Suppose, the field Eˆ2 is a coherent laser field (local oscillator) of a large amplitude α, whereas the signal (detected) field is weak. As a result, the terms proportional to 〈bˆ†bˆ〉 and 〈bˆ†〉, 〈bˆ〉 dominate over those without bˆ† and bˆ. Hence, the terms independent of the amplitude of the local oscillator can be discarded. Thus, we can ignore the term β〈aˆ†aˆ〉, and denoting 〈bˆ〉 = |α| exp(iφ), 〈bˆ†〉 = |α| exp(−iφ), the resultant light intensity at the detector is then I ≈ (1− β) |α|2 + 2 |α| √ β (1− β) 〈Eˆφ〉 , (6.102) where 〈 Eˆφ 〉 = 1 2i (〈aˆ〉e−iφ − 〈aˆ†〉eiφ) , (6.103) and φ is the phase of the laser. The first term on the right-hand side of Eq. (6.102) is equal to the intensity of the reflected coherent beam. The second term is an interference term between the coherent and the signals beams. This term contains the phase-dependent quadrature amplitude of the signal beam.
Suppose, the field Eˆ2 is a coherent laser field (local oscillator) of a large amplitude α, whereas the signal (detected) field is weak. As a result, the terms proportional to 〈bˆ†bˆ〉 and 〈bˆ†〉, 〈bˆ〉 dominate over those without bˆ† and bˆ. Hence, the terms independent of the amplitude of the local oscillator can be discarded. Thus, we can ignore the term β〈aˆ†aˆ〉, and denoting 〈bˆ〉 = |α| exp(iφ), 〈bˆ†〉 = |α| exp(−iφ), the resultant light intensity at the detector is then I ≈ (1− β) |α|2 + 2 |α| √ β (1− β) 〈Eˆφ〉 , (6.102) where 〈 Eˆφ 〉 = 1 2i (〈aˆ〉e−iφ − 〈aˆ†〉eiφ) , (6.103) and φ is the phase of the laser. The first term on the right-hand side of Eq. (6.102) is equal to the intensity of the reflected coherent beam. The second term is an interference term between the coherent and the signals beams. This term contains the phase-dependent quadrature amplitude of the signal beam.
However we have shown in Eqs.(18.29) and (18.30) that the square of the absolute value of a complex scalar wave function Ψ is a measure of the wave intensity, which is the quantity of primary interest for the theories of interference and diffraction for either mechanical or electromagnetic waves. The periodic spatial variation of intensity throughout a region of superposition of waves which originate in the same source is known as interference. Its theory is based essentially on the linearity of the wave equation, which allows a linear combination of partial waves to constitute the actual wave at any given point. Consider first the superposition of two monochromatic plane waves, having the complex representation (16.25), that is (18.1) Since we are usually interested in the relative variation of the intensity only, in a medium 0 ,Z and not in its absolute value, it is common practice to adopt simplified wave intensities, compared to those in Eqs.(18.29) or (18.30), which read ( ) ( )2 21 1 2 2, , ,I r t I r= Ψ = Ψ tG G (18.2) The superposition of monochromatic waves (18.1) results in an intensity ( ) ( ) ( )2 21 1 2, , ,tG ( )I r t r t r= Ψ = Ψ +ΨG G ( ) ( ) ( )2 2 *1 2 1 2, , 2Re ,r t r t r t r t,⎡ ⎤= Ψ + Ψ + Ψ Ψ⎣ ⎦G G G G ( )1 2 1 2 1 2 1 22 cosI I I I k r k r ϕ ϕ= + + ⋅ − ⋅ + −G GG G (18.3) where the last term is called the interference term. If the phase difference, given by ( )1 2 1k k r 2δ ϕ ϕ= − ⋅ + −G G G (18.4) is a constant, the two waves are said to be mutually coherent, and the interference term indicates a sinusoidal spatial variation between maxima and minima, of intensity max 1 2 1 22 ,I I I I I m2δ π= + + = (18.5) ( )min 1 2 1 22 , 2I I I I I m 1δ π= + − = + B A SS 21 Figure 18.1. Interference pattern produced by two monochromatic waves originating from sources 1 2andS S Hence, the stationary interference pattern consists of a succession of surfaces, defined by the condition Nδ π= where .12or2 += mmN It is usually produced by superposition of waves originating in two point sources , which can be assumed, for simplicity, to be of equal phase constants 21 and SS 1 2 0.ϕ ϕ− = If the two sources are far enough from a given point, at distances 1 2 1 2,r r r r>> − ,G G the incident waves at the point can be regarded as plane waves, as we shall show in Chapter 19. Under this assumption Eq.(18.4) reduces to ( ) ( )1 2 1 2 1 2or 2k k r k r r N r r N Nkπ λδ π= − ⋅ ≅ − = − = =G G G (18.6) Both the antinodal and nodal surfaces ( mN 2= ) ( )12 += mN are defined by Eq.(18.6) to be hyperboloids of revolution. Their intersection with a plane will produce alternate nodal and antinodal lines called fringes of the form of either hyperbolas or circular rings if the plane is parallel (A) or normal (B) to the line joining the sources (Figure 18.1). The interference fringes are not observed if the phase difference 1 2ϕ ϕ− has a random variation with time, since the interference term in Eq.(18.3) is averaged out. In this event the waves are said to be mutually uncoherent and do not interfere by superposition. Mutually coherent waves that originate in the same source can be obtained by splitting a single beam, so that on recombining partial waves interference effects occur.
Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.
In our semiclassical treatment of the laser in Chaps. 8-12, we “ Fourieranalyzed” the electric field. This frequency-domain method has much to commend it: explicit expressions for the field are often obtained, the calcu lations generally do not require much computer time, and considerable insight can be gained into the nature of the atom-field interaction which takes place in the laser. There are also shortcomings, however, to the frequency-domain picture. The number of modes which can be treated easily is small, perhaps as few as ten. In general, the mode amplitudes and phases cannot vary ap preciably in the atomic lifetimes (the total field, of course, can vary). Further more the strong-signal theories are rather complicated.
It is easy to show that the expectation value of the amplitude of the EM field in the photon number state |n〉 is equal to zero. Since 〈n| Eˆ x |n〉 = E 〈n|aˆ + aˆ†|n〉 sin(kz), (4.5) and aˆ |n〉 = √n |n− 1〉 , aˆ† |n〉 = √n+ 1 |n+ 1〉 , (4.6) with 〈n|m〉 = δnm, we obtain 〈n| Eˆ x |n〉 = 0. (4.7) Note that 〈n|Eˆ x |n〉 = 0 independent of n, that is, 〈n|Eˆ x |n〉 = 0 even if n → ∞. This is unusual property of the quantum field as in the limit of n → ∞ the properties of the photon-number state should convert into that corresponding to a classical state. However, we know that the classical field performs harmonic oscillations in time.
Starting from the well-known plane-wave decomposition of the operator associ- ated with the transverse part of the vector potential (Aˆ T) , and by making use of the operator relations defining the relations between and magnetic (Bˆ [with ∇ · Bˆ = 0]) field operators, the plane-wave mode expansions for Eˆ T and Bˆ are established. Subsequently, the general operator relation ∇ × Eˆ T = −∂ Bˆ/ ∂t follows. From the Heisenberg equation of motion for the magnetic field operator the inhomogeneous Maxwell operator equation ∇ × Bˆ = µ 0 Jˆsym + c − 2 ∂ Eˆ T/∂t is established. Here the operator Jˆsym T is the symmetrized transverse current den- sity operator in the space-time domain. The transverse set of Maxwell operator equations are of utmost importance for the subsequent photon energy operator wave function formalism. For globally coherent quantum fields, where the par- ticle source current density distribution is quasi-classical, dynamical equations for the two helicity species of Riemann-Silberstien field operators are obtained. Neglect of the quantum field fluctuations in the coherent state makes the bridge to photon wave mechanics in the Oppenheimer-Riemann-Silberstein-Bialynicki approach. The chapter is closed with a discussion of response theory and Ein- stein causality.