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A multi-mode optical fiber will only propagate light that enters the fiber within a certain range of angles, known as the acceptance cone of the fiber. The half-angle of this cone is called the acceptance angle, θmax. For step-index multimode fiber in a given medium, the acceptance angle is determined only by the indices of refraction of the core, the cladding, and the medium: n sin ⁡ θ max = n core 2 − n clad 2 , {\displaystyle n\sin \theta _{\max }={\sqrt {n_{\text{core}}^{2}-n_{\text{clad}}^{2}}},} where n is the refractive index of the medium around the fiber, ncore is the refractive index of the fiber core, and nclad is the refractive index of the cladding. While the core will accept light at higher angles, those rays will not totally reflect off the core–cladding interface, and so will not be transmitted to the other end of the fiber. The derivation of this formula is given below.

In air, the angular aperture of the lens is approximately twice this value (within the paraxial approximation). The NA is generally measured with respect to a particular object or image point and will vary as that point is moved. In microscopy, NA generally refers to object-space numerical aperture unless otherwise noted.

n sin ⁡ θ max = n core 2 − n clad 2 , {\displaystyle n\sin \theta _{\max }={\sqrt {n_{\text{core}}^{2}-n_{\text{clad}}^{2}}},}

When a light ray is incident from a medium of refractive index n to the core of index ncore at the maximum acceptance angle, Snell's law at the medium–core interface gives n sin ⁡ θ max = n core sin ⁡ θ r .   {\displaystyle n\sin \theta _{\max }=n_{\text{core}}\sin \theta _{r}.\ }

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Pupilaperture

The research work was published and featured as a cover article in top-tier international journal Science Robotics. Dr. ZHOU Yu (postdoc), Dr. SUN Zhibo (postdoc), and DING Yucheng (PhD student) are the co-first authors while Prof. Fan is the corresponding author.

Developed by the team led by Prof. FAN Zhiyong, Chair Professor at HKUST's Department of Electronic & Computer Engineering and Department of Chemical & Biological Engineering, this groundbreaking technology represents a significant leap forward in the field of biomimetic vision systems.

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The approximation holds when the numerical aperture is small, but it turns out that for well-corrected optical systems such as camera lenses, a more detailed analysis shows that N is almost exactly equal to 1/(2NAi) even at large numerical apertures. As Rudolf Kingslake explains, "It is a common error to suppose that the ratio [D/2f] is actually equal to tan θ, and not sin θ ... The tangent would, of course, be correct if the principal planes were really plane. However, the complete theory of the Abbe sine condition shows that if a lens is corrected for coma and spherical aberration, as all good photographic objectives must be, the second principal plane becomes a portion of a sphere of radius f centered about the focal point".[4] In this sense, the traditional thin-lens definition and illustration of f-number is misleading, and defining it in terms of numerical aperture may be more meaningful.

NA i = n sin ⁡ θ = n sin ⁡ [ arctan ⁡ ( D 2 f ) ] ≈ n D 2 f , {\displaystyle {\text{NA}}_{\text{i}}=n\sin \theta =n\sin \left[\arctan \left({\frac {D}{2f}}\right)\right]\approx n{\frac {D}{2f}},}

In photography, the factor is sometimes written as 1 + m, where m represents the absolute value of the magnification; in either case, the correction factor is 1 or greater. The two equalities in the equation above are each taken by various authors as the definition of working f-number, as the cited sources illustrate. They are not necessarily both exact, but are often treated as if they are.

where Nw is the working f-number, m is the lens's magnification for an object a particular distance away, P is the pupil magnification, and the NA is defined in terms of the angle of the marginal ray as before.[3][5] The magnification here is typically negative, and the pupil magnification is most often assumed to be 1 — as Allen R. Greenleaf explains, "Illuminance varies inversely as the square of the distance between the exit pupil of the lens and the position of the plate or film. Because the position of the exit pupil usually is unknown to the user of a lens, the rear conjugate focal distance is used instead; the resultant theoretical error so introduced is insignificant with most types of photographic lenses."[6]

Numericalaperture

Substituting cos θc for sin θr in Snell's law we get: n n core sin ⁡ θ max = cos ⁡ θ c . {\displaystyle {\frac {n}{n_{\text{core}}}}\sin \theta _{\max }=\cos \theta _{c}.}

In laser physics, numerical aperture is defined slightly differently. Laser beams spread out as they propagate, but slowly. Far away from the narrowest part of the beam, the spread is roughly linear with distance—the laser beam forms a cone of light in the "far field". The relation used to define the NA of the laser beam is the same as that used for an optical system,

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The working f-number is defined by modifying the relation above, taking into account the magnification from object to image:

The f-number describes the light-gathering ability of the lens in the case where the marginal rays on the object side are parallel to the axis of the lens. This case is commonly encountered in photography, where objects being photographed are often far from the camera. When the object is not distant from the lens, however, the image is no longer formed in the lens's focal plane, and the f-number no longer accurately describes the light-gathering ability of the lens or the image-side numerical aperture. In this case, the numerical aperture is related to what is sometimes called the "working f-number" or "effective f-number".

Aperture

In most areas of optics, and especially in microscopy, the numerical aperture of an optical system such as an objective lens is defined by

By squaring both sides n 2 n core 2 sin 2 ⁡ θ max = cos 2 ⁡ θ c = 1 − sin 2 ⁡ θ c = 1 − n clad 2 n core 2 . {\displaystyle {\frac {n^{2}}{n_{\text{core}}^{2}}}\sin ^{2}\theta _{\max }=\cos ^{2}\theta _{c}=1-\sin ^{2}\theta _{c}=1-{\frac {n_{\text{clad}}^{2}}{n_{\text{core}}^{2}}}.}

Aperturediaphragm

This has the same form as the numerical aperture in other optical systems, so it has become common to define the NA of any type of fiber to be N A = n core 2 − n clad 2 , {\displaystyle \mathrm {NA} ={\sqrt {n_{\text{core}}^{2}-n_{\text{clad}}^{2}}},}

Prof. Fan said, "This compound eye design is simple, light and cheap. Although it won't fully replace traditional cameras, it could be a huge boost in certain robotics applications, such as in a swarm of drones flying in close formation. By further miniaturizing the device size and increasing the number of ommatidia, imaging resolution, and response speed, this type of device can find broad applications in optoelectronics and robotics."

but θ is defined differently. Laser beams typically do not have sharp edges like the cone of light that passes through the aperture of a lens does. Instead, the irradiance falls off gradually away from the center of the beam. It is very common for the beam to have a Gaussian profile. Laser physicists typically choose to make θ the divergence of the beam: the far-field angle between the beam axis and the distance from the axis at which the irradiance drops to e−2 times the on-axis irradiance. The NA of a Gaussian laser beam is then related to its minimum spot size ("beam waist") by

where ncore is the refractive index along the central axis of the fiber. Note that when this definition is used, the connection between the numerical aperture and the acceptance angle of the fiber becomes only an approximation. In particular, "NA" defined this way is not relevant for single-mode fiber.[7][8] One cannot define an acceptance angle for single-mode fiber based on the indices of refraction alone.

This ratio is related to the image-space numerical aperture when the lens is focused at infinity.[3] Based on the diagram at the right, the image-space numerical aperture of the lens is:

Traditionally, roboticists have mainly focused on replicating the visual capabilities of insects, which offer a wide field of view and advanced motion-tracking capabilities. However, integrating compound eye systems into autonomous platforms like robots or drones has been challenging as these systems often suffer from issues related to complexity and stability during deformation, geometry constraints, as well as potential mismatches between optical and detector components.

A research team at the School of Engineering of the Hong Kong University of Science and Technology (HKUST) has recently developed a novel artificial compound eye system that is not only more cost-effective, but demonstrates a sensitivity at least twice that of existing market products in small areas. The system promises to revolutionize robotic vision, enhance robots' abilities in navigation, perception and decision-making, while promoting commercial application and further development in human-robot collaboration.

F-number

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where λ0 is the vacuum wavelength of the light, and 2w0 is the diameter of the beam at its narrowest spot, measured between the e−2 irradiance points ("Full width at e−2 maximum of the intensity"). This means that a laser beam that is focused to a small spot will spread out quickly as it moves away from the focus, while a large-diameter laser beam can stay roughly the same size over a very long distance. See also: Gaussian beam width.

Increasing the magnification and the numerical aperture of the objective reduces the working distance, i.e. the distance between front lens and specimen.

Numerical aperture is not typically used in photography. Instead, the angular aperture of a lens (or an imaging mirror) is expressed by the f-number, written f/N, where N is the f-number given by the ratio of the focal length f to the diameter of the entrance pupil D:

From the geometry of the above figure we have: sin ⁡ θ r = sin ⁡ ( 90 ∘ − θ c ) = cos ⁡ θ c {\displaystyle \sin \theta _{r}=\sin \left({90^{\circ }}-\theta _{c}\right)=\cos \theta _{c}}

1 2 NA i = N w = ( 1 − m P ) N , {\displaystyle {\frac {1}{2{\text{NA}}_{\text{i}}}}=N_{\text{w}}=\left(1-{\frac {m}{P}}\right)N,}

As a renowned researcher in biomimetic optoelectronics, Prof. Fan is keen on combining practical approach with daring imaginations to drive innovative research. This unique compound eye work marks another breakthrough in the field of vision and robotic systems following his development of the world's first spherical artificial eye with 3D retina in 2020.

Mimicking the visual capabilities of compound eyes, this innovative system can be applied in a wide range of scenarios, such as installing on drones to improve their accuracy and efficiency in tasks like irrigation or emergency rescue in disaster sites. With its high sensitivity, the system can also enable closer collaboration among robots and other connected devices. In the long term, the compound eye system will enhance autonomous driving safety and accelerate the adoption of intelligent transport systems, fostering the development of smart cities.

Conversely, the object-side numerical aperture is related to the f-number by way of the magnification (tending to zero for a distant object):

where n is the index of refraction of the medium in which the lens is working (1.00 for air, 1.33 for pure water, and typically 1.52 for immersion oil;[1] see also list of refractive indices), and θ is the half-angle of the maximum cone of light that can enter or exit the lens. In general, this is the angle of the real marginal ray in the system. Because the index of refraction is included, the NA of a pencil of rays is an invariant as a pencil of rays passes from one material to another through a flat surface. This is easily shown by rearranging Snell's law to find that n sin θ is constant across an interface.

In microscopy, NA is important because it indicates the resolving power of a lens. The size of the finest detail that can be resolved (the resolution) is proportional to ⁠λ/2NA⁠, where λ is the wavelength of the light. A lens with a larger numerical aperture will be able to visualize finer details than a lens with a smaller numerical aperture. Assuming quality (diffraction-limited) optics, lenses with larger numerical apertures collect more light and will generally provide a brighter image, but will provide shallower depth of field.

In multimode fibers, the term equilibrium numerical aperture is sometimes used. This refers to the numerical aperture with respect to the extreme exit angle of a ray emerging from a fiber in which equilibrium mode distribution has been established.

To address these challenges, Prof. Fan's team developed a pinhole compound vision system by adopting new materials and structures. This system features several key characteristics, including an inherent hemispherical perovskite nanowire array imager with high pixel density to enlarge the imaging field; and a 3D-printed lens-free pinhole array with a customizable layout to regulate incident light and eliminate the blind area between neighboring ommatidia (individual units within an insect's compound eye). Owing to its good angular selectivity, a wide field of view, wide spectrum response in monocular and binocular configurations, as well as its dynamic motion tracking capability, the pinhole compound eye not only can accurately locate targets but can also track a moving quadruped robot after incorporated onto a drone.

where θ c = arcsin ⁡ n clad n core {\displaystyle \theta _{c}=\arcsin {\frac {n_{\text{clad}}}{n_{\text{core}}}}}

In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another, provided there is no refractive power at the interface. The exact definition of the term varies slightly between different areas of optics. Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective (and hence its light-gathering ability and resolution), and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.