Diffraction-Fraunhofer Diffraction

A detailed presentation on fraunhofer diffraction and also an introduction to the concept of diffraction.There is also a brief discussion on fresnel diffraction and the difference between former and the latter.

1. Diffraction Fraunhofer Diffraction

2. Contents 1.Introduction 2.Examples 3.History 4.Mechanism-Ancient View 5.Mechanism-Modern Quantum Mechanical View 6.Fresnel Diffraction

3. 7.Distinction between fresnel and fraunhofer diffraction 8.Fraunhofer Diffraction 9.Far Field 10.Focal planeof a positive lens as the far field plane 11.Diffraction due to a Single Slit 12.Circular Aperture

4. 13.Limit of Resolution 14.Resolving Power of Grating 15.Determination of wavelength using Diffraction Grating 16.Fraunhofer Diffraction due to double slit 17.Fraunhofer diffraction due to n slits

5. 1.Introduction Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660.

6. 2.Examples The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern seen when looking at a disc. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example.

7. a).Diffraction in water c).Diffraction b). Infinitely many points (3 shown) form at one point

8. 3.History Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits.Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, made public in 1815 and 1818 and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory.

9. 4.Mechanism-Ancient View In traditional classical physics diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle and the principle of superposition of waves. The propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a point source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.

10. 5.Mechanism-Modern Quantum Mechanical View In the modern quantum mechanical understanding of light propagation through a slit (or slits) every photon has what is known as a wavefunction which describes its path from the emitter through the slit to the screen. The wavefunction — the path the photon will take — is determined by the physical surroundings such as slit geometry, screen distance and initial conditions when the photon is created. In important experimentsthe existence of the photon's wavefunction was demonstrated. In the quantum approach the diffraction pattern is created by the distribution of paths, the observation of light and dark bands is the presence or absence of photons in these areas (no interference!). The quantum approach has some striking similarities to the Huygens-Fresnel principle ; in that principle the light becomes a series of individually distributed light sources across the slit which is similar to the limited number of paths (or wave functions) available for the photons to travel through the slit.

11. 6.Fresnel Diffraction In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation.

12. The Fresnel Diffraction Integral

13. The Fresnel Approximation

14. Comparison between the diffraction pattern obtained with the Rayleigh-Sommerfeld equation, the (paraxial) Fresnel approximation, and the (far-field) Fraunhofer approximation.

15. 7.Distinction between Fresnel and Fraunhofer Diffraction

16. 8.Fraunhofer diffraction 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 . 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 A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer diffraction equation.

17. 9.Far Field 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

18. 10.Focal plane of a positive lens as the far field plane In the far field, propagation paths for individual wavelets from every point on the aperture to a point of observation are approximately parallel, and the positive lens (focusing lens) focuses parallel rays toward the lens to a point on the focal plane (the focus point position depends on the angle of the parallel rays with respect to the optical axis). So, if the focal length of the lens is sufficiently large such that differences between electric field orientations for wavelets can be ignored at the focus, then the lens practically makes the Fraunhofer diffraction pattern on its focal plane as the parallel rays meet each other at the focus

19. 11.Diffraction due to a Single Slit In the single slit diffraction experiment, we can observe the bending phenomenon of light or diffraction that causes light from a coherent source interfere with itself and produce a distinctive pattern on the screen called the diffraction pattern . Diffraction is evident when the sources are small enough that they are relatively the size of the wavelength of light . You can see this effect in the diagram below. For large slits, the spreading out is small and generally unnoticeable.

20. Case 1:Principal Maximum

21. Case 2:Minimum Intensity Positions

22. Case 3:Secondary Maximum

23. Intensity Distribution Graph

24. 12.Circular Aperture

25. When light from a point source passes through a small circular aperture, it does not produce a bright dot as an image, but rather a diffuse circular disc known as Airy's disc surrounded by much fainter concentric circular rings. This example of diffraction is of great importance because the eye and many optical instruments have circular apertures. If this smearing of the image of the point source is larger that that produced by the aberrations of the system, the imaging process is said to be diffraction-limited, and that is the best that can be done with that size aperture. This limitation on the resolution of images is quantified in terms of the Rayleigh criterion so that the limiting resolution of a system can be calculated.

26. The aperture diffraction pattern above was photographed with Fuji Sensia 100ASA slide film and then digitized. With the time exposure necessary to show the side lobes, the central peak was washed out nearly white. The only retouching of the digital image was to paint in the washed out part of the central maximum (Airy's disc). The pinhole was made by placing aluminum foil on a glass plate, sticking a straight pin into the aluminum foil, and then rotating the foil. Several pinholes were made, and this one was the closest to being round.

27. 13.Limit of Resolution The limit of resolution (or resolving power) is a measure of the ability of the objective lens to separate in the image adjacent details that are present in the object. It is the distance between two points in the object that are just resolved in the image. The resolving power of an optical system is ultimately limited by diffraction by the aperture. Thus an optical system cannot form a perfect image of a point. For resolution to occur, at least the direct beam and the first-order diffracted beam must be collected by the objective. If the lens aperture is too small, only the direct beam is collected and the resolution is lost.

28. Consider a grating of spacing d illuminated by light of wavelength λ , at an angle of incidence i .

29. Numerical Aperture

30. Airy Discs When light from the various points of a specimen passes through the objective and an image is created, the various points in the specimen appear as small patterns in the image. These are known as Airy discs. The phenomenon is caused by diffraction of light as it passes through the circular aperture of the objective. Airy discs consist of small, concentric light and dark circles. The smaller the Airy discs projected by an objective in forming the image, the more detail of the specimen is discernible. Objective lenses of higher numerical aperture are capable of producing smaller Airy discs, and therefore can distinguish finer detail in the specimen. The limit at which two Airy discs can be resolved into separate entities is often called the Rayleigh criterion. This is when the first diffraction minimum of the image of one source point coincides with the maximum of another.

31. From the equation it can be seen that the radius of the central maximum is directly proportional to λ / d . So, the maximum is more spread out for longer wavelengths and/or smaller apertures. The primary minimum sets a limit to the useful magnification of the objective lens. A point source of light produced by the lens is always seen as a central spot, and second and higher order maxima, which is only avoided if the lens is of infinite diameter. Two objects separated by a distance less than θ R cannot be resolved.

32. 14.Resolving Power of Grating The capacity of an optical instrument to show separate images of very closely placed two objects is called resolving power. The resolving power of a diffraction grating is defined as its ability to form separate diffraction maxima of two closely separated wave lengths.It is defined as the capacity of a grating to form separate diffraction maxima of two wavelengths which are very close to each other.

33. 15.Determination of wavelength of light using Diffraction Grating Young's Double-Slit Experiment verifies that light is a wave simply because of the bright and dark fringes that appear on a screen. It is the constructive and destructive interference of light waves that cause such fringes. Constructive Interference: The following two waves (Fig. 1) that have the same wavelength and go to maximum and minimum together are called coherent waves. Coherent waves help each other's effect, add constructively, and cause constructive interference. They form a bright fringe.

34. Destructive Interference of Waves In Fig. 2 however, the situation is different. When the wave with amplitude A 1 is at its maximum, the wave with amplitude A 2 is at its minimum and they work completely against each other resulting in a wave with amplitude A 2 - A 1 . These two completely out of phase waves interfere destructively. If A 2 = A 1 , they form a dark fringe. The bright and dark fringes in Young's experiment follow these formulas: Bright Fringes: d sinθ k = k λ where k = 0, 1, 2, 3, ... Dark Fringes: d sinθ k = (k - 1/2 ) λ where k = 1, 2, 3, ...

35. The above formulas are based on the following figures :

36. Check the following statement for correctness based on the above figure . Light rays going to D 2 from S 1 and S 2 are 3 ( 0.5λ ) out of phase (same as being 0.5λ out of phase) and therefore form a dark fringe . Light rays going to B 1 from S 1 and S 2 are 2(0.5λ) out of phase (same as being in phase) and therefore form a bright fringe . Note that SB o is the centerline . Going from a dark or bright fringe to its next fringe changes the distance difference by 0.5 λ.

37. Diffraction grating is a thin film of clear glass or plastic that has a large number of lines per (mm) drawn on it. A typical grating has density of 250 lines/mm. Using more expensive laser techniques, it is possible to create line densities of 3000 lines/mm or higher. When light from a bright and small source passes through a diffraction grating, it generates a large number of sources at the grating. The very thin space between every two adjacent lines of the grating becomes an independent source. These sources are coherent sources meaning that they emit in phase waves with the same wavelength. These sources act independently such that each source sends out waves in all directions. On a screen a distance D away, points can be found whose distance differences from these sources are different multiples of λ causing bright fringes. One difference between the interference of many slits (diffraction grating) and double-slit (Young's Experiment) is that a diffraction grating makes a number of principle maxima along with with lower intensity maxima in between. The principal maxima occur on both sides of the central maximum for which a formula similar to Young's formula holds true.

38. D = the distance from the grating to the screen d = the spacing between every two lines ( same as every two sources ) If there are N lines per mm of the grating, then d, the space between every two adjacent lines or ( every two adjacent sources ) is d=1/N or N=1/d The diffraction grating formula for the principal maxima is : d sin θ k = k λ where k = 1, 2, 3, ...

39. A.Determination of (Lines/mm) of the Diffraction Grating: a) Fix a laser pointer and the diffraction grating (placed in a target holder) on an optical bench as shown . Try to make a distance D (grating to wall) of about 1.5m.

40. b) Make sure that the direction of the optical bench is normal (at right angle) to the wall and that you are measuring the perpendicular distance D from the grating to the wall . c) Measure y 1 , y 2 , and D with the precision of mm and record the values . d)Angles θ 1 and θ 2 may now be calculated from the measured values as follows :

41. e) Use the tan -1 function (built-in in your calculator) to calculate θ 1 and θ 2 . f ) Use angles θ 1 and θ 2 along with the wavelength given on your laser pointer ( in meters ) and the diffraction grating formula to calculate d, the distance between adjacent spaces (sources) on the grating . Find d once on the basis of k = 1 and once on the basis of k =2 . Theoretically, the two values you obtain for d must be equal ; however , due to measurement errors, they might be slightly different . Find an average value for d in meters . g) From d, determine N, the number of lines per mm of the grating.

42. 2. Red and Violet Wavelengths : a) Hold a diffraction grating close to your eye and look at the objects around you . You will see a continuous spectrum of rainbow colors around bright objects . The diffraction grating separates the colors of white light similar to what a prism does . White light coming from a bright object separates into its constituent colors as it passes thru the grating and reaches your eyes . If you are looking through a grating at a bright spot such as the filament of a lit light bulb, you will be able to direct another person to move to the left or right and mark the ends of the spectrum you are observing . By measuring the distance between each end of the spectrum and the bright filament Y violet or Y red and D the distance from the filament to the grating ( held by you ), it is possible to calculate the angles θ violet and θ red . Then, by using the formula d sin θ k = k λ , the corresponding wavelengths for violet and red light can be determined . Note that through the grating you will see more than one rainbow band . You will see two or three bands on each side of the center . If you use the 1 st band to one side of the center, then k = 1 . For the 2 nd band k = 2, and for the 3 rd band k = 3 .

43. b) Place the optical bench near the board in your lab or class on a somewhat high table . c) Make sure that the optical bench stays at right angle to the board and mount a light-bulb so that it almost touches the board . Turn the light bulb on . d) Hold a diffraction grating at a fixed distance D from the lit bulb . When you look into the grating, your line of sight must be normal to the board . A diagram of the set-up is shown below :

44. where V (in the diagram) is the Violet End of the spectrum, and R the Red end of it . Also BV is the same as Y 1V , the distance from the bulb to the violet end of the first fringe . Similarly, BR is the same as Y 1R , the distance from the bulb to the red end of the first fringe .

45. a) While looking into the grating and observing the spectrum, guide your partner to the extreme ends of the spectrum so that he/she can mark those points on the board . Your partner must have previously observed the same spectrum and have a good understanding of the experimental procedure . b) When those points are marked, double-check their precision and measure distances BV and BR to the nearest cm as shown in the figure . Also measure D . c) From the data collected, calculate angles θ violet and θ red and use each in the above-mentioned formula separately to find the corresponding wavelengths.

46. 16.Fraunhofer Diffraction due to Double Slit 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.These fringes are often known as Young's fringes. The angular spacing of the fringes is given by:

47. The spacing of the fringes at a distance z from the slits is given by 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.

48. Semi-Quantitative Explanation of Double-Slit Fringes

49. Diffraction 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 θ n given by: This is known as the grating equation . The finer the grating spacing, the greater the angular separation of the diffracted beams.

50. If the light is incident at an angle θ , the grating equation is: 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.

51. 17.Fraunhofer Diffraction due to n slits(Grating) An arrangement consisting of large number of parallel slits of the same width and separated by equal opaque spaces is known as Diffraction grating. Gratings are constructed by ruling equidistant parallel lines on a transparent material such as glass, with a fine diamond point. The ruled lines are opaque to light while the space between any two lines is transparent to light and acts as a slit. This is known as plane transmission grating. When the spacing between the lines is of the order of the wavelength of light, then an appreciable deviation of the light is produced. Theory: A section of a plane transmission grating AB placed perpendicular to the plane of the paper is as shown in the figure.

52. Intensity Distribution

53. Hence if the value of N is larger, then the secondary maxima will be weaker and becomes negligible when N becomes infinity.

54. BY: P.JOHN ISAAC BSc(MPCs) 121418468027 St.JOSEPH’S DEGREE AND PG COLLEGE, KING KOTHI, HYDERABAD.