Depth of focus / depth of field

Depth of focus (DOF, image side) and depth of field (DOF, object side) are two related quantities that describe how far the imaging system can be from optimal focus before the image becomes unacceptably blurred. The two are connected by the longitudinal magnification of the system.

Image-side depth of focus. The longitudinal distance over which the image of a point source remains within an acceptable size (typically within the diffraction-limited spot):

$$\text{DOF}_\text{image} \;=\; \frac{\pm 2 \lambda \, n}{\text{NA}_\text{image}^2}$$

where NA is the numerical aperture in image space and $n$ is the index in image space (often $n = 1$).

For typical imaging systems:

• Microscope objective 100×/NA = 1.4: DOF$_\text{image}$ $\sim \pm 0.6$ μm
• Camera lens 50 mm f/2: DOF$_\text{image}$ $\sim \pm 10$ μm
• Telescope: DOF$_\text{image}$ $\sim \pm 100$ μm

Object-side depth of field. The longitudinal distance in object space over which an object remains within acceptable focus:

$$\text{DOF}_\text{object} \;=\; \frac{\pm 2 \lambda \, n}{\text{NA}_\text{object}^2}.$$

This is what photographers care about: the range of distances at which subjects appear acceptably sharp in the captured image.

Magnification relation. Longitudinal magnification is the square of transverse magnification:

$$m_\text{long} \;=\; m_\text{trans}^2.$$

So the image-space DOF and object-space DOF are related by:

$$\text{DOF}_\text{image} \;=\; m_\text{trans}^2 \cdot \text{DOF}_\text{object}.$$

For a 100× microscope: $\text{DOF}_\text{object}$ is $10^4 \times$ smaller than $\text{DOF}_\text{image}$. A 0.6 μm image-space DOF corresponds to a $6 \times 10^{-5}$ μm = 60 picometer object-space DOF — meaning the object position must be controlled to far better than wavelength precision to stay in focus.

In practice, microscopes use NA in object space (which is the natural way to describe the objective), and the formula becomes:

$$\text{DOF}_\text{object} \;=\; \frac{\pm \lambda n}{\text{NA}^2}$$

(with a factor of 2 difference from the image-side formula depending on convention).

Photographic convention. For cameras, DOF is usually expressed in terms of the "circle of confusion" (CoC) — the maximum acceptable blur diameter at the image sensor. Common conventions:

Sensor format	Acceptable CoC
Full-frame 35 mm	0.029 mm
APS-C	0.019 mm
Micro Four Thirds	0.015 mm
Mobile phone (~6.4 mm sensor)	0.005 mm

For a focal length $f$ and aperture diameter $D = f/\#$, the photographic DOF range about the focal distance $s$ is approximately:

$$\text{DOF} \;\approx\; \frac{2 (s/f)^2 \cdot N \cdot c}{1 - (s/(N \cdot s_\text{hyperfocal}))^2}, \quad s_\text{hyperfocal} \;=\; \frac{f^2}{N c},$$

where $N$ is the f-number and $c$ is the circle of confusion. The "hyperfocal distance" is the focus distance at which DOF extends from half that distance to infinity.

For a 50 mm lens at f/4 on full-frame ($c = 29$ μm):

• Hyperfocal distance $s_\text{hyperfocal} = 50^2 / (4 \times 0.029) = 21$ m
• DOF at focus distance = 10 m: $\sim 5.5$ m to 50 m (most of the image in focus)
• DOF at focus distance = 3 m: $\sim 2.7$ m to 3.4 m (narrow)
• DOF at focus distance = 0.5 m: $\sim 0.49$ m to 0.51 m (very narrow)

Why DOF matters in microscopy.

• 3D imaging: knowing DOF tells you the z-resolution. Confocal microscopy with NA = 1.4 has DOF $\sim 0.6$ μm; multi-photon microscopy achieves similar.
• Sample mounting: must hold samples flat within the DOF; deviations show as out-of-focus regions
• Focus stacking: image acquisition at multiple Z positions extends effective DOF for thick samples
• Lithography: features must be uniformly in-focus across the wafer; DOF sets the wafer-flatness tolerance

Lithography stepper DOF. Photolithography steppers at the bleeding edge of NA constraint:

Stepper generation	NA	$\lambda$	DOF (image)
KrF DUV (1990s)	0.6	248 nm	$\pm 0.7$ μm
ArF DUV (2000s)	0.9	193 nm	$\pm 0.24$ μm
ArF immersion (2010s)	1.35	193 nm	$\pm 0.11$ μm
EUV (2020s)	0.33	13.5 nm	$\pm 0.12$ μm
EUV high-NA (2025+)	0.55	13.5 nm	$\pm 0.045$ μm

The decreasing DOF at high NA is one of the major engineering challenges of modern lithography — wafer flatness, mask flatness, and lens depth-of-focus must all be controlled to within 50 – 100 nm for the latest generations.

DOF in laser focusing. For a focused Gaussian beam, the relevant longitudinal length is the Rayleigh range:

$$z_R \;=\; \frac{\pi w_0^2}{\lambda} \;\approx\; \frac{\lambda}{\pi \, \text{NA}^2}.$$

The Rayleigh range plays the role of DOF for Gaussian beams: over this range, the beam stays within $\sqrt{2}$ of its minimum waist. This is the natural figure of merit for laser machining, particle trapping, and nonlinear-optics interactions where the longitudinal interaction length matters.

Extended DOF techniques.

• Cubic phase mask (Dowski-Cathey): a phase mask plus computational deblurring extends DOF by 5 – 30× at the cost of slight loss of resolution
• Hyperbolic phase mask: similar concept, different mask profile
• Light-field imaging (Lytro, Raytrix): capture multiple focal planes simultaneously; reconstruct any focus after capture
• Time-multiplexed focus: ultrasound-driven focus scanning at video rate

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 3 (Gaussian beam Rayleigh range) and Ch. 4 (imaging system DOF); Hecht, Optics (5th ed., 2017), Ch. 5 for the imaging-optics treatment; Born & Wolf, Principles of Optics (7th ed.), Ch. 8 (aberration-broadened imaging).