Performance and Configuration
4.2 The Diffraction Limit
System Limits:
Performance and Configuration
4.1 Introduction
It is the intent of this chapter to briefly outline some limits to which all optical systems must conform. These are: (1) limits of performance or resolution, (2) limits on throughput, and (3) limits on the relation-ships between beam angles and sizes. In the initial stages of system layout it is essential that these limits be known and harmonized with what is expected of the optical system. Occasionally this first step simply proves that “it can’t be done.” But even a negative result like this can be worthwhile if it avoids a waste of time spent on the physi-cally impossible.
4.2 The Diffraction Limit
The image of a point source, formed by a perfect optical system with a uniformly transmitting circular aperture, is a diffraction pattern which consists of a circular central bright patch, called the Airy disk, surrounded by alternating dark and light concentric rings. The Airy disk contains 84 percent of the energy in the image. The first dark ring has a diameter given by
D 1.22 (4.1)
NA
first bright ring is only 1.7 percent of the peak illumination in the Airy disk, and the illumination level in the other rings falls off quite rapidly. Figures 4.1 and 4.2 describe the diffraction pattern.
Point resolution is the ability to distinguish the images of two adja-cent, closely spaced points. Obviously the diffraction blur affects this ability.
The Rayleigh criterion assumes that two points can be clearly resolved if they are separated by the radius of the first dark ring of the diffraction pattern, or
Rayleigh separation 0.61 (4.2)
NA
Figure 4.1 The diffraction pattern: the image of a point formed by a perfect lens with a circular aperture. The pattern consists of a bright central circular patch, called the
The Sparrow criterion postulates a limit of about a 20 percent smaller separation, or
Sparrow separation (4.3)
The Dawes criterion is an empirical one, derived from observations with the human eye as the sensor; it is only about 2 percent larger than the Sparrow criterion.
The resolution limit at the image is related to the resolution at the object by the magnification of the optical system. Since the magnifica-tion is equal to u/u′ or sin u/sin u′, one can determine the resolution limit at the object simply by using the object-side NA in Eq. (4.2) or (4.3). If the object is at infinity, the resolution must be expressed angu-larly, and the separation corresponding to the Rayleigh criterion is
Rayleigh angular separation (4.4) where P is the diameter of the entrance pupil of the system. A
com-1.22
P
0.50
NA
Figure 4.2 Tabulation of the characteristics of the diffraction pattern. Z is the radial dimension of the ring. Note that most (84 percent) of the energy is in the central patch, contained within the first dark ring, and the peak illumination in the first bright ring is only 1.7 percent of illumination at the center of the Airy disk.
peak of the human visual response), that P is in inches, and that the angular resolution is in seconds of arc.
Rayleigh angular separation (4.5) For the Sparrow criterion, the constant in Eq. (4.5) becomes 4.5; for Dawes it is 4.6.
Line resolution is the ability to separate or recognize the elements of a pattern of alternating high and low brightness parallel lines. An optical system is a low-pass filter, in that it cannot transmit informa-tion at a spatial frequency higher than the cutoff frequency, given (in cycles per unit length) by
vo (4.6)
This frequency corresponds to a line spacing equal to the Sparrow cri-terion in Eq. (4.3). It is an absolute cutoff, with zero contrast between the light and dark lines in the image. At a frequency corresponding to the Rayleigh criterion, the perfect lens image pattern has about 10 percent modulation [defined by Eq. (4.8) below]. In collimated space, i.e., with the object or image at infinity, the frequency is defined in cycles per radian, and the cutoff frequency is
Angular vo (4.7)
where P is the diameter of the entrance or exit pupil.
The modulation transfer function (MTF) describes the way that the optical system transfers contrast or modulation from object to image, as a function of spatial frequency. The modulation is defined as
M (4.8)
where max and min are, respectively, the maximum and minimum values of brightness (in the object) or illumination (in the image), and the object is a pattern of parallel lines whose brightness varies according to a sine function. The modulation transfer factor for a spe-cific frequency is the ratio of the modulation in the image to that in the object, or
MTF (4.9)
For a perfect optical system the modulation transfer function is given by Mi
MTF(v) (cos sin ) (4.10) and is defined as
arccos冢 冣 (4.11)
where v is the spatial frequency, is the wavelength, and NA n sin u is the numerical aperture. Note that the term within parentheses in Eq. (4.11), being a cosine, cannot exceed unity; this then is the source of Eq. (4.6) for the cutoff frequency vo.
Figure 4.3 shows the MTF plotted against spatial frequency (which has been normalized to unity for the cutoff frequency) for a perfect optical system (curve A), as well as for a perfect system defocused by various amounts. Note that curve B results from an amount of defocus which produces a wavefront deformation, or OPD, equal to a quarter wavelength (/4). This is the Rayleigh limit for a tolerable amount of aberration, yielding an image which is “sensibly perfect.” An optical system with this amount (OPD /4) of wavefront aberration is often described as diffraction limited (although it obviously is not).
When the pupil of the optical system is not uniformly illuminated (as assumed above) the diffraction pattern will differ from that described in Figs. 4.1 and 4.2. For example, if the wavefront intensity varies as a gaussian or exponential (as in a laser beam), the diffrac-tion pattern is also a gaussian distribudiffrac-tion.
The center of the pupil transmits the low spatial frequency infor-mation and the outer portions transmit the high spatial frequency information. In some optical systems where a high image modulation is required for the low spatial frequencies and a low modulation is acceptable for the high frequencies (as in microlithography) the illu-mination (condenser) system is designed so that only the center of the pupil is filled with light. Called semicoherent illumination, this pro-duces very high contrast images at low frequencies (at the expense of the high). Decentering the illuminated portion of the pupil can emphasize the contrast of features which have a strongly directional characteristic.