of the monochromator is achieved using monochromatic light of a known wavelengths and adjusting the rotation of the grating until the maximum intensity output is obtained.
Optical spectrum analyser
Figure 5.13: Photograph of an optical spectrum analyser in the laboratory setting.
The monochromator is the basis for the more sophisticated optical spec-trum analyser (OSA). At its heart, an OSA just has a diffraction grating (or set of gratings) mounted on an automated motor. Light is coupled into the device via fibre optic patch cable and focused onto the grating. As the grating is rotated, an optical detector measures the optical signal at some fixed angle and plots the intensity of the signal against wavelength (or fre-quency). Various additional facilities may also be integrated into the device.
Figure 5.13 shows an OSA as part of an experimental setup to characterise semiconductor lasers.
5.8 Diffraction around objects
We have seen how a wave may spread out as it is diffracted through a nar-row aperture. An analogous situation pertains when light encounters an opaque object: the wavefronts may now diffract around the object. Some
(a) (b)
Figure 5.14: (a) Diffraction around the side of an object according to Huy-gens’ Principle. (b) Including interference via the Huygens-Fresnel Princi-ple.
insight into this phenomenon is furnished by Huygens’ Principle, as illus-trated in Fig. 5.14 (a). Here we see how light at the edge of the object, acting as a point source of a wavelet, propagates around the object. This however, is not the full story. According to such a picture , all waveforms ought to be diffracted around the object by π/2. The fact that this does not happen is explained via interference.
In Fig. 5.14 (b) we see a similar picture to the earlier diagrams of light passing through a slit, in which the condition for constructive interference at P is met by the wavelength of light being greater than the maximum path difference
λ >
AB~
. (5.45)
Now as λ is reduced, so must the maximum path difference to maintain constructive interference. Hence, the angle θ will also tend to be reduced, limited the angular spread of the wavefront.
For an object in a train of wavefronts, this means that if the size of the object is comparable to the wavelength, the waves will tend to diffract right around the object, as illustrated in Fig. 5.15.
5.9 Summary
• For the diffraction pattern beyond an aperture in a screen through which light passes, we may define to distinct regions:
– Near field or Fresnel diffraction
The diffraction pattern varies considerably with increasing dis-tance from the aperture.
5.9. SUMMARY 101
(a) (b)
Figure 5.15: (a) Waves encountering an object with dimensions that are greater than the wavelength. (b) Diffraction around an object of dimensions comparable with the wavelength.
– Far field or Fraunhofer diffraction
The diffraction pattern settles down to a constant profile.
• Analysis of Fraunhofer diffraction – Single-slit
The intensity of the far-field diffraction pattern from a slit of width Din a barrier may be written as
I (θ) = I (0)
k is the wave-vector and θ is the angle from the normal to the barrier.
– The Fraunhofer condition
The condition for the validity of the Fraunhofer treatment is given by
D R λ
D. (5.48)
where λ is the wavelength and R is the radial distance from the slit.
• Fraunhofer diffraction from a circular aperture
– The Airy disc
The intensity for far-field diffraction from a circular aperture of diameter D is given by
I (θ) = I (0) 2J1(kD sin (θ/2)) kD sin (θ/2)
2
, (5.49)
where J1is the first order Bessel function.
The central peak of this diffraction pattern is known as the Airy disc and the bright interference rings around it are called Airy rings.
– The Rayleigh criterion
In order to resolve two points sources, then the angular separa-tion between them must be greater than
θmin ≈ 1.22λ
D . (5.50)
• Multiple slit diffraction
The intensity profile of multiple slit diffraction in the far field is given by
I (θ) = I (0) sin N α N sin α
2
sin β β
2
. (5.51)
where N is the number of slits,
α = ka
2 sin θ, (5.52)
β = kD
2 sin θ, (5.53)
kis the wave-vector, a is the slit spacing and D is the slit width.
• The grating equation
mλ = a (sin θm+ sin θi) . (5.54) where m is an integer and λ is the wavelength, gives the condition for the local maxima for multiple slit diffraction.
5.10. REFERENCES 103
• Resolving power of a grating
R = λ
∆λ. (5.55)
• Monochromators
Diffraction gratings may be used in monochromators, such as the Czerny-Turner design, to measure the frequency composition of poly-chromatic light.
• Optical spectrum analyser
The monochromator may be integrated into an automated unit known as an optical spectrum analyser.
5.10 References
[1] Correspondence of Scientific Men of the Seventeenth Century...., vol 2, Ed. Stephen Jordan Rigaud, Oxford, England: Oxford University Press (1841)
Part III
Electromagnetic Waves
105
6. Wave Solutions to Maxwell’s Equations
6.1 General remarks
Alongside the quantum mechanical explanation of light, Maxwell’s predic-tion of the generapredic-tion of electromagnetic (EM) waves is key to our under-standing of optical phenomena. Furnishing Thomas Young’s conclusive ob-servations of the wave nature of light with a physical explanation, Maxwell predicted [1] in 1865 that electric and magnetic fields would mutually in-duce each other, propagating with a constant speed in a vacuum given by c = (ε0µ0)−1/2, reproducing the measured speed of light. In turn, ε0 and µ0 are both fundamental physical constants, determining the response of the vacuum to electric and magnetic fields respectively. Maxwell’s predic-tion was then later confirmed by Hertz in a paper of 1892 [2] reporting the generation of radio waves in the laboratory setting.
Although it was initially believed that EM waves would require some kind of physical medium for their propagation (the ‘luminous ether’), this was later shown by Einstein to be superfluous to requirements in his 1905 paper on Special Relativity [3]. Einstein was also a leading figure in the early development of quantum theory. Despite the apparently contradictory nature of the so-called ‘wave-particle’ duality of light, the two pictures are, in fact, complimentary to one another. In this Chapter, we shall reiterate the relation between the classical theory of optical absorption and the quantum mechanical explanation.
A crucial aspect of this Chapter is the introduction to the electric suscep-tibility tensor. Although we shall currently limit our consideration to linear, isotropic and homogeneous media, the explanation of optical phenomena in anisotropic media (to be covered in the Chapters on Crystal Optics) will depend heavily on our understanding of the susceptibility tensor.
Lastly in this Chapter, we shall consider the flux of electromagnetic en-ergy. We specify this in terms of the Poynting vector. Again, we shall anticipate subtleties in our understanding of the energy flux to arise in the context of anisotropic media.
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6.2 Learning objectives
The aims of this section are to understand
• Wave solutions of Maxwell’s equations in free space and dielectric media
• The electric and magnetic susceptibility tensors
• Relative permittivity and permeability
• The refractive index of a medium
• Frequency dependence of the electric susceptibility
• Optical loss in media and the relation to the photon picture
• The Poynting vector for the electromagnetic energy flux