Overview of transverse effects in nonlinear-optical systems

Beams of light interacting with nonlinear-optical materials naturally form complex transverse profiles. Self-focusing, filamentation, modulational instabilities, spontaneous pattern formation and symmetry breaking, nonlinear waveguiding, and transverse mode excitation are but some of the many phenomena that are observed in lasers, in optical bistability, and in the free interaction of beams. As an introduction to a special issue of the Journal of the Optical Society of America B on transverse effects in nonlinear-optical systems, we provide this overview of history and current work.

[1]  G. Ernst,et al.  Mode structure of active resonators , 1973 .

[2]  Jerome V Moloney,et al.  Fixed Points and Chaotic Dynamics of an Infinite Dimensional Map , 1987 .

[3]  Lugiato,et al.  Spatial and temporal instabilities in a CO2 laser. , 1989, Physical review letters.

[4]  Hermann Haken,et al.  Analogy between higher instabilities in fluids and lasers , 1975 .

[5]  A. Godone,et al.  Routes to chaotic emission in a cw He-Ne laser , 1983 .

[6]  D. R. Heatley,et al.  Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides , 1990 .

[7]  N. Lawandy,et al.  Bistable direction switching in an off-axis pumped continuous wave ruby laser , 1988 .

[8]  H. Risken,et al.  Fokker-Planck equation, distribution and correlation functions for laser noise , 1966 .

[9]  The kinetics of the hysteresis change of the beamprofile in non-linear interferometers , 1981 .

[10]  D. Franzen,et al.  Radial gain profiles in CO 2 laser discharges , 1972 .

[11]  G. J. Ernst,et al.  Gain induced stability of active plane-parallel resonators , 1974 .

[12]  Hyatt M. Gibbs,et al.  Role of Diffractive Coupling and Self-Focusing or Defocusing in the Dynamical Switching of a Bistable Optical Cavity , 1982 .

[13]  B. Luther-Davies,et al.  Single-mode resonator incorporating an internal multimode optical fiber and a phase-conjugate reflector , 1990 .

[14]  Nasser N Peyghambarian,et al.  Continuous-wave off-resonance rings and continuous-wave on-resonance enhancement , 1984 .

[15]  J. Zondy,et al.  Linear analysis of single-feedback-mirror spatiotemporal instabilities , 1990 .

[16]  Nikolay N. Rosanov,et al.  Diffractive autosolitons in nonlinear interferometers , 1990 .

[17]  C. Paré,et al.  Transverse Modulational Instabilities in Kerr Media , 1990 .

[18]  K. Shore.  Non-linearities in laser diodes , 1988 .

[19]  Hyatt M. Gibbs,et al.  Optical bistability in semiconductors , 1979 .

[20]  Govind P. Agrawal,et al.  Transverse modulation instability of copropagating optical beams in nonlinear Kerr media , 1990 .

[21]  R. Lefever,et al.  Spatial dissipative structures in passive optical systems. , 1987, Physical review letters.

[22]  Bandy,et al.  Role of transverse effects in laser instabilities. , 1988, Physical review. A, General physics.

[23]  G. Magyar,et al.  Laser Q-switching by organic solvents , 1969 .

[24]  J. Buus.  Models of the static and dynamic behavior of stripe geometry lasers , 1983 .

[25]  Michael D. Duncan,et al.  Control of transverse spatial modes in transient stimulated Raman amplification , 1990 .

[26]  H. Risken,et al.  Self‐Pulsing in Lasers , 1968 .

[27]  Mario Dagenais,et al.  Low power transverse optical bistability near bound excitons in cadmium sulfide , 1984 .

[28]  H. Haken.  Theory of intensity and phase fluctuations of a homogeneously broadened laser , 1966 .

[29]  Herwig Kogelnik,et al.  On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss or Gain Variation , 1965 .

[30]  L. Hesselink,et al.  Conical emissions and phase conjugation in atomic sodium , 1989, Annual Meeting Optical Society of America.

[31]  Harrison,et al.  Experimental evidence of three-mode quasiperiodicity and chaos in a single longitudinal, multi-transverse-mode cw CO2 laser. , 1985, Physical review. A, General physics.

[32]  J. Dyment,et al.  Resonant modes of GaAs junction lasers - III: Propagation characteristics of laser beams with rectangular symmetry , 1970 .

[33]  H. Kimble,et al.  Observation of absorptive bistability with two-level atoms in a ring cavity , 1983 .

[34]  Brambilla,et al.  Multimode instability in optical bistability. , 1989, Physical review. A, General physics.

[35]  K. Shore,et al.  Analysis of a nonlinear rib-structure semiconductor optical waveguide , 1989 .

[36]  L W Casperson,et al.  Gaussian light beams in inhomogeneous media. , 1973, Applied optics.

[37]  H. Weber,et al.  Simple mathematical model describing multitransversal solid-state lasers , 1990 .

[38]  Injection‐Laser Far‐Field Patterns with Gaussian Profiles in the Junction Plane , 1968 .

[39]  L. Lugiato,et al.  Spontaneous breaking of the cylindrical symmetry in lasers , 1989 .

[40]  Lambertus Hesselink,et al.  Degenerate conical emissions in atomic-sodium vapor , 1990 .

[41]  V. M. Dwyer,et al.  The role of diffraction in dispersive optical bistability , 1985 .

[42]  Second threshold condition in the case of Q switching by self-focusing , 1986 .

[43]  R. Boyd,et al.  Four-wave mixing resonantly enhanced by ac-Stark-split levels in self-trapped filaments of light , 1984 .

[44]  Leonid A. Melnikov,et al.  Transverse pattern formation and spectral characteristics of cw light beams in resonant media: an improved numerical simulation technique and mode analysis , 1990 .

[45]  T. Dziura,et al.  Beyond mean field and plane wave theories of bistable semiconductor lasers , 1986 .

[46]  C. Tamm,et al.  Spontaneous breaking of cylindrical symmetry in an optically pumped laser , 1990 .

[47]  L. Lugiato,et al.  Transverse effects and self-pulsing in optical bistability , 1983 .

[48]  Sargent,et al.  Quantum theory of multiwave mixing. III. Averages over inhomogeneous broadening, spatial hole burning, and Gaussian beams. , 1985, Physical review. A, General physics.

[49]  Y. Liu,et al.  Theoretical and experimental studies of conical stokes emission , 1988 .

[50]  W. Firth,et al.  Modelling of cross talk in 2-D bistable arrays with carrier diffusion , 1987 .

[51]  K. Kohiyama,et al.  Self-locking of transverse higher-order modes in a He-Ne laser , 1968 .

[52]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[53]  D. Auston,et al.  Forced and spontaneous phase locking of the transverse modes of a He-Ne laser , 1968 .

[54]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[55]  Michael D. Feit,et al.  Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams , 1988 .

[56]  George I. Stegeman,et al.  Nonlinear codirectional guided wave mode conversion in grating structures , 1988 .

[57]  M. Sargent,et al.  Effects of Gaussian-beam averaging on phase conjugation and beat-frequency spectroscopy , 1984 .

[58]  K. Shore.  Amplification properties of dynamic instabilities: Hopf bifurcation in semiconductor lasers , 1988 .

[59]  K. Shore.  Optical and electronic control of Hopf bifurcation in twin-stripe semiconductor lasers , 1987 .

[60]  K. Shore.  Optical limiter action in twin-stripe-geometry lasers , 1982 .

[61]  C. Paré,et al.  Transverse instabilities due to counterpropagation in Kerr media , 1990 .

[62]  Luigi A. Lugiato,et al.  Cooperative frequency locking and stationary spatial structures in lasers , 1988 .

[63]  Transverse Effects in Nonlinear Optics Introduction , 1990 .

[64]  J. R. Morris,et al.  Perturbation growth by thermal blooming in turbulence , 1990 .

[65]  H. Mackenzie,et al.  Cross talk between bistable elements on an InSb étalon , 1988 .

[66]  G. Grynberg,et al.  Spatial Instability for a Standing Wave in a Nonlinear Medium , 1989 .

[67]  Quantitative test of the single-mode theory of optical bistability , 1987 .

[68]  Yuen-Ron Shen,et al.  Self-focusing: Experimental , 1975 .

[69]  T. Rozzi,et al.  Switching frequency for transverse modes in stripe-geometry injection lasers , 1983 .

[70]  David A. B. Miller,et al.  Optoelectronic applications of quantum wells , 1990 .

[71]  E. Wright.  Quantum theory of self-phase modulation , 1990 .

[72]  N. Yu,et al.  Picosecond optical properties of a grating surface emitting two-dimensional coherent laser array , 1989, IEEE Photonics Technology Letters.

[73]  William J. Firth,et al.  Spatial solitary wave optical memory , 1990, Annual Meeting Optical Society of America.

[74]  Eitan Abraham,et al.  Effect of pixelation on the switching speeds of InSb bistable elements , 1990 .

[75]  V. V. Sobolev,et al.  Behavior of Light Beams in Nonlinear Media , 1971 .

[76]  David A. B. Miller,et al.  Optical bistability and signal amplification in a semiconductor crystal: applications of new low‐power nonlinear effects in InSb , 1979 .

[77]  G. Ernst,et al.  The effect of radial radiation transport on intensity characteristics and oscillation frequency of homogeneously broadened lasers , 1974 .

[78]  Gian-Luca Oppo,et al.  Instabilities and spatial complexity in a laser , 1990 .

[79]  Wavefront measurements on semiconductor lasers , 1983 .

[80]  M. G. Galushkin,et al.  BRIEF COMMUNICATIONS: Self-interaction of radiation in a CO2 laser resonator , 1985 .

[81]  G. Ernst,et al.  On the saturation effects and start jump of Gaussian modes in oscillators , 1975 .

[82]  E. Arimondo,et al.  Microwave Dispersive Bistability in a Confocal Fabry-Perot Microwave Cavity , 1981 .

[83]  Gian-Luca Oppo,et al.  Characterization of Spatiotemporal Structures in Lasers: A Progress Report , 1989 .

[84]  L. Lugiato,et al.  Effects of the Radial Variation of the Electric Field on Some Instabilities in Optical Bistability and Lasers , 1984 .

[85]  J. Sipe,et al.  Solid–melt patterns induced on silicon by a continuous laser beam at nonnormal incidence , 1990 .

[86]  L. Ronchi,et al.  Self-guiding flashlamp-pumped dye lasers. , 1975, Applied optics.

[87]  J. Butler,et al.  The effect of device geometry on lateral mode content of stripe geometry lasers , 1979 .

[88]  Gibbs,et al.  Gain-feedback approach to optical instabilities in sodium vapor. , 1988, Physical review letters.

[89]  I. Galbraith,et al.  Diffusive transverse coupling of bistable elements - Switching waves and crosstalk , 1985, IEEE Journal of Quantum Electronics.

[90]  S. N. Liu,et al.  Route to mode locking in a three-mode He-Ne 3.39-μm laser including chaos in the secondary beat frequency , 1983 .

[91]  Lugiato,et al.  Single-mode and multimode instabilities in lasers and related optical systems. , 1985, Physical review. A, General physics.

[92]  Tamm.  Frequency locking of two transverse optical modes of a laser. , 1988, Physical review. A, General physics.

[93]  Two-dimensional periodic structures in a nonlinear resonator , 1990 .

[94]  R. Collins,et al.  A NEW METHOD OF GIANT PULSING RUBY LASERS , 1968 .

[95]  Michiharu Nakamura,et al.  Effects of lateral mode and carrier density profile on dynamic behaviors of semiconductor lasers , 1978 .

[96]  D. K. Bandy,et al.  Spatial pattern formation in lasers and cooperative frequency locking , 1988, Annual Meeting Optical Society of America.

[97]  C. Weiss,et al.  Oscillation period doubling chaos in a laser , 1982 .

[98]  K. D. De Valois,et al.  Orientation and spatial-frequency discrimination for luminance and chromatic gratings. , 1990, Journal of the Optical Society of America. A, Optics and image science.

[99]  H. Winful,et al.  Stability of phase locking in coupled semiconductor laser arrays , 1988 .

[100]  L. Casperson,et al.  Beam deflection and isolation in laser amplifiers , 1974 .

[101]  H. Maeda.  Amplification of Gaussian beams through a gaseous medium , 1976 .

[102]  Observation of beats between transverse modes in ruby lasers , 1963 .

[103]  Instabilities and chaotic behavior in a hybrid bistable system with a short delay , 1983 .

[104]  G. Giusfredi,et al.  Optical instabilities in sodium vapor , 1988 .

[105]  M. Belić,et al.  Calculation of transverse effects in optical bistability using fast Fourier transform techniques , 1982 .

[106]  Anthony E. Siegman,et al.  Nonlinear-optical calculations using fast-transform methods: Second-harmonic generation with depletion and diffraction , 1980 .

[107]  H. Gamo,et al.  Pulse propagation in a partially homogeneously broadened linear gas laser amplifier: Determination of line‐shape parameters , 1975 .

[108]  H. Winful,et al.  Spectral and picosecond temporal properties of flared guide Y‐coupled phase‐locked laser arrays , 1988 .

[109]  T. Rozzi,et al.  Intrinsic instabilities in narrow stripe geometry lasers caused by lateral current spreading , 1981, IEEE Journal of Quantum Electronics.

[110]  Robert L. Varwig,et al.  Competition of Longitudinal and Transverse Modes in a CW HF Chemical Laser , 1980 .

[111]  D. Welch,et al.  Observation of sustained self-pulsation in CW operated flared Y-coupled laser arrays , 1988 .

[112]  H. M. Gibbs,et al.  Differential Gain and Bistability Using a Sodium-Filled Fabry-Perot Interferometer , 1976 .

[113]  I. Freund.  SELF‐Q‐SWITCHING IN RUBY LASERS , 1968 .

[114]  J. Moloney,et al.  Instability of standing waves in nonlinear optical waveguides , 1986 .

[115]  Formation and dynamics of spatial structures in increasing-absorption bistable schemes , 1991 .

[116]  Influence of the refractive index nonlinearity on the dynamics of emission from semiconductor lasers , 1972 .

[117]  Kam Y. Lau,et al.  Bistability and pulsations in semiconductor lasers with inhomogeneous current injection , 1982 .

[118]  L. Lugiato,et al.  Effects of Gaussian-beam averaging on laser instabilities , 1985 .

[119]  Gibbs,et al.  cw conical emission: First comparison and agreement between theory and experiment. , 1990, Physical review letters.

[120]  T. Rozzi,et al.  Stability analysis of transverse modes in stripe - geometry injection lasers , 1981 .

[121]  K. Shore,et al.  Asymmetric semiconductor waveguide with defocusing nonlinearity , 1988 .

[122]  Neal B. Abraham,et al.  I Dynamical Instabilities and Pulsations in Lasers , 1988 .

[123]  M. G. Evtikhov,et al.  NONLINEAR OPTICS: Self-focusing effects in a bistable semiconductor interferometer , 1990 .

[124]  Weiss,et al.  Spatiotemporal chaos from a continuous Na2 laser. , 1989, Physical review. A, General physics.

[125]  Rose,et al.  Conjecture on the dimensions of chaotic attractors of delayed-feedback dynamical systems. , 1987, Physical review. A, General physics.

[126]  Firth Wj,et al.  Optical memory and spatial chaos. , 1988 .

[127]  John P. Goldsborough,et al.  Beat Frequencies Between Modes of a Concave-Mirror Optical Resonator , 1964 .

[128]  Colin J. McKinstrie,et al.  Nonlinear focusing of coupled waves. , 1988, Physical review letters.

[129]  J. Fleck,et al.  Coupled‐Mode Laser Oscillation , 1964 .

[130]  D. R. Heatley,et al.  Numerical calculations of spatially localized wave emission from a nonlinear waveguide: two-level saturable media , 1990 .

[131]  McLaughlin,et al.  New class of instabilities in passive optical cavities. , 1985, Physical review letters.

[132]  P. Milonni,et al.  Chaotic two-mode lasing , 1984 .

[133]  L. A. Melnikov,et al.  Nonlinear dynamics of beam parameters and intensity in a unidirectional ring laser with a homogeneously broadened line , 1990 .

[134]  P. Cheo,et al.  Gain characteristics of CO 2 laser amplifiers at 10.6 microns , 1967 .

[135]  Jerome V. Moloney,et al.  Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity , 1983 .

[136]  R. Ballagh,et al.  Dispersive self-focusing in atomic media , 1988 .

[137]  G. Stephan,et al.  Inhomogeneity effects in a gas laser , 1983 .

[138]  R. Pratesi,et al.  Graded-gain dye laser amplifiers: an experimental investigation. , 1975, Applied optics.

[139]  R. Hauck,et al.  Chaotic and periodic emission of high power solid state lasers , 1983 .

[140]  H. Kimble,et al.  Intrinsic Dynamical Instability in Optical Bistability with Two-Level Atoms , 1984 .

[141]  P. Drummond,et al.  Optical bistability in a radially varying mode , 1981 .

[142]  F. Nash,et al.  Gain−induced guiding and astigmatic output beam of GaAs lasers , 1975 .

[143]  S. Seshadri,et al.  Excitation of nonlinear surface polaritons supported by a metal–vacuum interface , 1990 .

[144]  I. Golub,et al.  Light focusing and guiding by an axicon-pair-generated tubular light beam , 1990 .

[145]  Stegeman,et al.  Multisoliton emission from a nonlinear waveguide. , 1986, Physical review. A, General physics.

[146]  G. Grynberg,et al.  Mirrorless four-wave mixing oscillation in atomic vapors , 1988 .

[147]  D. McLaughlin,et al.  Modulational-induced optical pattern formation in a passive optical-feedback system , 1990 .

[148]  C. Weiss.  Observation of instabilities and chaos in optically pumped far-infrared lasers , 1985 .

[149]  H. Maeda,et al.  Theory of a gas laser with a Gaussian field profile , 1975 .

[150]  H. Gibbs,et al.  Coherent On-Resonance Self-Focusing of Optical Pulses in Absorbers , 1976 .

[151]  E. Wright,et al.  Diffusion Effects in Bistable Optical Arrays , 1985, Topical Meeting on Optical Bistability (OB3).

[152]  B. Jiang,et al.  Optical–optical double-resonance multiphoton-ionization spectroscopy of ammonia-d 3 . I An investigation of the Jahn–Teller effect for the B˜ 1 E ″ vibronic levels , 1990 .

[153]  P. W. Smith,et al.  Reflection of a Gaussian beam at a nonlinear interface. , 1982, Applied optics.

[154]  N. Byer,et al.  Optical field distribution in close-confined laser structures , 1970 .

[155]  A. V. Grigor'yants,et al.  Multiple-order spatial hysteresis and radial variation of output in an indium antimonide bistable cavity , 1990 .

[156]  K. A. Shore,et al.  Static and dynamic bifurcations in semiconductor lasers for device applications , 1987 .

[157]  A. Yariv,et al.  Gain and dispersion focusing in a high gain laser. , 1972, Applied Optics.

[158]  Lorenzo M. Narducci,et al.  Spontaneous Oscillations In A Single Mode Co2 Laser In A Fabry-Perot Cavity , 1986, Other Conferences.

[159]  D. Weaire,et al.  Theory of spin-flip laser modes - I: Threshold modulation and tuning behavior , 1976, IEEE Journal of Quantum Electronics.

[160]  N. V. Vyssotina,et al.  Instability of guided-wave transverse structure in layered nonlinear systems , 1990 .

[161]  Martial Ducloy,et al.  Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium , 1988 .

[162]  R. S. Gioggia,et al.  Experimental evidence for self-pulsing and chaos in CW-excited lasers , 1983 .

[163]  D. Weaire,et al.  Dynamics of optical bistability: a simple model , 1990 .

[164]  J. W. Bennett,et al.  Hole Burning Effects in a He-Ne Optical Maser , 1962 .

[165]  H. Gibbs,et al.  Effects of transverse beam variation on bifurcations in an intrinsic bistable ring cavity , 1982 .

[166]  F. R. Nash,et al.  Mode guidance parallel to the junction plane of double-heterostructure GaAs lasers , 1973 .

[167]  W. J. Sandle,et al.  Optical bistability in a gaussian cavity mode , 1981 .

[168]  E. Chauchard,et al.  On the origin of the so called conical emission in laser pulse propagation in atomic vapor , 1984 .

[169]  M. Milani,et al.  Disappearance of laser instabilities in a Gaussian cavity mode , 1983 .

[170]  Jin-yue Gao,et al.  Higher-order bifurcations in a bistable system with delay , 1984 .

[171]  Paul Anthony Kirkby,et al.  Observations of self-focusing in stripe geometry semiconductor lasers and the development of a comprehensive model of their operation , 1977 .

[172]  A. Yariv,et al.  The Gaussian mode in optical resonators with a radial gain profile , 1968 .

[173]  C. Jung,et al.  Single-longitudinal-mode laser as a discrete dynamical system , 1985 .

[174]  Masanori Koshiba,et al.  Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems , 1990 .

[175]  W. Firth.  Optically bistable arrays and chaotic dynamics , 1987 .

[176]  E. Wright,et al.  Theory of Gaussian-beam optical bistability , 1982 .

[177]  Jan P. Allebach,et al.  Quantization and multilevel halftoning of color images for near-original image quality , 1990 .

[178]  R. Lefever,et al.  Phase dynamics of transverse diffraction patterns in the laser , 1989 .

[179]  G. Broggi,et al.  Spatial Symmetry Breaking in Optical Systems , 1988 .

[180]  L. Gagnon.  Exact solutions for optical wave propagation including transverse effects , 1990 .

[181]  H. Weber,et al.  Quasiperiodicity versus chaos in high power solid state lasers in multitransversal mode operation , 1990 .

[182]  L. Casperson.  Mode stability of lasers and periodic optical systems , 1974 .

[183]  C. McKinstrie,et al.  Transverse modulational instability of collinear waves , 1990 .

[184]  Space-temporal behavior of a light pulse propagating in a nonlinear nondispersive medium , 1986 .

[185]  Gibbs,et al.  High-dimension chaotic attractors of a nonlinear ring cavity. , 1986, Physical review letters.

[186]  E. Wright,et al.  BEAM PROPAGATION IN A MEDIUM WITH A DIFFUSIVE KERR-TYPE NONLINEARITY. , 1985 .

[187]  R. Nabiev,et al.  Angle-dependent nonlinear modes of a cylindrical waveguide , 1990 .

[188]  L. Hesselink,et al.  Sodium phase-conjugate oscillator in a combusting environment. , 1987, Optics letters.

[189]  Hermann Haken,et al.  Neural and Synergetic Computers , 1988 .

[190]  Colin J. McKinstrie,et al.  The modulational instability of coupled waves , 1989 .

[191]  E. Wright,et al.  Diffusion and diffraction in dispersive optical bistability , 1985 .

[192]  C. Paré,et al.  Transverse modulational instabilities for counterpropagating beams in Kerr media. , 1988, Optics letters.

[193]  C. O. Weiss,et al.  Bistability and optical switching of spatial patterns in a laser , 1990 .

[194]  H. Haus,et al.  Lamb dip in CO lasers , 1973 .

[195]  P. Wolff,et al.  Measurement of radial variation of 3.51-µm gain in xenon discharge tubes , 1977 .

[196]  K. Ikeda.  Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system , 1979 .

[197]  D. Auston,et al.  Transverse mode locking , 1968 .

[198]  R. Lefever,et al.  Instabilities in Passive Optical Systems: Temporal and Spatial Patterns , 1988 .

[199]  Peter W. E. Smith.  SIMULTANEOUS PHASE‐LOCKING OF LONGITUDINAL AND TRANSVERSE LASER MODES , 1968 .

[200]  Yu. A. Rzhanov,et al.  Spatial structure multistability in an array of optically bistable elements , 1989 .

[201]  A. Schmidt,et al.  UNDAMPED REGULAR SPIKING OF A RUBY LASER IN A MISALIGNED CAVITY , 1968 .

[202]  Ajoy K. Kar,et al.  Optical bistability in InSb at room temperature with two‐photon excitation , 1983 .

[203]  Excitation Mechanisms and Current Dependence of Population Inversion in He-Ne Lasers , 1963 .

[204]  Y. Silberberg,et al.  Stability of optical resonators with an active medium. , 1975, Applied optics.