# Diffraction Rectangular Aperture

Diffraction of light through a rectangular aperture is a rather straightforward extension of 1-dimensional diffraction from a slit, as shown in the diagrams below.

- Fraunhofer Diffraction Rectangular Aperture
- Diffraction Pattern Rectangular Aperture
- Diffraction From Rectangular Aperture

- Diffraction from a Square Aperture The diffracted field is a sinc function in both x 1 and y 1 because the Fourier transform of a rect function is sinc. Diffracted irradiance Diffracted field. Diffraction from a Circular Aperture A circular aperture yields a diffracted 'Airy Pattern,'.
- Rectangular apertures have been used as a simple means to approximate elliptical Gaussian beams in femtosecond direct writing systems to correct for the elongated focus inherent in low numerical aperture (NA) systems. In this work it is recognized that the rectangular aperture, more accurately functions as a diffractive element and hence redistributes the intensity gradient around the focus in.
- Diffraction from a single rectangular slit. Problem of the interference from neighboring elements of a finite aperture is called Fraunhofer diffraction, after the man who first described it. (And also, incidentally, was the first to measure the solar spectrum using a prism.

A circular aperture is qualitatively similar, but an accurate quantitative treatment of the pattern requires more complicated mathematics. The intensity pattern is called the 'Airy Disk'. The main features are shown in the diagram below. The first minimum occurs at an angle θ = 1.22 λ/D, where D is the diameter of the aperture. On a screen a distance L >> D from the aperture the minimum is seen at a radial distance r' = 1.22 λL/D from the center of the pattern

In this section we will look at the Fresnel diffraction for both circular apertures and rectangular apertures. To help our physical understanding we will begin our discussion by describing Fresnel zones. 13.1 Fresnel Zones In the study of Fresnel diffraction it is convenient to divide the aperture into regions called Fresnel zones. Feb 07, 2011 rectangular aperture, at a perpendicular distance Lm ( ) away, such that the conditions ba L,.and. Λ L both hold simultaneously, where λ (m) is the wavelength of the sound – this is the so-called “far-field” limit. The expression for diffraction through a rectangular aperture in this simplest theory is given by: 2 2 22 sin sin y x sinc sinc.

Producing a laser beam is an attempt to confine the light in the directions transverse to the direction of propagation. The light will spread out in the same way it does after passing through an aperture.

Assume that at z = 0 the diameter of a laser beam is restricted to a(0). The angle through which the light spreads is approximately θ ≈ λ/a(0). (For back-of-the-envelope calculations we often ignore the factor of 1.22.) Because the laser beam diameter is typically much larger the wavelength of light, or a(0) >> λ, θ is quite small. At a large distance z the diameter of the beam will have increased to a(z) ≈ z*2θ. Consider a HeNe laser, for which λ = 633 nm with a beam waist of ~ 0.6 mm. Then θ ~ 10^{-3} rad = 1 millirad. The beam must propagate ~ 3 m before the diameter increases by a factor of 10.

In geometrical optics we assume that an ideal, aberration-free lens focuses parallel rays to a single point one focal length away from the lens. But the lens itself acts like an aperture with diameter D for the incident light. The light passing through the lens therefore spread out. This yields a blurred spot at the focal point. Light near the focal point exhibits an Airy Disc pattern. The size of the Airy Disc is determined by the focal length f and diameter D of the lens. The radius r of the Airy Disc at the focal point of a lens is given by r = 1.22 λf/D.

If all ray aberrations in an optical system can be eliminated, such that all of the rays leaving a given object point land inside of the Airy Disc associated with the corresponding image point, then we have a diffraction-limited optical system. This is the absolute best we can do for an optical system that has lenses with finite diameters.

The **resolving power** of an optical instrument is its ability to separate the images of two objects, which are close together. Some binary stars in the sky look like one single star when viewed with the naked eye, but the images of the two stars are clearly resolved when viewed with a telescope.

If you look at a far-away object, then the image of the object will form a diffraction pattern on your retina. For two far-away objects, separated by a small angle θ, the diffraction patterns will overlap. You are able to resolve the two objects as long as the central maxima of the two diffraction patterns do not overlap. The two images are just resolved when one central maximum falls onto the first minimum of the other diffraction pattern. This is known as the **Rayleigh criterion**. If the two central maxima overlap the two objects look like one.

The width of the central maximum in a diffraction pattern depends on the size of the aperture, (i.e. the size of the slit). The aperture of your eye is your pupil. A telescope has a much larger aperture, and therefore has a greater resolving power. The minimum angular separation of two objects which can just be resolved is given by θ_{min} = 1.22 λ/D, where D is the diameter of the aperture. The factor of 1.22 applies to circular apertures like the pupil of your eye or the apertures in telescopes and cameras.

When light passes through an aperture with diameter D, then diffraction limits the resolution to θ = 1.22λ/D. If the angular separation of two sources is less than θ, they cannot be resolved. The closer you are to two objects, the greater is the angular separation between them. Up close, two objects are easily resolved. As your distance from the objects increases, their images become less well resolved and eventually merge into one image.

#### Problem:

A spy satellite travels at a distance of 50 km above Earth's surface. How large must the lens be so that it can resolve objects with a size of 2 mm and thus read a newspaper? Assume the light has a wavelength of 400 nm.

Solution:

- Reasoning:

The minimum angular separation of two points which can just be resolved by an optical instrument is given by θ_{min}= 1.22 λ/D, where D is the diameter of the aperture of the instrument. - Details of the calculation:

Diffraction limits the resolution according to θ = 1.22 λ/D = y/L. Here the height of the object to be resolved is y and the distance to the object is L. Solving for D we find D = 12.2 m.

#### Module 9, Question 2

The limit to the eye's acuity is actually related to diffraction by the pupil.

Estimate the angle between two just-resolvable points of light for the pupil in bright daylight. Make reasonable assumptions.

Given this angle, estimate the greatest possible distance you can resolve the two headlights of a car?

**Discuss your estimates with your fellow students in the discussion forum!**

### Optipedia • SPIE Press books opened for your reference.

DiffractionMTF is a wave-optics calculation for which the only variables (for a given aperture shape) are the aperture diameter *D*, wavelength λ, and focal length *f*. The MTF_{diffraction} is the upper limit to the system’s performance; the effects of optical aberrations are assumed to be negligible. Aberrations increase the spot size and thus contribute to a poorer MTF. The diffractionMTF is based on the overall limiting aperture of the system (the aperture stop). The diffraction effects are only calculated once per system and do not accumulate multiplicatively on an element-by-element basis.

## Calculation of diffraction MTF

The diffraction OTF can be calculated as the normalized autocorrelation of the exit pupil of the system. We will show that this is consistent with the definition of Eqs. (1.6), (1.7), and (1.10), which state that the OTF is the Fourier transform of the impulse response. For the incoherent systems we consider, the impulse response *h*(*x,y*) is the square of the two-dimensional Fourier transform of the diffracting aperture *p*(*x,y*). The magnitude squared of the diffracted electric-field amplitude **E** in V/cm gives the irradiance profile of the impulse response in W/cm^{2}:

From Eq. (1.23), we must implement a change of variables ξ = *x*/λ*f* and η = *y*/λ*f* to express the impulse response (which is a Fourier transform of the pupil function) in terms of image-plane spatial position. We then calculate the diffraction OTF in the usual way, as the Fourier transform of the impulse response *h*(*x*,*y*):

Because of the absolute-value-squared operation, the two transform operations of Eq. (1.24) do not exactly undo each other—the diffraction OTF is the two-dimensional autocorrelation of diffracting aperture *p*(*x,y*). The diffractionMTF is thus the magnitude of the (complex) diffraction OTF. As an example of this calculation, we take the simple case of a square aperture, seen in Fig. 1.24:

The autocorrelation of the square is a triangle-shaped function,

with cutoff frequency defined by

**Figure 1.24**Terms for calculating the MTF of a square aperture.

For the case of a circular aperture of diameter *D*, the system has the same cutoff frequency, ξ_{cutoff} = 1/(l *F*/#), but the MTF has a different functional form:

for ξ < ξ_{cutoff} and

for ξ > ξ_{cutoff}. These diffraction-limited MTF curves are plotted in Fig. 1.25. The diffraction-limited MTF is an easy-to-calculate upper limit to performance; we need only λ and the *F*/# to compute it. An optical system cannot perform better than its diffraction-limited MTF—any aberrations will pull the MTF curve down. It is useful to compare the performance specifications of a given system to the diffraction-limited MTF curve to determine the feasibility of the proposed specifications, to decide how much headroom has been left for manufacturing tolerances, or to see what performance is possible within the context of a given choice of λ and *F*/#.

### Fraunhofer Diffraction Rectangular Aperture

**Figure 1.25**Universal curves for diffraction-limited MTFs, for incoherent systems with circular or rectangular aperture.

As an example of the calculations, let us consider the square-aperture system of Fig. 1.26(a) with an object at finite distance. Using the object-space or image-space *F*/# as appropriate, we can calculate ξ_{cutoff} in either the object plane or image plane:

or

Because *p* < *q*, the image is magnified with respect to the object; hence a given feature in the object appears at a lower spatial frequency in the image, so the two frequencies in Eqs. (1.30) and (1.31) represent the same feature. The filtering caused by diffraction from the finite aperture is the same, whether considered in object space or image space. With the cutoff frequency in hand, we can answer questions such as for what image spatial frequency is the MTF 30%? We use Eq. (1.26) to find that 30% MTF is at 70% of the image-plane cutoff frequency, or 223 cy/mm. This calculation is for diffraction-limited performance. Aberrations will narrow the bandwidth of the system, so that the frequency at which the MTF is 30% will be lower than 223 cy/mm.

**Figure 1.26**(a)Rectangular-aperture finite-conjugate MTF example, and (b) rectangular-aperture infinite-conjugate MTF example.

The next example shows the calculation for an object-at-infinity condition, with MTF obtained in object space as well as image space. We obtain the cutoff frequency in the image plane using Eq. (1.27):

As in Fig. 1.26(a), a given feature in Fig. 1.26(b) experiences the same amount of filtering, whether expressed in the image plane or in object space. In the object space, we find the cutoff frequency is

Let us verify that this angular spatial frequency corresponds to the same feature as that in Eq. (1.32). Referring to Fig. 1.27, we use the relationship between object-space angle θ and image-plane distance *X*,

Inverting Eq. (1.34) to obtain the angular spatial frequency 1/θ:

### Diffraction Pattern Rectangular Aperture

Given that θ is in radians, if *X *and* f* have the same units, we can verify the correspondence between the frequencies in Eqs. (1.32) and (1.33):

**Figure 1.27**Relation between object-space angle and image-plane distance (adapted from Ref. 1).

It is also of interest to verify that the diffractionMTF curves in Fig. 1.25 are consistent with the results of the simple 84% encircled-power diffraction spot-size formula of 2.4 λ (*F*/#). In Fig. 1.28, we create a one-dimensional spatial frequency with adjacent lines and spaces. We approximate the diffraction spot as having 84% of its flux contained inside a circle of diameter 2.44 λ (*F*/#), and 16% in a circle of twice the diameter.

**Figure 1.28**Verification of the formula for diffractionMTF.

The fundamental spatial frequency of the above pattern is

and the modulation depth at this frequency is

in close agreement with Fig. 1.25 for a diffraction-limited circular aperture, at a frequency of ξ = 0.21 ξ_{cutoff} .

### Reference

- E. Dereniak and G. D. Boreman,
*Infrared Detectors and Systems*, Wiley, New York (1996), referenced figures reprinted by permission of John Wiley and Sons, Inc.

**Citation:**

G. D. Boreman, *Modulation Transfer Function in Optical and Electro-Optical Systems*, SPIE Press, Bellingham, WA (2001).