In a camera or imaging system an object far away gets imaged onto the film or detector plane by the objective lens, and the far field diffraction pattern is observed at the detector. The resulting image is a convolution of the ideal image with the Airy diffraction pattern due to diffraction from the iris aperture or due to the finite size of the lens. This leads to the finite resolution of a lens system described above.

The diffraction pattern resulting from a uniformly illuminated circular aperture has a bright region in the center, known as the Airy disc which together with a series of concentric rings is called the Airy pattern (both named after George Airy). The diameter of this disc is related to the wavelength of the illuminating light and the size of the circular aperture.

Far away from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by

For Spectral-Domain OCT systems, the A-Scan rate is determined by the speed of the camera in the detection spectrometer. For Swept-Source OCT systems, the A-Scan rate is determined by the sweep speed of the swept laser source. There is a tradeoff between A-Scan rate and the sensitivity of an OCT system: higher A-Scan rate results in lower sensitivity.

The length (L) and width (W) of the FOV are limited by the scan lens properties. Most of our OCT systems have a 10 mm x 10 mm (L x W) FOV while the GANYMEDE-II-HR offers 6 mm x 6 mm (L x W) FOV in the default configuration. The maximum depth (D) attainable is set by the design of the OCT system. The graphic to the right shows variation in depth among all of our OCT systems. However, the actual imaging depth will typically depend on the optical properties of the sample. Our standard OCT systems are designed to provide an optimized balance between imaging depth and axial resolution. For applications requiring greater depth or higher resolution, we offer custom configurations.

AiryDisk telescope

The sensitivity of an OCT system describes the largest permissible signal attenuation within a sample that can still be distinguished from the noise. In practice, higher sensitivity OCT systems are capable of providing higher contrast images. Since the sensitivity of an OCT system can be increased by increasing the integration time, there is usually a tradeoff between A-scan rate and sensitivity.

but is the f-number of a lens. A typical setting for use on an overcast day would be f/8.[1] For blue visible light, the wavelength λ is about 420 nanometers.[2] This gives a value for x of about 0.004 mm. In a digital camera, making the pixels of the image sensor smaller than this would not actually increase image resolution.

Thorlabs offers a wide variety of OCT Imaging Systems. To assist in narrowing down which OCT system(s) is best suited for your application, we have provided the guide below. We always encourage all customers to contact us to discuss specific imaging requirements.

Airydisk derivation

A Gaussian beam with 1 / e2 diameter of D focused through an aperture of diameter D will have a focal profile that is nearly Gaussian, and the intensity at the center of the focus will be 0.924 times I0,Airy.[7]

In practice, the conditions for uniform illumination can be met by placing the source of the illumination far from the aperture. If the conditions for far field are not met (for example if the aperture is large), the far-field Airy diffraction pattern can also be obtained on a screen much closer to the aperture by using a lens right after the aperture (or the lens itself can form the aperture). The Airy pattern will then be formed at the focus of the lens rather than at infinity.

A circular laser beam with uniform intensity across the circle (a flattop beam) focused by a lens will form an Airy disk pattern at the focus. The size of the Airy disk determines the laser intensity at the focus.

where I0 is the intensity in the center of the diffraction pattern, J1 is the Bessel function of the first kind of order one, k = 2π / λ is the wavenumber, a is the radius of the aperture, and θ is the angle of observation, i.e. the angle between the axis of the circular aperture and the line between aperture center and observation point. , where q is the radial distance from the optics axis in the observation (or focal) plane and N = R / d (d=2a is the aperture diameter, R is the observation distance) is the f-number of the system. If a lens after the aperture is used, the Airy pattern forms at the focal plane of the lens, where R = f (f is the focal length of the lens). Note that the limit for (or for ) is I(0) = I0.

In OCT, the axial (depth) and lateral resolutions are dependent on different factors. The axial resolution of the OCT system is proportionally dependent on the center wavelength of the source and inversely proportional to the source bandwidth. In practice, the axial resolution is also improved by the index of refraction of the sample. For example, the axial resolution of the CALLISTO OCT system is 7 ?m in air or 5.3 ?m in water-rich samples such as tissue (n=1.35).

OCT Cross-Sectional Image of a Human Finger. Layers of Skin: E-Epidermis; D-Dermis; BV-Blood Vessels. Image Size: 4.9 mm x 2.6 mm. Image Taken with a Telesto Series OCT System.

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Jun 29, 2023 — The limit for resolution is the focal ratio (f-number) which is the focal length divided by the working diameter. The enemy is diffraction.

Airydisk calculator

The intensity I0 at the center of the diffraction pattern is related to the total power P0 incident on the aperture by[4]

The spectral bandwidth of the OCT light source is indirectly proportional to the axial (depth) resolution of the imaging system. Therefore, broadband light sources are used to provide high axial resolution.

Airydisk radius

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The expression for I(θ) above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:

Thorlabs offers a wide variety of Optical Coherence Tomography (OCT) imaging systems. We recognize each imaging application has their specific needs. With the growing number of OCT systems available, it can be challenging to decide which system best meets your needs. Below we have put together a Selection Guide that outlines a few key technical specifications of each of our systems as well as some tips on how to choose the best OCT system for your application.

Airydisk Microscopy

Airy patterncalculator

The most important application of this concept is in cameras and telescopes. Due to diffraction, the smallest point to which one can focus a beam of light using a lens is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by this lens. An optical system in which the resolution is no longer limited by imperfections in the lenses but only by diffraction is said to be diffraction limited.

Thorlabs currently offers OCT systems that operate with a center wavelength of either 900 nm, 930 nm or 1300 nm. The center wavelength contributes to the actual imaging depth and resolution of the system. Shorter wavelength OCT systems, such as our 930 nm or 900 nm systems, are ideal for higher resolution imaging compared to systems with a center wavelength of 1300 nm. For imaging samples that have higher optical scattering properties, such as tissue, the longer wavelength systems are recommended. The longer center wavelength is not affected by scattering, and therefore, the light is able to penetrate deeper into the sample and return for detection.

Airydisc diffraction

where R is the distance from the aperture. The half maximum of the central Airy disk (where J1(x) = 1 / 2) occurs at x = 1.61633...; the 1/e2 point (where J1(x) = 1 / e2) occurs at x = 2.58383..., and the maximum of the first ring occurs at x = 5.13562....

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A circular laser beam with uniform intensity profile, focused by a lens, will form an Airy pattern at the focal plane of the disk. The intensity at the center of the focus will be I0,Airy = (P0A) / (λ2f2) where P0 is the total power of the beam, A = πD2 / 4 is the area of the beam (D is the beam diameter), λ is the wavelength, and f is the focal length of the lens.

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A single depth profile (Intensity vs Depth) is called an A-Scan. A B-Scan, or two-dimensional cross-sectional image, is created by laterally scanning the OCT beam and collecting sequential A-scans. The speed with which a B-scan is collected depends on the A-Scan or Line rate.

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As with general microscopy principles, the lateral resolution is dependent on the focusing objective in the imaging probe. All of Thorlabs’ OCT systems come with our specially designed OCT scan lens which provides telecentric scans across the entire field of view.

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Airydisk formula

Light from a uniformly illuminated circular aperture (or from a uniform, flattop beam) will exhibit an Airy diffraction pattern far away from the aperture due to Fraunhofer diffraction (far-field diffraction).

Therefore the smallest angular separation two objects can have before they significantly blur together is given as stated above by

The smallest f-number for the human eye is about 2.1.[3] The resulting resolution is about 1 μm. This happens to be about the distance between optically sensitive cells, photoreceptors, in the human eye.[citation needed]

The conditions for being in the far field and exhibiting an Airy pattern are: the incoming light illuminating the aperture is a plane wave (no phase variation across the aperture), the intensity is constant over the area of the aperture, and the distance R from the aperture where the diffracted light is observed (the screen distance) is large compared the aperture size, and the radius a of the aperture is not too much larger than the wavelength λ of the light. The last two conditions can be formally written as R > a2 / λ .

where Ε is the source strength per unit area at the aperture, A is the area of the aperture (A = πa2) and R is the distance from the aperture. At the focal plane of a lens, I0 = (P0A) / (λ2f2). The intensity at the maximum of the first ring is about 1.75% of the intensity at the center of the Airy disk.

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where J0 and J1 are Bessel functions. Hence the fractions of the total power contained within the first, second, and third dark rings (where J1(kasinθ) = 0) are 83.8%, 91.0%, and 93.8% respectively.[5]

The Airy disc (or Airy disk) is a phenomenon in optics. Owing to the wave nature of light, light passing through an aperture is diffracted and forms a pattern of light and dark regions on a screen some distance away from the aperture (see interference).

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Similar equations can also be derived for the obscured Airy diffraction pattern[6][7], which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center.

where ε is the annular aperture obscuration ratio, or the ratio of the diameter of the obscuring disk and the diameter of the aperture (beam). , and x is defined as above: where R is the radial distance in the focal plane from the optical axis, λ is the wavelength and N is the f-number of the system. The encircled energy (the fraction of the total energy contained within a circle of radius R centered at the optical axis in the focal plane) is then given by:

where λ is the wavelength of the light and d is the diameter of the aperture. The Rayleigh criterion for barely resolving two objects is that the center of the Airy disc for the first object occurs at the first minimum of the Airy disc of the second. This means that the angular resolution of a diffraction limited system is given by the same formula.

If two objects imaged by a camera are separated by an angle small enough that their Airy disks on the camera detector start overlapping, the objects can not be clearly separated any more in the image, and they start blurring together. Two objects are said to be just resolved when the maximum of the first Airy pattern falls on top of the first minimum of the second Airy pattern (the Rayleigh criterion).

where x is the separation of the images of the two objects on the film and f is the distance from the lens to the film. If we take the distance from the lens to the film to be approximately equal to the focal length of the lens, we find