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Second Harmonic Reflected LightPublished online: 11 Nov 2010.

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OPTICA ACTA, 1966, VOL. 13, NO. 4, 311-322

Second harmonic reflected lightt

N. BLOEMBERGEN

Harvard University, Cambridge, Massachusetts

(Received 4 May 1966)

Abstract. A review is given of the laws of non-linear optical reflection whichgovern the direction, polarization and intensity of second harmonic light generatedin reflection. The complex non-linear susceptibility of III-V and II-VIpiezoelectric compounds has been measured over a range of frequencies andshows characteristic dispersive properties. Both the amplitude and the phasecan be determined. When the medium has a centre of inversion, the reflectedsecond harmonic production is several orders of magnitude smaller. The effecthas magnetic dipole and electric quadrupole character and has been observedin silicon, germanium and several metals and alloys. In metals the contributionfrom core electrons and the conduction electron plasma have the same order ofmagnitude.

1. IntroductionThe behaviour of light waves at the boundary of a non-linear medium has

been analysed theoretically by Bloembergen and Pershan [1]. The incidentlaser beam or beams at frequency co create, after refraction, a polarization at theharmonic frequency 2 in the non-linear optical medium. This non-linearpolarization has a definite phase in each unit cell. For a piezoelectric crystalthe harmonic polarization is given by:

PN(2o) = XNL: EEjexp (2ik. r-2icot), (1)

where E is the field amplitude at the fundamental frequency inside the medium.The non-linear susceptibility is described by a third rank tensor which vanishesin media with inversion symmetry. The radiation from this phased array ofdipoles at 2 can be calculated [1, 2]. The harmonic radiation is characterizedby distinct direction, polarization and intensity characteristics. The equationsdescribing these may be regarded as generalizations to the non-linear case of thewell-known Fresnel formulae for linear optical reflection and transmission.

The direction of the harmonic reflected ray from a plane boundary is determinedby the condition that the tangential component of the wave vectors must beconserved as shown in figure 1. When the incident laser beam is split into twocomponents, which are redirected onto a surface of GaAs crystal, as shown infigure 2, three second harmonic beams emerge. The components of the wavevector parallel to the surface are given by k(2co)=2kg (1)(w), or 2kt,(2)(co) orki( l )(co) + k( 2 )(co). When the GaAs mirror is submerged in a linear dispersivefluid, the reflected second harmonic has a different direction from the reflectedfundamental beam, as shown by Ducuing [3].

t Invited paper presented at the CIO 7 Conference on Recent Progress in PhysicalOptics, Paris, 2-7 May 1966.

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x

Nonli

S

,_T

Figure 1. The creation of the sum frequency by two lightnon-linear medium.

Filtering andDetecting System

A_ ._ ~~~ KDPGoAs

waves at the boundary of a

Beam Splitter

Mirror

Filtering andDetecting Systemon rotatable arm

Figure 2. The experimental arrangement to detect the second harmonic generation by asplit-beam incident laser pulse.

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Second harmonic reflected light 313

For crystals with 43 m (Td) symmetry, to which most III-V semiconductorsbelong, the third rank tensor has the following non-vanishing component,XXVz=xz = XV XUZX = X14. If the incident laser field is polarized parallel to a cubicaxis, [0, 0, 1] direction, of the crystal, no second harmonic polarization is createdaccording to equation (1). If the incident electric field is parallel to a body

B

V//

7///

LaserA beam

V

Figure 3. The reflected harmonic ray from a' non-linear mirror' immersed in a dispersivelinear medium.

Ki

0w)[1-10] O:

Ga//As//////GaAsI

Itnol KR

a(2w) (i-il] XjI I s i (

////////,GaAs

//I/I//I

95 pNLS

Ks

[It'lu /I1VJKR

I ZKR

OR (2w)

p N LS

(2w)KS

~ri-1o]axis 2&[-11 ]axis (110)FACE

Theoretical

20 ° 400 60 ° 80°i (ANGLE OF INCIDENCE)

Figure 4. The ratio of the second harmonic intensities for two geometries shown at the top,as a function of the angle of incidence. The Fresnel transmission factor for thelaser beam is eliminated in this comparison for two harmonic polarizations.

GAs MA \

Benzene

CE

2i]-2

77//J//i/f////////////7/ZZ7..777;.

..,..........,.,.,............ //J/l//2,

78_= 710

-( I

I )/Y///////////A - \ V r/////////// X/I////////////// I~~~~% Y///////// //

b

�y V

_�Z

1, ^1

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diagonal, [1, 1,1] direction, the second harmonic light is polarized parallel to theincident laser light. If the incident laser light is parallel to a face diagonal, [1, 1, 0]direction, the second harmonic polarization is at right angles, along the [0, 0, 1]direction. These polarization properties have also been verified by Ducuing [4].

2. Fresnel's laws and Brewster angle for non-linear reflectionThe variation of the intensity of the reflected second harmonic with the angle

of incidence of the fundamental beam has been measured by Chang [5]. A largepart of the angular variation stems from the linear Fresnel factor describing thetransmission of the laser beam into the medium. In order to verify thecharacteristic non-linear Fresnel factor, it is best to compare the ratio of the secondharmonic intensity for two geometries, which are identical as far as the linearoptical properties are concerned, but have different non-linear characteristics.Note that the linear dielectric constant is a scalar for the cubic 43 m symmetry, butthe non-linear susceptibility has tensorial properties. In figure 4 the secondharmonic intensity, reflected from a [110] crystallographic plane surface, hasbeen compared for the two cases, that the second harmonic polarization is paralleland perpendicular to the incident fundamental polarization. The experimental

.110 K . -

Ki

,GaAs/////GaAs

K -) K r Kr

(2W) [-0 (2(2 wj)

> V.////////tz //////////, ,//////////

GaAs

P2to)p NLS

(2W) aKS 'Ks

100

80-

(110) FACE 2- (001) FACE [1-10]axis

(I110)FACE 2 = (2wC)-Sin2Oi(001) FACE [-lO]axis Sin29j

60-

40-

20,

mental

I I I I -20 ° 40° 60* 800 8i (ANGLE OF INCIDENCE)

(d)Figure 5. The ratio of the second harmonic intensities for two geometries shown at the top,

as a function of the angle of incidence. The Fresnel transmission factor for the laserbeam is eliminated in this comparison of two different crystallographic orientationsof the mirror face.

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Second harmonic reflected light 315

points agree well with the drawn theoretical curve for the non-linear Fresnelfactor ratio in this geometry. In figure 5 the second harmonic intensities,polarized perpendicular to the laser polarization but reflected from differentcrystallographic surfaces, are compared. The observed ratio again follows thetheoretical non-linear Fresnel formula for this geometry.

Ki

[1,1,0]\ e

/// 1/1///////////Ga As

E KR

[0,0,1]4!

I1 1 = 8"p /NLS

(2Ks)

Ks

Figure 6. The geometry used for the verification of Brewster's angle for second harmonicgeneration.

2,

OI0YM

Degree

(C)

Figure 7. The second harmonic intensity, for the geometry of figure 6, as a function of theangle of incidence for a neodymium glass laser beam.

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In the geometry of figure 6 the harmonic polarization makes always an angleof 8° with the normal of the reflecting surface, regardless of the angle of incidencei. Figure 7 shows that this polarization cannot radiate in a direction R = 38 °

outside the medium. It should be noted that this angle satisfies the conditionsin38°= e (2 w)sin8 ° . The harmonic polarization cannot radiate parallelto itself inside the linear dielectric medium. The second harmonic polarizationmust be considered as radiating inside the linear dielectric, whereas the linearpolarization radiates into the vacuum in the conventional derivation of the linearBrewster angle. The data in figure 7 are taken with a neodymium glass laser. Atthe harmonic ruby frequency the imaginary part of the dielectric constant is solarge that no zero results, although a minimum in the intensity is still discernible

[5].

RATIO OFNONLIN E

SUSCEPTIBI

PHOTON ENERGY OF FUNDAMENTAL WAVE

Figure 8. The modulus /Xx4NL(2wc)/ of the non-linear susceptibility in GaAs as a functionof frequency, relative to the value in KDP.

3. Dispersion of the complex non-linear susceptibilityThe intensity of the reflected harmonic is proportional to the square of the

absolute value of the non-linear susceptibility. The reflection method permits themeasurement of this quantity even when the medium is absorbing at the funda-mental and/or the harmonic frequency. In either case the non-linear susceptibilityis a complex quantity which shows a marked dispersion as a function of frequency.The absolute value of this quantity has been measured [6] in GaAs, ZnTe, InAsand InSb at seven different wavelengths, relative to the known value of the

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Second harmonic reflected light 317

non-linear susceptibility in a KDP crystal. The primary beam was obtainedfrom a Nd glass laser, and from this beam doubled in frequency by another KDPcrystal. Intermediate frequencies were obtained from a ruby laser beam, andfrom stimulated Raman emission by this beam in hydrogen gas, cyclohexane,nitrobenzene and benzene. The data obtained for GaAs are reproduced infigure 8. This is an early example of non-linear spectroscopy. The non-lineardispersion can be correlated with the linear dispersion. Theoretical considerationsshow that extrema, corresponding to critical points in the joint densities of statesfor (cw) and (2 co), should be evident also in the non-linear susceptibility.

MONITOR

EXPERIMENTAL SET-UP FOR PHASE MEASUREMENT

Figure 9. Diagram of the experimental apparatus to determine the relative phase of thesecond harmonic field.

~~2,~~~~~0[~ ~KOP AT 259*2011~ . GaAs (ITO) AXIS VERTICAL ,

I

1.4

2

, oI

a.

2 0.8

~0..

1P0.40 20 30 40 50 60

cm of MERCURY70

Figure 10. The interference between the second harmonic fields produced by the same laserbeam in a GaAs mirror and a KDP platelet, as a function of air pressure in the boxshown in figure 9.

>D

7

AVERAGE

123' 4'

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I I I I I I I I I I I I I I I I I I0

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The phase of this complex quantity is a rather elusive thing. It measures,according to equation (1), the phase of the second harmonic polarization relativeto the phase of the fundamental field squared. Since it relates phases of Fouriercomponents at different frequencies, the imaginary part cannot be identifiedwith an absorption mechanism. Nevertheless, it has been possible [7] to measurethe phase in this parametric process, by the experimental arrangement shown infigure 9. The same laser beam creates a second harmonic polarization in theIII-V semiconductor and in a crystal of KDP. The second harmonic radiationgenerated by the two samples can interfere, because they have both a well definedphase with respect to twice the phase of the fundamental field. One must keeptrack of all the phase shifts occurring on reflection, as well as in the dispersivelight paths of both the fundamental and harmonic beams. By variation of theair pressure, the dispersion in the air path between the two non-linear samplescauses the interference curve of figure 10. Since KDP is non-absorbing at coand 2co, its XNI' is real. The phase of the curve in figure 10 at zero pressure is ameasure for the phase of XNL(2w) of the material of the mirror. The phase andabsolute value of XNL(2wo) are related to each other by a relation of the Kramers-Kronig type [8].

4. Second harmonic light reflected from media with inversion symmetryWhen the optical medium possesses a centre of symmetry, the non-linear

polarization expressed by equation (1) vanishes. There are, however, stillsources of harmonic generation because equation (1) is only the pure electricdipole term in a multipole expansion [9]. To keep the following terms in itssimplest form, only isotropic media will be considered in this section. In liquidsand polycrystalline solids the sources for harmonic radiation may be written inthe form [10]:

PNL(2to) = E(w) x H((w), (2)

- V. QNL(2o) = -V . (' (co)E(to), (3 a)

where is a pseudoscalar and /' a scalar quantity. The divergence of thequadrupolarization may be transformed into a surface dipole term of the form:

PsurfNL(2) = E(V. E). (3 b)

The magnitude of these terms, given by equations (2) and (3) is ka times theelectric dipole term of equation (1), where k - 1 is (1 /27r) times the optical wavelengthor the absorption depth and a the atomic dimension. Whereas in GaAs, e.g.,the harmonic radiation originates from a layer of thickness k-1, the radiation inGe, which possesses a centre of symmetry, is equivalent to the electric dipoleradiation from one atomic layer at the surface. In agreement with these consider-ations the reflected harmonic radiation in Si and Ge is three to four orders ofmagnitudes smaller than in the III-V semiconductors. During a laser pulse con-taining about 101x incident quanta, about 3-30 harmonic quanta are typicallyemitted from these materials. Photon counting techniques must be employed tomeasure the weak signals accurately.

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Second harmonic reflected light 319

The reflected harmonics from isotropic materials were first considered inconnection with conduction electron plasma's. The term given by equation (2)was discussed by Kronig [11] and by the present author [12]. The surface termequation (3b) was first introduced by Jha [13]. The reflected harmonic frommetallic silver was first observed by Brown and co-workers [14]. The polarizationand directional properties are very distinct and qualitatively the same for metalsand for semi-conductors. They can be calculated by substituting the source termsof equations (2) and (3) into the appropriate expressions of reference [1].

I

zI-

Z

I4W>r

9o 60 30 0 3' 60 90'

Figure 11. The variation of the second harmonic generation from silver as a function ofthe angle (9 between the fundamental electric field vector and the plane of incidence.The angle of incidence is kept constant at 45° (after Brown et al.) [14].

The variation of the second harmonic intensity as a function of the angle ofthe electric field vector with the plane of incidence is shown in figure 11 for silver.The angle of incidence is kept constant at 45 °. Similar curves apply for Au, Cu,Si and Ge. The dependence of the harmonic intensity as a function of the angle

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of incidence, of the laser beam, polarised in the plane of incidence, is shown infigure 12. There is good agreement with theoretical curve both for Ag and Si.

The dispersion of the second harmonic generation in three metals [10] isshown in figure 13. The drawn curves are calculated on the basis that the coreelectrons, or interband transitions, do not contribute to the non-linearity. Themaximum in the curve for Ag is due to the plasma resonance, which is close tothe second harmonic frequency of ruby. Apparently there is destructiveinterference between the non-linearity of the interband and intraband transitionswhich obliterates this maximum. Data for the second harmonic generation inAg-Au alloys show the marked decrease which arises from the change in theFresnel factor at the second harmonic frequency, as a few per cent Au washes outthe plasma resonance. Proceeding to the pure gold composition the experimentalpoints display the difference in the non-linear interband contribution for Au andAg. The conclusion can be drawn that in metals interband (core) and intraband(plasma) contributions to the non-linearity have the same order of magnitude.In silicon and germanium only interband transitions play a role, as the plasmadensity is negligible [10].

DEGREEANGLE OF INCIDENCE

Figure 12. The variation of the second harmonic intensity in Si and Ag as a function ofthe angle of incidence. The incident polarization is kept in the plane of incidence.

5. ConclusionThe reflected harmonic light intensity is always very weak compared to what

can be achieved in transmission, especially under phase-matched conditions.The effect is, however, important for the measurement of non-linearities inabsorbing materials. It has also permitted a rather detailed verification of the lawsof non-linear optics. Since the second harmonic generation from atomic surface

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Second harmonic reflected light 321

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e e e

0 Q

. ~ 0cd

a ~ v

0 .q 0

U 0 '

.~ ~ 0 ~

O 'C : '7i

-o

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zLU

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Second harmonic reflected light

layers is observable, it may become a new tool in surface studies. The measurementof non-linear optical properties of materials supplements our knowledge aboutelectronic structure obtainable from linear optical constants. Extension of thesemeasurements to ultra-violet and infra-red frequencies is both feasible and useful.

On presente une revue des lois de la rflexion optique non-lineaire, qui rgissent ladirection, la polarisation et l'intensite du second harmonique engendr6 par rflexion. Lasusceptibility complexe non-lin6aire des composes piezo-6lectriques II-V et II-VI a temesur6e sur un certain domaine de frequences et montre des proprietes dispersives caracteris-tiques. On peut determiner la fois l'amplitude et la phase. Lorsque le milieu a uncentre d'inversion, la production du second harmonique par reflexion est de quelques ordresde grandeur plus faible. I1 a le caractere d'un dip6le magnetique et d'un quadrup6ole6lectrique et il a et6 observe dans le silicium, le germanium et plusieurs metaux et alliages.Dans les m6taux, les contributions des 61ectrons lis et des electrons de conduction sont dumeme ordre de grandeur.

Es wird ein berblick fiber die Gesetze der nicht-linearen optischen Reflektion gegeben,die die Richtung, Polarisation und Intensitit der zweiten, bei der Reflektion auftretendenharmonischen Oberwelle des Lichtes beschreiben. Die komplexe nicht-lineare Suszepti-bilitat der piezoelektrischen Komponenten aus den III-V- und II-V-Gruppen wurde uibereinen Frequenzbereich hin gemessen und zeigt charakteristische Dispersionseigenschaften.Man erhalt dabei beides, Amplitude und Phase. Wenn die Substanz ein Inversionszentrumbezitzt, so ist das Auftreten der zweiten harmonischen Oberwelle um mehrere Gr6-ssenordnungen geringer. Die Welle ist magnetisch ein Dipol und elektrisch ein Quadrupol.Sie wurde an Silkion, Germanium und einigen Metallen und Legierungen beobachtet. InMetallen haben die Anteile der gebundenen Elektronen und des Plasmas der Leitungselek-tronen die gleiche Gr6ssenordnung.

REFERENCES

[1] BLOEMBERGEN, N., and PERSHAN, P. S., 1962, Phys. Rev., 128, 606.[2] BLOEMBERGEN, N., 1965, Non-linear Optics (New York: W. A. Benjamin, Inc.).[3] BLOEMBERGEN, N., and DUcUING, J., 1963, Phys. Letters, 6, 5.[4] DuCUING, J., and BLOEMBERGEN, N., 1963, Phys. Rev. Letters, 10, 474.[5] CHANG, R. K., and BLOEMBERGEN, N., 1966, Phys. Rev., 144, 775.[6] CHANG, R. K., DuCUING, J., and BLOEMBERGEN, N., 1965, Phys. Rev. Letters, 15, 415.[7] CHANG, R. K., DUCUING, J., and BLOEMBERGEN, N., 1965, Phys. Rev. Letters, 15, 6.[8] CASPERS, W. J., 1964, Phys. Rev. A, 133, 1249.[9] PERSHAN, P. S., 1963, Phys. Rev., 130, 919.

[10] BLOEMBERGEN, N., CHANG, R. K., and LEE, C. H., 1966, Phys. Rev. Letters, 16,986.[11] KRONIG, R., and BOUKEMA, J. I., 1963, Koninkl. Ned. Akad. Wetensch, Amsterdam,

Proc. B, 66, 8.[12] BLOEMBERGEN, N., 1963, Lectures in Theoretical Physics, Boulder, 1962, Vol. V (New

York: Wiley), p. 253; 1963, Proc. IEEE, 51, 124.[13] JHA, S. S., 1965, Phys. Rev. Letters, 15, 412; 1965, Phys. Rev. A, 140, 2020.[14] BROWN, F., PARKS, R. E., and SLEEPER, A. M., 1965, Phys. Rev. Letters, 14, 1029.

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