (Zernike, 1953). He received the 1953 Nobel Prize in Physics for the phase contrast method, especially the phase contrast microscope (NobelPrize.org, 1953). Georges Nomarski, at the Institut d’Optique, devised his modified Wollaston prism in 1952, the basis of the DIC equipment that microscope makers offer (Sauer & Surrel, 2017). Digital cameras later turned these instruments into sensors whose output is analyzed by computer, and the far-field limit itself was circumvented by the fluorescence-based super-resolution methods that bioimage analysis summarizes.

Comparison with Other Sensors

• Ordinary cameras follow the same image formation chain. In a microscope, the blur is not mainly a matter of imperfect lenses but a limit set by the nature of light (Bankhead, 2022–2025), and the specimen is usually transparent.
• Electron microscopes use electron beams with far shorter wavelengths and reach resolutions of about 0.2 nm in transmission, but work in vacuum on dead specimens and need complicated preparation (Mualla et al., 2018).
• Confocal and light-sheet microscopes, fluorescence variants that bioimage analysis describes, reject out-of-focus light and give optical sections, at the cost of scanning or special illumination.
• Whole-slide scanners are microscopes automated to image an entire slide as tiles that are stitched together, which makes shading between tiles visible unless it is corrected (Peng et al., 2017).