COHERENCE AND INTERFERENCE

Transcription

1 COHERENCE AND INTERFERENCE - An interference experiment makes use of coherent waves. The phase shift (Δφ tot ) between the two coherent waves that interfere at any point of screen (where one observes the interference) remains constant during observation time and the result is a stable pattern. This model assumes the superposition of two (a) infinite long waves in time and space with (b) equal frequency and (c) constant phase shift all time. But, one knows that there is no source that can emit waves indefinitely and this means that the coherent wave is an ideal model. Meanwhile the equal frequency of superposing waves and their constant phase shift are essential for interference. So, let s verify the limits of our assumptions for light waves and get a more realistic definition for coherence. a) Experiments have shown that the visible light waves (λ av ~ 550nm) are emitted by the atoms and molecules during τ ~10-8 sec while they pass from higher to lower energy levels. The light wave is emitted randomly in time and space direction. The period of this wave is T = λ/c = 550*10-9 / 3*10 8 = 1.833* sec. So, there are / T 10 /1.833*10 5.5*10 full periods contributing in a wave train emitted by a single atom. For one period, the wave front advances in space by one wavelength. This means that for t= / T periods(end of emission) the wave front has travelled in space a distance of 6 9 ( / T)* 5.5*10 *550*10 3m (1) ( / T)* * / T * * c L By transforming the left side of expression (1), one get w light C This calculations give an initial idea about the spatial length of a wave train or wavelet of light emitted by a single free atom or molecule in sample ; L c parameter is called COHERENCE LENGTH of wave train. Actually, due to collisions with other atoms in sample, L c is smaller (~ 3mm for a Na discharge lamp and 0-30cm for the best conventional monochromatic light sources or common He-Ne lasers), but it can get till 3km for some special lasers. So, the coherence length, i.e. the spatial length of light waves that interfere is not infinity. The extension of light wave train in time domain is measured by another parameter, COHERENCE TIME (see fig.1.a) which is defined as τ = L c /c () Fig 1 - The coherence length L c limits the extension of region occupied by fringes in Young s experiment. The interference related calculations refer to the path length difference (δ = r 1 r ) and one assumes that the two wave trains (wavelets) superpose. But if δ > L c the wavelets cannot superpose (see fig 1.b) and there is not interference. So, L c of interfering wavelets restricts the interference order of fringes one may see on a screen to a certain maximum M max ; this effect happens in all interference experiments. Ex. The path length difference in a thin film is δ =nl. If the light from a Na source is used to observe interference, the restriction δ < L c requests nl < 3mm which means that one may expect to observe thin film interference on glass films (n~ 1.5) only if the film thickness l is smaller than 1mm. b) Same frequency is a precise modeling for two atoms of the same element. We will explain this later. c) Constant phase shift all time between any two wavelets emitted by source is essentially impossible because the process of irradiation is random. At Young s experiment, this problem is avoided by using 1

2 two slits that produce two in phase Huygens wavelets born from the same plane wave front. The phase of the wave front that hits on two slits plane changes randomly in time but, at a given moment, it is the same at all points across the wave front and consequently on the two slits (two coherent sources). So, even though the phase at the entrance of slits is random, the phase difference between the two sources that emit interfering wavelets is zero, all time. -As a big number of atoms participate in the process of emission, the light wave produced by the source is constituted by a big number of wavelets. The average coherence length L c of those wavelets is larger if the number of random collisions between the atoms is smaller. From this point of view, a low pressure gas source does offer advantages. Meanwhile, to emit a coherent light wave with considerable intensity one needs the emission along the same space direction of a big number of equal wavelets IN PHASE to each other. This requests a simultaneous emission of identical wavelets along the same direction by atoms inside the source of light. The laser sources meet very well these requirement; besides, they offer biggest values of coherence length Lc for wavelets. Lasers are the best sources of coherent light waves. One distinguishes two kind of coherence characteristics for the emitted wave trains (fig ) : 1) The TEMPORAL COHERENCE defined by τ parameter. A good temporal coherence requires large τ (= L c /c) values and this mean large coherence length L C, too. ) The SPATIAL COHERENCE of a SET OF WAVE TRAINS requires all atoms to emit wavelets simultaneously, in phase and along the same direction in space. Fig (Short Lc) (Longt Lc) - When calculating the interference from two light waves (Young s exp.) we found the resultant intensity I 4I I E (3) 0 cos ( / ) _ remember _ 0 0 Note that this result and its graph(fig.3) assume a constant phase shift φ between interfering waves. For a random φ (non coherent superposing waves) one has to refer to the average of function cos / in the 1 range φ [0,π]. This average is calculated as cos ( / ) d 1/ and expression (3) becomes I 4Io *1/ I0 I0 I0 0 (4) which corresponds to a uniform illumination of the screen by the superposition of two completely incoherent waves. Fig.3

3 LIGHT DIFFRACTION -As mentioned previously, the term diffraction concerns the effect of borders of objects (or apertures) on a wave front falling on them. In optics, the diffraction produces light bending around borders and its result is the presence of some light in regions where geometrical optics predicts shadow. Actually, there are diffraction effects wherever there is a cut of wave front. As all optical devices select (cut at input aperture) only a limited portion of wave front, they cannot avoid diffraction. -The figure 4 shows the diffraction pattern formed on screen by a circular object at moderate distance from screen. A set of successive dark/bright fringes around the image borders and a central bright spot (Poisson spot) appear on screen. This effect is known as NEAR FIELD or FRESNEL" diffraction: the light source or the screen or both are close to the obstacle (or aperture). In these circumstances, the wave fronts are spherical (not planes) and the calculations go beyond the limits of this course. Fig.4 -We will study the diffraction pattern when the light source and the screen are far away from the aperture (or object). In this case, the incident light on aperture and incident light on screen 1 can be modelled by plane waves. This situation is known as FAR FIELD or FRAUNHOFER diffraction. SINGLE SLIT FRAUNHOFER DIFFRACTION - This is the situation for each of slits in Young s experiment. In the following, we consider the effects of diffraction produced on slit borders over far field interference if a monochromatic light with plane wave fronts hits at slits input. - Consider the division of slit aperture a into four equal thin strips parallel to the slit. Each strip element (its center is presented by a dot in fig.5) emits one Huygens wavelet of light. All those wavelets are considered coherent to each other and having the same coherence length (L C ). Along the direction θ = 0 0, all the wavelets interfere in phase and produce a maximum of interference because the path length difference between any two of them is zero; their phasors are aligned along the same direction all the time (fig.6.a). a δ 1- δ 1-3 δ 1-4 Figure 5 θ Figure 6.a Sum of four phasors in phase Ph_3 Ph_1 Ph_ Ph_4 Ph_sum 1 In optical devices, those waves pass through a convergent lens before illuminating the screen placed at lens focal plan. 3

4 - For directions θ 0 o there is a path length difference between interfering wavelets. Consider the angle θ 1 along which the path length difference between the first and the second wavelet is δ 1- = λ/4 (fig.5). Along this direction, the phase shift between the two first wavelets is 1 ( / ) * / 4 /. Along the same space direction θ 1, the path length difference between the first and the third wavelet (see fig.5) is δ 1-3 = δ 1- = λ/ and the corresponding phase shift is 1 3 ( / )* /. The path length difference between the first and the fourth wavelet is δ 1-4 = 3 δ 1- =3* λ/4 and the corresponding phase shift is 1 4 ( / ) *3 / 4 3 /. By using these values of phase shift versus the first wavelet one finds out that the sum of all four phasors is zero (see fig. 6.b) which means that the superposition of these four wavelets will produce a minimum of interference along the space direction θ 1. Ph_3 Ph_4 Ph_ Ph_1 Fig. 6.b Fig. 6.c - One may improve the calculation accuracy by increasing the number "n" of stripes and simultaneously decreasing their thickness. Noting that for n = 4, δ 1- = λ/4 and δ 1-4 = 3λ/4, for n 4 one might get δ 1- = λ / n and δ 1-n = (n-1)*λ / n (4) n 1 n 1 and 1 n ( / )* 1n * n n (5) Figure 6.c presents the sum of phasors for n=1. One may see that the tip of last phasor (n=1) fits to the tail of first phasor (n=1). This means that the sum of all the phasors gives zero. For larger "n" values the phasors will have extremely small magnitude and will be aligned around a perfect circle. Yet, the tip of the last phasor will meet the tail of the first phasor and will produce a zero sum. Finally, when one tents n-value to infinity expression (4,5) give respectively δ 1-n (limit) = λ and Δφ 1-n = π. - So, along the direction θ 1, on which the path length difference between the two extreme wavelets (emitted at slit borders, see fig.5) 1 n asin1 is equal to λ, the diffraction produces a zero net phasor and this means a minimum of interference along direction θ 1 a sin (6) A similar mathematical procedure based on the phasor modeling shows that the diffraction produces a minimum of light intensity along each direction θs that corresponds to the general condition a* sin s s; _ s 1,, 3,... (7) Note that: a) s = 0 is excluded in relation (7) because it corresponds to the central maximum which is large and becomes larger with the decrease of slit thickness (a parameter). (Why?) b) If a is very small, sinθ 1 (= λ/a) may be out of small angle approx range even for s =1. 1 The way how to get this formula is given in Benson textbook, p

5 c) Between any two consecutive minima of diffraction there is a maximum due to single slit diffraction. The directions of these secondary maxima are found from the expression a* sin S (S 1) /;_ S 1,, 3,... (8) d) The intensity of single-slit diffraction pattern decreases fast (see fig.7) with distance from screen center and it is described by the expression (8) (I max stands for the top intensity of central maximum) I I max a sin( sin ) a sin (9) Fig.7 Evolution of light intensity for single slit diffraction -The formulas (7, 8) tell that even a single slit produces maxima-minima patterns in the conditions of Fraunhofer diffraction. Similar calculations show that tiny objects (hair, blood cells, ) can produce diffraction related patterns, too. One uses these patterns to estimate the dimension "a" of small objects. Actually, the single slit diffraction related patterns are visible only when a λ. If a >>> λ the angles θ s of minima directions are so small that fringes superpose to each other (i.e. disappear) and the image on the screen takes the slit s shape as predicted by geometrical optics. CORRECTIONS TO THE RESULTS OF YOUNG S EXPERIMENT -When considering the interference conditions for Young s experiment we did not account for the diffraction effect of each slit individually. For two slits interference, we found the direction of: a) the maxima by condition d * sin M M _ M 0, 1,, 3,... (10) b) the minima by condition d * sin m (m 1) / _ m 0, 1,, 3,... (11) Note: Do not mix slits distance d in (10,11) with slit " thickness" a in expressions (6,7,8). -Suppose that along the same direction θ the conditions (7) and (10) are fulfilled simultaneously, i.e. θ s = θ M = θ for two slits with equal aperture a at a distance d. The condition (7) tells that along this direction each slit produces a diffraction minimum; i.e. no light sent along this direction. As there are no light waves to superpose, the maximum predicted from two slits interference will be missing. -Suppose that along another direction θ the conditions (8) and (11) are fulfilled simultaneously. The relation (8) tells that, along this direction, each slit alone would produce a diffraction maximum of light. But the eq. (11) fixes a minimum for this direction and the light brought along this direction by each single slit is removed. The two slits interference effect takes the light out of this direction (fig. 8). 5

6 1 slit diffraction pattern works as envelope of -slits system of fringes Fig.8 The interference-diffraction pattern from two slits (a < d) 1-slits diffraction removes light from here -slits interference removes light from here As the same light must obey simultaneously to two minima rules it will be a minimum along each direction θ for which is met one of minima condition. So, some maxima predicted by two-slits interference are cancelled due to one slit diffraction mechanism. The intensities of two slits maxima pattern are modified in conformity to one-slit diffraction pattern, too. Notes: a) In general, the distance d between two slits is larger than the slit aperture a. It comes out that λ/d < λ/a which means that sinθ M=1 < sinθ m=1. So, several maxima of two slit interference pattern will fall between the two first order minima (s = +/-1) of "single slit diffraction pattern. b) In general, the small angle approximation does work for two slits interference but, it does not work always for single slit diffraction pattern. So, when considering effects related to single slit diffraction, before using the small angle approximation, one must verify whether the conditions for small angle approximation are met. THE RAYLEIGH CRITERION FOR OPTICAL RESOLUTION -The sharpness of images recorded by a camera is defined by its spatial resolution, i.e. the capacity to discern two neighbour points on the object surface. This parameter is limited by the light diffraction. -In any optical system there is an input aperture that allows (cuts) only a portion of wave front to pass through lenses. But, any cut of wave front is associated with light diffraction. This means that the light wave emitted by a point on object produces a Fraunhoffer 3 diffraction pattern on recording screen. -In figure 9, two points on object surface are emitting monochromatic light(same λ) as non-coherent point sources. Each of these points produces a similar diffraction pattern on screen plane but there is no interference between them to redistribute the light because they are not coherent waves. Meanwhile, each of these waves will get diffracted at input aperture of camera. It can be proved that when a light wave is diffracted by a circular aperture with diameter a, the direction of first minimum in the diffracted light is defined by expression (proportionality coefficient changes from 1 to 1. in expression "6"): 1. sin s 1 (1) a 3 We consider a plane front wave at system input and recording plane. Lens brings the image form infinity to focal distance. 6

7 The aperture of an optical device a >> λ. So, θ 1 is very small and, by using the small angle approximation the relation (11) is written 1. s 1 a (13) This expression gives the direction of first minimum with respect to the center of diffraction pattern for light emitted by a point source far away from the aperture. Taking into account that a similar diffraction pattern is produced by the other point source, one gets the light intensity distribution shown in figure 9. Two non-coherent point sources can be easily resolved as far as their diffraction patterns do not overlap significantly. θ θ Fig. 9 -When the angular separation θ between the two points on the object surface is small, the two diffraction patterns on screen plane overlap and may not allow to record two clearly distinct image points. How to judge whether the images are well separated or not enough separated on the screen? Rayleigh criterion offers a precise answer to this question. RAYLEIGH CRITERION: Two images are barely resolved when the center of central maximum of one diffraction pattern coincides with the first diffraction minimum of the other one (see fig. 10b). Following this criterion, the critical angular separation is θ cr = θs= 1 cr 1. a (14) For smaller angular separation between the two points (or any two light sources) θ < θ cr, the recording device of optical system cannot distinguish between the two images. As seen from relation (14), one may increase the spatial resolution (decrease θ Cr ) by using lenses with bigger diameters 4 (the increase of a produces a decrease of θ cr ). This way, the condition θ > θ cr is met for smaller θ-values, and this allows to distinguish the images of two closer points (sources) on the object (see fig. 9). Note that, in general, the light that falls on a photographic device is not monochromatic and the angle θ cr corresponds to largest wavelengths. 4 Photographs use this method to improve the quality of recorded pictures. 7

8 Fig.10 a) The diffraction pattern of two point sources of monochromatic light on a circular aperture. b) The two images can be just resolved if the central maximum of one patter coincides with the first order minimum of diffraction in the other pattern. c) If the angular separation between sources is reduced further, their images cannot be resolved. 8