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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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".

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

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}}},}

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}},}

Sperian has four models of Laser-Gard; two for daylight use (bronze-colored sunglasses that also have laser protection), and two for nighttime use (salmon-colored glasses with maximum light transmission except for the laser wavelength). Within each group, there is green-only protection, and green-and-red protection. The cost for the nighttime, green-only Laser-Gard glasses we examined is $99. One U.S. source is Rockwell Laser Industries.

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.

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.

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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 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.

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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.

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

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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.

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

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.

As of July 2011, there are three models. PBG reduces green by 99.5%, blue by 97%, has 49% transmittance for low-light conditions, and has “good” instrument panel visibility. AG2 reduces green by 99%, has 53% transmittance for low-light conditions, and has “excellent” instrument panel visibility. BGR is for full sun conditions; it reduces green, blue and red by 90-97%, with 29% transmittance for full sun conditions.

The eyewear is said to have an Optical Density between 2 and 5, meaning that it will attenuate the amount of 532nm light reaching eyes by 100 to 100,000 times. The frame is wider at top, sides and bottom, to block light coming from directions other than the front.MTI originally developed the holographic laser-reflecting technology for use in windscreens. The goal was to protect pilots without the need for eyewear. However, windscreen modifications require a slow, multi-year process of obtaining governmental and airline approval. In addition, the cost to modify windscreens is much higher than the cost of one or two pairs of eyewear. So eyewear was a natural choice for the first commercially available laser protection product from MTI. According to its metaAIR website, the company may also produce glare shields using the same technology.As of late February 2019, MTI is taking pre-orders for its eyewear, distributed by Satair, which will be available "spring 2019."

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,}

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:

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}.}

June 2018 guidance from an aviation advisory committee, SAE Aerospace Recommended Practice 6378, discusses Laser Glare Protection. An appendix lists LGP products known to the SAE G10-OL committee at time of writing the document.“Eyewear Helps Thwart Laser-Pointer Attacks”, by Jason Palidwar, Iridian Spectral Technologies. Reprinted from Photonics Spectra, January 2014.The Air Force Aircrew Laser Eye Protection (ALEP) program. This unclassified document describes many of the characteristics of anti-laser glasses used by the Air Force. It does not list specific wavelengths or protection factors (e.g., optical density) as these are classified. From the abstract: “ It describes design criteria used in the development of laser protective devices, operational considerations, training, fit, inspection, maintenance and immediate response procedures to follow after suspected laser exposure. Procedures outlined in this document apply to all Air Force personnel….” Block 0, 0+ and 1 versions are discussed.ALEP Program Status: Below is the bottom half of slide 4, in an Air Force briefing about Aircrew Performance Programs. This has more information about ALEP Block 2.

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,

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

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They began work on the “laser dazzle visor” in 2014 and hope to start selling to military, airline suppliers and emergency services in 2018. A March 2017 article in the Express gives some additional information.

Simulated use of Laser Glare Protection glasses. The laser location is still visible, but glare is significantly reduced. Note that green cockpit lights are not adversely affected -- they can still be seen.

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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

ST Laserstrike has two levels of protection available:Type 1 – General laser flash protection (suitable for most operations)Allow normal operation with laser pointers <200mW* Provide eye damage protection from sources up to 2W, based on pilot being >200m from source & typical beam divergence. To comply with minimum EN207 DIR LB2 (400-445nm blue, 532nm green, 808-1064nm Infrared). Provide approx. 40% Visible Light TransmissionType 2 – High powered laser attack protection (suitable for extreme environments)Allow normal operation with laser pointers <2W, and eye damage protection from sources up to 20W, based on pilot being >200m from source & typical beam divergence. To comply with minimum EN207 DIR LB3 (400-445nm blue, 532nm green, 808-1064nm Infrared). Provide approx. 30% Visible Light Transmission

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.

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:

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]

This chart from PerriQuest shows how colors shift when seen through the lens. For example, the bright green at a*=-70, b*=+80 shifts to a more yellow green; reds shift to orange. But because the shifted colors remain visible and stay within the same general hue, the manufacturer says this “eliminates the possibility of color confusion.”

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}}}.}

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

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}}

Weather radar colors. Top: Normal colors as seen without glasses. Bottom: Simulated view through PerriQuest Laser Defense Eyewear. (The manufacturer did not specify if this simulates the Night or Day version.)

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Simulated use of general-purpose green-blocking glasses,where 50% of broadband green light is blocked.Note significant change in instrument panel colors.Obviously, these should not be used for anti-laser purposes.

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.

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

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.

For each frame style, there are three different lens types available:Green Beam Reduction: 190-400 nm (UV), OD 5+. 532 nm (green), OD 2.5+. Visible light transmission: 33%. Orange/Peach lens color.Blue/Green Beam Reduction: 190-400 nm (UV), OD 5+. 445-450 nm (blue), OD 2+. 532 nm (green), OD 2.5+. Visible light transmission: 43%. Orange/Peach lens color.Blue/Green/Red Beam Reduction: 190-400 nm (UV), OD 5+. 440-470 nm (blue), OD 1.5+. 532 nm (green), OD 1.5+. 633-640 nm (red), OD 1.2+. Visible light transmission: 23.3%. Green lens color.The cost is $150 in single quantities.