University of LjubljanaFaculty of Mathematics and Physics

Seminar - 1. letnik, II. stopnja

Phase Conjugate Mirrors

Author: Žiga PušavecMenthor: dr. Mojca Vilfan

Ljubljana, May 2018

AbstractA Phase Conjugate Mirror reflects an incoming wave so that the re-

flected wave exactly follows the path it has previously taken - this can beinterpreted as rewinding the time. This unintuitive phenomenon has manyinteresting applications in the correction of aberration due to imperfectoptical elements in fields of high energy lasers, optical communications andoptical circuits. The origin of this phenomenon lies in the third order non-linear optical response of the matter. Two main mechanisms are discussed:degenerate four-wave mixing and stimulated Brillouin backscattering.

1 IntroductionThe basic concept of light reflecting from a mirror is well established. The an-gle between the reflected wave and the normal to the surface is the same as theangle between the incident wave and the normal. Light will reflect back to itsorigin only for a wave that is perpendicular to the surface. This is described bySnell’s law and has been known for a long time.[1]

In the late 1970s a new and very different kind of mirror has been invented,based on the principles of third-order nonlinear optics. If one would look intosuch a mirror, one would see absolutely nothing1. This is a consequence of thefact that reflection from such a mirror has a different physical origin comparedto a normal one. Due to the nonlinear phenomena, the phase of the reflectedlight is the exact conjugate of the incident one. Such mirrors are called phaseconjugate mirrors (PCM). The main consequence of this is that light, reflectedfrom such mirrors, will exactly follow the path it had made before hitting themirror, matching in phase at all points along the way.[2] This basic concept hasmany useful applications in fields of high energy lasers, communication signalsand adaptive optics.[3]

2 Basic Concept of Phase Conjugation [4, 5]To study various applications of PCM, the knowledge of underlying non-linearmechanism(s) is not required. We will discuss the later in Section 4, when thecontext will already be well established.

The general concept of phase conjugation is very simple – the phase of theincident light gets conjugated on reflection. To describe this let us define 2 theincident wave in the direction ez with:

E3(x, y, z; t) = E0(x, y, z) ei(Φ(x,y,z)−ωt) + c.c. , (1)

where Φ(x, y, z) = kz + ϕ(x, y, z) is the dependance of the space component ofthe wave’s phase. The ϕ(x, y, z) describes the distortion of the wave front (itsphase) by the medium while passing through it. We can now rewrite this as:

E3(x, y, z; t) = E0(x, y, z) ei(kz+ϕ(x,y,z)) e−iωt + c.c.

= A3(x, y, z) e−iωt + c.c. ,(2)

where we have defined A3 as the complex amplitude of the incident wave. Nowwe can write the reflected wave in the same way, while taking into accountalso that its phase gets conjugated. This means that both components of the

1 In fact, one does see a little bit of light, but only the few rays that reflect from his eyedirectly onto the mirror. The intensity of such light, however, is very low and difficult todetect.

2 The choice of indices 3 and 4 will become clear later in Section 4.3.

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propagation vector will change in sign as k → −k and the same for its phasedistortion: ϕ→ −ϕ. Reflected wave is then:

E4(x, y, z; t) = E0(x, y, z) ei(−Φ(x,y,z)−ωt) + c.c.

= E0(x, y, z) ei(−kz−ϕ(x,y,z)) e−iωt + c.c.

= A4(x, y, z) e−iωt + c.c. ,

(3)

where we have again defined A4 as the complex amplitude of the wave. Com-paring the two expressions we see that A4 = A∗

3. To better understand this, itis worth writting down the real parts of the waves:

Re[E4(x, y, z; t)] = 2 E0 cos(kz + ϕ(x, y, z) + ωt)

= 2 E0 cos(−(−kz − ϕ(x, y, z)− ωt))= 2 E0 cos(kz + ϕ(x, y, z)− ω(−t))) .

(4)

Thus we see:Re[E4(x, y, z; t)] = Re[E3(x, y, z;−t)] . (5)

From this, we can interpret the reflected wave as the incident wave, reversed asa wave going back in time, exactly following the path it previously took. If weilluminate such mirror with light from a point source, the reflected light will notexpand after reflection, but rather focus back to its source.

Figure 1: The wave reflects differently from PCM than from an ordinarymirror.[5]

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3 Applications of Phase Conjugate MirrorsPhase conjugation gives us a very different and unintuitive look at reflectionand as such offers various new and interesting applications, which will now bepresented.

3.1 Compensation for AberrationOne of the main applications of PCM is in compensation for imperfect opticalelements. While travelling through an nonideal medium, different rays of lighttake different optical paths. This causes the distortion of the plane wavefront.The phase distortion caused by the medium is described with ϕ(x, y, z; t). Be-cause of the inversion of the wavefront on reflection from PCM, we can view theconjugated wave E4 as going back in time. This means that on the second passthrough the distorting medium, the phase distortion gets repaired, and when itre-emerges from the ϕ(x, y, z; t) medium, its wavefront will again be plane.[6]This concept is the basis of the applications that are going to be discussed here.It is worth emphasising two main limitations to this process: first, in the timeit takes the light wave to travel there and back again through the distortingmedium, ϕ(x, y, z; t) must not change significantly. Secondly, light must notalter the physical properties of the medium while passing through it.[6]

Figure 2: The originally plane wave, that is reflected from PCM, reemerges asan undistorded plane wave when exiting the distorting medium.[7]

The fundamental example of application is in long range communications. If wesend a light signal to a detector relatively far away, its wavefront will get dis-torted due to the density variations in the atmosphere. If we send this alreadydistorted signal back, it would get additionally distorted. But if PCM is usedfor reflection, it would be almost undistorted. The original receiver would alsobe automaticaly targeted, as discused in Subsection 3.3. Information in suchsystem would be stored in signal’s intensity3, not phase.[4]

3 Amplitude or the position of the pulse.[3]

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3.2 Optical FibersThe common way of information transfer nowadays is via optical fibers. Themain limitation of optical fibers are the different dispersions. While some ofthem can be reduced by fiber design4, there are still some which cannot beavoided, like the dispersion due to multiple possible wave modes in fibers. WithPCM we can reduce some of them.

In multimode optical fiber, dispersion is a consequence of the fact that differentallowed wave modes travel with different group velocities. The pulse length thusexpands. One way to reduce such effect is by using single mode optical fibers.Another way is by using PCM. If we put PCM on one end of the optical fiber,the wave’s phase will get reversed on reflection. When such reflected wave issent back trough an identical optical fiber, its dispersion will get reduced whiletravelling back. The previously faster mode is now slowed down.[3] This enablescommunications with a distant source while in return receiving an undistortedand low-noise signal. Information is again stored in the intensity of the signal(amplitude, length of the pulse). On the receiving end information can be ma-nipulated using electrooptic effect. Acording to Pepper et al., such manipulationcauses only insignificant loss of intesity.[3] Thus, two way communication withvery low levels of noise can be achieved.

It is worth emphasising that due to PCM, not only single mode but also mul-timode optical fibers can be used for communication, which can significantlyincrease the amount of transfered data. Again, Pepper etal. state that the up-per data bandwith limit in such optical fibers will be determined by the qualityof electronics, not PCM.[3]

3.3 High-Intensity LasersThe aberration compensation can be very effectively used in high-intensitylasers. Building a good laser is an art form in itself, but the main components arefairly simple: the amplifying medium, through which the energy is pumped intothe system (by the principle of inverted population) and a resonator consistingof two mirrors, which allows light beams to pass through the amplifier multipletimes.[1] One of the engineering limitations is that while passing through theamplification medium, the beam gets diffracted/distorted because the refractiveindex changes with space due to inhomogenous material and thermal fluctua-tions. This can be solved by replacing one of the resonator mirrors with PCM.This results in a beam, that has been amplified twice, but has not divergedduring the amplification.[10] The first experimental realizations of this conceptin the 1970s have already proven usefulness of this concept when a decrease ofbeam divergence from 2,5 to 0,15 mrad has been achieved by Pepper et al.[3]

4 For example, we can compensate the effect of the chromatic dispersion with the waveguidedispersion.[14]

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This kind of application also solves another problem. To achive stable oper-ation of an ordinary laser, only certain curvatures of mirrors are allowed.[14]With PCM, this is no longer needed, since light beam does not diverge whendouble passing through the amplification medium.[2] As such, PCM can be usedto create new types of high energy lasers which can achive greater beam qualityand stability than ordinary ones.[6]

In practice, PCM are most frequently found in Master Oscilator Power Ampli-fier (MOPA) laser setup [5], used to achieve high intensities of lasers in severalsteps. In ordinary lasers, output light intensity is limited by the fact that theeffective power per volume of the amplification medium is upward-limited. Itis then more effective to have several smaller amplifiers/resonators coupled to-gether. The PCM is here used to coherently combine a set of parallel amplifiers,while compensating for individual phase shifts between individual beams.[3] Ex-perimentaly, H. J. Eichler, has shown that the power of a base oscillator hasbeen increased from 1 W to 130 W using this technique.[5]

If phase conjugation is achived by degenerate four-wave mixing (Section 3.3 ),one can also externaly control the frequency of the laser output.[6]

3.4 Laser Tracking/Automatic AlignmentPCM can also be used for alignment of laser beams. It has been suggested thatit could be useful for targeting of fusion pallets in fusion reactors, where thereis a problem of focusing light in the reactor core, where it otherwise gets toodistorted before even reaching the target.[10]

The general concept is again fairly simple. If a (moving) target is illuminatedwith a low energy laser, light will reflect from it in all directions. A small partof intensity of light will reflect in the direction of our laser tracking sistem. Thislight will first pass through a strong amplifying medium and then reflect from aPCM. Because of the basic properties of PCM, the reflected light will directlyreturn to the target without any divergence, the only difference being that itwill have twice passed the (strong!) amplification medium. Thus, a very stronglaser beam can be automaticaly targeted on a target, which would otherwise behard to hit.[10]

3.5 Other ApplicationsOnly several most basic applications have been discussed in order to illustratethe concept of phase conjugation in optical systems. The use of PCM is howevernot limited to these examples - many additional applications can be found infields such as image projection [2], spectroscopy [2], photolitography in produc-tion of silicon chips [11], pattern recognition [7], parallel processing in opticalcomputing [3] and other similiar ideas.

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4 Mathematical Description of Phase Conjuga-tion

The physical process of phase conjugation result from different physical phenom-ena. The two most important ones, degenerate four-wave mixing and stimulatedBrillouin backscattering will now be presented, but first some general contexthas to be established about nonlinear optics, as well as the concept of hologra-phy.

4.1 Nonlinear OpticsThe discovery of the nonlinear optical response is fairly recent discovery[14] ashigh light intensities are required. This can only be achieved with lasers, whichhave only been available for the last 50 years or so. The nonlinear response issimiliar to the one in magnetic systems. For low intensities of external excita-tions we can approximate the response of the medium (magnetization) M to belinear with the excitation H: M(H) ≈ χmH. In optics, we can do the same forpolarization P(E) ≈ χe E. In the linear approximation electric suscetibility χe

is:χe = ε− 1 . (6)

This is of course only true for low intensities of incident light. To describe thenonlinear response at higher intesities we must write χe as a tensor up to itsfourth rank:

P = Plinear + Pnonlinear

= ε0 χ(1) E + ε0 χ

(2) : EE + ε0 χ(3)

... EEE ...(7)

A given component is written as:

(Pnonlinear)i = ε0 χ(2)ijk EjEk + ε0 χ

(3)ijkl EjEkEl + ... (8)

It is worth noting that χ(2) is only non-zero for materials without a centre ofinversion, while χ(3) is always present.5 Because of this, third order nonlinearphenomenon is observable in all states of matter – PCM can be constructedusing any of them.[3]

4.2 HolographyOne of the limitations of classical light detection is that only information aboutthe amplitude of the wave is detected and information about phase is lost. Anexample of this is classical X-ray crystalography. This has inspired the inventionof holography. The hologram is created in two separate steps. First, the initial

5This is due to the fact that the polarization P is not the same for the different directionsof external field E and −E. This means that the structure of the crystal is not the same inall directions and therefore lacks symetry.

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beam is split into two parts. One part, so called reference wave illuminates thephotoactive film directly. Second signal wave is shone onto the observed object,from which it reflects. The reflected wave has a modified wavefront, which canbe described by a varying phase of the wave. When the two waves meet atthe point of the photoactive film they interfere. Due to the different phases adiffractive grating appears in the film emulsion, which stores the informationabout the object. The second step consist of reading this information by usinga third reconstructive wave, which diffracts on the grating. In observing thisdifracted wave the 3D image of the original object becomes visible.[8]

Figure 3: A typical setup for recording a hologram.[15]

This is the origin of PCM. It comes from the discovery that if you illuminatethe film grating with a beam that has the same characteristics as the referencebeam, but travels in the opposite direction, a real wave is produced, that is theconjugate of the original one.[2]

4.3 Degenerate Four-Wave MixingThe practical use of the above mentioned method was in practice very limiteddue to its nature as a two-step procedure. The next step, which made the phaseconjugation useful in practice, was that the refractive grating was succssefullymade to change in real-time. This could be achieved with third order nonlin-ear effects. Information from incoming signal wave could thus be stored in thegrating by the third order field coupling of a signal wave with one of the pumpwaves. At the same time, this information could be read from the grating bythe other pump wave, whose scattered light would exit the medium as a wave,

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which would be the exact phase conjugate of the original6. This is the mainconcept behind the real-time holography, better known as degenerate four-wavemixing (DFWM).[6]

Figure 4: The most common sheme for DFWM. We will describe pump waveswith E1 and E2, the signal wave with E3 and the phase conjugated wave withE4.[5]

The most common setup for DFWM is shown in the Figure 3, with three in-coming waves E1, E2 and E3 and one outgoing wave E4. It is important thatthe pump waves E1 and E2 are facing opposite directions (k1 = −k2), the am-plitudes of the pump waves are much greater then the one of the signal waveE3 and that all four waves have the same frequency ω. We will see that fromthis conditions we get the outgoing phase conjugated wave E4.[4]

To get the general solution for four-wave mixing we have to solve the nonlinearwave equation:

∇2E + εω2

c20E = µ0

∂2Pnonlinear

∂t2. (9)

In general, the outgoing wave is given by [8]:

(P(3)nonlinear)i = ε0 χ

(3)ijkl(ω1, ω2, ω3, ω4) Ej(ω1)Ek(ω2)El(ω3) + c.c. . (10)

In the most general case, 384 terms appear in this expansion.[2] This number canbe vastly reduced if we take into accout the specifics of DFWM. The orientationof pump waves (k1 = −k2) reduces the problem to one dimension and ensuresphase matching at all times.[8] The outgoing frequency is then ω4 = ω1+ω2−ω3.If we now considere the degenerate case where ω1 = ω2 = ω3 = ω we see that alsoω4 = ω. By chosing the same polarization for all four waves we can reduce thetensor χ(3)

nonlinear to only one element χ(3).[8] By assuming that the amplitudes

6 We now see that PCM is not an actuall mirror, since the light does not really reflect fromit. The light that we percieve as reflected actually originates from the pump waves, whichscatters on the diffractive grating.

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of the pump waves are much greater than of the signal wave and by insertingthe ansatz for plane waves into Eq.(9), we obtain [7]:

dE4

dt= i

ω0

2

õ

εχ(3)E1E2E

∗3 (11)

dE3

dt= − i

ω0

2

õ

εχ(3)E1E2E

∗4 . (12)

These two equations describe the process of phase conjugation by DFWM inPCM. We describe the effect of the medium as well as of the two pump waveswith a new constant κ, defined as:

κ =ω0

2

õ

εχ(3)E1E2 . (13)

This represents the spatial dependence of the refractive index, which results inthe formation of the diffraction grating.[8] After some calculation we get thesolutions to these equations [2, 14]:

E4(z = 0) = E∗3 (z = 0)

κ

|κ|tan(|κ|L) (14)

E3(z = L) = E∗4 (z = L)

1

cos(|κ|L)(15)

where ez is the direction of incident wave and L is the length of the nonlinearsystem. The z = 0 indicates the begining of the system and z = L its end.This is how we phase conjugate an incoming wave using DFWM.

4.3.1 Reflective Coefficient

One of the interesting consequences of DFWM is that the reflective coefficientR = tan2(|κ|L) can be larger than 100 %. This is because energy can betransfered from the strong pump waves to the signal wave. In experiment,the reflectivity up to 10.000 % was achieved.[11] This property can be used indesigning lasers with PCM.[3]

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4.4 Stimulated Brillouin BackscatteringOne of the main disadvantages of DFWM is that we need two strong match-ing pump lasers, which can be practicaly hard to achieve on a small scale. Adifferent kind of process can be used to solve this. In fact, it was through thisprocess that the first real PCM was constructed in the 1972.[3]

Figure 5: The general scheme of SBS. [2]

Due to thermal fluctuations, acoustic waves (phonons) can be formed in matter.[2]This creates a moving diffractive grating, from which the incoming wave E3 isBrillouin-scattered.[8] If it is of sufficient intensity, the backwards going E4 theninterferes with E3 and creates even more sound waves due to the electrostrictiveforce7.[6] The trick for construction of PCM is in successfully creating a positivefeedback loop between these two processes.[6] This can be achieved at certainfrequencies and leads to an exponential growth of E4. Even though the initialwave scatters in all directions, it can be shown that the exactly phase conjugatewave gets amplified the most.[6]

This leads to a more practical approach to creating a PCM. The main downsideof this process is that the created wave is not ideal. Due to the reflection froma moving sound wave, some energy is lost to the phonon and the frequencyof E4 gets Doppler shifted.[2] The reflectivity of such mirrors can reach reach100 %.[11]

5 ConclusionPhase Conjugate Mirrors offer an interesting example of what can be achievedin the very rich field of nonlinear optics. It also serves as an example of howcomplex industry designs can originate from a single very simple concept. In-spired by this, one cannot imagine what new inventions will be made in thecoming years, that will alter our conception of how the universe should workeven further.

7 This is a force that points in the direction of the increasing electric field in a dielectricmaterial.[6]

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References[1] B. E. A. Saleh & M. C. Teich. Fundamentals of Photonics, 2nd Edition.

John Wiley & Sons, Inc., USA, 2007.

[2] M.C. Gower. The Physics of Phase Conjugate Mirrors. Progress in Quan-tum Electronics, Vol. 9, p. 101-147, 1984.

[3] D. M. Pepper, David A. Rockwell & Gilmore J. Dunning. Nonlinear OpticalPhase Conjugation. IEEE Circuits and Devices Magazine, Vol. 7, Issue: 5,p. 21-34, 1991.

[4] G. Grynberg, A. Aspect & C. Fabre. Introduction to Quantum Optics. Cam-bridge University Press, New York, 2010.

[5] H. J. Eichler & O. Mehl. Phase Conjugate Mirrors. Journal of NonlinearOptical Physics & Materials, Vol. 10, No. 1, p. 43-52, 2001.

[6] B. J. Feldman, et al. Through the Looking Glass with phase conjugation.Los Alamos Science, Fall 1982.

[7] J. P. Huignard & A. Brignon. Overview of Phase Conjguation; Phase Con-jugate Laser Optics. John Wiley & Sons, Inc, 2004.

[8] R. Guenther. Modern Optics. John Wiley & Sons, Inc., USA, 1990.

[9] M. Čopič, A. Petelin & M. Vilfan. Fotonika. FMF, 2018.

[10] Y. R. Shen. The Principles of Nonlinear Optics. John Wiley & Sons, Inc.,USA, 2003.

[11] M. Gower. Phase Conjugate Mirrors – Mirrors that reflect time. Nature,Vol. 308, 8, 1984.

[12] P. Meystre & M. Sargent III. Elements of Quantum Optics. Springer-Verlag,Berlin, 1990.

[13] R. Kowarschik, et al. Optical Measurements with phase-conjugate mirrors;Applied Physics B, Lasers & Optics. Scpringer-Verlag, 1999.

[14] P.A. Franken, A.E. Hill, C.W. Peters & G. Weinreich. Generation of OpticalHarmonics. Physical Review Letters, vol. 7, Issue 4, pp. 118-119, 1961

[15] G.R. Fowles. Introduction to Modern Optics. Dover Publication, Inc., NewYork, 1975

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