Mathematically, the diffraction pattern is characterized by the wavelength of light illuminating the circular aperture, and the aperture's size. The appearance of the diffraction pattern is additionally characterized by the sensitivity of the eye or other detector used to observe the pattern. This experimental system is designed to quantitatively investigate diffraction patterns of various apertures.
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.[1][2] In contrast, the diffraction pattern created near the object, in the near field region, is given by the Fresnel diffraction equation.The equation was named in honor of Joseph von Fraunhofer although he was not actually involved in the development of the theory.[3]
This article explains where the Fraunhofer equation can be applied, and shows the form of the Fraunhofer diffraction pattern for various apertures. A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer diffraction equation.
Equation
- Main page: Fraunhofer diffraction equation
When a beam of light is partly blocked by an obstacle, some of the light is scattered around the object, light and dark bands are often seen at the edge of the shadow – this effect is known as diffraction.[4] These effects can be modelled using the Huygens–Fresnel principle. Huygens postulated that every point on a primary wavefront acts as a source of spherical secondary wavelets and the sum of these secondary wavelets determines the form of the proceeding wave at any subsequent time. Fresnel developed an equation using the Huygens wavelets together with the principle of superposition of waves, which models these diffraction effects quite well.
It is not a straightforward matter to calculate the displacement (amplitude) given by the sum of the secondary wavelets, each of which has its own amplitude and phase, since this involves addition of many waves of varying phase and amplitude. When two waves are added together, the total displacement depends on both the amplitude and the phase of the individual waves: two waves of equal amplitude which are in phase give a displacement whose amplitude is double the individual wave amplitudes, while two waves which are in opposite phases give a zero displacement. Generally, a two-dimensional integral over complex variables has to be solved and in many cases, an analytic solution is not available.[5]
The Fraunhofer diffraction equation is a simplified version of the Kirchhoff's diffraction formula and it can be used to model the light diffracted when both a light source and a viewing plane (the plane of observation) are effectively at infinity with respect to a diffracting aperture.[6] With the sufficiently distant light source from the aperture, the incident light to the aperture is a plane wave so that the phase of the light at each point on the aperture is the same. The phase of the contributions of the individual wavelets in the aperture varies linearly with position in the aperture, making the calculation of the sum of the contributions relatively straightforward in many cases.
With a distant light source from the aperture, the Fraunhofer approximation can be used to model the diffracted pattern on a distant plane of observation from the aperture (far field). Practically it can be applied to the focal plane of a positive lens.
Far field
Fraunhofer diffraction occurs when: [math]displaystyle{ frac{W^2}{Llambda} ll 1 }[/math] |
[math]displaystyle{ W }[/math] – aperture or slit size, [math]displaystyle{ lambda }[/math] – wavelength, [math]displaystyle{ L }[/math] – distance from the aperture |
When the distance between the aperture and the plane of observation (on which the diffracted pattern is observed) is large enough so that the optical path lengths from edges of the aperture to a point of observation differ much less than the wavelength of the light, then propagation paths for individual wavelets from every point on the aperture to the point of observation can be treated as parallel. This is often known as the far field and is defined as being located at a distance which is significantly greater than W2/λ, where λ is the wavelength and W is the largest dimension in the aperture. The Fraunhofer equation can be used to model the diffraction in this case.[7]
For example, if a 0.5 mm diameter circular hole is illuminated by a laser with 0.6 μm wavelength, the Fraunhofer diffraction equation can be employed if the viewing distance is greater than 1000 mm.
Focal plane of a positive lens as the far field plane
Plane wave focused by a lens.
In the far field, propagation paths for wavelets from every point on an aperture to a point of observation are approximately parallel, and a positive lens (focusing lens) focuses parallel rays toward the lens to a point on the focal plane (the focus point position on the focal plane depends on the angle of the parallel rays with respect to the optical axis). So, if a positive lens with a sufficiently long focal length (so that differences between electric field orientations for wavelets can be ignored at the focus) is placed after an aperture, then the lens practically makes the Fraunhofer diffraction pattern of the aperture on its focal plane as the parallel rays meet each other at the focus.[8]
Examples of Fraunhofer diffraction
In each of these examples, the aperture is illuminated by a monochromatic plane wave at normal incidence.
Diffraction by a slit of infinite depth
Graph and image of single-slit diffraction
The width of the slit is W. The Fraunhofer diffraction pattern is shown in the image together with a plot of the intensity vs. angle θ.[9] The pattern has maximum intensity at θ = 0, and a series of peaks of decreasing intensity. Most of the diffracted light falls between the first minima. The angle, α, subtended by these two minima is given by:[10]
- [math]displaystyle{ alpha approx {frac{2 lambda}{W}} }[/math]
Thus, the smaller the aperture, the larger the angle α subtended by the diffraction bands. The size of the central band at a distance z is given by
- [math]displaystyle{ d_f = frac {2 lambda z}{W} }[/math]
For example, when a slit of width 0.5 mm is illuminated by light of wavelength 0.6 μm, and viewed at a distance of 1000 mm, the width of the central band in the diffraction pattern is 2.4 mm.
The fringes extend to infinity in the y direction since the slit and illumination also extend to infinity.
If W < λ, the intensity of the diffracted light does not fall to zero, and if D << λ, the diffracted wave is cylindrical.
Semi-quantitative analysis of single-slit diffraction
Geometry of single-slit diffraction
We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. Consider the light diffracted at an angle θ where the distance CD is equal to the wavelength of the illuminating light. The width of the slit is the distance AC. The component of the wavelet emitted from the point A which is travelling in the θ direction is in anti-phase with the wave from the point B at middle of the slit, so that the net contribution at the angle θ from these two waves is zero. The same applies to the points just below A and B, and so on. Therefore, the amplitude of the total wave travelling in the direction θ is zero. We have:
- [math]displaystyle{ theta_text{min} approx frac {CD} {AC} = frac{lambda}{W}. }[/math]
The angle subtended by the first minima on either side of the centre is then, as above:
- [math]displaystyle{ alpha = 2 theta_text{min} = frac{2lambda}{W}. }[/math]
There is no such simple argument to enable us to find the maxima of the diffraction pattern.
Single-slit diffraction of Electric Field using Huygen's Principle
Continuous broadside array of point sources of length a.
We can develop an expression for the far field of a continuous array of point sources of uniform amplitude and of the same phase. Let the array of length a be parallel to the y axis with its center at the origin as indicated in the figure to the right. Then the differential field is:[11]
[math]displaystyle{ dE=frac{A}{r_1}e^{j omega [t-(r_1/c)]}dy=frac{A}{r_1}e^{j(omega t-beta r_1)}dy }[/math]
where [math]displaystyle{ beta=omega/c=2pi /lambda }[/math]. However [math]displaystyle{ r_1=r-ysintheta }[/math]and integrating from [math]displaystyle{ -a/2 }[/math] to [math]displaystyle{ a/2 }[/math],
[math]displaystyle{ E=A^'intlimits_{-a/2}^{a/2} e^{jbeta y sintheta} dy }[/math]
where [math]displaystyle{ A^'=frac{Ae^{j(omega t-beta r)}}{r_1} }[/math].
Integrating we then get
[math]displaystyle{ E=frac{2A^'}{beta sin theta}sin(frac{beta a}{2} sin theta) }[/math]
Letting [math]displaystyle{ psi^'=beta a sin theta = alpha_r sin theta }[/math]where the array length in rad[math]displaystyle{ a_r=beta a=2pi a/lambda }[/math], then,
[math]displaystyle{ E=frac{sin(psi^'/2)}{psi^'/2} }[/math][11]
Diffraction by a rectangular aperture
Computer simulation of Fraunhofer diffraction by a rectangular aperture
The form of the diffraction pattern given by a rectangular aperture is shown in the figure on the right (or above, in tablet format).[12] There is a central semi-rectangular peak, with a series of horizontal and vertical fringes. The dimensions of the central band are related to the dimensions of the slit by the same relationship as for a single slit so that the larger dimension in the diffracted image corresponds to the smaller dimension in the slit. The spacing of the fringes is also inversely proportional to the slit dimension.
If the illuminating beam does not illuminate the whole vertical length of the slit, the spacing of the vertical fringes is determined by the dimensions of the illuminating beam. Close examination of the double-slit diffraction pattern below shows that there are very fine horizontal diffraction fringes above and below the main spot, as well as the more obvious horizontal fringes.
Diffraction by a circular aperture
Computer simulation of the Airy diffraction pattern
The diffraction pattern given by a circular aperture is shown in the figure on the right.[13] This is known as the Airy diffraction pattern. It can be seen that most of the light is in the central disk. The angle subtended by this disk, known as the Airy disk, is
- [math]displaystyle{ alpha approx frac {1.22 lambda} {W} }[/math]
where W is the diameter of the aperture.
The Airy disk can be an important parameter in limiting the ability of an imaging system to resolve closely located objects.
Diffraction by an aperture with a Gaussian profile
Intensity of a plane wave diffracted through an aperture with a Gaussian profile
The diffraction pattern obtained given by an aperture with a Gaussian profile, for example, a photographic slide whose transmissivity has a Gaussian variation is also a Gaussian function. The form of the function is plotted on the right (above, for a tablet), and it can be seen that, unlike the diffraction patterns produced by rectangular or circular apertures, it has no secondary rings.[14] This technique can be used in a process called apodization—the aperture is covered by a Gaussian filter, giving a diffraction pattern with no secondary rings.
The output profile of a single mode laser beam may have a Gaussian intensity profile and the diffraction equation can be used to show that it maintains that profile however far away it propagates from the source.[15]
Diffraction by a double slit
Double-slit fringes with sodium light illumination
In the double-slit experiment, the two slits are illuminated by a single light beam. If the width of the slits is small enough (less than the wavelength of the light), the slits diffract the light into cylindrical waves. These two cylindrical wavefronts are superimposed, and the amplitude, and therefore the intensity, at any point in the combined wavefronts depends on both the magnitude and the phase of the two wavefronts.[16] These fringes are often known as Young's fringes.
The angular spacing of the fringes is given by
- [math]displaystyle{ theta_text{f} = lambda/d. }[/math]
The spacing of the fringes at a distance z from the slits is given by[17]
- [math]displaystyle{ w_text{f} = z theta_f = z lambda/d, }[/math]
where d is the separation of the slits.
The fringes in the picture were obtained using the yellow light from a sodium light (wavelength = 589 nm), with slits separated by 0.25 mm, and projected directly onto the image plane of a digital camera.
Double-slit interference fringes can be observed by cutting two slits in a piece of card, illuminating with a laser pointer, and observing the diffracted light at a distance of 1 m. If the slit separation is 0.5 mm, and the wavelength of the laser is 600 nm, then the spacing of the fringes viewed at a distance of 1 m would be 1.2 mm.
Semi-quantitative explanation of double-slit fringes
Geometry for far-field fringes
The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves.
If the viewing distance is large compared with the separation of the slits (the far field), the phase difference can be found using the geometry shown in the figure. The path difference between two waves travelling at an angle θ is given by
- [math]displaystyle{ d sin theta approx d theta. }[/math]
When the two waves are in phase, i.e. the path difference is equal to an integral number of wavelengths, the summed amplitude, and therefore the summed intensity is maximal, and when they are in anti-phase, i.e. the path difference is equal to half a wavelength, one and a half wavelengths, etc., then the two waves cancel, and the summed intensity is zero. This effect is known as interference.
The interference fringe maxima occur at angles
- [math]displaystyle{ d theta_n = n lambda,quad n = 0, 1, 2, ldots, }[/math]
where λ is the wavelength of the light. The angular spacing of the fringes is given by
- [math]displaystyle{ theta_text{f} approx lambda/d. }[/math]
When the distance between the slits and the viewing plane is z, the spacing of the fringes is equal to zθ and is the same as above:
- [math]displaystyle{ w = zlambda/d. }[/math]
Diffraction by a grating
Diffraction of a laser beam by a grating
A grating is defined in Born and Wolf as 'any arrangement which imposes on an incident wave a periodic variation of amplitude or phase, or both'.
A grating whose elements are separated by S diffracts a normally incident beam of light into a set of beams, at angles θn given by:[18]
- [math]displaystyle{ ~ sin theta_n = n lambda /S, n = 0, pm 1, pm 2 ...... }[/math]
This is known as the grating equation. The finer the grating spacing, the greater the angular separation of the diffracted beams.
If the light is incident at an angle θ0, the grating equation is:
- [math]displaystyle{ sin theta_n = frac {n lambda} {S} + sin theta_0, n=0, pm1, pm2.... }[/math]
The detailed structure of the repeating pattern determines the form of the individual diffracted beams, as well as their relative intensity while the grating spacing always determines the angles of the diffracted beams.
The image on the right shows a laser beam diffracted by a grating into n = 0, and ±1 beams. The angles of the first order beams are about 20°; if we assume the wavelength of the laser beam is 600 nm, we can infer that the grating spacing is about 1.8 μm.
Semi-quantitative explanation
A simple grating consists of a series of slits in a screen. If the light travelling at an angle θ from each slit has a path difference of one wavelength with respect to the adjacent slit, all these waves will add together, so that the maximum intensity of the diffracted light is obtained when:
- [math]displaystyle{ W sin theta = n lambda, n=0, pm 1, pm 2, ..... }[/math]
This is the same relationship that is given above.
See also
- Fraunhofer diffraction (mathematics)
- Airy disc
References
- ↑Born & Wolf, 1999, p. 427.
- ↑Jenkins & White, 1957, p288
- ↑http://scienceworld.wolfram.com/biography/Fraunhofer.html
- ↑Heavens and Ditchburn, 1996, p. 62
- ↑Born & Wolf, 1999, p. 425
- ↑Jenkins & White, 1957, Section 15.1, p. 288
- ↑Lipson, Lipson and Lipson, 2011, p. 203
- ↑Hecht, 2002, p. 448
- ↑Hecht, 2002, Figures 10.6(b) and 10.7(e)
- ↑Jenkins & White, 1957, p. 297
- ↑ 11.011.1Kraus, John Daniel; Marhefka, Ronald J. (2002) (in en). Antennas for all applications. McGraw-Hill. ISBN9780072321036. https://books.google.com.ph/books/about/Antennas_for_all_applications.html?id=NRxTAAAAMAAJ&redir_esc=y.
- ↑Born & Wolf, 1999, Figure 8.10
- ↑Born & Wolf, 1999, Figure 8.12
- ↑Hecht, 2002, Figure 11.33
- ↑Hecht, 2002, Figure 13.14
- ↑Born & Wolf, 1999, Figure 7.4
- ↑Hecht, 2002, eq. (9.30).
- ↑Longhurst, 1957, eq.(12.1)
Sources
- Born M & Wolf E, Principles of Optics, 1999, 7th Edition, Cambridge University Press, ISBN:978-0-521-64222-4
- Heavens OS and Ditchburn W, Insight into Optics, 1991, Longman and Sons, Chichester ISBN:978-0-471-92769-3
- Hecht Eugene, Optics, 2002, Addison Wesley, ISBN:0-321-18878-0
- Jenkins FA & White HE, Fundamentals of Optics, 1957, 3rd Edition, McGraw Hill, New York
- Lipson A., Lipson SG, Lipson H, Optical Physics, 4th ed., 2011, Cambridge University Press, ISBN:978-0-521-49345-1
- Longhurst RS, Geometrical and Physical Optics,1967, 2nd Edition, Longmans, London
External links
- Fraunhofer diffraction on ScienceWorld
- Fraunhofer diffraction on HyperPhysics
(0 votes) Original source: https://en.wikipedia.org/wiki/ Fraunhofer diffraction. Read more |
- ↑Goodman, Joseph W. (1996). Introduction to Fourier Optics (second ed.). Singapore: The McGraw-HillCompanies, Inc. p. 73. ISBN0-07-024254-2.
Retrieved from 'https://handwiki.org/wiki/index.php?title=Physics:Fraunhofer_diffraction&oldid=33062'
Diffraction is an optical effect which limits the total resolution of your photography — no matter how many megapixels your camera may have. It happens because light begins to disperse or 'diffract' when passing through a small opening (such as your camera's aperture). This effect is normally negligible, since smaller apertures often improve sharpness by minimizing lens aberrations. However, for sufficiently small apertures, this strategy becomes counterproductive — at which point your camera is said to have become diffraction limited. Knowing this limit can help maximize detail, and avoid an unnecessarily long exposure or high ISO speed.
BACKGROUND
Light rays passing through a small aperture will begin to diverge and interfere with one another. This becomes more significant as the size of the aperture decreases relative to the wavelength of light passing through, but occurs to some extent for any aperture or concentrated light source.
Since the divergent rays now travel different distances, some move out of phase and begin to interfere with each other — adding in some places and partially or completely canceling out in others. This interference produces a diffraction pattern with peak intensities where the amplitude of the light waves add, and less light where they subtract. If one were to measure the intensity of light reaching each position on a line, the measurements would appear as bands similar to those shown below.
For an ideal circular aperture, the 2-D diffraction pattern is called an 'airy disk,' after its discoverer George Airy. The width of the airy disk is used to define the theoretical maximum resolution for an optical system (defined as the diameter of the first dark circle).
3-D Visualization
When the diameter of the airy disk's central peak becomes large relative to the pixel size in the camera (or maximum tolerable circle of confusion), it begins to have a visual impact on the image. Once two airy disks become any closer than half their width, they are also no longer resolvable (Rayleigh criterion).
No Longer Resolved
Diffraction thus sets a fundamental resolution limit that is independent of the number of megapixels, or the size of the film format. It depends only on the f-number of your lens, and on the wavelength of light being imaged. One can think of it as the smallest theoretical 'pixel' of detail in photography. Furthermore, the onset of diffraction is gradual; prior to limiting resolution, it can still reduce small-scale contrast by causing airy disks to partially overlap.
VISUAL EXAMPLE: APERTURE VS. PIXEL SIZE
The size of the airy disk is primarily useful in the context of pixel size. The following interactive tool shows a single airy disk compared to pixel size for several camera models:
Note: above airy disk will appear narrower than its specified diameter (since this is defined by where it reaches its first minimum instead of by the visible inner bright region).
As a result of the sensor's anti-aliasing filter (and the Rayleigh criterion above), an airy disk can have a diameter of about 2-3 pixels before diffraction limits resolution (assuming an otherwise perfect lens). However, diffraction will likely have a visual impact prior to reaching this diameter.
As two examples, the Canon EOS 20D begins to show diffraction at around f/11, whereas the Canon PowerShot G6 begins to show its effects at only about f/5.6. On the other hand, the Canon G6 does not require apertures as small as the 20D in order to achieve the same depth of field (due to its much smaller sensor size).
Since the size of the airy disk also depends on the wavelength of light, each of the three primary colors will reach its diffraction limit at a different aperture. The calculation above assumes light in the middle of the visible spectrum (~550 nm). Typical digital SLR cameras can capture light with a wavelength of anywhere from 450 to 680 nm, so at best the airy disk would have a diameter of 80% the size shown above (for pure blue light).
Another complication is that sensors utilizing a Bayer array allocate twice the fraction of pixels to green as red or blue light, and then interpolate these colors to produce the final full color image. This means that as the diffraction limit is approached, the first signs will be a loss of resolution in green and pixel-level luminosity. Blue light requires the smallest apertures (highest f-stop) in order to reduce its resolution due to diffraction.
Other Technical Notes:- The physical pixels do not actually occupy 100% of the sensor area, but instead have gaps in between. This calculation assumes that microlenses make these gaps negligible.
- Some cameras have pixels which are slightly rectangular, in which case diffraction will reduce resolution more in one direction than the other.
- The above chart approximates the aperture as being circular (a common approximation), but in reality these are polygonal with 5-8 sides.
- The calculation for pixel area assumes these extend all the way to the edge of each sensor, and all contribute to the final image. In reality, camera manufacturers leave some pixels unused around the edge of the sensor. Since not all manufacturers specify the number of used vs. unused pixels, only used pixels were considered when calculating the fraction of total sensor area. The pixel sizes above are thus slightly larger than if measured (but by no more than 5%).
WHAT IT LOOKS LIKE
Although the above diagrams help give a feel for the concept of diffraction, only real-world photography can show its visual impact. The following series of images were taken on the Canon EOS 20D, which typically exhibits softening from diffraction beyond about f/11. Move your mouse over each f-number to see how these impact fine detail:
No Overlap of Airy DisksPartial Overlap of Airy Disks
Note how most of the lines in the fabric are still resolved at f/11, but have slightly lower small-scale contrast or acutance (particularly where the fabric lines are very close). This is because the airy disks are only partially overlapping, similar to the effect on adjacent rows of alternating black and white airy disks (as shown on the right). By f/22, almost all fine lines have been smoothed out because the airy disks are larger than this detail.
CALCULATING THE DIFFRACTION LIMIT
The form below calculates the size of the airy disk and assesses whether the camera has become diffraction limited. Click on 'show advanced' to define a custom circle of confusion (CoC), or to see the influence of pixel size.
Note: CF = 'crop factor' (commonly referred to as the focal length multiplier);assumes square pixels, 4:3 aspect ratio for compact digital and 3:2 for SLR.*Calculator assumes that your camera sensor uses the typical bayer array.
This calculator shows a camera as being diffraction limited when the diameter of the airy disk exceeds what is typically resolvable in an 8x10 inch print viewed from one foot. Click 'show advanced' to change the criteria for reaching this limit. The 'set circle of confusion based on pixels' checkbox indicates when diffraction is likely to become visible on a computer at 100% scale. For a further explanation of each input setting, also see the depth of field calculator.
In practice, the diffraction limit doesn't necessarily bring about an abrupt change; there is actually a gradual transition between when diffraction is and is not visible. Furthermore, this limit is only a best-case scenario when using an otherwise perfect lens; real-world results may vary.
NOTES ON REAL-WORLD USE IN PHOTOGRAPHY
Even when a camera system is near or just past its diffraction limit, other factors such as focus accuracy, motion blur and imperfect lenses are likely to be more significant. Diffraction therefore limits total sharpness only when using a sturdy tripod, mirror lock-up and a very high quality lens.
Some diffraction is often ok if you are willing to sacrifice sharpness at the focal plane in exchange for sharpness outside the depth of field. Alternatively, very small apertures may be required to achieve sufficiently long exposures, such as to induce motion blur with flowing water. In other words, diffraction is just something to be aware of when choosing your exposure settings, similar to how one would balance other trade-offs such as noise (ISO) vs shutter speed.
This should not lead you to think that 'larger apertures are better,' even though very small apertures create a soft image; most lenses are also quite soft when used wide open (at the largest aperture available). Camera systems typically have an optimal aperture in between the largest and smallest settings; with most lenses, optimal sharpness is often close to the diffraction limit, but with some lenses this may even occur prior to the diffraction limit. These calculations only show when diffraction becomes significant, not necessarily the location of optimum sharpness (see camera lens quality: MTF, resolution & contrast for more on this).
Are smaller pixels somehow worse? Not necessarily. Just because the diffraction limit has been reached (with large pixels) does not necessarily mean an image is any worse than if smaller pixels had been used (and the limit was surpassed); both scenarios still have the same total resolution (even though the smaller pixels produce a larger file). However, the camera with the smaller pixels will render the photo with fewer artifacts (such as color moiré and aliasing). Smaller pixels also give more creative flexibility, since these can yield a higher resolution if using a larger aperture is possible (such as when the depth of field can be shallow). On the other hand, when other factors such as noise and dynamic range are considered, the 'small vs. large' pixels debate becomes more complicated...
Technical Note: Independence of Focal LengthSince the physical size of an aperture is larger for telephoto lenses (f/4 has a 50 mm diameter at 200 mm, but only a 25 mm diameter at 100 mm), why doesn't the airy disk become smaller? This is because longer focal lengths also cause light to travel farther before hitting the camera sensor -- thus increasing the distance over which the airy disk can continue to diverge. The competing effects of larger aperture and longer focal length therefore cancel, leaving only the f-number as being important (which describes focal length relative to aperture size).
Fraunhofer Diffraction Integral
For additional reading on this topic, also see the addendum:Digital Camera Diffraction, Part 2: Resolution, Color & Micro-Contrast