1 Intrinsic Optical Properties Of Diamond
11Intrinsic Optical Properties of DiamondRichard P. MildrenDiamond comprises the lowest mass element that can form a stable covalentlybonded crystal lattice, and this lattice is highly symmetric and tightly bound. Itsresulting extreme properties, along with the recent developments in its synthesis,have led to an explosion of interest in the material for a diverse range of opticaltechnologies including sensors, sources, and light manipulators. The optical properties in many respects sit well apart from those of other materials, and thereforeoffer the tantalizing prospect of greatly enhanced capability. A detailed knowledgebase of the interaction of electromagnetic radiation with the bulk and the surfaceof diamond is of fundamental importance in assisting optical design.For any material, the dataset characterizing optical performance is large anddiamond is no exception despite its inherent lattice simplicity. The properties ofinterest extend over a large range of optical frequencies, intensities and environmental parameters, and for many variants of the diamond form including defectand impurity levels, crystal size, and isotopic composition. Over and above thefascination held for this ancient material, its highly symmetric structure and purenatural isotopic content (98.9% 12C) provides an outstanding example for underpinning solid-state theory. As a result, diamond has been extensively studied andits optical properties are better known than most other materials.Many excellent reviews of optical properties have been reported previously (seee.g. Refs [1–3]). These concentrate mainly on linear optical properties, often focuson extrinsic phenomena, and are written from perspectives outside of the field ofoptics, such as electronics and solid-state physics. Consequently, there is a needto consolidate the data from the perspective of optical design. Furthermore, thenonlinear optical properties of diamond have not to date been comprehensivelyreviewed. The aim of this chapter is to do this, with emphasis placed on the intrinsic properties of single-crystal diamond (i.e., pure Type IIa diamond1)). The chapter1) Type IIa represents the most pure form;other categories (Types Ia, Ib, and IIb) havesubstantial levels of nitrogen (Type Ia andIb) and/or boron impurity (Type IIb). Notethat the delineations between types are notwell defined. Type IIa are rarely found aslarge homogeneous crystals in nature asnitrogen aids the formation process. It isthus for historical reasons that nitrogendoped diamonds, which provide the majorsource of natural gemstones, arecategorized as Type I.Optical Engineering of Diamond, First Edition. Edited by Richard P. Mildren and James R. Rabeau. 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.
1 Intrinsic Optical Properties of Diamondalso includes the dependence of optical properties on basic variables such aswavelength, temperature, and isotopic composition. Although the scope is limitedto bulk intrinsic properties, the intention is to stimulate a further expansion of theknowledge base as the limits of measurement resolution and performance areextended, and as more detailed investigations emerge into areas such as surfaceoptics, crystal variants, and nano-optical effects.The chapter focuses on the optical properties spanning from ultraviolet (UV) toinfrared (IR). It should be noted that, throughout the chapter, Système Internationale (SI) units have been used, apart from some exceptions to stay with conventions. The data provided refer to diamond with the naturally occurring isotopicratio, unless specifically stated otherwise.1.1TransmissionDiamond has a wide bandgap and lacks first-order infrared absorption, whichmakes it one of the most broadly transmitting of all solids. As shown in Figure1.1, the transmission spectrum for a diamond window is featureless for wavelengths longer than approximately 225 nm (α 1 cm 1 for λ 235 nm), apart froma moderate absorption in the range 2.6 to 6.2 μm and extending weakly outsideeach side. Indeed, there is no absorption in the long-wavelength limit, which is acharacteristic of the Group IV elements (e.g., Si and Ge) that share the diamond10080Transmission (%)2Transmission Fresnel loss onlyInfrared ngth (μm)Figure 1.1 Transmission spectrum for aType IIa diamond window (“Type IIIa,”Element 6) of 1 mm thickness. The spectrumwas measured using a Cary 5000 spectrometer (UV-near IR) and Bruker Zertex 80( 2 μm; resolution 4 cm 1). The transmissionfor Fresnel loss only (dashed curve) wascalculated using the relation described in thetext and in Equation (1.6). The smalldifference between the dashed and measuredcurves in the regions away from the UV-edgeand lattice absorption is largely attributed tothe combination of spectrometer calibrationerror and scatter in the sample.
1.2 Lattice Absorptionlattice symmetry. UV-edge absorption, infrared lattice absorption and Fresnelreflection dominate the wavelength dependence for transmission. The Fresnelreflection at each diamond–air interface is approximately 17% in the visible(R 17.1% at 632 nm), and when accounting for multiple reflections betweeneach surface this leads to a maximum transmission of (1 R)2/(1 R2) 70.8%.Using dispersion data for the refractive index (see Section 1.4), the transmissionupper limit (no absorption) is shown as a function of wavelength (dashed line inFigure 1.1).1.2Lattice AbsorptionThe absorption in the mid-IR, which is most prominent in the range 2.6 to 6.2 μm,arises due to the coupling of radiation with the movement of nuclei, and is oftenreferred to as “lattice” or “multiphonon” absorption. The magnitude and shape ofthe absorption spectrum is a consequence of the vibrational properties of thecrystal lattice, which are governed by the forces between neighboring atoms andthe symmetry of collective vibrations. The theoretical framework that most successfully describes the spectrum has been developed since the 1940s, stimulatedby the pioneering work of Sir C.V. Raman on diamond’s optical properties andMax Born on the quantum theory of crystals. It is interesting to note that, althoughdiamond’s lattice is one of the most simple, there have been substantial controversies in explaining the spectrum (see e.g., Ref. [4]) and there are on-going challenges to thoroughly explain some of the features.A brief and qualitative summary of the theory of lattice absorption is providedhere to assist in an understanding of the IR spectrum’s dependence on importantmaterial and environmental parameters such as impurity levels, isotopic content,and temperature. A greatly simplifying and important aspect is that there is noone-phonon absorption in pure, defect-free diamond (which would appear moststrongly near 7.5 μm for diamond), as also for other monatomic crystals withinversion symmetry such as Si and Ge. The movement of nuclei in vibrationalmodes of the lattice are countered by equal and opposite movement of neighbors,so that no dipole moment for coupling with radiation is induced. One-phononabsorption may proceed by spoiling the local symmetry through, for example,lattice imperfections (impurities and defects) or by the application of electricfield. Dipole moments may also be induced in the crystal via interaction of theincident photon with more than one phonon, although with reduced oscillatorstrength; this is the origin of lattice absorption in pure diamond. From a classicalviewpoint, the absorption mechanism can be qualitatively understood as onephonon inducing a net charge on atoms, and a second phonon (or more) vibratingthe induced charge to create a dipole moment [5]. The maximum phonon frequency that can be transmitted by the lattice is 1332 cm 1 (which correspondsto the zero-momentum optical phonon and the Raman frequency), and integermultiples of this frequency at 3.75 μm (2665 cm 1) and 2.50 μm (3997 cm 1) mark3
41 Intrinsic Optical Properties of DiamondAbsorption coefficient 1100015002000250030003500400045005000Wave number (cm–1)Figure 1.2 Two-, three-, and four-phononlattice absorption bands. The underlyingfigure showing the calculated (smooth solidcurve) and measured absorption spectra isreprinted with permission from Ref. [6]; 1994, SPIE. The measurements werecollated from several sources, as detailed inthe reference.the short-wavelength limits for two- and three-phonon absorption regions. Thedemarcations between two- and three-phonon absorption are clearly evident in thetransmission spectrum of Figure 1.1 and the logarithmic plot of lattice absorptionin Figure 1.2 [6]. Between wavelengths 3.75 and 6 μm, the lattice absorption atroom temperature is strongest with a peak of approximately 10 cm 1 at 4.63 μm,and is primarily attributable to two-phonon absorption.1.2.1The Two-Phonon RegionAbsorption may involve the creation and destruction of phonons, which are constrained to certain energies and wavevectors as a result of the symmetry andinteratomic forces. For two-phonon creation, the absorption at a given frequencyis proportional to the number of pairs of modes of the created phonons and atransition probability that takes into consideration allowed phonon combinations(e.g., longitudinal or transverse) and the transition oscillator strength. The numberof allowed combinations of a given energy is usually highest for phonon wavevectors along directions of high symmetry in the lattice, and with momenta that
1.2 Lattice Absorptioncorrespond to phonon wavelengths of the order of the atomic spacing in thatdirection (i.e., at the edge of the Brillouin zone). Along with the generally higherdensity of modes at the Brillouin zone edge, the transition probability is alsohigher as the largest charge deformations are induced. For diamond, there is alsoa peak in the density of modes in one symmetry direction ( 110 ) for momentacorresponding to wavevectors at approximately 70% of the Brillouin zone. Thesepeaks in the density of states are the so-called “critical points” in the lattice phonondispersion curves.The primary directions of symmetry for the diamond lattice, along with thefrequency of critical point phonons, are listed in Table 1.1. Also listed are the critical points derived from the relatively recent studies of Vogelsegang et al. [7] andKlein et al. [8], for data derived using a combination of neutron-scattering data,second-order Raman spectra2) [9] and, in the case of Ref. [8], by using impurityspectra to access single phonon information. There remains significant uncertainty in the frequency of many critical points, however, and as a result there isdisagreement between some phonon assignments.For temperatures below 1000 K, the populations of critical-point phonons aresmall (due to their characteristically high energies in diamond), and only phononsummations appear strongly in the spectrum. Momentum–energy conservationand symmetry impose selection rules for the type of phonons created. As thephoton momentum is negligible compared to the Brillouin zone-edge phononsand to conserve crystal momentum, the wavevector of each phonon is equal inmagnitude and opposite in direction. The character of the phonons must alsobe different; that is, they should correspond to different dispersion branches(either optical or acoustic phonon), or have a different polarization (longitudinalor transverse direction). The resulting absorption features are referred to as“combination lines.” Pairs of phonons of the same type (overtones) are absentor weak.Due to the large number of possible phonon modes, the spectrum appears as afairly smooth continuum extending to wavelengths that extend beyond 10 μm.A joint-density-of-states calculation [10] was successful in reproducing the grossfeatures of the two-phonon spectrum, as shown in Figure 1.3, including the broadpeak near 2500 cm 1, the region of highest absorption from 1800–2300 cm 1, thelocal minimum at 2100 cm 1, and the tail at frequencies less than 1750 cm 1. Animproved agreement would require a better knowledge of the dispersion curvesand transition probabilities. Unfortunately, however, there are large uncertaintiesin the phonon dispersion data obtained by neutron scattering data, due to thelack of test samples of sufficient size. The more recently published critical pointvalues [7, 8], obtained with the assistance of optical spectra of variable impurityand isotopic content, have enabled some features to be assigned to critical point2) The second-order Raman scattering involvestwo phonons. The second-order spectrumcontains a peak at twice the Ramanfrequency that is more than two orders ofmagnitude weaker than the first-order peak,and a broad feature extending to lowerfrequencies (see e.g., Ref. [9]).5
61 Intrinsic Optical Properties of DiamondDirections of high symmetry, critical point phonons, the corresponding frequencies and twophonon combinations identified in the IR absorption spectrum. Combinations corresponding to the majorpeaks are shown in bold. L longitudinal, T transverse, O optical, and A acoustic. Note that the use ofL and T labels for the K symmetry points is not conventional and should not be taken as indicating branchesof purely transverse or longitudinal character.Table 1.1Crystal direction (asviewed in perspectivethrough the 4 4unit cell)Crystal direction,K-space symmetrylabelCritical point frequencies (cm 1)Ref. [7]Ref. [8] 100 , X1170 (L)a)1088 (TO)786 (TA)1191 3 (L)a)1072 2 (TO)829 2 (TA)2260 6 L TO (X)1895 6 TO TA (X)c) 110 , K1236 (LO)1112 (TO1)1051 (TO2)b)986 (LA)982 (TA1)b)748 (TA2)1239 2 (LO)1111 1 (TO1)1042 2 (TO2)b)992 3 (LA)978 1 (TA1)b)764 4 (TA2)1977 2 LA TA1 (Σ)2005.5 2 LO TA2 (Σ)2029 2 TO2 (?)d) (Σ)2097 2 TO1 LA (Σ)2160 2 TO1 TO2 (Σ)2293 8 LO TO2 (Σ) 111 , L124512081009572125612201033553 210 , W1164 (L)a)1012 (TO)915 (TA)(LO)(TO)(LA)(TA) 4222(LO)(TO)(LA)(TA)1146 1 (L)a)1019 3 (TO)918 12 (TA)Observed spectrumfeatures in thetwo-phonon region(see Figure 1.4) withassigned phononcombinations (cm 1)1777.5 4 TO TA (L)1816.5 4 TA LO (L)2260 6 LA (?)e) (L)2293 6 LO LA (L)2175 5 L TO (W)a) The LO and LA modes are degenerate.b) “Accidental” critical points occur for the TO2- and TA1-labeled phonons for wavevectors nearby the symmetrypoint.c) Assignment agrees with Ref. [8] data only.d) Ambiguous assignment – Ref. [8] suggests LA, whereas Ref. [7] suggests TA1.e) Ambiguous assignment – Ref. [8] suggests LO, whereas Ref. [7] suggests TO.
1.2 Lattice Absorptionε′′(Arbitrary 00Frequency (cm–1)Figure 1.3 Comparison of two-phononabsorption band with the joint density ofstates (JDS) calculation of Wehner et al. [10],showing qualitative agreement with severalof the main features in the measured25002750spectrum. The stepped appearance of thecalculated spectrum is a consequence of thedigitizing procedure used to sample thephonon dispersion curves. Reproduced withpermission from Ref. [10]; 1967, Elsevier.combinations with greater confidence. The last column in Table 1.1 lists the frequencies of features that can be readily seen in the spectrum of Figure 1.4, alongwith their suggested assignments of the likely critical point phonons. The majorpeaks at 4.63 μm (2160 cm 1), 4.93 μm (2030 cm 1) and 5.06 μm (1975 cm 1) all correspond to phonons in the 110 symmetry direction, for which there is an “accidental” critical point in the dispersion curves for one or both of the phononsinvolved in the assigned combinations. A similar type of analysis can be performed, at least in principle, for the third- and higher-order phonon bands atwavelengths 3.75 μm ( 2665 cm 1); however, such a procedure is very difficultdue to the greatly increased number of phonon combinations, the lack of detailedknowledge on transition probabilities, and the poor visibility of critical pointlocations.1.2.2Absorption at Wavelengths Longer than 5 μmIn the range 5 to 10 μm, lattice absorption decreases approximately exponentiallyfrom 10 cm 1 to approximately 0.05 cm 1. The weaker longer-wavelength absorption results primarily from combination pairs of low-energy acoustic phononsaway from the phonon dispersion critical points. A calculation for multiphononabsorption using polynomial fits to the acoustic phonon densities of states [6],reproduces this trend satisfactorily (as seen Figure 1.2). At wavelengths longerthan 8 μm ( 1250 cm 1), a significant departure of experiment from theory isobserved as the weaker absorption approaches the level of impurity absorptionand the resolution of the measurement. Due to interest in diamond as a window7
81 Intrinsic Optical Properties of DiamondDetail of the two-phononabsorption region (using the data of Figure1.1) with identification of major features inthe spectrum with suggested critical pointFigure 1.4phonon summations (refer to Table 1.1). Thevertical line at 2665 cm 1 corresponds totwice the Raman frequency, and indicates theupper limit for two-phonon absorption.material for missile domes and high-power CO2 lasers, absorption in the longwavelength atmospheric window at 8–12 μm and at 10.6 μm has been studied indepth (see e.g., Refs [6, 11, 12]). Absorptions as low as 0.03 cm 1 at 10.6 μm havebeen measured for single-crystal and polycrystalline material [13]. Intrinsic absorption is expected to decrease monotonically at longer wavelengths ad infinitum dueto the diminishing number of phonon modes and, indeed, low-loss material hasbeen observed up to and beyond 500 μm [13, 14].1.2.3Temperature DependenceTemperature affects lattice absorption via changes in the phonon ambient population density and shifts in phonon mode frequency. The effect of the thermalphonons in the material is to stimulate absorption events coinciding with theincident photon. As described by Lax [15], the relationship between two-phononabsorption coefficient α and temperature is given byα (n1 1)(n2 1)(1.1)where ni are the occupation numbers at thermal equilibrium of the final statephonons (also called Bose–Einstein factors)n i (ω ,T ) (exp[ ω /kBT ] 1) 1(1.2)
1.2 Lattice Absorptionwhere kB is Boltzmann’s constant and ħ is Planck’s constant divided by 2π. Afurther consideration noted by Lax was that of induced emission, which involvesthe annihilation of thermal phonons and the creation of an IR photon at thefrequency sum. This is proportional to n1n2 for two phonons, so that the netabsorption becomesα (n1 1)(n2 1) n1n2 n1 n2 1(1.3)However, as n1n2 can be neglected for temperatures less than approximately1000 K, Equation (1.1) holds for most temperatures of interest. At room temperature and below, the density of thermal phonons is small so that absorption isessentially constant and reflects the spontaneous component (i.e., the componentcontribution caused by quantum fluctuations). At elevated temperatures, however,absorption increases notably (as shown in Figure 1.5a [16]) for temperatures upto 800 K and for wavelengths spanning the two- and three-phonon bands (from2.5 μm to beyond 20 μm). The temperature dependence of a given feature in theabsorption band varies slightly according to the thermal populations of the responsible phonons of the feature combination. Although the agreement with Equations(1.1) and (1.2) above is quite good for some spectral features [17], for others – suchas those near the 4.9 μm peak – the increase in absorption exceeds the prediction.A more thorough treatment would need to consider contributions from higherorder multiphonon processes and also two-phonon difference bands (of frequencyωi – ωj, where the phonon ωj is absorbed and ωi phonon is emitted) which play anincreasingly important role at higher temperatures and longer wavelengths [18].Lattice absorption spectra between 2 and 20 μm for several temperatures up to500 K are shown in Figure 1.
nonlinear optical properties of diamond have not to date been comprehensively reviewed. The aim of this chapter is to do this, with emphasis placed on the intrin-sic properties of single - crystal diamond (i.e., pure Type IIa diamond 1) ). The chapter 1 Optical Engineering of Diamond, First Edition. Edited by Richard P. Mildren and James R. Rabeau.
Fortran Top 95—Ninety-Five Key Features of Fortran 95 vii 48 Integer Type and Constants 96 49 INTENT Attribute and Statement 98 50 Interfaces and Interface Blocks 100 51 Internal Procedures 102 52 INTRINSIC Attribute and Statement 104 53 Intrinsic Function Overview 106 54 Intrinsic Functions: Array 108 55 Intrinsic Functions: Computation 110 56 Intrinsic Functions: Conversion 112
Semiconductor Optical Amplifiers (SOAs) have mainly found application in optical telecommunication networks for optical signal regeneration, wavelength switching or wavelength conversion. The objective of this paper is to report the use of semiconductor optical amplifiers for optical sensing taking into account their optical bistable properties .
A novel all-optical sampling method based on nonlinear polarization rotation in a semiconductor optical amplifier is proposed. An analog optical signal and an optical clock pulses train are injected into semiconductor optical amplifier simultaneously, and the power of the analog light modulates the intensity of the output optical pulse through
Mar 14, 2005 · Background - Optical Amplifiers zAmplification in optical transmission systems needed to maintain SNR and BER, despite low-loss in fibers. zEarly optical regeneration for optic transmission relied on optical to electron transformation. zAll-optical amplifiers provide optical g
1.1 Classification of optical processes 1 1.2 Optical coefficients 2 1.3 The complex refractive index and dielectric constant 5 1.4 Optical materials 8 1.5 Characteristic optical physics in the solid state 15 1.6 Microscopic models 20 Fig. 1.1 Reflection, propagation and trans mission of a light beam incident on an optical medium.
Intrinsic brand fungicides Insignia SC Intrinsic and Honor Intrinsic, belongs to the strobilurin class of fungicides. In addition to excellent, broad spectrum disease control, research has shown pyraclostrobin-based fungicides also provide additional plant health benefits. Pyraclostrobin-based fungicides control foliar fungal diseases by
motivation for all of their daily activities - including reading! Motivation can be intrinsic or extrinsic. Intrinsic motivation: behavior that is driven by internal rewards. Intrinsic motivation is a type of motivation that arises from within the individual because the activity is naturally satisfying to him/her. Examples of intrinsic .
2 Tomlinson, rian and Milano, Gregory Vincent and Yiğit, Alexa and Whately, Riley, The Return on Purpose: efore and during a crisis (October 21, 2020). 3 SEC.gov The Importance of Disclosure –For Investors, Markets and Our Fight Against COVID-19 Hermes Equity Ownership Services Limited 150 Cheapside, London EC2V 6ET United Kingdom 44 (0)20 7702 0888 Phone www.FederatedHermes.com . Hermes .
