APPLICATIONS OF NONLINEAR FIBER OPTICS
CONTENTS FIBER GRATINGS BASIC CONCEPTS BRAGG DIFFRACTION PHOTOSENSITIVITY FABRICATION TECHNIQUES SINGLE-BEAM INTERNAL TECHNIQUE DUAL-BEAM HOLOGRAPHIC TECHNIQUE PHASE MASK TECHNIQUE POINT-BY-POINT FABRICATION TECHNIQUE GRATING CHARACTERISTICS COUPLED-MODE EQUATIONS CW SOLUTION IN THE LINEAR CASE PHOTONIC BANDGAP, OR STOP BAND GRATING AS AN OPTICAL FILTER EXPERIMENTAL VERI?CATION CW NONLINEAR EFFECTS NONLINEAR DISPERSION CURVES OPTICAL BISTABILITY MODULATION INSTABILITY LINEAR STABILITY ANALYSIS EFFECTIVE NLS EQUATION EXPERIMENTAL RESULTS NONLINEAR PULSE PROPAGATION BRAGG SOLITONS RELATION TO NLS SOLITONS FORMATION OF BRAGG SOLITONS NONLINEAR SWITCHING EFFECTS OF BIREFRINGENCE RELATED PERIODIC STRUCTURES LONG-PERIOD GRATINGS NONUNIFORM BRAGG GRATINGS PHOTONIC-CRYSTAL FIBERS PROBLEMS REFERENCES FIBER COUPLERS COUPLER CHARACTERISTICS LOW-POWER OPTICAL BEAMS LINEAR PULSE SWITCHING NONLINEAR EFFECTS QUASI-CW SWITCHING NONLINEAR SUPERMODES ULTRASHORT PULSE PROPAGATION NONLINEAR SWITCHING OF OPTICAL PULSES VARIATIONAL APPROACH COUPLER-PAIRED SOLITONS EXTENSIONS AND APPLICATIONS ASYMMETRIC COUPLERS ACTIVE COUPLERS GRATING-ASSISTED COUPLERS BIREFRINGENT COUPLERS MULTICORE COUPLERS FIBER INTERFEROMETERS FABRY–PEROT AND RING RESONATORS TRANSMISSION RESONANCES NONLINEAR DYNAMICS AND CHAOS ULTRAFAST NONLINEAR EFFECTS SAGNAC INTERFEROMETERS NONLINEAR TRANSMISSION APPLICATIONS MACH–ZEHNDER INTERFEROMETERS NONLINEAR CHARACTERISTICS MICHELSON INTERFEROMETERS FIBER AMPLI?ERS PUMPING AND GAIN COEF?CIENT AMPLI?ER GAIN AND BANDWIDTH AMPLI?ER NOISE ERBIUM-DOPED FIBER AMPLI?ERS GAIN SPECTRUM AMPLI?ER GAIN DISPERSIVE AND NONLINEAR EFFECTS MAXWELL–BLOCH EQUATIONS GINZBURG–LANDAU EQUATION DISTRIBUTED AMPLI?CATION PERIODIC LUMPED AMPLI?CATION NOISE AMPLI?CATION OPTICAL SOLITONS AUTOSOLITONS MAXWELL–BLOCH SOLITONS PULSE AMPLI?CATION PICOSECOND PULSES ULTRASHORT PULSES FIBER LASERS PUMPING AND OPTICAL GAIN CAVITY DESIGN LASER THRESHOLD AND OUTPUT POWER CW FIBER LASERS ND-DOPED FIBER LASERS ERBIUM-DOPED FIBER LASERS OTHER FIBER LASERS SELF-PULSING AND CHAOS SHORT-PULSE FIBER LASERS PHYSICS OF MODE LOCKING ACTIVE MODE LOCKING HARMONIC MODE LOCKING OTHER TECHNIQUES PASSIVE MODE LOCKING SATURABLE ABSORBERS NONLINEAR FIBER-LOOP MIRRORS NONLINEAR POLARIZATION ROTATION HYBRID MODE LOCKING OTHER MODE-LOCKING TECHNIQUES ROLE OF FIBER NONLINEARITY AND DISPERSION SATURABLE-ABSORBER MODE LOCKING ADDITIVE-PULSE MODE LOCKING SPECTRAL SIDEBANDS POLARIZATION EFFECTS PULSE COMPRESSION PHYSICAL MECHANISM GRATING-FIBER COMPRESSORS GRATING PAIR OPTIMUM COMPRESSOR DESIGN PRACTICAL LIMITATIONS SOLITON-EFFECT COMPRESSORS COMPRESSOR OPTIMIZATION HIGHER-ORDER NONLINEAR EFFECTS FIBER BRAGG GRATINGS GRATINGS AS A COMPACT DISPERSIVE ELEMENT GRATING-INDUCED NONLINEAR CHIRP BRAGG-SOLITON COMPRESSION CHIRPED-PULSE AMPLI?CATION DISPERSION-DECREASING FIBERS COMPRESSION MECHANISM OTHER COMPRESSION TECHNIQUES CROSS-PHASE MODULATION GAIN-SWITCHED SEMICONDUCTOR LASERS OPTICAL AMPLI?ERS FIBER COUPLERS AND INTERFEROMETERS FIBER-OPTIC COMMUNICATIONS SYSTEM BASICS LOSS MANAGEMENT DISPERSION MANAGEMENT STIMULATED BRILLOUIN SCATTERING BRILLOUIN THRESHOLD CONTROL OF SBS STIMULATED RAMAN SCATTERING RAMAN CROSSTALK POWER PENALTY SELF-PHASE MODULATION SPM-INDUCED FREQUENCY CHIRP LOSS AND DISPERSION MANAGEMENT XPM-INDUCED PHASE SHIFT FOUR-WAVE MIXING FWM EF?CIENCY FWM-INDUCED CROSSTALK SYSTEM DESIGN NUMERICAL MODELING DESIGN ISSUES SYSTEM PERFORMANCE SOLITON LIGHTWAVE SYSTEMS PROPERTIES OF SOLITONS SOLITON BIT STREAM SOLITON INTERACTION EFFECT OF FIBER LOSS LOSS-MANAGED SOLITONS LUMPED AMPLI?CATION CHIRPED SOLITONS ASE-INDUCED FLUCTUATIONS TIMING JITTER CONTROL OF TIMING JITTER DISPERSION-MANAGED SOLITONS PERIODIC DISPERSION MAPS WDM SOLITON SYSTEMS INTERCHANNEL COLLISIONS EFFECT OF LUMPED AMPLI?CATION CHAPTER BASIC CONCEPTS DIFFRACTION GRATINGS CONSTITUTE A STANDARD OPTICAL COMPONENT AND ARE USED ROU- TINELY IN VARIOUS OPTICAL INSTRUMENTS SUCH AS A SPECTROMETER THE UNDERLYING PRINCIPLE WAS DISCOVERED MORE THAN 200 YEARS AGO [1] FROM A PRACTICAL STAND- POINT, A DIFFRACTION GRATING IS DE?NED AS ANY OPTICAL ELEMENT CAPABLE OF IM BRAGG DIFFRACTION THE DIFFRACTION THEORY OF GRATINGS SHOWS THAT WHEN LIGHT IS INCIDENT AT AN AN- GLE ?I (MEASURED WITH RESPECT TO THE PLANES OF CONSTANT REFRACTIVE INDEX), IT IS DIFFRACTED AT AN ANGLE ?R SUCH THAT [1] SIN?I SIN?R = M? =(¯N?); (1 1 1) WHERE ? IS THE GRATING PERIOD, ? =¯N IS THE WAVELENGTH OF LIGHT INSIDE THE MEDIUM WITH AN AVERAGE REFRACTIVE INDEX ¯N, AND M IS THE ORDER OF BRAGG DIFFRACTION THIS CONDITION CAN BE THOUGHT OF AS A PHASE-MATCHING CONDITION, SIMILAR TO THAT OCCURRING IN THE CASE OF BRILLOUIN SCATTERING OR FOUR-WAVE MIXING, AND CAN BE WRITTEN AS KI KD = MKG ; (1 1 2) WHERE KI AND KD ARE THE WAVE VECTORS ASSOCIATED WITH THE INCIDENT AND DIFFRACTED LIGHT THE GRATING WAVE VECTOR KG HAS MAGNITUDE 2? =? AND POINTS IN THE DI- RECTION IN WHICH THE REFRACTIVE INDEX OF THE MEDIUM IS CHANGING IN A PERIODIC MANNER IN THE CASE OF SINGLE-MODE ?BERS, ALL THREE VECTORS LIE ALONG THE ?BER AXIS AS A RESULT, KD = KI AND THE DIFFRACTED LIGHT PROPAGATES BACKWARD THUS, AS SHOWN SCHEMATICALLY IN FIG 1 1, A ?BER GRATING ACTS AS A RE?ECTOR FOR A SPECI?C WAVELENGTH OF LIGHT FOR WHICH THE PHASE-MATCHING CONDITION IS SATIS?ED IN BASIC CONCEPTS TERMS OF THE ANGLES APPEARING IN EQ (1 1 1), ?I = ? =2 AND ?R = ? =2 IF M = 1, THE PERIOD OF THE GRATING IS RELATED TO THE VACUUM WAVELENGTH AS ? = 2¯N? THIS CONDITION IS KNOWN AS THE BRAGG CONDITION, AND GRATINGS SATISFYING IT ARE REFERRED TO AS BRAGG GRATINGS PHYSICALLY, THE BRAGG CONDITION ENSURES THAT WEAK RE?ECTIONS OCCURRING THROUGHOUT THE GRATING ADD UP IN PHASE TO PRODUCE A STRONG RE?ECTION FOR A ?BER GRATING RE?ECTING LIGHT IN THE WAVELENGTH REGION NEAR 1 5 µM, THE GRATING PERIOD ? 0:5 µM BRAGG GRATINGS INSIDE OPTICAL ?BERS WERE ?RST FORMED IN 1978 BY IRRADIATING A GERMANIUM-DOPED SILICA ?BER FOR A FEW MINUTES WITH AN INTENSE ARGON-ION LASER BEAM [2] THE GRATING PERIOD WAS ?XED BY THE ARGON-ION LASER WAVE- LENGTH, AND THE GRATING RE?ECTED LIGHT ONLY WITHIN A NARROW REGION AROUND THAT WAVELENGTH IT WAS REALIZED THAT THE 4% RE?ECTION OCCURRING AT THE TWO ?BER–AIR INTERFACES CREATED A STANDING-WAVE PATTERN AND THAT THE LASER LIGHT WAS ABSORBED ONLY IN THE BRIGHT REGIONS AS A RESULT, THE GLASS STRUCTURE CHANGED IN SUCH A WAY THAT THE REFRACTIVE INDEX INCREASED PERMANENTLY IN THE BRIGHT REGIONS ALTHOUGH THIS PHENOMENON ATTRACTED SOME ATTENTION DURING THE NEXT 10 YEARS [3]–[15], IT WAS NOT UNTIL 1989 THAT ?BER GRATINGS BECAME A TOPIC OF INTENSE INVESTIGATION, FUELED PARTLY BY THE OBSERVATION OF SECOND-HARMONIC GENERATION IN PHOTOSENSI- TIVE ?BERS THE IMPETUS FOR THIS RESURGENCE OF INTEREST WAS PROVIDED BY A PHOTOSENSITIVITY THERE IS CONSIDERABLE EVIDENCE THAT PHOTOSENSITIVITY OF OPTICAL ?BERS IS DUE TO DEFECT FORMATION INSIDE THE CORE OF GE-DOPED SILICA ?BERS [28]–[30] AS MENTIONED IN SECTION A 1 2, THE ?BER CORE IS OFTEN DOPED WITH GERMANIA TO INCREASE ITS REFRACTIVE INDEX AND INTRODUCE AN INDEX STEP AT THE CORE-CLADDING INTERFACE THE GE CONCENTRATION IS TYPICALLY 3–5% FIBER GRATINGS THE PRESENCE OF GE ATOMS IN THE ?BER CORE LEADS TO FORMATION OF OXYGEN- DE?CIENT BONDS (SUCH AS SI–GE, SI–SI, AND GE–GE BONDS), WHICH ACT AS DEFECTS IN THE SILICA MATRIX [49] THE MOST COMMON DEFECT IS THE GEO DEFECT IT FORMS A DEFECT BAND WITH AN ENERGY GAP OF ABOUT 5 EV (ENERGY REQUIRED TO BREAK THE BOND) SINGLE-PHOTON ABSORPTION OF 244-NM RADIATION FROM AN EXCIMER LASER (OR TWO-PHOTON ABSORPTION OF 488-NM LIGHT FROM AN ARGON-ION LASER) BREAKS THESE DEFECT BONDS AND CREATES GEE 0 CENTERS EXTRA ELECTRONS ASSOCIATED WITH GEE 0 CENTERS ARE FREE TO MOVE WITHIN THE GLASS MATRIX UNTIL THEY ARE TRAPPED AT HOLE-DEFECT SITES TO FORM COLOR CENTERS KNOWN AS GE(1) AND GE(2) SUCH MODI?- CATIONS IN THE GLASS STRUCTURE CHANGE THE ABSORPTION SPECTRUM ? (? ) HOWEVER, CHANGES IN THE ABSORPTION ALSO AFFECT THE REFRACTIVE INDEX SINCE ?? AND ?N ARE RELATED THROUGH THE KRAMERS–KRONIG RELATION [51] ?N(? ) = C ? Z ? (1 1 3) EVEN THOUGH ABSORPTION MODI?CATIONS OCCUR MAINLY IN THE ULTRAVIOLET REGION, THE REFRACTIVE INDEX CAN CHANGE EVEN IN THE VISIBLE OR INFRARED REGION MORE- OVER, SINCE INDEX CHANGES OCCUR ONLY IN THE REGIONS OF ?BER CORE WHERE THE ULTRAVIOLET LIGHT IS ABSORBED, A PERIODIC INTENSITY PATTERN IS TRANSFORMED INTO AN INDEX GRATING TYPICALLY, INDEX CHANGE ?N IS 4 IN THE 1 3- TO 1 6- µM WAVELENGTH RANGE, BUT CAN EXCEED 0 001 IN ?BERS WITH HIGH GE CONCENTRA- TION [34] THE PRESENCE OF GEO DEFECTS IS CRUCIAL FOR PHOTOSENSITIVITY TO OCCUR IN OPTICAL ?BERS HOWEVER, STANDARD TELECOMMUNICATION ?BERS RARELY HAVE MORE THAN 3% OF GE ATOMS IN THEIR CORE, RESULTING IN RELATIVELY SMALL INDEX CHANGES THE USE OF OTHER DOPANTS SUCH AS PHOSPHORUS, BORON, AND ALUMINUM CAN EN- HANCE THE PHOTOSENSITIVITY (AND THE AMOUNT OF INDEX CHANGE) TO SOME EXTENT, BUT THESE DOPANTS ALSO TEND TO INCREASE ?BER LOSSES IT WAS DISCOVERED IN THE EARLY 1990S THAT THE AMOUNT OF INDEX CHANGE INDUCED BY ULTRAVIOLET ABSORPTION CAN BE ENHANCED BY TWO ORDERS OF MAGNITUDE (?N > 0:01) BY SOAKING THE ?BER IN HYDROGEN GAS AT HIGH PRESSURES (200 ATM) AND ROOM TEMPERATURE [39] THE DENSITY OF GE–SI OXYGEN-DE?CIENT BONDS INCREASES IN HYDROGEN-SOAKED ?BERS BECAUSE HYDROGEN CAN RECOMBINE WITH OXYGEN ATOMS ONCE HYDROGENATED, THE ?BER NEEDS TO BE STORED AT LOW TEMPERATURE TO MAINTAIN ITS PHOTOSENSITIVITY HOWEVER, GRATINGS MADE IN SUCH ?BERS REMAIN INTACT OVER LONG PERIODS OF TIME, INDICATING THE NEARLY PERMANENT NATURE OF THE RESULTING INDEX CHANGES [46] HYDROGEN SOAKING IS COMMONLY USED FOR MAKING ?BER GRATINGS FABRICATION TECHNIQUES SINGLE-BEAM INTERNAL TECHNIQUE IN THIS TECHNIQUE, USED IN THE ORIGINAL 1978 EXPERIMENT [2], A SINGLE LASER BEAM, OFTEN OBTAINED FROM AN ARGON-ION LASER OPERATING IN A SINGLE MODE NEAR 488 NM, IS LAUNCHED INTO A GERMANIUM-DOPED SILICA ?BER THE LIGHT RE?ECTED FROM THE NEAR END OF THE ?BER IS THEN MONITORED THE RE?ECTIVITY IS INITIALLY ABOUT 4%, AS EXPECTED FOR A ?BER–AIR INTERFACE HOWEVER, IT GRADUALLY BEGINS TO INCREASE WITH TIME AND CAN EXCEED 90% AFTER A FEW MINUTES WHEN THE BRAGG GRATING IS COMPLETELY FORMED [4] FIGURE 1 2 SHOWS THE INCREASE IN RE?ECTIVITY WITH TIME, OBSERVED IN THE 1978 EXPERIMENT FOR A 1-M-LONG ?BER HAVING A NUMERICAL APERTURE OF 0 1 AND A CORE DIAMETER OF 2 5 µM MEASURED RE?ECTIVITY OF 44% AFTER 8 MINUTES OF EXPOSURE IMPLIES MORE THAN 80% RE?ECTIVITY OF THE BRAGG GRATING WHEN COUPLING LOSSES ARE ACCOUNTED FOR GRATING FORMATION IS INITIATED BY THE LIGHT RE?ECTED FROM THE FAR END OF THE ?BER AND PROPAGATING IN THE BACKWARD DIRECTION THE TWO COUNTERPROPAGAT- ING WAVES INTERFERE AND CREATE A STANDING-WAVE PATTERN WITH PERIODICITY ? =2¯N, WHERE ? IS THE LASER WAVELENGTH AND ¯N IS THE MODE INDEX AT THAT WAVELENGTH THE REFRACTIVE INDEX OF SILICA IS MODI?ED LOCALLY IN THE REGIONS OF HIGH INTENSITY, DUAL-BEAM HOLOGRAPHIC TECHNIQUE THE DUAL-BEAM HOLOGRAPHIC TECHNIQUE, SHOWN SCHEMATICALLY IN FIG 1 3, MAKES USE OF AN EXTERNAL INTERFEROMETRIC SCHEME SIMILAR TO THAT USED FOR HOLOGRAPHY TWO OPTICAL BEAMS, OBTAINED FROM THE SAME LASER (OPERATING IN THE ULTRAVIOLET REGION) AND MAKING AN ANGLE 2? ARE MADE TO INTERFERE AT THE EXPOSED CORE OF AN FABRICATION TECHNIQUES FIGURE 1 3 SCHEMATIC ILLUSTRATION OF THE DUAL-BEAM HOLOGRAPHIC TECHNIQUE OPTICAL ?BER [16] A CYLINDRICAL LENS IS USED TO EXPAND THE BEAM ALONG THE ?BER LENGTH SIMILAR TO THE SINGLE-BEAM SCHEME, THE INTERFERENCE PATTERN CREATES AN INDEX GRATING HOWEVER, THE GRATING PERIOD ? IS RELATED TO THE ULTRAVIOLET LASER WAVELENGTH ?UV AND THE ANGLE 2? MADE BY THE TWO INTERFERING BEAMS THROUGH THE SIMPLE RELATION ? = ?UV =(2SIN? ): (1 2 1) THE MOST IMPORTANT FEATURE OF THE HOLOGRAPHIC TECHNIQUE IS THAT THE GRAT- ING PERIOD ? CAN BE VARIED OVER A WIDE RANGE BY SIMPLY ADJUSTING THE ANGLE ? (SEE FIG 1 3) THE WAVELENGTH ? AT WHICH THE GRATING WILL RE?ECT LIGHT IS RELATED TO ? AS ? = 2¯N? SINCE ? CAN BE SIGNI?CANTLY LARGER THAN ?UV, BRAGG GRATINGS OPERATING IN THE VISIBLE OR INFRARED REGION CAN BE FABRICATED BY THE DUAL-BEAM HOLOGRAPHIC METHOD EVEN WHEN ?UV IS IN THE ULTRAVIOLET REGION IN A 1989 EXPERIMENT, BRAGG GRATINGS RE?ECTING 580-NM LIGHT WERE MADE BY EXPOS- ING THE 4 4-MM-LONG CORE REGION OF A PHOTOSENSITIVE ?BER FOR 5 MINUTES WITH 244-NM ULTRAVIOLET RADIATION [16] RE?ECTIVITY MEASUREMENTS INDICATED THAT THE REFRACTIVE INDEX CHANGES WERE 5 IN THE BRIGHT REGIONS OF THE INTERFERENCE PATTERN BRAGG GRATINGS FORMED BY THE DUAL-BEAM HOLOGRAPHIC TECHNIQUE WERE STABLE AND REMAINED UNCHANGED EVEN WHEN THE ?BER WAS HEATED TO ÆC BECAUSE OF THEIR PRACTICAL IMPORTANCE, BRAGG GRATINGS OPERATING IN THE 1 55- µM REGION WERE MADE IN 1990 [17] SINCE THEN, SEVERAL VARIATIONS OF THE BASIC TECHNIQUE HAVE BEEN USED TO MAKE SUCH GRATINGS IN A PRACTICAL MANNER AN INHERENT PROBLEM FOR THE DUAL-BEAM HOLOGRAPHIC TECHNIQUE IS THAT IT REQUIRES AN ULTRAVIOLET LASER WITH EXCELLENT TEMPORAL AND SPATIAL COHERENCE EXCIMER LASERS COMMONLY USED FOR THIS PURPOSE HAVE RELATIVELY POOR BEAM QUALITY AND FIBER GRATINGS REQUIRE SPECIAL CARE TO MAINTAIN THE INTERFERENCE PATTERN OVER THE ?BER CORE OVER A DURATION OF SEVERAL MINUTES IT TURNS OUT THAT HIGH-RE?ECTIVITY ?BER GRATINGS CAN BE WRITTEN BY USING A SINGLE EXCIMER LASER PULSE (WITH TYPICAL DURATION OF 20 NS) IF THE PULSE ENERGY IS LARGE ENOUGH [31]–[34] EXTENSIVE MEASUREMENTS ON GRATINGS MADE BY THIS TECHNIQUE INDICATE A THRESHOLD-LIKE PHENOMENON NEAR A PULSE ENERGY LEVEL OF ABOUT 35 MJ [34] FOR LOWER PULSE ENERGIES, THE GRATING IS RELATIVELY WEAK SINCE INDEX CHANGES ARE ONLY ABOUT 5 BY CONTRAST, INDEX CHANGES OF ABOUT 3 ARE POSSIBLE FOR PULSE ENERGIES ABOVE 40 MJ BRAGG GRATINGS WITH NEARLY 100% RE?ECTIVITY HAVE BEEN MADE BY USING A SINGLE 40-MJ PULSE AT THE 248-NM WAVELENGTH THE GRATINGS REMAINED STABLE AT TEMPERATURES AS HIGH AS