Spatial Frequency and Image Resolution

This foundational knowledge article explores how spatial frequency impacts the resolution of structures in microscopy. Learn how to observe diffraction patterns and understand the relationship between specimen periodicity and diffraction pattern maxima.

Key Learnings:

• Observe a microscopically small object's diffraction pattern in the back focal plane of the microscope objective.
• To resolve a diffraction grating image, at least two diffraction orders must be captured and focused in the back focal plane (usually the zero and first order).
• Fine structures diffract the light stronger, resulting in maxima with larger diffraction angles (distances) in the back focal plane.

Microscopic observation of the diffraction pattern produced by an object of known spatial frequency can help to understand the image formation theory of the microscope developed by Prof. Ernst Abbe in 1873 in Jena. An important part of this is to understand how the finest structures of a specimen can be resolved by the light microscope. The diffraction pattern caused by such structures of a specimen can be seen by looking into the back focal plane of the microscope objective using a Bertrand lens system or an auxiliary microscope. The appearance of the diffraction pattern depends on the distance between the structures of a specimen: Coarser structures which are further apart from each other (i.e. lower spatial frequency) diffract the light less and therefore produce smaller diffraction angles. Finer structures that are closer together (i.e. higher spatial frequency) diffract the light more and therefore produce larger diffraction angles.

In the objective´s back focal plane, different images of the light source (e.g. the condenser aperture diaphragm) can be seen: the undiffracted light spot in the center and additional diffracted spots form a pattern. This object-specific pattern is called the diffraction pattern. It is induced by the small diffracting structures of a microscopical specimen. The diffraction pattern consists of the undiffracted “lamp” light (principal maximum or zero order maximum) surrounded by the diffracted “sample” light (1st, 2nd, 3rd etc. order maxima). Usually, the zero order maximum is the central image of the light source as represented by the size and shape of the condenser aperture diaphragm. If white light is used, it will be displayed as a white central spot. The maxima of higher orders then appear as overlapping images of the light source, separated into all spectral colors from violet/blue to red. The blue light image is more inward, the red light image more outward, and the green light image is in between.
In other words: diffraction is wavelength dependent. A given spacial frequency of sample structures diffracts blue light less but red light more.

The diffraction pattern image in the back focal plane of the objective is oriented perpendicular to the structure orientations in the specimen: That means in case of a line grating as a specimen, the long axis of its diffraction pattern in the back focal plane is perpendicular to the orientation of the long axis of the lines of this periodic grating. If the grating has very large spacings between adjacent lines (lower spatial frequency), more images of the light source (namely higher order diffraction maxima) will appear within the back focal plane. If the grating has very small spacings between adjacent lines (higher spatial frequency), fewer light source images appear within the back focal plane. In cases where the aperture diaphragm is not closed to its smallest possible size, these images may overlap. The diameter of the back focal plane increases with the numerical aperture. This tutorial explores the relationship between the periodic spacing (spatial frequency) of the lines in the specimen (here: a microscopically small line grating) and the number of and distance between the maxima that form the diffraction pattern in the back focal plane of the objective.

This tutorial displays a ray-tracing drawing of the optical beam path, showing the diffraction angles (ψ) of the rays and the resulting diffraction pattern in the back focal plane of the objective, caused by a fine line grating used as a sample (left side). The objective´s back focal plane with the diffraction pattern and the intermediate image of the line grating appear on the right.
To operate the tutorial, use the Spatial Frequency radio buttons to change the spatial frequency of the line grating, measured in lines/mm. At the beginning of the tutorial, the radio button with the highest spatial frequency (250 lines/mm) is activated for the line grating, which corresponds to the largest spacing between adjacent images of the condenser aperture diaphragm (i.e. the diffraction orders). Activating one of the other two radio buttons will decrease the spatial frequency (number of lines) of the line grating, resulting in a simultaneous decrease of the spacing (S) of the condenser aperture diaphragm images at the back focal plane. This tutorial shows that there is a reciprocal relationship between the line spacings in the specimen (D) (line pairs/mm) and the separation distance (S) of the diffraction pattern maxima in the back focal plane of the objective.

The ray-tracing scheme (on the left side of the animation below) depicts the zero and higher order diffracted light rays focused in the back focal plane of the objective. The condenser and the objective are symbolized each by a single lens element. The condenser aperture stop opening is shown at the bottom of the figure. The specimen (a line grating) is shown as a dashed line through which illuminating light rays from the condenser pass. 
Equation (1) shows the reciprocal relation of the spacing (S) between diffraction orders in the back focal plane of the objective and the distance (D) between the lines of the line grating in the sample plane. Furthermore, it shows the connection between the diffraction angles (sin(φ)) and the spacing (S) and the line distance (D), as well as the orientation of the diffracted rays:

S/f ≈ λ/D = sin(φ)(1)

In addition, the absolute numbers depend on the wavelength (λ) of the illuminating light and the focal length (f) of the objective. The back focal plane image displays the distance between the focused diffraction orders, (S), the number of orders is proportional to the numerical aperture of the microscope objective.

Equation (2) shows the direct proportionality between the diffraction order spacing (S) and the diffracted light angle (sin(φ)). However, the spacing (S) depends in addition on the refractive index (n) of the medium between the specimen and the objective’s front lens.

S/f ≈ n(sin(φ))(2)

n is the refractive index of the medium between the specimen and the objective’s front lens. Ernst Abbe demonstrated that in order to resolve the specimen’s line grating, at least two adjacent diffraction orders (usually the zero and the first) must be captured by the microscope objective and be focused in the back focal plane. As the numerical aperture increases, additional higher order rays are included in the diffraction pattern and the resolved object details (line grating) become much clearer and more realistic in the image. In addition, finer gratings with higher spatial frequency forming wider diffraction angles can now be resolved because their first order maxima can be captured by such objectives of higher numerical aperture (NA). These fine gratings could not be resolved by the previous objective with lower NA, because their first order maxima could not be captured by that objective. Note that when only the zero and first orders are captured, the lines are resolved, but poorly and not very realistically imaged. This is what happens at the resolution limit.