Polarization beating with random electromagnetic beams

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POLARIZATION BEATING WITH RANDOM

ELECTROMAGNETIC BEAMS

Jehona Salaj

∆Ω1

∆Ω2

ω ω+ ∆ω

Master Thesis May 2019

Department of Physics and Mathematics

University of Eastern Finland

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Jehona Salaj Polarization beating with random electromagnetic beams, 55 pages University of Eastern Finland

Master’s Degree Programme in Photonics Supervisors Prof. Tero Set¨al¨a

Ph.D. Hanna Lajunen

Abstract

Although optical beating can be encountered in many situations, the literature usually covers only the beating with scalar monochromatic light. In this thesis the understanding of the problem is deepened to cover random electromagnetic quasi- monochromatic beams and their polarization modulation during beating. Non-stationary beating with quasi-monochromatic light is examined and the other possibilities are treated as special cases. Beating is then compared to interference, and similari- ties and differences are discussed. A situation of self-beating between one beam and its frequency-shifted counterpart is described and examined in detail as an illustration of a typical case. The added understanding of the case prepares the ground for using the presence of beating when necessary.

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Preface

The time during which I have studied here and worked on this thesis has been interesting although quite unusual. I had envisioned it very different. I would like to thank everybody involved with the Photonics programme. Special thanks go to my supervisor, Prof. Tero Set¨al¨a, for his patience and help. On the personal side I am grateful to my aunt Lumi, my aunt Dija, my sister, my boyfriend, and his adorable dog for the emotional support they have given me.

Joensuu, the 28th of May 2019 Jehona Salaj

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Chapter I

Introduction

When an electron moves to a lower energy level, during the collision or decay of particles, as well as during a range of light-matter interactions, the energy difference is released in the form of photons. These processes are random and therefore the light generated by any source has a random amplitude and phase. How random it is, depends on the type of the source, thermal sources being typically the ones with the highest degree of randomness in their properties and lasers the ones with the lowest.

It is common to use monochromatic light as an approximation because it works well for simplified situations in which our intention is a general understanding of a phenomenon. However, by omitting properties like spectral width, partial coherence, and partial polarization, we fail to explain real life situations in a form that is useful for new discoveries and future applications.

The statistical explanation given by coherence theory is more suitable for understanding nondeterministic fields. Coherence theory is the study of the consequences of partial correlation of light examined at two or more space-time points. Its foun- dations lie in the work of Verdet [1,2], von Laue [3], Stokes [4], and Michelson [5–10].

Its main construction was made by Wiener [11], Khintchine [12], van Cittert [13] and Zernike [14]. The modern form of coherence theory was constructed: in spatiotemporal domain by Wolf [15–18], Blanc-Lapierre and Dumontet [19]; and in frequency domain by Wolf [20–22], Mandel [23,24] and Agarwal [25]. Higher order correlations were used by Hanbury Brown and Twiss in their famous experiment [26, 27]. The quantum version of coherence theory was formulated by Glauber [28]. Most of the research has been concentrated on the stationary fields, while non-stationary fields are covered in fewer papers [29, 30]. Similarly, there has been less research studying

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correlations between two fields [31] than dealing with correlations in a single field.

Interference is an important property of waves. In fact, it played a major role in determining that light propagates in the form of transverse waves. Since Young’s experiment [32–34] it has been extensively studied and used as a mean for studying the intensity and polarization of different fields. Beating, on the other hand, has been less studied. Beating or temporal interference happens when fields with a very small difference in their central frequencies propagate close to each other in space.

It is better known for acoustic waves, but it is present in the case of light waves as well.

In this work, the intensity and polarization modulation during beating between fields are studied. The polarization and coherence matrices, as well as the Stokes parameters in space-time domain, are presented. The beating of fields is studied both in scalar and electromagnetic case. It is also compared to interference in the space- frequency domain, presented in the context of Young’s interference experiment. A typical situation in which beating between two quasi-monochromatic beams takes place is illustrated and explained. The equations we derived during the theoretical study are applied to it and conclusions are drawn.

In Chapter 2 is presented the concept of monochromatic light as an idealization and beating with monochromatic light both in the scalar and electromagnetic case.

Chapters 3 and 4 underlie the basics of coherence and polarization formalisms used in the rest of the thesis, where, in the former is presented the case of scalar light, while in the latter that of electromagnetic light beams. In Chapter 5, the derived results for the beating of quasi-monochromatic scalar beams are presented, while the results of the beating of electromagnetic light beams are shown in Chapter 6. The results are initially given for non-stationary beams, and then the case of stationary light is discussed. Chapter 7 covers the focused discussion of a specific case. The results derived for beating are compared to the similar cases for spatial interference in Chapter 8. The findings are summarized in Chapter 9.

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Chapter II

Beating with monochromatic light

One of the basic properties of waves is their ability to interfere with each other when overlapping spatially and having a high enough degree of mutual coherence.

However, if these waves are spectrally separated, another property arises. This property is the beating or temporal interference. It is usually manifested in the form of temporally varying intensity. In this Chapter, the basics of optical beating are given. The concepts of scalar and electromagnetic monochromatic light are briefly explained and the results for the total fields and intensities are presented.

2.1 Scalar monochromatic light

The electromagnetic wave which has only one constant frequency is called monochromatic light [35]. The monochromatic light would correspond to the radiation from an atom with strictly fixed energy levels Em and En, radiating with angular frequency ωmn = 2πνmn = (Em−En)/~, where ~= h/2π, h being Planck’s constant, and νmn is the frequency. However, according to the uncertainty principle for the energy and lifetime of a quantum state (∆E∆t ≥ ~/2), the energy of a state m can have any value between Em+¹₂∆E and Em−¹₂∆E. Therefore, monochromatic light can be no more than idealization. This idealization is mathematically useful in many cases, as it helps us to treat a problem in its simplest form, allowing to acquire basic understanding of it before dwelling on the details.

A monochromatic plane wave can be represented, using real quantities, as UR(r, t) =Acos(k·r−ωt), (2.1) where A is the amplitude, k is the wave vector, ω is the angular frequency in

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rad/s, while r denotes position, and t denotes time. This wave satisfies the wave equation [36]

∇²U(r, t)− 1 c²

∂²U(r, t)

∂t² = 0, (2.2)

where ∇² is the Laplacian operator and c is the speed of light.

In the case of monochromatic light, the wave is represented by a strictly harmonic function of time. Its amplitude is constant and there are no random fluctuations in the field. Such wave, if it would exist, would last forever. Using real quantities for the representation of a wave can sometimes lengthen and complicate the calculations. It is for this reason that complex quantities are commonly used. The complex representation also enables us to combine the amplitude and the phase into a complex amplitude of the form A = |A|exp(iarg[A]). Thus, we use the complex representation

U(r, t) =Aexp[i(k·r−ωt)], (2.3)

where A is the complex amplitude whose argument contributes to the phase. The complex form given by Eq. (2.3) satisfies the wave equation of Eq. (2.2) as well. The real field of the form (2.1) can always be obtained by taking the real part of the complex field representation.

2.2 Beating with scalar monochromatic light

Let us consider the case of two monochromatic beams with frequenciesωandω+∆ω with corresponding wave vectors kandk+ ∆k, interfering at some moment in time t. We assume ∆ω ≪ ω and |∆k| ≪ |k|. Using the previously introduced complex representation in Eq. (2.3), we can write these beams as

U1(r, t) =A1exp[i(k·r−ωt)],

U₂(r, t) =A₂exp{i[(k+ ∆k)·r−(ω+ ∆ω)t]}. (2.4) Because of the linearity of the wave equation, the principle of superposition applies.

U(r, t) =U1(r, t) +U2(r, t)

=|A1|exp[i(k·r−ωt+ϕ1)]

+|A2|exp{i[(k+ ∆k)·r−(ω+ ∆ω)t+ϕ2]}

= exp[i(k·r−ωt)]

× {|A1|exp[iϕ1] +|A2|exp[i(∆k·r−∆ωt+ϕ2)]},

(2.5)

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where ϕ1 = arg[A1] and ϕ2 = arg[A2] are the phases of A1 and A2, respectively. In the case of|A1|=|A2|=|A|, Eq. (2.5) takes the form

U(r, t) =|A|exp

i

k+∆k 2

·r−

ω+ ∆ω 2

t+ ϕ1+ϕ2

2

×

exp

−i ∆k

2 ·r− ∆ω

2 t− ϕ1−ϕ2

2

+ exp

i ∆k

2 ·r− ∆ω

2 t− ϕ1−ϕ2

2

= 2|A|exp

i

k+∆k 2

·r−

ω+ ∆ω 2

t+ ϕ1+ϕ2

2

×cos ∆k

2 ·r− ∆ω

2 t− ϕ1−ϕ2

2

.

(2.6)

To simplify the situation further we setϕ1 =ϕ2 =ϕ and the total field becomes U(r, t) = 2|A|exp

i

k+∆k 2

·r−

ω+ ∆ω 2

t

+ϕ

×cos ∆k

2 ·r− ∆ω 2 t

.

(2.7)

From the previously stated conditions we can neglect the terms for ∆k/2 and ∆ω/2 in the exponential without major consequences. In that case, we can write

U(r, t)≈2|A|exp[i(k·r−ωt)] cos ∆k

2 ·r− ∆ω 2 t

. (2.8)

From this we see that the amplitude of the resulting field varies sinusoidally in time as well, depending on the difference between the frequencies of the two fields. In Fig. 2.1 is shown the superposition of two monochromatic waves corresponding to Eq. (2.8), at a fixed point in space.

However, the principle of superposition is not valid for the intensity [36], which can instead be found by squaring the total field

I(r, t) =|U(r, t)|² = 2|A|²[1 + cos(∆k·r−∆ωt)]. (2.9) It is clearly visible that at a fixed position in space which for simplicity is taken to be r= 0, the intensity is

I(t) = 2|A|²[1 + cos(∆ωt)]. (2.10)

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0 20 40 60 80 100 -1

-0.5 0 0.5 1

t [fs]

ℜ[U]/2|A|

Figure 2.1: Normalized magnitude of the superposition of two scalar monochromatic waves with wavelengths λ1 = 632 nm and λ2 = 652 nm as a function of time.

This is a periodic function oft with period 2π/∆ω specified by the frequency difference of the beams. The angular frequency of the variation is ∆ω which therefore is called ”beat frequency”. Eq. (2.10) describes the temporal interference or beating.

It is obvious that beating does not occur between two monochromatic beams with the same frequency, for the sum of which the intensity will be 4|A|². For any other value, the intensity varies in time between zero and 4|A|². This is analogous with the spatial interference, the difference being its dependence on ∆k against the dependence on ∆ω of the temporal case. Figure 2.2 illustrates the relationship between the superposition field of two monochromatic waves of Figure 2.1 and its intensity.

2.3 Electromagnetic monochromatic light

An electromagnetic field is described by two vector functions, electric and magnetic fields (E and H), the relations between which are given by the Maxwell equations.

Each of their components satisfies Eq. (2.2). For monochromatic light in a medium

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0 20 40 60 80 100 -1

-0.5 0 0.5 1 1.5 2

-1 -0.5 0 0.5 1 1.5 2

t [fs]

ℜ[U]/2|A| I/2|A|2

Figure 2.2: Intensity (red) related to the superposition of two monochromatic waves (blue) as a function of time.

with no free charges and currents, we can write:

∇ ×H=iωD,

∇ ×E=−iωB,

∇ ·D = 0,

∇ ·B= 0,

(2.11)

where D and B are the electric and the magnetic flux densities. The constitutive relations characterizing the response of the (non-magnetic) medium are

D =ǫ0E+P,

B=µ0H, (2.12)

where Pis the polarization density.

In the case of electromagnetic monochromatic waves, all the components of electric and magnetic fields will be harmonic functions of the same frequency [36]. The electric and magnetic field can therefore be expressed as

E(r, t) =E₀exp[i(k·r−ωt)],

H(r, t) =H₀exp[i(k·r−ωt)]. (2.13) Figure 2.3 shows the components of the electromagnetic wave as a function of time.

It follows from Maxwell’s equations that E and H are perpendicular to each other and tok.

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-1 1 -0.5

0.5 20

0

15 0.5

0 10

1

-0.5 5

-1 0 t[fs]

ℜ[E(t)]/|E₀| ℜ[H(t)]/|H0|

Figure 2.3: A monochromatic light’s electric (blue) and magnetic (red) components as a function of time.

2.4 Beating with electromagnetic monochromatic light

Let us now consider two monochromatic electromagnetic beams with frequencies ω and ω + ∆ω and corresponding wave vectors k and k+ ∆k, and investigate the beating between them. We can write

E₁(r, t) =A₁exp[i(k·r−ωt)],

E₂(r, t) =A₂exp{i[(k+ ∆k)·r−(ω+ ∆ω)t)]}, (2.14) where A₁ and A₂ are complex amplitudes, |∆k| ≪ |k|, and |∆ω| ≪ |ω| as before.

The total field is

E(r, t) = E₁(r, t) +E₂(r, t)

=A₁exp[i(k·r−ωt)] +A₂exp{i[(k+ ∆k)·r−(ω+ ∆ω)t]}

= exp[i(k·r−ωt)]{A₁+A₂exp[i(∆k·r−∆ωt)]}.

(2.15)

The behaviour of E(r, t) is illustrated in Fig. 2.4. The fields E₁(r, t) and E₂(r, t) with unit amplitudes are taken to be orthogonally polarized along thexand yaxes.

E(r, t) is elliptically polarized at each instant of time but the polarization ellipse changes orientation and ellipticity over time, as can be seen in Fig. 2.4(a). When the orientation of the ellipse reaches 45^◦ or 135^◦, E(r, t) changes its rotation from anticlockwise to clockwise and vice versa. This is illustrated in Fig. 2.4(b) by using

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different colours for clockwise and anticlockwise rotation given by right hand rule for their propagation in time. The intensity can be written as

(a) front view (b) angle view

-1 -0.5 0

0.5 1

0.5

0

-0.5

1 -1

E1

E2

-1 1 -0.5 0

0.5 0.5

0 1

-0.5 10 15

-1 0 5

E1

E2

t[fs]

Figure 2.4: Superposition of the electric components of two waves as a function of time. The blue and red parts of the plot represent the clockwise and anticlockwise rotation of the vector, respectively.

I(r, t) =E^∗(r, t)·E(r, t),

={A^∗₁+A^∗₂exp[−i(∆k·r−∆ωt)]} · {A₁+A₂exp[i(∆k·r−∆ωt)]},

=A^∗₁·A₁ +A^∗₁·A₂exp[i(∆k·r−∆ωt)]

+A^∗₂·A₁exp[−i(∆k·r−∆ωt)] +A^∗₂·A₂,

(2.16) where the asterisk denotes the complex conjugate. By replacingA₁andA₂according to

A₁ =p I1ˆe₁, A₂ =p

I2ˆe₂, (2.17)

whereˆe₁ andˆe₂ are unit vectors representing the respective directions of the amplitudes, and I₁ and I₂ represent the intensities of fields E₁ and E₂, for which

ˆ

e^∗₁·ê₂ =|ê^∗₁·ê₂|exp(iφ), (2.18) where φ is the phase between them, we get

I(r,t) =I1+I2+p

I1I2{ˆe^∗₁·ˆe₂exp[i(∆k·r−∆ωt)]

+ˆe₁·ˆe^∗₂exp[−i(∆k·r−∆ωt)]}

=I1+I2+ 2p

I1I2|ˆe₁ ·ˆe₂|cos (φ+ ∆k·r−∆ωt).

(2.19)

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If we consider the intensity at a fixed point in space, it is clear from Eq. (2.19) that it is a sinusoidally varying function ofr andt. Similarly as in the scalar case, ∆ω is the difference between the frequencies of the two fields, but here, when ∆ω = 0 and we observe the field at a fixed point in space, the only phase dependence is viaφ, the phase between the two amplitudes. Equation (2.19) shows that the intensity depends also on the polarization states of the fields. In the case ofˆe₁ ⊥ˆe₂ the interference term vanishes, leading to a constant intensity equal to the sum of intensities of the two fields, I1 and I2.

The Stokes parameters for monochromatic plane waves represent one convenient way of describing the polarization of a field, and they are measurable quantities [35].

The Stokes parameters can be written as

S₀ =|A_x|²+|A_y|², S1 =|Ax|²− |Ay|², S₂ = 2|A_xA_y|cosϑ, S3 = 2|AxAy|sinϑ,

(2.20)

whereAx and Ay are the Cartesian xand y components of A, which represents the amplitude of the total field and ϑ is the phase difference between the components.

The first Stokes parameter gives the intensity of the field and the other three give the state of polarization. They correspond to the intensity differences between the linearly polarized light along thexandyaxes (Ix−Iy), linearly polarized light along a direction making a 45^◦ angle with x-axis and the direction making a −45^◦ angle with it (I45^◦−I−45^◦), and right hand and left hand polarized light (IRCP−ILCP).

The Stokes parameters obey

S₀² =S₁²+S₂²+S₃². (2.21) It is also possible to write [37]

S1 =S0cos 2χcos 2ψ, S₂ =S₀cos 2χsin 2ψ, S3 =S0sin 2χ,

(2.22)

whereψ (0≤ψ < π) gives the orientation of the polarization ellipse andχ(−π/4≤ χ≤π/4) defines the ellipticity.

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Chapter III

Concepts of coherence theory of scalar fields

In this Chapter, the main concepts of coherence theory for the scalar fields are introduced. We start by recalling random light, quasi-monochromatic light, and the field envelope representation. Then, we introduce the statistical properties of scalar fields, such as coherence functions and the degree of coherence, and finish with the reduction to stationary fields and how it affects the aforementioned concepts.

3.1 Random light

We usually assume the light to be deterministic, although natural light is always random. Natural light is a superposition of emissions from a large number of atoms, which radiate independently of each other, having different frequencies and phases [36]. The spontaneous emissions occurring in a source, contribute to the level of randomness of its radiation. However, different sources have different levels of disorder in their emissions, resulting in emitted light, which is more random in some cases and less in others [38, 39]. But, the randomness of light can also be altered by phenomena other than the level of disorder in their emission, phenomena which introduce random variations to its optical wavefront [36].

In order to study random light without making any simplifications, we use statistical approach, in which case we consider it a stochastic process (depends on time in a non-deterministic way) [40]. To keep the discussion closer to reality we do not assume stationarity in this part of the discussion. In Fig. 3.1 is shown how a non- stationary field varies over time. Three realizations of the field are given. It is clear that the averages of its properties change as a function of time.

This is different, however, in the case of stationary fields. While not very realistic,

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0 1 2 3 4 5 0

0 0

t[ps]

I[A.U]

Figure 3.1: Three realizations of a non-stationary field. Their intensity as a function of time is given in arbitrary units.

stationarity is a very useful simplification and is usually correct for short periods of time. Stationary process approximation in coherence theory was first introduced by Zernike [14]. It assumes the average characteristics of the process to be constant in time. This will be useful later when we reduce our discussion, for simplicity, to such fields.

It is convenient to use the complex representation to describe the field. In the case of partially coherent, randomly fluctuating fields, complex analytic signal [41] is used. We use Fourier transform formalism to model the non-stationary time-varying fields [42]. So, for any real field Ur(r, t) we can write the complex analytic signal such that

Ur(r, t) =ℜ{U(r, t)}= 1

2[U(r, t) +U^∗(r, t)], (3.1) whereℜ denotes the real part and the asterisk denotes the complex conjugate. The real (square integrable) field can be written as the temporal Fourier integral

Ur(r, t) = Z ∞

−∞

Uer(r, ω) exp (−iωt)dω, (3.2) which describes the field as superposition of the monochromatic components with angular frequency ω. The information about the amplitudes and phases of the harmonic field components is given by the weighting coefficients determined by the

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Fourier transform

Uer(r, ω) = 1 2π

Z ∞

−∞

Ur(r, t) exp (iωt)dt, (3.3) which, because of Ur(r, t) being real, satisfies the condition

Ue_r(r,−ω) = [Ue_r(r, ω)]^∗. (3.4) This indicates that the negative frequency components contain no information that is not already contained in the positive frequency components. Therefore, we can define the complex analytic signal as

Ue(r, ω) =





2Uer(r, ω) ifω ≥0

0 if ω <0, (3.5)

and write

U(r, t) = Z ∞

0

Ue(r, ω) exp (−iωt)dω, (3.6) with

Ue(r, ω) = 1 2π

Z ∞

−∞

Ur(r, t) exp (iωt)dt. (3.7) In this form we can include all the information about the real field into a simple complex analytic version which contains only the positive frequencies.

We have, thus, described the random, partially coherent light with equations which can simplify our calculations, without oversimplifying the problem and risk of loosing information in the process. Next, we explore the case of a signal with narrow power spectrum.

3.2 Quasi-monochromatic light and its envelope representation

Real sources, apart from having randomly fluctuating phase and amplitude, have also a finite emitting surface. Furthermore, they can never produce monochromatic light. If the power spectrum of the light is so narrow that

∆Ω ≪Ω, (3.8)

where ∆Ω is the bandwidth and Ω represents the center frequency of the spectrum, it can be considered quasi-monochromatic [39, 43].

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The quasi-monochromatic light is illustrated in Fig. 3.2. Quasi-monochromatic light has a range of frequencies distributed around a center frequency. This center frequency is used in calculations in a way that is analogous to the frequency of monochromatic beams.

Ω

∆Ω

Figure 3.2: A quasi-monochromatic light spectrum with the center frequency Ω and bandwidth ∆Ω. The bandwidth is chosen to be much smaller than the center frequency satisfying the condition given in Eq. 3.8.

For a quasi-monochromatic light, in terms of amplitude A(r, t) and phase ϕ(t), we can write

U_R(r, t) =A(r, t) cos[ϕ(t)−Ωt]. (3.9) Both A(r, t) and ϕ(t) are slowly varying real functions. By choosing a suitable imaginary part [44] it is possible to write

U(r, t) =A(r, t) exp[iϕ(t)] exp(−iΩt). (3.10) The complex analytic signal is closely related to the envelope representation as shown by Mandel [44]. We can define the complex envelope as

A(r, t)≡A(r, t) exp[iϕ(t)]. (3.11)

It is sometimes referred to as the time-varying phasor amplitude [39]. We can now rewrite the complex analytic signal in the form

U(r, t) =A(r, t) exp(−iΩt). (3.12)

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Figure 3.3 illustrates how the envelope is related to the field amplitude. By in- cluding the phase in the complex envelope we simplify the representation of the quasi-monochromatic light in a way that the resulting representation is analogous to the case of monochromatic light. Throughout this work, we use the complex envelope when we discuss the behaviour of the quasi-monochromatic beams.

0 20 40 60 80 100

-1 -0.5 0 0.5 1

ℜ[U]/|A|

t [fs]

Figure 3.3: The complex envelope (blue) of a field (black).

3.3 Mutual coherence function

A field is considered coherent if its fluctuations remain proportional to each other during its propagation in space (spatial coherence) or in time (temporal coherence).

Figure 3.4 shows examples of highly incoherent, partially coherent, and completely coherent fields. The incoherent light in the upper part of the figure has very little correlation between different points in time. Such light is usually produced by broad spectrum sources such as thermal lamps or LEDs. The partially coherent light represented in the middle of the figure has some correlation between different points in time. This kind of light is the most realistic since most light sources exhibit some degree of coherence. The lower part shows the idealized case of completely temporally coherent light. While such light is difficult to produce, some lasers, such as single-mode lasers, come very close.

The correlations between two fields in space-time domain can be thought of as a generalization and connection of spatial and temporal coherence. The correlation

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0 5 10 15 20 25 30 0

0 0

U

t [fs]

Figure 3.4: Illustrations of incoherent (black), partially coherent (blue), and fully coherent (red) fields.

function of nonstationary partially coherent fields

Γ(r1,r₂, t1, t2) =hU^∗(r1, t1)U(r2, t2)i, (3.13) can be used to describe the correlations between two points at two different time instants, where the first point would be considered at t1 and the second at t2. The angle brackets in Eq. (3.13) denote the ensemble average, which for a random process f(t) is defined as

hf(t)i ≡ lim

N→∞

1 N

XN n=1

fn(t) (3.14)

where fn(t) is one realization. The correlation function in optics was first introduced by Wolf [17] for stationary fields and named mutual coherence function. It is Hermitian in the sense that

Γ^∗(r1,r₂, t1, t2) = Γ(r2,r₁, t2, t1), (3.15) and the non-negative definiteness conditions apply [40]. In the special case in which r₁ =r₂ =r and t1 =t2 =t we see that

I(r, t) = Γ(r,r, t, t) = h|U(r, t)|²i ≥0, (3.16)

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which gives us the intensity. The correlations between two waves are given by Γ^(pq)(r1,r₂, t1, t2) = hU_p^∗(r1, t1)Uq(r2, t2)i, (3.17) where p, q ∈ (1,2) denote the different waves, with p 6= q. In the same way as Eq. (3.13), this function is Hermitian and non-negative definite.

3.4 Degree of coherence

By normalizing the cross-correlation function we get the complex degree of coherence between the two scalar waves

γ(r1,r₂, t1, t2) = Γ(r1,r₂, t1, t2)

pI(r₁, t1)I(r₂, t2). (3.18) It has values

0≤ |γ(r1,r₂, t1, t2)| ≤1, (3.19) for every combination of arguments.

The physical meaning of this normalized function is very special. We can discuss it by splitting the inequality in three parts. For |γ| = 0 the fields at r₁ and r₂ at t1 and t2 are not correlated and the field is completely incoherent. The upper limit |γ|= 1, is achieved in the case of complete coherence. The values in between, 0<|γ|<1, correspond to partially coherent fields.

3.5 Reduction to stationary field

Stationary processes make up a subset of stochastic processes. However, approximat- ing the fields as stationary processes usually simplifies the calculations and therefore they are widely used in coherence theory. A stationary field is independent of time.

It depends instead on the time-difference between two points which does not change.

So, it is not important at which time instant the measurement is made, as the coherence functions will remain constant for a given time delay. We can denote this time delay byτ =t2−t1. A stationary wave is represented in Fig. 3.5. The random character of the wave in the figure does not vary with time and therefore correlations between two points in time separated by an arbitrary time difference τ will always be the same.

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0 5 10 15 20 25 30 0

0 0

I[A.U.]

t [fs]

Figure 3.5: Illustration of three realizations of a stationary field.

For an ergodic process, all time averages of different realizations, as well as their functions, are identical and equal to ensemble averages [39]. Defining the time average of a random time-dependent processf(t) as

hf(t)i^t≡ lim

T→∞

1 2T

Z T

−T

f(t)dt, (3.20)

we can replace the ensemble cross-correlation function by the corresponding temporal correlation function [40] and set

Γ(r1,r₂, τ) =hU^∗(r1, t)U(r2, t+τ)i^t= lim

T→∞

1 2T

Z T

−T

U^∗(r1, t)U(r2, t+τ)dt. (3.21) Analogously to Eq. (3.16) the intensity can be written as

I(r) = Γ(r,r,0), (3.22)

and the complex degree of coherence transforms to γ(r₁,r₂, τ) = Γ(r1,r₂, τ)

pΓ(r1,r₁,0)Γ(r2,r₂,0) = Γ(r1,r₂, τ)

pI1(r1)I2(r2). (3.23) The assumption of stationarity simplifies the calculations but it is not always ap- plicable and therefore a general description for non-stationary light is necessary as well.

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Chapter IV

Basics of electromagnetic coherence theory of beams

In this Chapter we go through the main concepts of the electromagnetic coherence theory. In the following discussion we consider the fields beam-like, with two transverse electric components. The concepts explained in this chapter are later used to derive the properties of beating for quasi-monochromatic electromagnetic fields. We start by introducing the coherence and polarization matrices, and then we introduce the concepts of the electromagnetic degree of coherence and degree of polarization.

In the end, the Stokes parameters and the two-time Stokes parameters are given.

4.1 Coherence matrices

In most of the cases, when dealing with the description of various properties of fields or their interactions with different structures, we cannot rely anymore on the scalar approach. Electromagnetic approach, on the other hand, gives more information because it includes all the components of the field. The broadest description of the field would be done using the 3×3 correlation matrices which are associated with a set of second-rank correlation tensors, describing every field component (E, H, D, B) and their correlations [40, 45]. However, when dealing with partially polarized and partially coherent beams the analysis can be carried on using the 2×2 correlation matrices [46]. We do not consider the magnetic component of the field because it is not of interest.

For a single non-stationary beam with the above given description we can write

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the mutual coherence matrix

Γ(r1,r₂, t1, t2) =

"

Γxx Γxy

Γ_yx Γ_yy

#

, (4.1)

such that its elements are given by

Γij(r1,r₂, t1, t2) = hE_i^∗(r1, t1)Ej(r2, t2)i, (4.2) where Ei(r1, t1) and Ej(r2, t2), with i, j ∈ (x, y), are the complex analytic signals (Sec. 3.2) of the components of the field at time instantst1 andt2, respectively. The coherence matrix is Hermitian so that

Γ^†(r1,r₂, t1, t2) =Γ(r2,r₁, t2, t1), (4.3) where the dagger is used to denote the conjugate transpose. Same as in the case of scalar correlation function, also the coherence matrix is non-negative definite [40]

XN p=1

XN q=1

a^†(rp, tp)Γ(rp,r_q, tp, tq)a(rq, tq)≥0, (4.4) where a(r, t) is an arbitrary complex vector.

Let us now define the mutual coherence matrix of two beams as Γ⁽¹²⁾(r₁,r₂, t₁, t₂) =

"

Γ⁽¹²⁾xx Γ⁽¹²⁾xy

Γ⁽¹²⁾yx Γ⁽¹²⁾yy

#

, (4.5)

whose elements are given by

Γ⁽¹²⁾_ij (r₁,r₂, t1, t2) =hE_1i^∗(r₁, t1)E2j(r₂, t2)i, (4.6) where E_1i(r₁, t₁) and E_2j(r₂, t₂), with i, j ∈ (x, y), are the complex analytic signals of the optical fields at the pointsr₁ andr₂at the time instantst₁ andt₂, respectively.

The matrix is Hermitian

Γ^(12)†(r₁,r₂, t₁, t₂) =Γ⁽²¹⁾(r₂,r₁, t₂, t₁), (4.7) and non-negative definite

XN p=1

XN q=1

a^†(r_p, tp)Γ⁽¹²⁾(r_p,r_q, tp, tq)a(r_p, tq)≥0. (4.8) It is clear that Eqs. (4.2) and (4.6) are mathematically very similar and have the same properties. Physically, though, they are different. While the first one gives us the correlation for two different positions and time instants for one field only, in the latter we consider correlation between two separate fields.

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4.2 Polarization matrix

In order to describe the polarization properties of a beam, we use the polarization matrix [46] (also called equal-time coherence matrix [40] or coherence matrix [35,40, 46])

J(r, t) =

"

Jxx Jxy

J_yx J_yy

#

, (4.9)

where the elements are

J_ij(r, t) =hE_i^∗(r, t)E_j(r, t)i, (4.10) with i, j ∈(x, y). The polarization matrix is purely Hermitian and

J_xy^∗ (r, t) =Jyx(r, t), (4.11) holds. It is also non-negative definite [47] such that

a^†(r, t)J(r, t)a(r, t)≥0. (4.12) The polarization matrix can also be written as

J(r, t) =Γ(r,r, t, t), (4.13) which describes the coherence properties of the beam in a single spatiotemporal point, hence the above mentioned names.

The diagonal elements add up to represent (disregarding a proportionality factor) the intensity

I(r, t) = Jxx(r, t) +Jyy(r, t). (4.14) Furthermore, each one of them represents the intensity of the beam with only one component present. The off-diagonal elements give the correlation between the orthogonal components at a single point.

4.3 Degree of polarization

A way to get information about whether a beam is polarized or not, without caring about its state of polarization is to study the degree of polarization, which is used to describe polarization quantitatively. A quasi-monochromatic partially polarized

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beam can be considered a sum of a fully polarized and a fully unpolarized beam.

This can be presented in terms of the polarization matrix [35]

J(r, t) =J^(pol)(r, t) +J^(unpol)(r, t), (4.15) where the polarized part can be represented by

J^(pol)(r, t) =

"

B(r, t) D(r, t) D^∗(r, t) C(r, t)

#

, (4.16)

where B(r, t) and C(r, t) are non-negative, and

B(r, t)C(r, t)−D(r, t)D^∗(r, t) = 0. (4.17) The unpolarized part is

J^(unpol)(r, t) = A(r, t)

"

1 0 0 1

#

, (4.18)

where alsoA(r, t)≥0. Using these, we can define the degree of polarization [35, 48]

to be

P(r, t) = trJ^(pol)(r, t) trJ(r, t) =

s

1− 4detJ(r, t) tr²J(r, t) =

s

2trJ²(r, t)

tr²J(r, t)−1. (4.19) The value of the degree of polarization varies between zero for an unpolarized beam and one for a completely polarized beam. If the two orthogonal components of the beam are fully correlated, the beam is fully polarized. For the unpolarized beam, on the other hand, the orthogonal components are uncorrelated and have the same intensity.

4.4 Degree of coherence

A form of calculating the degree of coherence for electromagnetic beams, based on the scalar case and valid for paraxial stationary fields, was proposed by Kar- czewski [49]

ζ(r1,r₂, τ) = h[E(r1, t+τ)·E^∗(r2, t)]i ph|E(r1, t)|²ip

h|E(r2, t)|²i. (4.20)

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ζ contains information about the visibility of the interference fringes, like in the scalar case.

However, this quantity was shown to fail to correctly describe the spacial coherence properties of the field and therefore a different method, which properly takes into consideration the vectorial nature of light, was introduced [50]

γ_E²(r1,r₂, τ) = tr[Γ(r1,r₂, τ)Γ(r1,r₂,−τ)]

hI(r₁, t)ihI(r₂, t)i . (4.21) For the non-stationary fields, this has the form

γ_E²(r₁,r₂, t₁, t₂) = tr[Γ(r₁,r₂, t₁, t₂)Γ^†(r₁,r₂, t₁, t₂)]

trΓ(r1,r₁, t1, t1)trΓ(r2,r₂, t2, t2), (4.22) and is related to the two-dimensional degree of polarization as follows

P²(r, t) = 2

γ_E²(r,r, t, t)−1 2

. (4.23)

4.5 Stokes parameters

One of the most usual descriptions of the polarization of the field is done by using the Stokes parameters introduced by Stokes as early as 1852 [4]. A unique set of Stokes parameters was introduced by Wolf [51] using the complex analytic signal representation. In terms of the elements of the polarization (coherency) matrix for non-stationary light in space-time domain, the parameters have the forms

S₀(r, t) =J_xx(r, t) +J_yy(r, t), S1(r, t) =Jxx(r, t)−Jyy(r, t), S₂(r, t) =J_xy(r, t) +J_yx(r, t), S3(r, t) =i[Jyx(r, t)−Jxy(r, t)].

(4.24)

We can also represent the polarization matrix in terms of Stokes parameters [40]

Jxx(r, t) = 1

2[S0(r, t) +S1(r, t)], Jyy(r, t) = 1

2[S0(r, t)−S1(r, t)], Jxy(r, t) = 1

2[S2(r, t) +iS3(r, t)], Jyx(r, t) = 1

2[S2(r, t)−iS3(r, t)].

(4.25)

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In a more compact form we can write for the polarization (coherency) matrix [51]

J(r, t) = 1 2

X3 p=0

S_p(r, t)σ_p, (4.26)

and for the Stokes parameters [40]

Sp(r, t) = tr[J(r, t)σp], (4.27) where p∈(0, . . . ,3). In both cases we make use of the unit matrix

σ₀ =

"

1 0 0 1

#

, (4.28)

and the Pauli spin matrixes σ₁ =

"

1 0

0 −1

#

, σ₂ =

"

0 1 1 0

#

, σ₃ =

"

0 −i

i 0

#

. (4.29)

The Stokes parameters define the polarization state based on the difference of the intensities of orthogonal states, described by the components of the polarization matrix. To eliminate this dependence and to get information on the polarization state without depending on the intensity of the beam, we use the normalized Stokes parameters [40].

sn(r, t) = Sn(r, t)

S0(r, t), (4.30)

where n∈(1,2,3).

The degree of polarization can be used to represent the polarization of the beam graphically in the so-called Poincar´e sphere, where it represents the radius length while the coordinates are given by the normalized Stokes parameters. We can write in terms of normalized Stokes parameters

P²(r, t) = X3 n=1

s²_n(r, t), (4.31)

and for the sum of squared Stokes parameters X3

n=1

S_n²(r, t) =S₀²(r, t)P²(r, t). (4.32)

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Figure 4.1: The Poincar´e sphere with the normalized Stokes parameters for axes. The degree of polarization is given by the length of the vector P(r, t).

The Poincar´e vector is of the form

P(r, t) =s1(r, t)û₁ +s2(r, t)û₂+s3(r, t)û₃, (4.33) whereû₁,uˆ₂anduˆ₃are the unit vectors of the axes. In Fig. 4.1 is shown the Poincaré sphere. Its axes are the normalized Stokes parameters. The Poincaré vector gives the degree of polarization in each state. The sphere is centered at zero, with its surface corresponding to fully polarized light.

4.6 Two-point Stokes parameters

In order to describe also the properties of the correlations between two electromagnetic fields, the Stokes parameters have been generalized to the so called two-point Stokes parameters. They were first introduced for the space-time domain by El- lis and Dogariu [52]. In a more compact format, Korotkova and Wolf introduced them in the space-frequency domain [53]. In order to draw analogy with the original Stokes parameters and discuss their physical properties, in a compact form for the

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space-time domain we formulate the Stokes parameters for the interaction between two beams as

S₀⁽ⁿ⁾(r₁,r₂, t₁, t₂) = Γ⁽ⁿ⁾_xx(r₁,r₂, t₁, t₂) + Γ⁽ⁿ⁾_yy(r₁,r₂, t₁, t₂), S₁⁽ⁿ⁾(r1,r₂, t1, t2) = Γ⁽ⁿ⁾_xx(r1,r₂, t1, t2)−Γ⁽ⁿ⁾_yy (r1,r₂, t1, t2), S₂⁽ⁿ⁾(r₁,r₂, t₁, t₂) = Γ⁽ⁿ⁾_xy(r₁,r₂, t₁, t₂) + Γ⁽ⁿ⁾_yx(r₁,r₂, t₁, t₂), S₃⁽ⁿ⁾(r1,r₂, t1, t2) =i[Γ⁽ⁿ⁾_yx(r1,r₂, t1, t2)−Γ⁽ⁿ⁾_xy(r1,r₂, t1, t2),

(4.34)

wheren ∈(12,21). If we set r₁ =r₂ =rand disregard the position dependence, we can reduce them to ”two-time Stokes parameters”.

S₀⁽ⁿ⁾(t1, t2) = Γ⁽ⁿ⁾_xx(t1, t2) + Γ⁽ⁿ⁾_yy (t1, t2), S₁⁽ⁿ⁾(t1, t2) = Γ⁽ⁿ⁾_xx(t1, t2)−Γ⁽ⁿ⁾_yy (t1, t2), S₂⁽ⁿ⁾(t1, t2) = Γ⁽ⁿ⁾_xy(t1, t2) + Γ⁽ⁿ⁾_yx(t1, t2), S₃⁽ⁿ⁾(t1, t2) = i[Γ⁽ⁿ⁾_yx(t1, t2)−Γ⁽ⁿ⁾_xy (t1, t2)].

(4.35)

We can also express the mutual coherence matrix in analogy to Eq. (4.26) Γ(t1, t2) = 1

2 X3 p=0

Sp(t1, t2)σp, (4.36) and the Stokes parameters in analogy to Eq. (4.27) as

Sp(r, t) = tr[J(r, t)σp], (4.37) where p∈(0, . . . ,3) and σ_p are given by the matrices in Eqs. (4.28) and (4.29).

The normalized two-time Stokes parameters are of the form [53]

ς_p⁽¹²⁾(t1, t2) = Sp⁽¹²⁾(t1, t2) q

S₀⁽¹⁾(t₁)S₀⁽²⁾(t₂)

, (4.38)

where p∈(1,2,3).

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Chapter V

Beating with scalar quasi-monochromatic light

In this Chapter are presented and discussed the findings on the beating of quasi- monochromatic scalar light. The total field and the intensity are presented first.

After showing the correlation between the fields, the intensity modulation is shown again in another formulation.

5.1 The total field and the intensity

We can now move on to the beating of two quasi-monochromatic random stationary beams with center frequencies ω andω+ ∆ωand corresponding wave vectors kand k+∆k, taking place at timet. We assume the frequencies to be such that ∆ω≪ω and the wave vectors to obey |∆k| ≪ |k|. We can write the fields as

U1(r, t) =A1(t) exp[i(k·r−ωt)], (5.1) U₂(r, t) =A₂(t) exp{i[(k+∆k)·r−(ω+ ∆ω)t]}, (5.2) where A₁(t) and A₂(t) are the random complex envelopes. They are slowly varying functions in the sense that their Fourier transforms are narrow functions of frequency around 0. The two quasi-monochromatic peaks between which beating can occur are illustrated in Fig. 5.1. In both these peaks, the bandwidth is small, surrounding a central frequency, just like it was described for Fig. 3.2. The central frequencies denoted as Ω in Fig. 3.2 are different for the two peaks and therefore we will consider them to be Ω₁ = ω for the first peak and Ω₂ = ω+ ∆ω for the second. We will denote the spectral width of the beams as ∆Ω₁ and ∆Ω₂.

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∆Ω1

∆Ω2

ω ω+ ∆ω

Figure 5.1: Spectra of two quasi-monochromatic fields between which beating can take place. The peak frequencies of the fields are ω and ω+ ∆ω and the widths of the spectra are ∆Ω₁ and ∆Ω₂, respectively.

The superposition of the beams can be written as U(r, t) =U1(r, t) +U2(r, t)

= exp[i(k·r−ωt)]{A1(t) +A2(t) exp[i(∆k·r−∆ωt)]}, (5.3) Figure 5.2 shows the total field due to interference of two quasi-monochromatic fields given in Eq. 5.3 at a fixed point in spacer= 0. For simplification we have also considered A1 = A2. By comparing Fig. 5.2 to Fig. 2.1 we can see the differences between the behaviour of ideal monochromatic field and the more realistic quasi- monochromatic one. While the superposition of monochromatic fields is a simple repeatable pattern to infinity, the inclusion of the finite band of frequencies creates a pattern that changes over time.

The corresponding intensity is

I(r, t) =h|U(r, t)|²i=hU^∗(r, t)U(r, t)i

=h|A1(t)|²i+hA^∗₁(t)A2(t)iexp[i(∆k·r−∆ωt)]

+hA1(t)A^∗₂(t)iexp[−i(∆k·r−∆ωt)]i+h|A2(t)|²i.

(5.4)

The two fields interfere as they propagate in time, creating a temporal pattern that mirrors the spatial interference pattern. It can be seen from Eq. (5.4) that the

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(a) (b)

0 1 2 3 4 5 6 7 8

-1 -0.5 0 0.5 1

ℜ[U]/2|A|

t[ps]

0 1 2 3 4 5 6 7 8

-1 -0.5 0 0.5 1

ℜ[U]/2|A|

t[ps]

(c) (d)

0 1 2 3 4 5 6 7 8

-1 -0.5 0 0.5 1

ℜ[U]/2|A|

t[ps]

0 1 2 3 4 5 6 7 8

-1 -0.5 0 0.5 1

ℜ[U]/2|A|

t[ps]

Figure 5.2: Four realizations of the superposition of two quasi- monochromatic beams with center frequencies ω = 28,90×10¹⁴ rad/s and ω+ ∆ω= 29,78×10¹⁴ rad/s.

intensity maxima and minima depend on the amplitudes of the beams as well as on the correlations hA^∗₁A2i and hA1A^∗₂i between the amplitudes of the two beams.

For the four realizations of the superposition presented in Fig. 5.2 the intensity as given in Eq. 5.4 is shown in Fig. 5.3. As in the case of the superposition field, we can see how the resulting intensity changes if we include the spectra of the two quasi-monochromatic fields instead of simplifying it by considering the sources to be monochromatic like we did in Fig. 2.2.

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0 1 2 3 4 5 6 7 8 0

0.2 0.4 0.6 0.8 1

t[ps]

I/4|A|2

Figure 5.3: Intensity modulation during beating of the superposition of two quasi-monochromatic beams. This intensity corresponds to the four superposition realizations shown in Fig. 5.2.

5.2 The correlations between beams

The correlation functions for the envelopes can be written Γ⁽¹⁾_A (t₁, t₂) =hA^∗₁(t₁)A₁(t₂)i, Γ⁽²⁾_A (t1, t2) =hA^∗₂(t1)A2(t2)i, Γ⁽¹²⁾_A (t₁, t₂) =hA^∗₁(t₁)A₂(t₂)i, Γ⁽²¹⁾_A (t1, t2) =hA^∗₂(t2)A1(t1)i.

(5.5)

The first two of Eqs. (5.5) are the autocorrelation functions for the amplitude envelopes of the beams 1 and 2, respectively, at two different time instants t₁ and t₂, while the second two are the cross-correlation functions between the amplitudes of the beams. The autocorrelation and the cross-correlation functions are Hermitian in the sense that

Γ^(1)∗_A (t1, t2) = Γ⁽¹⁾_A (t2, t1), Γ^(2)∗_A (t1, t2) = Γ⁽²⁾_A (t2, t1), Γ^(12)∗_A (t1, t2) = Γ⁽²¹⁾_A (t2, t1).

(5.6)

The correlation functions for the beams at two space-time points can be written

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as

Γ⁽¹⁾_U (r₁,r₂, t₁, t₂) =hU₁^∗(r₁, t₁)U₁(r₂, t₂)i

= Γ⁽¹⁾_A (t1, t2) exp[i(k·∆r−ω∆t)], Γ⁽²⁾_U (r₁,r₂, t₁, t₂) =hU₂^∗(r₁, t₁)U₂(r₂, t₂)i

= Γ⁽²⁾_A (t1, t2) exp{i[(k+ ∆k)·∆r−(ω+ ∆ω)∆t]}, Γ⁽¹²⁾_U (r₁,r₂, t₁, t₂) =hU₁^∗(r₁, t₁)U₂(r₂, t₂)i

= Γ⁽¹²⁾_A (t1, t2) exp[i(k·∆r+ ∆k·r₂−ω∆t−∆ωt2)], Γ⁽²¹⁾_U (r₁,r₂, t₁, t₂) =hU₂^∗(r₁, t₁)U₁(r₂, t₂)i

= Γ⁽²¹⁾_A (t2, t1) exp[−i(k·∆r+ ∆k·r₂−ω∆t−∆ωt2)],

(5.7)

where ∆r=r₂−r₁ and ∆t=t₂−t₁. Again, the first two are autocorrelation functions, while the last two are cross-correlation functions. In all the above equations, the statistical information is contained in the functions corresponding to the amplitudes, while the exponential term introduces a deterministic space-time modulation.

If we consider the beating at a particular point in space r at a fixed time t, the correlation functions take the form

Γ⁽¹⁾_U (r,r, t, t) = Γ⁽¹⁾_A (t, t), Γ⁽²⁾_U (r,r, t, t) = Γ⁽²⁾_A (t, t),

Γ⁽¹²⁾_U (r,r, t, t) = Γ⁽¹²⁾_A (t, t) exp[i(∆k·r−∆ωt)], Γ⁽²¹⁾_U (r,r, t, t) = Γ⁽²¹⁾_A (t, t) exp[−i(∆k·r−∆ωt)].

(5.8)

It is clear that the first two equations lead to the expectation values for the instan- taneous intensities

I⁽¹⁾(t) =h|U1(t)|²i= Γ⁽¹⁾_U (r,r, t, t) = Γ⁽¹⁾_A (t, t),

I⁽²⁾(t) =h|U2(t)|²i= Γ⁽²⁾_U (r,r, t, t) = Γ⁽²⁾_A (t, t), (5.9) respectively, for each of the beams.

Polarization beating with random electromagnetic beams