Since we live in a three dimensional world, it is necessary to control the attitude or orientation of a flying aircraft in all three dimensions. In flight, any aircraft will rotate about its center of gravity, a point which is the average location of the mass of the aircraft. We can define a three dimensional coordinate system through the center of gravity with each axis of this coordinate system perpendicular to the other two axes. We can then define the orientation of the aircraft by the amount of rotation of the parts of the aircraft along these principal axes.

HighNA lens

The pitch axis is perpendicular to the yaw axis and is parallel to the plane of the wings with its origin at the center of gravity and directed towards the right wing tip. A pitch motion is an up or down movement of the nose of the aircraft.

θ is the objective’s angular aperture and n is the refractive index of the medium (air, water, or oil) between the objective and the specimen.

Numerical aperture of optical fiber

The image formed by a perfect, aberration-free objective lens at the intermediate image plane of a microscope is a diffraction pattern with a very specific intensity distribution. This tutorial explores the effects of the objective´s numerical aperture (NA) on the diffraction pattern and the resolution of a microscope. The three-dimensional representation of the diffraction pattern is the Point-Spread-Function (PSF) which, in a coma- and/or astigmatism-free system, is symmetrically periodic both along the optical axis, and radially across the image plane. This diffraction pattern can be sectioned in the focal plane to produce a two-dimensional diffraction pattern, having a bright circular disk surrounded by an alternating series of bright and dark higher-order diffraction rings whose intensity decreases with distance from the central disk, the so-called Airy disk. Under visual microscopical observation, only two or three of the circular luminous rings are usually visible in the intermediate image plane.

In flight, the control surfaces of an aircraft produce aerodynamic forces. These forces are applied at the center of pressure of the control surfaces which are some distance from the aircraft cg and produce torques (or moments) about the principal axes. The torques cause the aircraft to rotate. The elevators produce a pitching moment, the rudder produces a yawing moment, and the ailerons produce a rolling moment. The ability to vary the amount of the force and the moment allows the pilot to maneuver or to trim the aircraft.

The roll axis is perpendicular to the other two axes with its origin at the center of gravity, and is directed towards the nose of the aircraft. A rolling motion is an up and down movement of the wing tips of the aircraft.

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Numerical aperture formula

r(Airy) is the Airy radius, λ is the wavelength of the illuminating light, and NA(Obj) is the objective´s numerical aperture (objective aperture = condenser aperture). The numerical aperture depends on the aperture angle of the illumination entering the objective aperture, as well as the refractive index of the imaging medium:

Numerical aperture of objectivelens

Numerical aperture of microscope

The tutorial starts with a pattern of Airy disks appearing in the focal plane of the microscope and the point-spread function / three dimensional of a corresponding, single Airy disk pattern shown on the right. To operate the tutorial, use the Numerical Aperture slider to change the objective´s numerical aperture and the resolution of the Airy patterns. The left position of the slider shows the pattern at the lowest objective numerical aperture (= 0.20), and the right position illustrates the highest degree of resolution (numerical aperture = 1.30). As the slider is moved from left to right, the objective’s numerical aperture increases and the complex Airy pattern, as visible in the image, results in a progressively increased resolution of image detail. Correspondingly, the central peak and higher-order diffraction rings in the three-dimensional Airy pattern drawing grow smaller in diameter.

Numerical Aperture calculator

Na lensformula

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This foundational knowledge article explores the effects of the numerical aperture (NA) of an objective lens on the resolution of images produced by a microscope. It explains the diffraction pattern produced by an objective lens and how increasing the NA results in higher resolution images. The tutorial demonstrates the changes in image structure as the NA is adjusted.

Resolution is clearly influenced by the objective’s numerical aperture. Note that lower values of D indicate higher resolution. In the tutorial, the Numerical Aperture slider is used to control how the image structure evolves as the objective’s numerical aperture is increased. At the lowest numerical aperture value (0.20), the image details visible in the microscope are poorly defined and surrounded by diffraction fringes. As the slider is moved to higher numerical aperture values (0.50-0.80), the structural outline of the image becomes sharper and higher-order diffraction rings begin to emerge. At the highest numerical apertures (1.00-1.30), the diffraction disks are resolved individually as discrete luminous points surrounded by alternating series of bright and dark higher-order diffraction rings of decreasing intensity.

The resolving power of an objective determines the size of the formed Airy diffraction pattern: The radius of the central disk is determined by the combined numerical apertures of the objective and condenser. When condenser and objective have equivalent numerical apertures or the objective acts also as the condenser like in an inverted fluorescence microscope, the Airy pattern radius from the central peak to the first minimum is given by the equation:

The yaw axis is defined to be perpendicular to the plane of the wings with its origin at the center of gravity and directed towards the bottom of the aircraft. A yaw motion is a movement of the nose of the aircraft from side to side.