Tài liệu Nguyên tắc cơ bản của lượng tử ánh sáng P20 pptx

Consider an acoustic plane wave traveling in the x direction in a medium with velocity v,, frequency f, and wavelength A = us/f.. Consider now an optical plane wave traveling in this med

CHAPTER

20 ACOUSTO-OPTICS

20.1 INTERACTION OF LIGHT AND SOUND

D Filters, Frequency Shifters, and Isolators

*20.3 ACOUSTO-OPTICS OF ANISOTROPIC MEDIA

799

Fundamentals of Photonics

Bahaa E A Saleh, Malvin Carl Teich

Copyright © 1991 John Wiley & Sons, Inc

ISBNs: 0-471-83965-5 (Hardback); 0-471-2-1374-8 (Electronic)

therefore modifies the effect of the medium on light; i.e., sound can contro2 light (Fig 20.0-l) Many useful devices make use of this acousto-optic effect; these include optical modulators, switches, deflectors, filters, isolators, frequency shifters, and spectrum analyzers

Sound is a dynamic strain involving molecular vibrations that take the form of waves which travel at a velocity characteristic of the medium (the velocity of sound) As an example, a harmonic plane wave of compressions and rarefactions in a gas is pictured

in Fig 20.0-2 In those regions where the medium is compressed, the density is higher and the refractive index is larger; where the medium is rarefied, its density and refractive index are smaller In solids, sound involves vibrations of the molecules about their equilibrium positions, which alter the optical polarizability and consequently the refractive index

An acoustic wave creates a perturbation of the refractive index in the form of a wave The medium becomes a dynamic graded-index medium-an inhomogeneous medium with a time-varying stratified refractive index The theory of acousto-optics deals with the perturbation of the refractive index caused by sound, and with the propagation of light through this perturbed time-varying inhomogeneous medium The propagation of light in static (as opposed to time-varying) inhomogeneous (graded-index) media was discussed at several points in Chaps 1 and 2 (Sets 1.3 and 2.4C) Since optical frequencies are much greater than acoustic frequencies, the variations of the refractive index in a medium perturbed by sound are usually very slow

in comparison with an optical period There are therefore two significantly different time scales for light and sound As a consequence, it is possible to use an adiabatic approach in which the optical propagation problem is solved separately at every instant

of time during the relatively slow course of the acoustic cycle, always treating the material as if it were a static (frozen) inhomogeneous medium In this quasi-stationary approximation, acousto-optics becomes the optics of an inhomogeneous medium (usu- ally periodic) that is controlled by sound

Sound

Figure 20.0-I Sound modifies the effect of an optical medium on light

(20.04) Bragg Condition

where A is the wavelength of light in the medium (see Exercise 2.53) This form of light-sound interaction is known as Bragg diffraction, Bragg reflection, or Bragg scattering The device that effects it is known as a Bragg reflector, a Bragg deflector, or

Bragg cells have found numerous applications in photonics This chapter is devoted

to their properties In Sec 20.1, a simple theory of the optics of Bragg reflectors is presented for linear, nondispersive media Anisotropic properties of the medium and the polarized nature of light and sound are ignored Although the theory is based on wave optics, a simple quantum interpretation of the results is provided In Sec 20.2, the use of Bragg cells for light modulation and scanning is discussed Section 20.3 provides a brief introduction to anisotropic and polarization effects in acousto-optics

The effect of a scalar acoustic wave on a scalar optical wave is described in this section

We first consider optical and acoustic plane waves, and subsequently examine the interaction of optical and acoustic beams

Consider an acoustic plane wave traveling in the x direction in a medium with velocity v,, frequency f, and wavelength A = us/f The strain (relative displacement) at position x and time t is

s(x,t) = S,cos(~t - qx), (20.1-l)

where S, is the amplitude, s1 = 2rf is the angular

wavenumber The acoustic intensity (W/m2) is

frequency, and q = 277/A is the

where Q is the mass density of the medium

The medium is assumed to be optically transparent and the refractive index in the absence of sound is n The strain s(x, t) creates a proportional perturbation of the refractive index, analogous to the Pockels effect in (18.1-4),

INTERACTION OF LIGHT AND SOUND 803

is proportional to the square root of the acoustic intensity,

(20.1-6)

where

p2n6 A=-

ev,"

(20.1-7)

is a material parameter representing the effectiveness of sound in altering the refrac- tive index J&’ is a figure of merit for the strength of the acousto-optic effect in the material

EXAMPLE 20.1-l Figure of Merit In extra-dense flint glass Q = 6.3 x lo3 kg/m3,

u, = 3.1 km/s, 12 = 1.92, p = 0.25, so that L#’ = 1.67 x lo-l4 m2/W An acoustic wave of intensity 10 W/cm2 creates a refractive-index wave of amplitude An,, = 2.89 X 10e5

Consider now an optical plane wave traveling in this medium with frequency u, angular frequency o = 27’rv, free-space wavelength A, = c,/u, wavelength in the unperturbed medium h = h,/n corresponding to a wavenumber k = no/co, and wavevector k lying in the x-z plane and making an angle 0 with the z axis, as illustrated in Fig 20.0-3

Because the acoustic frequency f is typically much smaller than the optical fre- quency v (by at least five orders of magnitude), an adiabatic approach for studying light-sound interaction may be adopted: We regard the refractive index as a static

“frozen” sinusoidal function

n(x) = n - An,,cos(qx - cp), (20.1-8) where cp is a fixed phase; we determine the reflected light from this inhomogeneous (graded-index) medium and track its slow variation with time by taking <p = fit

To determine the amplitude of the reflected wave we divide the medium into incremental planar layers orthogonal to the x axis The incident optical plane wave is partially reflected at each layer because of the refractive-index change We assume that the reflectance is sufficiently small so that the transmitted light from one layer approximately maintains its original magnitude (i.e., is not depleted) as it penetrates through the following layers of the medium

If AY = (d//h) Ax is the incremental complex amplitude reflectance of the layer

at position X, the total complex amplitude reflectance for an overall length L (see Fig 20.1-l) is the sum of all incremental reflectances,

dv Y= ej2kxsinOp *

The phase factor ej2kxsin e is included since the reflected wave at a position x is

T L

-L

Figure 20.1-l Reflections from layers of an inhomogeneous medium

advanced by a distance 2x sin 8 (corresponding to a phase shift 2kx sin 0) relative to the reflected wave at x = 0 (see Fig 20.1-l) The wavenumbers for the incident and reflected waves are taken to be the same, for reasons that will be explained later

An expression for the incremental complex amplitude reflectance AY in terms of the incremental refractive-index change An between two adjacent layers at a given position x may be determined by use of the Fresnel equations (see Sec 6.2) For TE (orthogonal) polarization, (6.2-4) is used with ni = n + An, n2 = n, 8, = 90” - 8, and Snell’s law n, sin 8, = n2 sin 8, is used to determine 0, When terms of second order

in An are neglected, the result is

In most acousto-optic devices 8 is very small, so that cos28 = 1, making (20.1-10) approximately applicable to both polarizations

Using (20.1-8) and (20.1-lo), we obtain

INTERACTION OF LIGHT AND SOUND 805 time, depending on the angle 8 For reasons to become clear shortly, the conditions 2k sin 0 = 4 and 2k sin 8 = -q are called the upshifted and downshifted reflections, respectively We first consider the upshifted condition, 2k sin 0 = q, for which the second term is negligible, and comment on the downshifted case subsequently Per- forming the integral in the first term of (20.1-13) and substituting cp = flit, we obtain

where sine(x) = sin(rx)/(rx) We proceed to discuss several important conclusions based on (20.1-14)

Bragg Condition

The sine function in (20.1-14) has its maximum value of 1.0 when its argument is zero, i.e., when q = 2 k sin 19 This occurs when 8 = 8,, where 8, = sin- ‘(q/2k) is the Bragg angle Since q = 27~/A and k = 27r/h,

(20.04) Bragg Angle

The Bragg angle is the angle for which the incremental reflections from planes separated by an acoustic wavelength A have a phase shift of 27r so that they interfere constructively (see Exercise 2.5-3)

the sound velocity is U, = 3 km/s and the refractive index is n = 1.95 The Bragg angle for reflection of an optical wave of free-space wavelength A, = 633 nm (A = A,/n = 325 nm) from a sound wave of frequency f = 100 MHz (A = us/f = 30 pm) is 0u = 5.4 mrad = 0.31” This angle is internal (i.e., inside the medium) If the cell is placed in air, 8, corresponds to an external angle 0b = n0u = 0.61” A sound wave of 10 times greater frequency (f = 1 GHz) corresponds to a Bragg angle 8, = 3.1”

The Bragg condition can also be stated as a simple relation between the wavevectors

of the sound wave and the two optical waves If q = (q, 0, 0), k = (-k sin 8,0, k cos e), and k, = (k sin 8,0, k cos 0) are the components of the wavevectors of the sound wave, the incident light wave, and the reflected light wave, respectively, the condition

q = 2k sin en is equivalent to the vector relation

illustrated by the vector diagram in Fig 20.1-2

Figure 20.1-2 The Bragg condition sin fIB = q/2k i s equivalent to the vector relation

k, = k + q

Tolerance in the Bragg Condition

The dependence of the complex amplitude reflectance on the angle 0 is governed by the symmetric function sinc[(q - 2k sin 8)L/2~] = sinc[(sin 8 - sin 8,)2L/A] in (20.1-14) This function reaches its peak value when 8 = 8, and drops sharply when 8 differs slightly from en When sin 0 - sin 0, = h/2L the sine function reaches its first zero and the reflectance vanishes (Fig 20.1-3) Because en is usually very small, sin 8 = 8, and the reflectance vanishes at an angular deviation from the Bragg angle of approximately 8 - 8, = h/2L Since L is typically much greater than A, this is an extremely small angular width This sharp reduction of the reflectance for slight deviations from the Bragg angle occurs as a result of the destructive interference between the incremental reflections from the sound wave

Doppler Shift

In accordance with (20.1-141, the complex amplitude reflectance Y is proportional to exp( jnt) Since the angular frequency of the incident light is o [i.e., E a exp(jot)], the reflected wave E, =rE a exp[ j(o + Ln>t] has angular frequency

(20.146)

The process of reflection is therefore accompanied by a frequency shift equal to the

Figure 20.1-3 Dependence of the reflectance [vi2 on the angle 0 Maximum reflection occurs

at the Bragg angle eB = sin-‘(h/2A)

INTERACTION OF LIGHT AND SOUND 807 frequency of the sound This can also be thought of as a Doppler shift (see Exercise 2.6-l and Sec 12.2D) The incident light is reflected from surfaces that move with a velocity u, Its Doppler-shifted angular frequency is therefore w, = ~(1 + 2u, sin e/c>, where u, sin 8 is the component of velocity of these surfaces in the direction of the incident and the reflected waves Using the relations sin 8 = h/2A, v, = Rfi/2~, and

c = ho/27r, (20.1-16) is reproduced The Doppler shift equals the sound frequency Because R K w, the frequencies of the incident and reflected waves are approxi- mately equal (with an error typically smaller than 1 part in 105) The wavelengths of the two waves are therefore also approximately equal In writing (20.1-9) we have implicitly used this assumption by using the same wavenumber k for the two waves, Also, in drawing the vector diagram in Fig 20.1-2 it was assumed that the vectors k, and k have approximately the same length no/c,

Reflectance

The reflectance 2 = 1~1~ is the ratio of the intensity of the reflected optical wave to that of the incident optical wave At the

1/‘12L2/4 Substituting for r’ from (20.1-121,

Bragg angle 6 = 8,, (20.1-14) gives 3 =

and using (20.1-Q, we obtain

(20.147)

rr2 L 2 9=2A2 0 - sin 19 1 /&?I, Reflectance (20.1-18)

The reflectance 9 is therefore proportional to the intensity of the acoustic wave I,, to the material parameter J&+’ defined in (20.1-7) and to the square of the oblique distance L/sin 8 of penetration of light through the acoustic wave

Substituting sin 8 = h/2R into (20.1-181, we obtain

Thus the reflectance is inversely proportional to At (or directly proportional to 04) The dependence of the efficiency of scattering on the fourth power of the optical frequency is typical of light-scattering phenomena

The proportionality between the reflectance and the sound intensity poses a prob- lem As the sound intensity increases, %? would eventually exceed unity, and the reflected light would be more intense than the incident light! This unacceptable result

is a consequence of violating the assumptions of this approximate theory It was assumed that the incremental reflection from each layer is too small to deplete the transmitted wave which reflects from subsequent layers Clearly, this assumption does not hold when the sound wave is intense In reality, a saturation process occurs, ensuring that 9 does not exceed unity A more careful analysis (see Sec 2O.lC), in which depletion of the incident optical wave is included, leads to the following

2

of the Bragg reflector on the intensity of sound 1,

When 1, is small Se = 9, which is a linear function >

expression for the reflectance:

(20.1-19)

where 9 is the approximate expression (20.1-18) and LZ’~ is the exact expression This relation is illustrated in Fig 20.1-4 Evidently, when 9 -C 1, sin fi = g, so that

se =L%?

material parameter M = 1.67 X lo- l4 m2/W (see Example 20.1-l) If A, = 633 nm (wavelength of the He-Ne laser), the sound intensity 1, = 10 W/cm2, and the length of penetration of the light through the sound is L/sin 0 = 1 mm, then ~4’ = 0.0206 and

Se = 0.0205, so that approximately 2% of the light is reflected If the sound intensity is increased to 100 W/cm*, then 9 = 0.206, Se = 0.192 (i.e., the reflectance increases to

= 19%)

Downshifted Bragg Diffraction

Another possible geometry for Bragg diffraction is that for which 2k sin 8 = -4 This

is satisfied when the angle 8 is negative; i.e., the incident optical wave makes an acute angle with the sound wave as illustrated in Fig 20.1-5 In this case, the second term of (20.1-13) has its maximum value, whereas the first term is negligible The complex amplitude reflectance is then given by

Figure 20.1-5 Geometry of

reflected wave is downshifted

downshifted reflection of light from sound The frequency of the

illustrated in Fig 20.1-5 Equation (20.1-22) is a phase-matching condition, ensuring that the reflections of light add in phase The frequency downshift in (20.1-21) is - consistent with the Doppler shift since the light and sound waves travel in the same direction

Interaction of light and sound occurs when a photon combines with a phonon to generate a new photon of the sum energy and momentum An incident photon of frequency o and wavevector k interacts with a phonon of frequency fi and wavevector

q to generate a new photon of frequency o, and wavevector k,, as illustrated in Fig 20.1-6 Conservation of energy and momentum require that Ao, = tie + Afl and

Ak, = Ak + Aq, from which the Doppler shift formula w, = o + fl and the Bragg condition, k, = k + q, are recovered

Bragg Diffraction as a Scattering Process

Light propagation through an inhomogeneous medium with dynamic refractive index perturbation An(x, t) my also be regarded as a light-scattering process and the Born approximation (see Sec 19.1) may be used to describe it A perturbation A9 of the

i

Figure 20.1-6 Bragg diffraction: a photon combines with a phonon to generate a new photon of different frequency and momentum

electric polarization density acts as a source of light

(20.1-23)

[see (5.2-19) and the discussion following (19.1-7)] Since L@ = E~x&? = E,(E/E, - l),%

E (n2 - l&Z’, where Z’ is the electric field, the perturbation An corresponds to Tp”= E, A(n2 - l)Z? = 2~,n A&?‘, so that

(20.1-24)

Thus the source 9 is proportional to the second derivative of the product Ang

To determine the scattered field we solve the wave equation (19.1-6), V2~ - (1/c2)&Y/d2t = -9, t ogether with (20.1-24) and An = -An, cos(ICZt - q * r) The idea of the first Born approximation is to assume that the source 9 is created

by the incident field only and to solve the wave equation for the scattered field Substituting G5’ = Re{A exp[j(ot - k * r)]} into (20.1-24), where A is a slowly varying envelope, we obtain

s;;“= - Aexp[jbf - Qr)]} + kzRe{Aexp[ j(m,t - k;r)]}),

(20.1-25) where cr), = o + CR, k, = k + q, k, = 0,/c; and o, = w - Cl, k, = k - q, k, = 0,/c

We thus have two sources of light of frequencies w I~I LR, and wavevectors k f q, that may emit an upshifted or downshifted Bragg-reflected plane wave Upshifted reflection occurs if the geometry is such that the magnitude of the vector k + q equals 0,./c = w/c, as can easily be seen from the vector diagram in Fig 20.1-2 Downshifted reflection occurs if the vector k - q has magnitude w,/c = o/c, as illustrated in Fig 20.1-5 Obviously, these two conditions may not be met simultaneously

We have thus independently proved the Bragg condition and Doppler-shift formula using a scattering approach Equation (20.1-25) indicates that the intensity of the emitted light is proportional to w: = 04, so that the efficiency of scattering is inversely proportional to the fourth power of the wavelength This analysis can be pursued further to derive an expression for the reflectance by determining the intensity of the wave emitted by the scattering source (see Problem 20.1-2)

Coupled- Wave Equations

To go beyond the first Born approximation, we must include the contribution made by the scattered field to the source 9 Assuming that the geometry is that of up- shifted Bragg diffraction, the field G? is composed of the incident and Bragg-reflected waves: &Y = Re{E exp(jot)} + Re{E, exp(jo,t)} With the help of the relation An = -An, cos(nt - q* r), (20.1-24) gives

Y = Re{ S exp( jwt) + S, exp( jw,t)} + terms of other frequencies,

where

INTERACTION OF LIGHT AND SOUND 811 Comparing terms of equal frequencies on both sides of the wave equation, V*E - (l/c*)J*Z/a*t = -9, we obtain two coupled Helmholtz equations for the incident wave and the Bragg-reflected wave,

(V* + k*)E = -S, (V* + k;)E, = -S, (20.1-27) These equations, together with (20.1-26), may be solved to determine E and E, Consider, for example, the case of small-angle reflection (f3 -K l), so that the two waves travel approximately in the z direction Assuming that k = k,, the fields E and

E, are described by E = A exp( -jkz) and E, = A, exp( -jkz), where A and A, are slowly varying functions of z Using the slowly varying envelope approximation (see Sec 2.2C), (V* + k*)A exp(-jkz) = -j2k(dA/dz) exp( -jkz), (20.1-26) and (20.1-27) yield

where

dA -=

dz j&4, d4 -=

d

Figure 20.1-7 Variation of the intensity of the incident optical wave (solid curve) and the intensity of the Bragg-reflected wave (dashed curve) as functions of the distance traveled through the acoustic wave

(20.1-29), we obtain L% = (7r2/A$)An@2 This is exactly the expression for the weak-sound reflectance in (20.1-17) with d = L/sin 0

It has been shown so far that an optical plane wave of wavevector k interacts with an acoustic plane wave of wavevector q to produce an optical plane wave of wavevector

k, = k + q, provided that the Bragg condition is satisfied (i.e., the angle between k and

q is such that the magnitude k, = Ik + ql = k = 27r/A) Interaction between a beam

of light and a beam of sound can be understood if the beam is regarded as a superposition of plane waves traveling in different directions, each with its own wavevector (see the introduction to Chap 4)

Diffraction of an Optical Beam from an Acoustic Plane Wave

Consider an optical beam of width D interacting with an acoustic plane wave In accordance with Fourier optics (see Sec 4.3A), the optical beam can be decomposed into plane waves with directions occupying a cone of half-angle

(20.1-31)

There is some arbitrariness in the definition of the diameter D and the angle 68, and a multiplicative factor in (20-l-31) is taken to be 1.0 If the beam profile is rectangular of width D, the angular width from the peak to the first zero of the Fraunhofer diffraction pattern is 68 = h/D; for a circular beam of diameter D, 88 = 1.22A/D; for a Gaussian beam of waist diameter D = 2W,, 68 = h/rWo = (2/~)h/D = 0.64h/D [see (3.1-19)] For simplicity, we shall use (20.1-31)

Although there is only one wavevector q, there are many wavevectors k (all of the same length 27r/A) within a cone of angle 68 As Fig 20.1-8 illustrates, there is only one direction of k for which the Bragg condition is satisfied The reflected wave is then

a plane wave with only one wavevector k,

Diffraction of an Optical Beam from an Acoustic Beam

Suppose now that the acoustic wave itself is a beam of width Ds If the sound frequency is sufficiently high so that the wavelength is much smaller than the width of

Figure 20.1-8 Diffraction of an optical beam from an acoustic plane wave There is only one plane-wave component of the incident light beam that satisfies the Bragg condition The diffracted light is a plane wave

INTERACTION OF LIGHT AND SOUND 813

the medium, sound propagates as an unguided (free-space)

analogous to those of optical beams, with angular divergence

If the acoustic-beam divergence is greater than the optical-beam divergence (60, z== 60) and if the central directions of the two beams satisfy the Bragg condition, every incident optical plane wave finds an acoustic match and the reflected light beam has the same angular divergence as the incident optical beam 68 The distribution of acoustic energy in the sound beam can thus be monitored as a function of direction, by using a probe light beam of much narrower divergence and measuring the reflected light as the angle of incidence is varied

Diffraction of an Optical Plane Wave from a Thin Acoustic Beam;

Raman-Nath Diffraction

Since a thin acoustic beam comprises plane waves traveling in many directions, it can diffract light at angles that are significantly different from the Bragg angle correspond- ing to the beam’s principal direction Consider, for example, the geometry in Fig 20.1-10 in which the incident optical plane wave is perpendicular to the main direction

of a thin acoustic beam The Bragg condition is satisfied if the reflected wavevector k, makes angles +8, where

.e A

sm z = 2h - (20.1-33)

Figure 20.1-10 An optical plane wave incident normally on a thin-beam acoustic standing wave

is partially deflected into two directions making angles = *h/R

If 8 is small, sin(0/2> = 8/2 and

(20.1-34)

The incident beam is therefore deflected into either of the two directions making angles f 8, depending on whether the acoustic beam is traveling upward or downward For an acoustic standing-wave beam the optical wave is deflected in both directions The angle 8 = h/A is the angle by which a diffraction grating of period A deflects

an incident plane wave (see Exercise 2.4-S) The thin acoustic beam in fact modulates the refractive index, creating a periodic pattern of period A confined to a thin planar layer The medium therefore acts as a thin diffraction grating This phase grating diffracts light also into higher diffraction orders, as illustrated in Fig 20.1-11(a) The higher-order diffracted waves generated by the phase grating at angles +28, +3e, may also be interpreted using a quantum picture of light-sound interaction One incident photon combines with two phonons (acoustic quantum particles) to form

a photon of the second-order reflected wave Conservation of momentum requires that

k, = k + 2s This condition is satisfied for the geometry in Fig 20.1-11(b) The second-order reflected light is frequency shifted to w, = w + 2fl Similar interpreta- tions apply to higher orders of diffraction

Figure 20.1-I 1 (a) A thin acoustic beam acts as a diffraction grating (b) Conservation-of- momentum diagram for second-order acousto-optic diffraction