Kitabı oku: «Optical Engineering Science», sayfa 14

6.8 Fresnel Diffraction

The study of Gaussian beam propagation has provided us with a more quantitative description of near field and far field propagation and where the boundary between the two zones occurs. In our analysis of Fraunhofer diffraction, we considered only the far field approximation. Related to the concept of the Rayleigh distance for Gaussian beam propagation is a dimensionless parameter called the Fresnel number. If the near field is defined by some aperture with a radial dimension of a, then the Fresnel number, F, at a propagation distance of L from the aperture is given by:

(6.48)

Referring to Eq. (6.36), then the Gaussian beam equivalent of the Fresnel number is the ratio of the Rayleigh distance, Z_R, to the beam propagation distance. For Fresnel numbers much less than one, then the diffraction pattern may be considered as a far field pattern and the Fraunhofer approximation applies. Where the Fresnel number is much greater than one, then one is in the near field.

The analysis of Fresnel diffraction is derived from the Rayleigh diffraction formulae (Eqs. [6.8] and [6.9]). The key assumption in the Fresnel analysis relates to an approximation of the propagation distance, s. If one assumes that the near field object is located at z = 0, then the propagation distance may be approximated in the following manner:

(6.49)

In making the above approximation, based on a Taylor series expansion, we are choosing to ignore terms of fourth order in x and y. These terms cannot be permitted to make a significant contribution to the phase when the approximation is applied to the Rayleigh formulae. Setting out the fourth order terms more explicitly, it is straightforward to delineate the approximation more clearly:

If we re-cast z as the propagation distance, L, and represent the ratio x/z as θ, the angular size of the near field and also denominate the near field radius as a, then the Fresnel condition is given by:

(6.50)

The value of the Fresnel approximation is that it now permits us to treat the axial propagation distance, z, as a constant and to remove it from the integral in the diffraction equation, producing a more simple expression involving integration with respect to x and y. This is the so-called Fresnel integral and it is set out below.

(6.51)

It would be useful, at this point, to illustrate the assumptions underlying Fresnel diffraction with a practical example. An optical system populated with components with a standard diameter of 25 mm would have an effective radius of 12.5 mm. For a wavelength of 500 nm, the Fresnel approximation applies to distances much greater than 250 mm. At that distance, the Fresnel number is about 1000, so we are clearly in the near field zone.

To illustrate the application of Fresnel diffraction, we might now apply it to a uniformly illuminated slit of width w. Without loss of generality, this provides a simple illustration of the application of Fresnel diffraction in one dimension. For a given source point, e.g. x = 0, then the phase of the sinusoidal component of Eq. (6.51) is of critical interest. In particular, we are concerned with points where the phase expressed in Eq. (6.51) is a half period number of waves. That is to say:

(6.52)

The effect of diffraction at an edge or an aperture is to produce an alternating series of light and dark rings. The disposition of these rings is affected by the relative phases of contributions from the source. As such, Eq. (6.52) provides some indication of the location of these rings. The locations of these points, as set out in Eq. (6.52) are referred to as the Fresnel zones. Based on application of Eq. (6.51), for the one dimension, the diffracted amplitude from the slit is proportional to:

(6.53)

We make the substitution s = x − x′ and make the further assumption that the diffraction pattern is symmetrical about the centre of the slit. In doing this, we may be permitted, without loss of generality in assuming that x > 0. The integral now becomes:

(6.54)

We will now refer to the quantity w/2 − x′ as Δ. The quantity Δ now represents the distance in x from the positive edge of the slit.

(6.55)

The sign of the first and third terms in Eq. (6.55) is dependent upon the sign of Δ. If Δ is greater than 0, then the sign is negative and vice versa. The structure of the integral above is of great importance, as it can be decomposed into two relatively simple integrals of the form:

(6.56)

The above integral is of great importance and is known as the Fresnel integral. Plotting both components of amplitude in Figure 6.15, we produce the familiar form of the Cornu spiral.

Progression around the Cornu spiral in Figure 6.15 is marked by increasing values of Δ, the distance from the slit boundaries. Each successive Fresnel zone is marked in Figure 6.15 and the numbering of the zones is as per Eq. (5.52). Most importantly, it is clear from Figure 6.15 that an asymptote is reached for large values of Δ. At large values of Δ, the integral tends to 0.25 + 0.25i. If, in Eq. (6.55), one assumes that w-Δ is large, then this asymptotic value must be added to the integral. In this case, we can now reasonably approximate the integral expression in Eq. (6.55) in the following manner:

(6.57)

When Δ is large and positive, the integral part of Eq. (6.57) cancels out the constant asymptotic values, so the amplitude is zero. Of course, that the amplitude is zero away from the illuminated portion of the slit is to be expected. In the opposite scenario, where a position within the illuminated area is viewed, then the flux levels tend to a uniform value. Around the edge position, and towards the illuminated area, a series of light and dark bands emerge. One can see from the disposition of the Cornu spiral, that the contrast of these bands diminishes as the effective Fresnel zone number increases and, from Eq. (6.52), they also become more tightly packed.

We can now illustrate this process by considering a slit with a width of 2 mm which is illuminated by a 500 nm source. By reference to the example of Gaussian beam propagation, we assume in this analysis that the illumination is significantly spatially coherent. This does not necessarily imply the use of a laser beam; in practice it means that the slit is illuminated by a parallel beam with very small angular divergence. We now view the slit at a distance of 100 mm. The Fresnel number is 20 and, as the effective angle, θ, is 0.01 rad, the Fresnel approximation is clearly justified by applying Eq. (6.50). Applying the Fresnel integral to this specific problem, we obtain the flux distribution described by Figure 6.16.

Figure 6.15 Fresnel integral and Cornu spiral.

As previously indicated, the illuminated portion is described a series of fringes characterised by the spacing of the Fresnel zones. In the obscured region, the flux tails off to zero. At the slit boundary, the flux is one half of the nominal. Of course, if the set up were reversed, and an obscuration substituted for the slit, then the pattern in Figure 6.16 would be reversed.

Generally, in problems associated with Fresnel diffraction, the diffraction pattern produced by sharp edges broadly follows that illustrated in Figure 6.16. The characteristic diffraction pattern away from the sharp edge feature consists of a series of ripples denominated by the relevant Fresnel zone.

6.9 Diffraction and Image Quality

6.9.1 Introduction

The analysis of image quality is central to any analysis of an imaging system. Where the wavefront error of a system is rather larger than the operating wavelengths of the system, the performance of the system may be adequately described by geometrical optics. Metrics such as geometrical spot size, as derived directly from ray tracing, prevail in this instance. However, where the wavefront errors are very much less than this, then diffraction effects prevail. Indeed, where effects other than diffraction may be legitimately ignored, the image is said to be diffraction limited. Overall, there are a number of metrics that quantitatively describe image quality and these are summarised below:

● Geometric spot size (rms spot size, 90% encircled energy etc.) – Geometric optics

● Point spread function (rms spot size, 90% encircled energy etc.) – Wave optics

● Strehl Ratio – Wave optics

● Modulation Transfer Function (MTF)

Figure 6.16 Fresnel diffraction at 100 mm from 2 mm Slit – λ = 500 nm.

6.9.2 Geometric Spot Size

This is perhaps the most straightforward of the image quality metrics to visualise. By virtue of ray tracing, for example, using ray tracing software, a number of representative rays that uniformly illuminate the entrance pupil are traced to the image plane. Of course, in an ideal image formation system, all rays would be traced to a common image point. However, deviation from this ideal behaviour is a measure of the image quality. Furthermore, this process would be attempted for a number of different field positions and, inevitably, for a dispersive system, for a number of different wavelengths. An example geometric spot is shown in Figure 6.17, illustrating the impact of spherical aberration and coma.

In order to quantify the data depicted in Figure 6.17, a number of different measures may be adopted. Measurements are characterised typically with respect to some central location. This central location may either be the intersection of the chief ray at the image plane or the weighted mean location of all intersecting rays – the centroid. That the two conventions might produce different answers is evident from the depiction of the comatic spot diagram in Figure 6.17, where the chief ray intersection corresponds to the apex at the bottom of the spot. Whichever convention is used, the size of the spot may be described in the following ways:

Figure 6.17 Geometric spots for spherical aberration and coma.

● Full width half maximum (FWHM) – the physical width in one dimension at which the flux density falls to half of the maximum

● Root mean square (rms) spot size

● Encircled energy – the physical radius within which some fixed proportion (e.g. 50% or 80%) of all rays lie.

● Ensquared energy – the size of the square within which some fixed proportion (e.g. 50% or 80%) of all rays lie

● Enslitted energy – the width of the slit within which some fixed proportion (e.g. 50% or 80%) of all rays lie

The FWHM is a useful description of the width of a sharp geometrical peak. On the other hand, the rms spot size is more mathematically useful, but not always universally applicable. In the case of an Airy disc, the rms spot size is actually infinite and the rms spot size is not so useful in situations where an image is attended by a large background signal. Encircled energy is useful to gauge the amount of light passing through a small circular aperture. Its equivalent for a rectangular geometry, the ensquared energy, is particularly useful for pixelated detectors whose sensor elements are naturally either square or rectangular. Similarly, for slitted instruments, such as spectrometers, enslitted energy is a useful metric.

Where the overall wavefront error is significantly larger than the wavelength, this geometric description of image quality is perfectly adequate. However, where this is not the case, we must look to a new approach.

6.9.3 Diffraction and Image Quality

In Section 6.5 we examined briefly the diffraction pattern produced by a uniformly illuminated pupil. This is the so-called Airy disc. The Airy disc is the diffraction pattern that one would obtain in the absence of any system aberration. In Section 6.6, we included the effect of system aberration, introducing the Huygens point spread function. The Huygens point spread function is the flux distribution pattern produced at the image plane produced by a point source at the object plane. Figure 6.18 shows an example of an aberrated pupil, where the OPD is mapped in two dimensions across a circular pupil.

The Huygens point spread function (PSF) of the same system is shown in Figure 6.19.

The PSF shown in Figure 6.19 shows much deeper rippling when compared to the Airy distribution and, unlike the geometrical analysis, represents an accurate solution for the local flux distribution at the image. In analysing the PSF, one can use similar metrics as for the geometric spot size, with the addition of the Strehl ratio:

Figure 6.18 OPD map across pupil.

Figure 6.19 Huygens point spread function.

● Strehl Ratio

● Full width half maximum

● Root mean square (rms) spot size

● Encircled energy

● Ensquared energy

● Enslitted energy

As mentioned previously, the Strehl ratio describes the ratio of the aberrated peak flux to the unaberrated peak flux. A ratio of 0.8 or greater, by virtue of the Maréchal criterion, is considered to be ‘diffraction limited’. This is consistent with an rms wavefront error of lambda/14 or a peak to valley wavefront error of lambda/4. This measure was introduced earlier in Section 6.6, and is an exceptionally important metric to keep in mind when designing a system that is diffraction limited or near diffraction limited.

6.9.4 Modulation Transfer Function

The MTF expresses the ability of an imaging system to replicate the contrast of a specific object pattern. In the case of the MTF, the object is represented by a sinusoidally varying pattern of light and dark, described by some kind of spatial frequency, k_o. That is to say, the spatial variation of the object illumination is represented by:

(6.58)

The contrast ratio of the illumination is defined as the ratio of the difference of the maximum and minimum fluxes to the sum of those fluxes. The illumination pattern represented in Eq. (6.58) has a contrast ratio of unity, with the minimum flux being zero. However, because of imaging imperfections, this is not fully represented at the image plane and the contrast is somewhat reduced. Assuming the system magnification is M, then

(6.59)

For an input contrast ratio of unity, the MTF is defined as the contrast ratio of the image:

(6.60)

Figure 6.20 MTF pattern.

A typical example of an MTF pattern is shown in Figure 6.20.

The MTF response shown in Figure 6.20 illustrates different final contrast levels, varying from 2% to 100%. In addition, a range of input spatial frequencies is shown. In practice, there is a tendency for the contrast ratio to reduce at higher spatial frequencies; a typical imaging system has a reduced capacity for replicating fine details. A typical MTF plot against spatial frequency is shown in Figure 6.21.

Figure 6.21 Typical MTF plot.

It is evident, from Figure 6.21, that the MTF declines with spatial frequency. Also included in the plot is the MTF of the diffraction limited system. In fact, the MTF is the absolute value of the complex optical transfer function (OTF). The OTF of a system is related to the Fourier transform of the point spread function. In fact, for the diffraction limited system, the MTF follows a fairly simple mathematical prescription. There is some maximum spatial frequency, υ_max, above which the MTF is zero and this is defined by the system numerical aperture, NA. In this case, the diffraction limited MTF is simply given by:

(6.61)

The MTF is widely used in the testing and analysis of camera systems. One particular attribute of the MTF is especially useful. For a system composed of a number of subsystems, the MTF of the system is simply given by the product of the individual MTFs:

(6.62)

Analysis of the MTF is also useful in incorporating the behaviour of the detector. In a traditional context, where photographic film had been used, the contrast provided by the film media would be defined by the spatial frequency at which its effective MTF fell to 50%. For high contrast black and white film, this spatial frequency might have been of the order of 100 cycles per mm, although this would vary with film type and sensitivity. On the whole, colour film had poorer contrast with the equivalent spatial frequency being less than 50 cycles per mm. Of course, modern cameras base their detection upon pixelated sensors. In this instance, the characteristic spatial frequency is defined by Nyquist sampling where the equivalent spatial frequency covers two whole pixels. That is to say, for a pixel spacing of 5 μm, the equivalent spatial frequency is 100 cycles per mm.

Worked Example 6.6 We are designing a camera system to give an MTF of 0.5 at 100 cycles per mm. The camera has a pixelated detector with a pixel spacing of 5 μm. It may be assumed that the effective MTF of this detector is 0.75. The remainder of the system may be assumed to be diffraction limited. For a working wavelength of 500 nm, what is the minimum numerical aperture that the system needs to have to fulfil its requirement?

From Eq. (6.62) – the MTF of the remainder of the system is equal to 0.5/0.75 = 0.67.

Using this MTF figure we can calculate υ/υ_max from Eq. (6.61) and this amounts to 0.265, giving υ_max as 377 cycles per mm. Again from Eq. (6.61), given a wavelength of 500 nm, we can calculate the minimum numerical aperture of the system as 0.095 or about f#5.

6.9.5 Other Imaging Tests

The MTF provides a clearly mathematically defined test pattern for testing and subsequent analysis. However, there are other image resolution tests based upon the replication of reticulated patterns, often consisting of sharply delineated features, such as lines or line pairs. One example of this is the 1951 USAF resolution test chart which is a standard reticle placed at the object location. Broadly speaking, this consists of a set of line features whose characteristic size reduces by the sixth root of two when progressing from feature to feature. Visual inspection of the final image enables determination of the minimum line spacing resolution. The standard USAF pattern is illustrated in Figure 6.22.

Although, these types of test are inherently simpler and less capital intensive, the reliance on human visual inspection is, in itself, a weakness. Where, at one time, analytical complexity precluded the widespread use of MTF and other more abstruse processes, the availability of high performance computational power overcomes this obstacle nowadays.

Figure 6.22 1951 USAF resolution test chart.

Source: Image Provided by Thorlabs Inc.

Further Reading

Bleaney, B.I. and Bleaney, B. (1976). Electricity and Magnetism, 3e. Oxford: Oxford University Press. ISBN: 978-0-198-51141-0.

Born, M. and Wolf, E. (1999). Principles of Optics, 7e. Cambridge: Cambridge University Press. ISBN: 0-521-642221.

Lipson, A., Lipson, S.G., and Lipson, H. (2011). Optical Physics. Cambridge: Cambridge University Press. ISBN: 978-0-521-49345-1.

Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. New York: Wiley. ISBN: 978-0-471-35832-9.

Wolf, E. (2007). Introduction to the Theory of Coherence and Polarisation of Light. Cambridge: Cambridge University Press. ISBN: 978-0-521-82211-4.

Yariv, A. (1989). Quantum Electronics, 3e. New York: Wiley. ISBN: 978-0-471-60997-1.

7
Radiometry and Photometry

7.1 Introduction

In the preceding chapters, we have been concerned with the general behaviour of light in an optical system, as described by ray and wave propagation. Hitherto, there has been no interest in the absolute magnitude of the wave disturbance. On the other hand, radiometry and photometry is intimately concerned with the absolute flux of light within a system, its analysis and, above all, its measurement.

At this point we will make a distinction between the two terms, radiometry and photometry. Radiometry relates to analysis of the absolute magnitude of optical flux, as defined by the relevant SI unit, e.g. Watts or Watts per square metre. In contrast, photometry is concerned with the measurement of flux as mediated by the sensitivity of some detector. Most notably, although not exclusively, the detector in question might be the human eye. So, from a radiometric perspective 1 W of ultraviolet or infrared emission is worth 1 W of visible emission. However, from a photometric view (as referenced to the human eye) the ultraviolet and infrared emissions are worthless.

In the study of radiometry, we are interested in the emission of light from a physical source that might have some area dS and subtend some solid angle, dΩ. The light may either be directly emitted from a luminous source, such as a lamp filament, or scattered indirectly. The generic geometry for this is illustrated in Figure 7.1.

The geometry above may be applied both to the emission of light from a surface or to the absorption/scattering of light at a surface. The distinction between these two scenarios simply implies a reversal of the direction of travel of the rays.