Performance Factors

Diffraction

In all light beams, some energy is spread outside the region predicted by rectilinear propagation. This effect, known as diffraction, is a fundamental and inescapable physical phenomenon. Diffraction can be understood by considering the wave nature of light. Huygen's principle, illustrated in the image below, states that each point on a propagating wavefront is an emitter of secondary wavelets. The combined locus of these expanding wavelets forms the propagating wave. Interference between the secondary wavelets gives rise to a fringe pattern that rapidly decreases in intensity with increasing angle from the initial direction of propagation. Huygen's principle nicely describes diffraction, but rigorous explanation demands a detailed study of wave theory.
Huygen's principle
Diffraction effects are traditionally classified into either Fresnel or Fraunhofer types. Fresnel diffraction is primarily concerned with what happens to light in the immediate neighborhood of a diffracting object or aperture. It is thus only of concern when the illumination source is close to this aperture or object. Consequently, Fresnel diffraction is rarely important in most optical setups. Fraunhofer diffraction, however, is often very important. This is the light-spreading effect of an aperture when the aperture (or object) is illuminated with an infinite source (plane-wave illumination) and the light is sensed at an infinite distance (far-field) from this aperture. From these overly simple definitions, one might assume that Fraunhofer diffraction is important only in optical systems with infinite conjugate, whereas Fresnel diffraction equations should be considered at finite conjugate ratios. Not so. A lens or lens system of finite positive focal length with plane-wave input maps the far-field diffraction pattern of its aperture onto the focal plane; therefore, it is Fraunhofer diffraction that determines the limiting performance of optical systems. More generally, at any conjugate ratio, far-field angles are transformed into spatial displacements in the image plane. Diffraction, poses a fundamental limitation on any optical system. Diffraction is always present, although its effects may be masked if the system has significant aberrations. When an optical system is essentially free from aberrations, its performance is limited solely by diffraction, and it is referred to as diffraction limited. In calculating diffraction, we simply need to know the focal length(s) and aperture diameter(s); we do not consider other lens-related factors such as shape or index of refraction. Since diffraction increases with increasing f-number, and aberrations decrease with increasing f-number, determining optimum system performance often involves finding a point where the combination of these factors has a minimum effect.
Fraunhofer Diffraction at a Circular Aperture
Fraunhofer diffraction at a circular aperture dictates the fundamental limits of performance for circular lenses. It is important to remember that the spot size, caused by diffraction, of a circular lens is
where d is the diameter of the focused spot produced from plane-wave illumination and l is the wavelength of light being focused. Notice that it is the f-number of the lens, not its absolute diameter, that determines this limiting spot size. The diffraction pattern resulting from a uniformly illuminated circular aperture is shown in the image below. It consists of a central bright region, known as the Airy disc, surrounded by a number of much fainter rings.
Diffraction pattern for a circular aperture
Each ring is separated by a circle of zero intensity. The irradiance distribution in this pattern can be described by
where I0 = peak irradiance in the image
J1(x) is a Bessel function of the first kind of order unity, and
where l is the wavelength, D is the aperture diameter, and q is the angular radius from pattern maximum. This useful formula shows the far-field irradiance distribution from a uniformly illuminated circular aperture of diameter, D.
Fraunhoffer Diffraction at a Slit Aperture
A slit aperture, which is mathematically simpler, is useful in relation to cylindrical optical elements. The irradiance distribution in the diffraction pattern of a uniformly illuminated slit aperture is described by
Where I0 is the peak irradiance of the image, and
where l is the wavelength, w is the slit width, and q is the angular radius from pattern maximum.
Energy Distribution Table
The table below shows the major features of pure (unaberrated) Fraunhofer diffraction patterns of circular and slit apertures. The table shows the position, relative intensity, and percentage of total pattern energy corresponding to each ring or band. It is especially convenient to characterize positions in either pattern with the same variable x. This variable is related to field angle in the circular aperture case by
where D is the aperture diameter. For a slit aperture, this relationship is given by
where w is the slit width, p has its usual meaning, and D, w, and l are all in the same units (preferably millimeters). Linear instead of angular field positions are simply found from
where s" is the secondary conjugate distance. This last result is often seen in a different form, namely the diffraction-limited spot-size equation. For a circular lens that was stated at the outset of this section:
This value represents the smallest spot size that can be achieved by an optical system with a circular aperture of a given f-number.

Energy Distribution in the Diffraction Pattern of a Circular or Slit Aperture
	Circular Aperture				Slit Aperture
Ring or Band	Position (x)	Relative Intensity (Ix/I0)	Energy in Ring (%)		Position (x)	Relative Intensity (Ix/I0)	Energy in Band (%)
Central Maximum	0.0	1.0	83.8		0.0	1.0	90.3
First Dark	1.22p	0.0			1.00p	0.0
First Bright	1.64p	0.0175	7.2		1.43p	0.0472	4.7
Second Dark	2.23p	0.0			2.00p	0.0
Second Bright	2.68p	0.0042	2.8		2.46p	0.0165	1.7
Third Dark	3.24p	0.0			3.00p	0.0
Third Bright	3.70p	0.0016	1.5		3.47p	0.0083	0.8
Fourth Dark	4.24p	0.0			4.00p	0.0
Fourth Bright	4.71p	0.0008	1.0		4.48p	0.0050	0.5
Fifth Dark	5.24p	0.0			5.00p	0.0

The graph below shows the form of both circular and slit aperture diffraction patterns when plotted on the same normalized scale. Aperture diameter is equal to slit width so that patterns between x values and angular deviations in the far field are the same.

Fraunhofer diffraction pattern of a singlet slit superimposed on the Fraunhofer diffraction pattern of a circular aperture

Gaussian Beams

Apodization, or nonuniformity of aperture irradiance, alters diffraction patterns. If pupil irradiance is nonuniform, the formulas and results given previously do not apply. This is important to remember because most laser-based optical systems do not have uniform pupil irradiance. The output beam of a laser operating in the TEM00 mode has a smooth Gaussian irradiance profile. Formulas to determine the focused spot size from such a beam are discussed in Gaussian Beam Optics. Furthermore, when dealing with Gaussian beams, the location of the focused spot also departs from that predicted by the paraxial equations for geometric optics.

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Optics Guide Copyright 2002 Melles Griot Inc.