Numerical Aperture, Resolution, and Magnification

Table of Contents

• What Is Numerical Aperture in Microscopy?
• Diffraction-Limited Resolution and the Role of Wavelength
• Magnification vs. Resolution: Avoiding Empty Magnification
• Depth of Field, Depth of Focus, and Working Distance
• Immersion Media, Refractive Index, and High-NA Objectives
• Illumination, Coherence, and Köhler Alignment
• Camera Pixel Size, Sampling, and Nyquist Criteria
• Choosing Objectives: NA, Magnification, and Trade-Offs
• Practical Calculations and Realistic Expectations
• Frequently Asked Questions
• Final Thoughts on Choosing the Right NA and Magnification

What Is Numerical Aperture in Microscopy?

Numerical aperture (NA) is a core specification of microscope objectives and condensers that sets the stage for what the optical system can resolve. In simple terms, NA answers: how much diffracted light from fine specimen detail can the lens accept? Its definition, in object space, is
NA = n D7 sin(B8), where n is the refractive index of the medium between the front lens and the specimen, and B8 is the half-angle of the maximum cone of light that can enter (for an objective) or leave (for a condenser) the lens.

A few immediate consequences follow from this definition:

• Higher NA means the objective accepts light at larger angles, collecting higher spatial frequencies that encode finer details.
• Changing the immersion medium (for example, air to oil) changes n and can raise the achievable NA, provided the lens is designed for it.
• NA is independent of magnification: a 40D7 objective can have low or high NA depending on its optical design.

On most objectives you will see markings like 40D7/0.65 or 100D7/1.25 Oil. The number after the slash is the NA, and it conveys more about resolving power than magnification alone. While magnification enlarges the image, NA drives resolution and influences contrast, brightness, depth of field, and working distance.

The condenser also has an NA, and its setting determines the illumination cone that reaches the specimen. The condenser NA works in tandem with the objective NA. To get the best resolution in brightfield, the effective illumination NA provided by the condenser should be high and well-matched to the objective (see Illumination, Coherence, and Köhler Alignment).

Diffraction-Limited Resolution and the Role of Wavelength

All lenses are ultimately limited by diffraction when aberrations are corrected well. Diffraction spreads a point source into an Airy pattern, setting a fundamental limit to how close two features can be and still be distinguished as separate. In microscopy, several related metrics are used to describe resolution. They are numerically similar, but they answer slightly different questions. The two most common are:

• Rayleigh criterion (point resolution, incoherent imaging): 94 248 0.61 D7 BB / NA. Here, 94 is the minimum center-to-center separation between two point-like emitters so that the first minimum of one Airy disk falls at the maximum of the other, and BB is the wavelength of light.
• Abbe limit (periodic structures, incoherent imaging): the highest transmitted spatial frequency is fc = 2 NA / BB. The corresponding minimum resolvable grating period is pmin = 1 / fc = BB / (2 NA).

These expressions emphasize two levers for resolution:

• Shorter wavelengths resolve more detail. Blue light (e.g., ~450–500 nm) provides better resolution than red light (~620–700 nm) when other factors are constant.
• Higher NA resolves more detail. Increasing NA reduces both Rayleigh and Abbe limits, improving lateral resolution.

For axial (depth) resolution in widefield imaging, a common diffraction-limited estimate is:

Axial resolution (widefield, incoherent): 94z 248 2 n D7 BB / NA2

This expression shows that axial resolution improves steeply with increasing NA and with shorter wavelengths, and it also depends on the refractive index n of the immersion medium.

It is useful to clarify terms often mixed in discussion:

• Resolution is the smallest separable detail the system can distinguish reliably.
• Contrast transfer describes how strongly different spatial frequencies are transmitted. Even below the theoretical cutoff 2 NA / BB, very fine features may have low contrast and be difficult to see.
• Sampling (see Camera Pixel Size, Sampling, and Nyquist Criteria) must be adequate to record the resolvable detail. If sampling is too coarse, fine structure is lost or aliased regardless of optical resolution.

Coherence also matters. In coherent imaging (e.g., illumination with a point-like source and suitable pupil), the cutoff frequency is different: fc,coh = NA / BB. Most brightfield microscopy is partially coherent; with properly adjusted Köhler illumination and a sufficiently open condenser, the behavior approaches the incoherent limit for small features, making 2 NA / BB a practical reference for maximum spatial frequency in intensity images.

To summarize, lateral resolution improves linearly with NA and inversely with wavelength, whereas axial resolution improves with the square of NA. These relationships underlie many of the trade-offs discussed next.

Magnification vs. Resolution: Avoiding Empty Magnification

Magnification enlarges an image, but does not create new information. If the optical system cannot resolve a detail, increasing magnification only makes the blur larger. Empty magnification occurs when image scale exceeds the optical (and sampling) resolution.

A practical way to connect magnification to NA is through the lens of sampling and observer acuity. For visual observation, a traditional heuristic is to use useful magnification in the range of roughly 500 to 1000D7 the NA of the objective. For example:

• With an NA 0.65 objective, a total magnification in the range of ~325D7 to ~650D7 is generally sufficient to render resolvable detail at a comfortable visual scale.
• With an NA 1.25 oil-immersion objective, ~625D7 to ~1250D7 is a reasonable range.

These are not hard limits, but they help avoid needlessly high magnifications that do not reveal additional structure. For digital imaging, the notion of useful magnification translates to effective sampling at the specimen plane (see Camera Pixel Size, Sampling, and Nyquist Criteria).

It is also crucial to remember that NA and condenser setup set the resolution limit (see Diffraction-Limited Resolution and Illumination). If you view with 20D7 eyepieces and a 100D7 objective, achieving 2000D7 total magnification, that does not bypass an NA-limited resolution of, say, ~0.28A0µm at 550A0nm.

Key takeaways to avoid empty magnification:

• Let NA and wavelength set expectations for resolvable detail.
• Match magnification to the resolution so that resolvable features occupy multiple pixels (digital) or a comfortable visual size (visual).
• Use higher magnification when it increases sampling adequacy or visualization comfort, not as a substitute for insufficient NA.

Depth of Field, Depth of Focus, and Working Distance

Resolution is just one piece of image quality. Microscopy often involves three-dimensional specimens. Two related but distinct concepts describe axial tolerance:

• Depth of field (DOF): the object-space range over which the specimen appears acceptably sharp.
• Depth of focus: the image-space range over which the sensor or image plane can move while maintaining acceptable sharpness.

For diffraction-limited imaging, a commonly used approximation for the diffraction contribution to object-space DOF is proportional to BB / NA2, with a factor that depends on refractive index and definition of acceptable blur. The key trends are robust:

• Increasing NA decreases DOF rapidly (approximately as 1 / NA2).
• Shorter wavelengths reduce DOF.
• Immersion in a higher-index medium increases DOF slightly via the refractive index term.

Depth of focus in image space also scales inversely with NA2. As NA grows, the image becomes more sensitive to focus errors, mechanical vibrations, and cover glass mismatch. This is one reason why high-NA oil objectives demand careful focusing and stable stands.

Working distance is the physical clearance between the front of the objective and the sample when in focus. Higher NA typically correlates with shorter working distance, because achieving a larger acceptance angle at the front lens requires lens elements close to the specimen. Long working distance objectives exist, but at a given magnification they often trade some NA to gain clearance.

These relationships mean you rarely maximize everything simultaneously. For thick samples or specimens that cannot lie perfectly flat, a moderate-NA objective may deliver more interpretable images than a very high-NA lens, because the extended DOF keeps more of the structure in acceptable focus. Conversely, for resolving the finest possible lateral details on thin specimens, prioritize NA, accept the reduced DOF, and employ precise focusing. When appropriate, focus stacking or optical sectioning methods can mitigate DOF limits, but those are separate techniques beyond the scope of this fundamentals article.

Finally, cover glass and immersion choices influence both DOF and resolution through spherical aberration. We return to this in Immersion Media, Refractive Index, and High-NA Objectives.

Immersion Media, Refractive Index, and High-NA Objectives

Because NA = n D7 sin(B8), raising the refractive index n at the specimen boosts NA. Air has n 248 1.0. Standard borosilicate cover glass and many oils are around n 248 1.51 at visible wavelengths and room temperature. Water is ~1.33. Objectives are designed for a specific immersion medium; using the correct one is essential for performance and to avoid damage.

Three common categories are:

• Air objectives: No immersion medium (specimen is in air). Typical NA ranges up to ~0.95 for very high-end dry objectives, but many standard dry objectives are in the 0.10–0.95 NA range depending on magnification and design. Dry objectives are convenient, require no cleanup, and often provide longer working distances, but their NA ceiling limits the finest achievable resolution compared with immersion designs.
• Water-immersion objectives: Designed to operate with water between the front lens and the specimen or coverslip. These are valuable for imaging aqueous specimens and for reducing refractive index mismatch in live or thick samples. They can offer relatively high NA with reduced spherical aberration when focusing into water-based media.
• Oil-immersion objectives: Use immersion oil matched to the refractive index of cover glass. These achieve the highest NAs in routine light microscopy. Correct index matching reduces refraction at interfaces and supports larger acceptance angles at the objective pupil.

A critical factor for high NA is cover glass thickness. Many high-NA transmitted-light objectives are corrected for a standard cover glass thickness of approximately 0.17A0mm (commonly marked as #1.5 or #1.5H). Departures from the design thickness introduce spherical aberration that degrades resolution and contrast, especially at high NA. Some objectives include a correction collar, allowing limited adjustment to compensate for thickness variations or different media. See the discussion in Frequently Asked Questions for more on cover glass effects.

Immersion choice also interacts with DOF and working distance (see Depth of Field, Depth of Focus, and Working Distance). Oil immersion objectives that push NA high tend to have the shallowest DOF and shortest working distances. Water immersion can be a pragmatic compromise in aqueous environments where index matching to the specimen reduces aberrations when focusing below the surface.

Practical guidance for immersion use:

• Use only the medium specified for the objective (air, water, oil, or other designated immersion), and apply the appropriate amount to avoid bubbles.
• For oil immersion, ensure the cover glass is clean and of the correct type. Bubbles or contaminants degrade NA and contrast.
• Refocus gently as you change immersion or media; refractive index changes alter optical path lengths noticeably at high NA.

Illumination, Coherence, and Köhler Alignment

Even a superb objective cannot deliver high resolution if the illumination is poorly set. Brightfield microscopy with Köhler illumination aims to provide even illumination and adjustable control of the illumination NA via the condenser aperture diaphragm. Proper alignment has two key benefits:

• Uniform field brightness and minimized artifacts from the lamp filament or LED emitter.
• Controlled illumination cone and hence controlled coherence, which influences the system’s transfer of spatial frequencies.

In brightfield, as you open the condenser aperture diaphragm, you increase the illumination NA. This affects both contrast and resolution:

• Higher illumination NA (wider condenser aperture) generally increases the range of spatial frequencies transmitted as intensity, improving potential resolution up to the objective’s NA limit (see Diffraction-Limited Resolution).
• Lower illumination NA (narrow condenser aperture) increases image contrast for low spatial frequencies and reduces glare, but at the expense of high-frequency transfer and resolution.

A practical rule of thumb in brightfield is to set the condenser aperture diameter to roughly 70–90% of the objective’s NA. This typically balances contrast and resolution while maintaining adequate depth of field. The exact setting depends on specimen transparency and the features you wish to emphasize.

Coherence influences the theoretical cutoff frequency: for incoherent imaging the cutoff spatial frequency is approximately 2 NA / BB, while for coherent imaging it is NA / BB. Typical Köhler brightfield is partially coherent; with a well-opened condenser, practical performance approaches the incoherent case for small features. If the condenser is stopped down strongly, the effective coherence increases, and the system behaves more like a coherent imaging system for which the cutoff is lower, reducing how fine a grating can be transferred as intensity.

Finally, illumination wavelength couples back to resolution and contrast. Using shorter-wavelength filters (blue-green) can enhance resolution modestly in brightfield, provided the specimen absorbs or scatters enough at those wavelengths and the camera or eye sensitivity is adequate. Keep in mind that shorter wavelengths may increase phototoxicity or reduce signal in some modalities; for non-live, non-fluorescent brightfield this is usually not a concern.

Camera Pixel Size, Sampling, and Nyquist Criteria

Optical resolution is not the final word. To capture the detail the optics provide, the camera must sample the image finely enough. The central parameter is the effective pixel size at the specimen plane:

sspecimen = psensor / Mtotal

where psensor is the camera pixel pitch (µm) and Mtotal is the total magnification from specimen to sensor. In infinity-corrected systems, Mtotal equals the objective magnification multiplied by any intermediate optics (for example, a 0.5D7 or 1D7 camera coupler).

To avoid aliasing and capture the finest spatial frequencies transmitted by the optics, Nyquist sampling theory provides a guideline. For incoherent imaging (typical brightfield intensity images with a sufficiently open condenser), the optical transfer function has a cutoff at fc = 2 NA / BB. The Nyquist sampling interval 94x should satisfy:

94x 264 1 / (2 fc) = BB / (4 NA)

Equivalently, the specimen-plane pixel size should meet sspecimen 264 BB / (4 NA) to sample up to the incoherent cutoff. In practice, recommendations often range between BB / (4 NA) (stricter) and about 248 0.3 D7 (BB / NA) (a more relaxed, Rayleigh-based sampling) depending on contrast and noise considerations. The stricter criterion ensures that even the highest transmitted frequencies are represented without aliasing.

For example, with green light BB 248 550A0nm and NA = 0.95, a strict Nyquist limit yields sspecimen 264 0.55 / (4 D7 0.95) 248 0.145A0B5m. If your camera has psensor = 6.5A0B5m pixels and you use a 40D7 objective with a 1D7 coupler (total 40D7), then sspecimen = 6.5 / 40 248 0.1625A0B5m. That is close, but slightly above the strict Nyquist criterion; a 1.1–1.3D7 intermediate magnification or a higher objective magnification would bring sampling into stricter compliance if needed.

Practical sampling considerations:

• Monochrome vs. color sensors: Bayer color filter arrays distribute color samples over multiple pixels and rely on demosaicing. For critical resolution, monochrome sensors sample luminance more directly and may be preferred. If using color sensors, slightly finer sampling (smaller specimen-plane pixel size) helps maintain detail post-demosaicing.
• Signal-to-noise trade-offs: Finer sampling spreads photons over more pixels. Ensure exposure and illumination are sufficient to maintain signal quality while respecting specimen health for live samples.
• Objective NA first: Do not compensate poor NA with finer sampling. If the optics cannot pass the spatial frequency, no sampling can recover it.

When tuning a system, use the relationships among NA, wavelength, and sampling to co-design optical and digital parameters. For instance, if you know you must resolve ~0.3A0µm features at 550A0nm, you can estimate the NA required (see Practical Calculations and Realistic Expectations), then choose magnification and camera coupling such that sspecimen comfortably meets the Nyquist target for that NA.

Choosing Objectives: NA, Magnification, and Trade-Offs

Objective selection is where theoretical relationships meet practical constraints. Each objective balances NA, magnification, working distance, field flatness, chromatic correction, and cost. Here are the principal decision criteria and trade-offs:

1) Define the smallest feature you must resolve

Start with the finest details that matter for your application. If you need to separate features around 0.5A0µm at 550A0nm in brightfield, a rough target via Rayleigh is NA 248 0.61 D7 BB / B4. Solving gives NA 248 0.61 D7 0.55 / 0.5 248 0.67. If periodic features dominate, the Abbe criterion suggests pmin = BB/(2 NA). For NA = 0.67 and 550A0nm, pmin 248 0.41A0B5m.

Keep in mind that contrast at cutoff frequencies is low. If features are faint, you may benefit from slightly higher NA than the bare formula indicates, assuming your specimen and budget permit it.

2) Match magnification to NA and sampling

Once NA is set, pick magnification so that the camera’s specimen-plane pixel size meets the Nyquist target (see Sampling) and that the visual scale is comfortable. Ensure that you are not massively oversampling, which can dilute signal, nor undersampling, which can throw away real detail.

3) Balance working distance and DOF

High NA usually shortens working distance and reduces DOF. If your specimen is three-dimensional, uneven, or easily damaged by proximity, consider a slightly lower NA objective with longer working distance. Specialized long-working-distance designs exist and can be appropriate if resolution needs are moderate.

4) Consider immersion and index matching

If your sample is aqueous or you must image below the surface, water-immersion objectives can reduce spherical aberration due to index mismatch. Oil immersion maximizes NA for thin, coverslipped samples. Always use the immersion the objective was designed for to realize the stated NA and correction quality (see Immersion Media).

5) Think about field of view and flatness

High-NA, high-magnification objectives often have smaller fields of view and are specified for a certain image circle. Ensure the objective and tube lens support your camera sensor size without excessive vignetting or field curvature. If you need to image large areas, a lower magnification objective with good field flatness, combined with tiling, may be more efficient.

6) Chromatic performance and application-specific needs

For white-light brightfield, apochromatic correction can improve color fidelity and sharpness across the visible spectrum, but it can be costly. If you primarily use a narrow wavelength band (e.g., green illumination), achromats may suffice. Mirrors and reflective objectives avoid chromatic aberration, but are specialized. Always cross-check your application with the objective class rather than relying on magnification alone.

In sum, choose the lowest NA that reliably resolves your features of interest with adequate contrast and DOF, and then match magnification and sampling to that NA. When in doubt, test with a stage micrometer or a known fine grating to verify system resolution and sampling empirically.

Practical Calculations and Realistic Expectations

To ground the concepts above, this section walks through practical calculations. The goal is not to memorize numbers, but to develop intuition for how NA, wavelength, and sampling interact in real setups.

Scenario A: Determining NA for a target resolution

Suppose you need to distinguish two features about 0.4A0µm apart in brightfield at around 550A0nm (green). Using the Rayleigh criterion:

B4 248 0.61 D7 BB / NA 192 NA 248 0.61 D7 BB / B4

With BB = 0.55A0µm and B4 = 0.4A0µm: NA 248 0.61 D7 0.55 / 0.4 248 0.84

An objective around NAA00.85 or higher would be a good target. If your specimen is thin and under a proper cover glass, a 60–80D7 dry or water-immersion objective near this NA might work; an oil immersion lens could push NA higher if needed.

As a cross-check with periodic structures, the Abbe limit gives pmin = BB/(2 NA). With NAA00.85 at 550A0nm, pmin 248 0.323A0B5m. That suggests a grating with ~0.32A0µm period would be at the high-frequency edge of transfer; isolated points remain governed by the Rayleigh estimate around 0.39A0µm. Both metrics paint a similar resolution picture.

Scenario B: Matching camera sampling to optics

Suppose your camera has 3.45A0µm pixels. You plan to use a 60D7 objective (no camera coupling optics). The specimen-plane pixel size is:

sspecimen = 3.45 / 60 248 0.0575A0B5m

For NAA00.95 and BB = 0.55A0µm, the strict Nyquist criterion is BB/(4 NA) 248 0.145A0B5m. With 0.0575A0B5m sampling, you are oversampling considerably, which is generally fine if exposure and noise are adequate. If your camera had 6.5A0µm pixels, sspecimen = 6.5 / 60 248 0.108A0B5m, still satisfying the strict criterion.

Scenario C: Avoiding empty magnification visually

If you observe through eyepieces with a 40D7, NAA00.65 objective, total magnification might be 400D7 (with 10D7 eyepieces) or 800D7 (with 20D7 eyepieces). The useful range of ~500–1000D7 NA translates to ~325–650D7 as a comfortable envelope. Jumping to 800D7 can be acceptable for some observers, but moving further provides little added detail and may even accentuate vibration and eye strain.

Scenario D: DOF and high NA

Consider switching from NAA00.65 to NAA01.25 (oil) at the same wavelength. The diffraction term of DOF scales as 1/NA2. That means DOF at NAA01.25 is roughly a quarter of that at NAA00.65 (since (0.65/1.25)2 248 0.27). Expect significantly less axial tolerance, requiring careful focusing and flat, well-mounted specimens.

Scenario E: Cover glass mismatch

With a high-NA oil immersion objective corrected for 0.17A0mm cover glass, using a cover slip that is substantially thicker or thinner introduces spherical aberration. The image will appear soft even after careful focusing, and contrast of fine details will suffer. If the objective has a correction collar, adjust it to compensate within its specified range. Otherwise, use the specified coverslip type to achieve the stated resolution.

Scenario F: Illumination NA

If you close the condenser aperture too far (for instance, to less than half the objective NA), you may see higher contrast in coarse features, but the finest details disappear or look washed out. Reopening the condenser to ~70–90% of the objective NA restores high-frequency transfer. This aligns the illumination with the objective to approach the incoherent resolution limits discussed in Diffraction-Limited Resolution.

These examples reinforce that high fidelity in microscopy emerges from consistency across objective NA, illumination NA, wavelength, and sampling. A deficiency in any one link constrains the whole chain.

Frequently Asked Questions

Does a 100D7 objective always resolve more than a 40D7?

No. Resolution depends primarily on NA, not magnification. A 100D7 objective with NAA00.90 will generally resolve more than a 40D7 with NAA00.65, but a 40D7 objective with NAA00.95 can outperform a 100D7 objective with NAA00.80 in terms of the smallest resolvable detail. Magnification enlarges features; NA determines how fine those features can be in the first place. To decide objectively, compare the NA values and consider the wavelength and illumination setup (see Illumination, Coherence, and Köhler Alignment).

How does cover glass thickness affect resolution?

Many high-NA objectives are corrected for a specific cover glass thickness, commonly around 0.17A0mm. If the actual cover glass differs substantially, spherical aberration increases. This broadens the point spread function, reducing contrast and effective resolution, especially at high NA. If your objective has a correction collar, use it to compensate within its designed range. If not, use the specified cover glass and ensure proper immersion to realize the stated NA and approach the diffraction limits outlined in Diffraction-Limited Resolution.

Final Thoughts on Choosing the Right NA and Magnification

Optical microscopy is a balancing act. To obtain crisp, information-rich images:

• Start with NA: Choose an objective with enough NA to resolve your target features at the wavelengths you use.
• Set illumination correctly: Use Köhler illumination and match the condenser aperture to the objective NA to support high-frequency transfer.
• Match sampling: Ensure your camera’s specimen-plane pixel size meets Nyquist for your NA and wavelength, typically no larger than BB / (4 NA) for incoherent imaging.
• Respect trade-offs: Higher NA shortens DOF and working distance; immersion choices affect aberrations and practicality.
• Avoid empty magnification: Increase magnification to aid viewing and sampling, not to compensate for insufficient NA.

With these principles, you can plan a system or session that reliably reaches the limits of what your optics and specimens can reveal. If you found this article useful, explore our other fundamentals, types, and application guides, and consider subscribing to our newsletter to receive future deep dives on microscopy techniques and technology.