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PHYS20312 Wave Optics -‐ Section 3: Interference PHYS20312 Wave Optics - Section 3: Interference 3.1 Conditions for Interference Consider the superposition of two plane waves: � = �!exp � �!. � − �!� + �! +��exp � �!. � − �!� + �! but we observe �. �∗ � = 2� where ∗ � � �. � = �� + �� + 2�!. �� cos �! − �! . � − �! − �! � + �! − �! constants interference term ��. �� = 0 if wave are orthogonally polarised Hence, we have our first condition for interference i.e. �!. �� ≠ 0 Now let’s concentrate on the interference term (and call it �!"#$% �, � for brevity) and re-write it as using the trigonometrical identity for the cosine of two angles: �!"#$% �, � = cos ∆�. � + ∆�] cos ∆�� + sin [∆�. � + ∆�] sin[∆�� where ∆� = �! − �!; ∆� = �! − �!; ∆� = �! − �!. Let’s now see what happens when we take the time average over a period � ∗ � � �. � = �� + �� + 2�!. �� �!"#$% �, � !! ∗ but sin �� = cos �� = 0 for � ≫ � then �. � = 0 unless ∆� = �! − �! ≈ 0 Therefore, our second condition for interference is �! ≈ �! If �! ≈ �! then cos �� ≈ 1 and sin �� ≈ 0 ∴ �!"#$% �, � = cos ∆�. � + ∆� = cos ∆�. � cos ∆� − sin ∆�. � sin ∆� If ∆� = � � then �. �∗ = 0 as above and so our third condition for interference is ∆� ≈ constant 1 | Page PHYS20312 Wave Optics - Section 3: Interference over �. Such beams are ‘coherent’ (more on this later) and can be achieved experimentally if the two beam originate from the same source. In contrast, it is very hard to synchronise to independent sources so that ∆� ≈ constant. Summary of the conditions of interference: 1. �!. �� ≠ 0 i.e. non-orthogonal polarization 2. ∆� < ~�!! i.e. narrow bandwidth 3. ∆� ≈ constant over � i.e. coherent sources 3.2 Coherence In general , waves that maintain a fixed phase relationship are coherent but we can divide this into two types of coherence: temporal and spatial 3.2.1 Temporal coherence If ∆� ≈ 0 the the phase relationship will change over a period ∆�!! , as shown below for two similar frequencies: Figure 1 Two similar get progressively out of phase as time advances (Image from Wikipedia). For polychromatic light with a bandwidth, ∆�, we can define a coherence time: 1 � = ! ∆� (3.1) and also a coherence length: �! = ��! (3.2) which is the characteristic path length difference between two beams that still allows interference to be observed. 2 | Page PHYS20312 Wave Optics - Section 3: Interference 3.2.2 Spatial coherence. Spatially coherence light maintains a fixed phase relationship along a wavefront, as illustrated in Fig 2 below: Figure 2 Spatial coherence (Image adapted from Wikipedia). Spatial coherence is important if different parts of a wavefront are to interfere such as for ‘wavefront division interference’ but is less important for ‘amplitude division interference’ – see Fig 3 below. We will now look at each of these cases in turn. Figure 3 Comparison of wavefront and amplitude division inteference. 3 | Page PHYS20312 Wave Optics - Section 3: Interference 3.2.3 Fringes and Fringe Visibility For parallel polarized, monochromatic, coherent beams: ∗ � � �. � = �� + �� + 2�!�! cos � or � = �! + �! + 2 �!�! cos � where � = �! − �! . � + Δ� if cos � > 0 then we have constructive interference if cos � < 0 then we have destructive interference As cos � varies between -1 and +1, we get a series of bright and dark band, which are known as “interference fringes”, as shown in the figure below. Figure 4 Interference fringes. If �! = �! = �! then � � = 2� + 2� cos � = 4� cos! ! ! ! 2 If �! ≠ �! we can define a measure of the depth of modulation of the light intensity which is known as the ‘fringe visibility’, �, (or alternatively as the ‘contrast’ or ‘modulation’): �!"# − �!"# 2 �!�! � = = �!"# + �!"# �! + �! (3.3) 4 | Page PHYS20312 Wave Optics - Section 3: Interference 3.3 Wavefront Division Interference. 3.3.1 Young slits. This was a very important experiment historically because it was the first demonstration of the wave nature of light. Young used a candle flame as his light source which produced broadband, unpolarized and incoherent light i.e. met none of the conditions for interference identified above. How did he get his experiment, shown below, to work? • The first slit ensured good spatial coherence at the double slits so that they become coherent sources • The path length difference (pld) to the screen was very small and so: o instantaneous polarization was the same o the pld < coherence length Figure 5 Young's double slit experiment. The double slits produce cylindrical wavefronts and so the combined field is given by 1 1 � = �! exp � ��! − �!� + �! + �! exp � ��! − �!� + �! �! �! and so � � ∗ �� �� 2�!. �� �. � = + + cos � �! − �! − ∆�� + ∆� �! �! �!�! 5 | Page PHYS20312 Wave Optics - Section 3: Interference For coherent sources with �! > �! − �! and for �! ≈ �! ≈ � ! � �! + �� 2�!. �! �. �∗ = + cos � � − � � � ! ! Moreover, if the slits are equal width then �! = �! and hence �! = �! = �! then �! � � = 4 cos! � 2 where � = � �! − �! . We can find an expression for � by considering figure 6: Figure 6 Calculating � for Young's slits. Hence � 2� � � = ��� = �� sin � ≈ ��� ≈ �� = � � � � Thus, �! ��� � = 4 cos! � �� (3.4) 6 | Page PHYS20312 Wave Optics - Section 3: Interference 3.3.2 Lloyd’s Mirror. Consider the interference between a direct beam and its reflection, as shown in Figure 7. Figure 7 Lloyd's mirror. This is a variation on Young’s slits, with one source and its reflection replacing the two slits. However, a difference now arises because one of the two rays undergoes a reflection. The Fresnel reflection coefficients for s- and p-polarized light are (see Hecht Chapter 4): sin �! − �! �! = − sin �! + �! tan �! − �! �! = tan �! + �! for �!~90° and �! < �! then �! < 0 and �! < 0 (See Hecht Ch 4) i.e. �� and �� are of opposite sign for both polarizations. In other words, a � phase change occurs on reflection under these conditions (N.B. exp �� =-1). Thus, now � = � �! − �! + � Hence, �! ��� �! ��� � = 4 cos! + � = 4 sin! � �� � �� (3.5) 7 | Page PHYS20312 Wave Optics - Section 3: Interference 3.3.3 Multiple slits. Let’s now extend our analysis to � slits, as in Figure 8: Figure 8 Interference from N slits. Consider the path lengths from each of the slits to �: �!� = �! �!� = �! + �� �!� = �! + 2�� etc etc until: �!!!� = �! + � − 1 �� Thus for the �th path length, �!, we have �! = �! + ��� for � = 0,1 ⋯ � − 1 but �� ≈ �� ∴ �! = �! + ��� for � ≫ �, �! ≈ � ∴ �! = � + ��� 8 | Page PHYS20312 Wave Optics - Section 3: Interference The field from the �th slit at P is 1 �! = �! exp ���! �! and hence the total field at P is !!! 1 � = �! exp ���! � !!! ! For � ≫ � then �! ≈ � for all �. We can thus replace �! with � in the denominator but not in the argument of the exponential because that is sensitive to changes in �! of the order of �. Hence, !!! �! � = exp ���! � !!! !!! �! � = exp �� � + ��� � !!! !!! �! � = exp ��� exp ����� � !!! This is a geometric series so �! 1 − exp ����� � = exp ��� � 1 − exp ���� ���� ���� ���� exp −� exp � − exp −� �! 2 2 2 � = exp ��� ��� ��� ��� � exp −� exp � − exp −� 2 2 2 which yields ���� sin! �! 2 � = ��� � sin! 2 (3.6) Figure 9 shows equation (3.6) plotted out for various values of � 9 | Page PHYS20312 Wave Optics - Section 3: Interference Figure 9 Inteference patterns for N slits. 3.4 Amplitude Division Interference. 3.4.1 The Michelson Interferometer. In this device, light from a source is split into two parts, each of which travel down a separate arm before being recombined so that they interfere – see Figure 10. Figure 10 The Michelson Interferometer. 10 | Page PHYS20312 Wave Optics - Section 3: Interference Figure 11 The compensator plate. Without the compensator plate, the two beams pass through unequal thicknesses of glass – see Figure 11. This means the that the optical path length difference is a function of wavelength because the refractive index of glass depends on wavelength. The compensator plate equalises the path through glass and this removes the wavelength dependence of the path length difference. We can analyse the Michelson Interferometer by first drawing an equivalent diagram – see Figure 12. Figure 12 Equivalent diagram for the Michelson Interferometer. 11 | Page PHYS20312 Wave Optics - Section 3: Interference Path length difference �� = � cos � Phase difference � = ��� + � where an additional phase lag, �, has been included because one beam undergoes more reflections than the other. Bright fringes will occur when � = 2��, where � is an integer i.e. ��cos�! + � = �2� where �! is the angle at which the �th bright fringe occurs. Hence, 2� �cos� = �2� − � � ! �cos�! = �� − �� where �� �� = 2� • These equations correspond to a set of fringes (one fringe for each value of �). • These fringes will form rings if the mirrors are well-aligned due to symmetry. • The greatest value of � corresponds to the central fringe (NB �cos�! is a maximum for �! = 0°). • If � is decreased then the rings will shrink towards the centre. ! • The central ring vanishes when ∆� = ! • At � = 0 the central fringe covers the entire screen For the central fringe, �! = 0°: � = �!� − �� For the �th fringe from the centre: �cos�! = �! − � � − �� Combining these together: � � = � 1 − cos�! !! for small �, cosθ ≈ 1 − ! 2�� ∴ � = ! � (3.7). 12 | Page PHYS20312 Wave Optics - Section 3: Interference 3.4.2 The Fourier Transform Spectrometer. The Michelson interferometer can also be used as a spectrometer. In this case, a detector is used to monitor the intensity at the centre of the fringe pattern (i.e. for � = 0°) whilst one of the mirrors is moved along the beam direction – see Figure 13. Figure 13 The Fourier
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