The numerical aperture of a microscope objective characterizes its capability to gather light and resolve fine specimen details when working at a fixed object (or specimen) distance. Imaging light waves pass through the specimen and enter the objective in the form of an inverted cone, as shown in Figure 1(a). White light consists of electromagnetic waves with a broad spectrum, with wavelengths ranging from 400 nm to 700 nm. For reference: 1 millimetre equals 1000 micrometres, and 1 micrometre equals 1000 nanometres. Green‑light wavelengths centre at 550 nm, equivalent to 0.55 μm. When observing tiny objects under a microscope (e.g., typical stained specimens mounted on microscope slides), light incident on these micro‑objects undergoes diffraction and deviates from its original path (Figure 1(a)). Smaller objects produce more pronounced diffraction of incident light. A higher numerical aperture allows more oblique light rays to enter the objective’s front lens, yielding higher‑resolution images and enabling clearer visualization of finer structures.

Figure 1(a) illustrates a basic microscope system composed of an objective and a specimen illuminated by collimated light, a scenario occurring when no condenser is deployed. Light diffracted by the specimen forms an inverted cone with a half‑angle, representing the limiting angle of light admissible into the objective. To increase the effective aperture and resolving power of the microscope, a condenser is added (Figure 1(b)) to generate a ray cone on the illumination side of the specimen. This enables the objective to collect light produced at larger diffraction angles, thereby improving the resolution of the microscope system. The sum of the aperture angles of the objective and the condenser is termed the working aperture. Maximum resolution is achieved when the condenser aperture angle matches that of the objective.

To compare different objectives and obtain quantitative resolution metrics, numerical aperture — a measure of the solid angle subtended by the objective — is defined as:

Where α equals half the objective aperture angle, and η denotes the refractive index of the immersion medium between the objective and the specimen‑covering coverslip (η = 1 for air; η = 1.5 for oil or glass). From this formula, the refractive index is clearly the limiting factor for achieving a numerical aperture greater than 1.0. Accordingly, to attain a higher working numerical aperture, the refractive index of the medium between the objective front lens and the specimen coverslip must be increased. The theoretical maximum aperture angle for standard microscope objectives is 180°, corresponding to a half‑angle of 90° in the numerical‑aperture formula. Since the sine of 90° equals 1, numerical aperture is constrained not only by aperture angle but also by the refractive index of the imaging medium. In practice, only high‑end objectives can reach aperture angles exceeding 70‑80°, and such objectives typically cost thousands of US dollars.

The resolution of an optical microscope is defined as the minimum distance separating two points on a specimen that can still be distinguished as separate entities. Resolution is directly related to the effective magnification of the microscope and the perceptual limit for specimen detail. It remains a somewhat subjective metric in microscopy: even at high magnification, an image may appear blurry while the objective and auxiliary optics are resolving detail to their full capacity. Due to the wave nature of light and associated diffraction phenomena, the resolution of a microscope objective is governed by the angles of light waves admitted into the front lens; hence the instrument is described as diffraction‑limited. This limitation is purely theoretical. Even a theoretically perfect objective free of optical aberrations has finite resolution.

If an objective projects fine detail onto the intermediate image plane at a resolution finer than human‑eye resolution (common with low magnification and high numerical aperture), the observer will fail to distinguish subtle image features. Empty magnification occurs when an image is magnified beyond its physical resolution limit. For these reasons, useful magnification for human observation should preferably lie between 500 × NA and 1000 × NA of the objective’s numerical aperture.

One method to improve the optical resolving power of a microscope is to use immersion fluid between the objective front lens and the coverslip. Most objectives with magnifications of 60 ×, 100 × or higher are designed for oil immersion. Optimal performance is achieved using oil with a refractive index η = 1.51, precisely matched to glass. This eliminates reflections along the light path from specimen to objective. Without this measure, reflections always cause light loss at the coverslip or at the front lens for high‑angle rays (Figure 2).

Such reflections degrade the usable numerical aperture and resolving power of the objective. An objective’s numerical aperture also depends partly on the degree of optical‑aberration correction. Highly corrected objectives tend to deliver larger numerical apertures at their respective magnifications, as shown in Table 1.

When light rays originating from each point of the specimen pass through the objective and recombine to form an image, each specimen point is reproduced in the image not as a sharp dot but as a small pattern known as an Airy disk. This phenomenon arises from diffraction and scattering as light passes through micro‑features and gaps within the specimen and through the objective’s circular rear aperture. The limiting distance at which two small objects can still be discriminated as separate entities serves as a measure of microscope resolving power. This limiting distance is defined as the effective microscope resolution, denoted \(d_0\). Resolution can be theoretically derived from the instrument’s optical parameters and the mean wavelength of illumination.

Importantly, objectives and tube lenses do not image a point on the specimen (e.g., a tiny aperture in metal foil) as a bright disc with crisp edges. Instead, they produce a slightly blurred spot surrounded by diffraction rings — the Airy disk (see Figure 3(a)). The three‑dimensional representation of the diffraction pattern near the intermediate image plane is the point‑spread function (Figure 3(b)). The Airy disk is the region enclosed by the first minimum of the Airy pattern, containing approximately 84 % of the total light energy, as illustrated in Figure 3(c). The point‑spread function is the three‑dimensional counterpart of the Airy disk.

Resolution can be calculated using the renowned formula proposed by Ernst Abbe in the late 19th century, which describes image sharpness for optical microscopes:

Where λ is the wavelength of light, η is the refractive index of the imaging medium as described above, and the combined term η · sin(α) is the objective numerical aperture (NA). For common microscope objectives, NA is below 1.5, restricting α to less than 70° (though modern high‑performance objectives approach this limit). Accordingly, for an objective with NA = 1.40 and the shortest practical wavelength (~400 nm), the theoretical lateral resolution limit is approximately 150 nm, while axial resolution approaches 400 nm. Structures separated by distances smaller than this cannot be resolved in the lateral plane using light microscopy. Given the critical interplay between imaging‑medium refractive index and objective angular aperture, Abbe introduced the concept of numerical aperture in his explanation of microscope resolution.

Diffraction rings within the Airy pattern originate from the limiting aperture of the objective, which acts as an aperture stop generating these rear‑focal‑plane diffraction rings. Larger apertures for both objective and condenser yield smaller \(d_0\). Therefore, higher overall‑system numerical aperture delivers superior resolution. One of the equations associated with the original Abbe formula expresses the relationship among numerical aperture, wavelength and resolution:

Where λ is the imaging wavelength, \(NA_{con}\) is the condenser numerical aperture, and \(NA_{obj}\) is the objective numerical aperture. The coefficient 1.22 derives from calculations for two approaching Airy disks with superimposed intensity distributions, as illustrated in the referenced figure. When two image points are widely separated, they are readily recognised as distinct objects. As the distance between Airy disks decreases, the limiting condition is reached when the primary maximum of the second Airy disk coincides with the first minimum of the first Airy disk. The superimposed intensity profile exhibits two brightness maxima separated by an intensity valley roughly 20 % lower than the peak values. This intensity drop is just sufficient for the human eye to resolve two separate points; this threshold is known as the Rayleigh criterion.

An analogy helps illustrate this principle. Telephone lines are poorly suited for electronically transmitting the subtle tones of violin music due to limited channel bandwidth. Far better results are obtained with high‑quality microphones and amplifiers whose frequency range matches human hearing. In music, fundamental information resides within mid‑range audio frequencies, yet fine tonal detail is encoded in high‑order harmonics. In microscopy, fine structural details are encoded within diffracted light. To visualise these details in the image space behind the objective, the objective must first collect that diffracted light. Larger aperture angles and higher numerical apertures facilitate this collection process.