Lecture 6. (Some) nonlinear optics in fibers and fiber components. Walter Margulis. Laserphysics KTH / Fiber optics Acreo

Transcription

1 Lecture 6 (Some) nonlinear optics in fibers and fiber components Walter Margulis Laserphysics KTH / Fiber optics Acreo

2 Rio, today

3 Lecture 6 Too much to be covered Qualitative approach

4 Lecture 6 Too much to be covered Qualitative approach you may remember afterwards easier for me

5 A word about Acreo (Fiber Optics) Fiber solutions to industrial problems

6 High precision machining Pallet Chuck

7 High precision machining Low cost interferometer D D N max ( N) 2

8 MEFOS laboratory: blast furnace Steel industry ~10% CO2 emission in Sweden, ~1.5 ton CO2 / ton steel Measurements of temperature and distribution in raceway in blast furnace Experimental blast furnace (MEFOS, Luleå)

9 High temperature sensing Planck radiation from hot material Spectrometer Silica fiber

10 Fiber production Research Development Production Hermetic fibers Harsh environments Microstructured fibers Large core fibers Specialty solutions

11 Making fibers

12 How to make an optical fiber Heat the preform until it softens and then draw it as a fiber

13 How to make an optical fiber Start with a fiber preform = a 1000 times expanded copy of the fiber 12 cm diameter preform: 9.2 km fiber/cm Reseach fiber 25 mm 1.6 mm

14 Raw material: silica tubes and gases

15 Preform fabrication

16 Liquid doping Oxidation (400 o C, 10 min) Drying (800 o C, 60 min) Sintering (1800 o C) Collapse (2200 o C) Doping solution (e.g., ErCl3)

17 Preform collapse

18 Drawing fibers

19 Ready for use (or cabling)

20 Fiber anatomy

21 Exemples of special fibers made at Acreo Microstructured fibers

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24 NLO in fiber Best waveguide, no diffraction, propagation Fiber loss P t = P o exp (-αl) At 1.5 µm: 3 db after 15 km α = 2 x 10-5 /cm

25 Nonlinear coefficient in silica extremely low! n 2 = 3.2 x m 2 /W Figure of merit: Intensity x Interaction length = Enhancement in a fiber: 10 7 (0.53 µm) and 10 9 (1.5 µm) Improvement: highly nonlinear fibers 1) Small radius (I = P/A eff ) (e.g., heavy doping, PCF, nanowire) 2) Special index profile (e.g., depressed cladding) 3) Large n 2 (nonlinear coefficient) (Pb, Ba, Te, Chalcogenides)

26 Refractive index of bulk silica (material dispersion) Group velocity maximum

27 Refractive index of bulk silica (material dispersion) n Normal dispersion Anomalous dispersion n decreases with ω n increases with λ Normal: n blue >n red Normal dispersion λ ω ωα/c ω o ω ω Refractive index is well described far from resonances by: Sellmeier equation

28 Phase velocity: c/n(ω) Short pulse Wide spectrum Envelope: group v g v Normal: v > v g Anomalous: v g > v β 2 : group velocity dispersion (GVD) parameter ~20 ps 2 /km

29 D β 2 Dispersion: Walk-off

30 Problem: describe pulse propagation in fiber knowing input pulse E (z=0, t) Answer: determine E / z (i.e., how E evolves along z) E = - B/ t H = J f + D/ t D = ρ f = 0 B = 0 Constitutive equations: D = ε o E + P B = µ o H + M D, H electric and magnetic flux P describes the response of the material to the presence of an electric field Eliminating magnetic terms B and H ( E) = - / t B = - / t ( µ o ( E) = - 1/c 2 2 E / t 2 - μ o 2 P / t 2 ( E) = ( E) - H ) = - µ o 2 (ε / t 2 o E + P) 2 E

31 ( E) = - 1/c 2 2 E / t 2 - μ o 2 P / t 2 ( E) = ( E) - 2 E Pulse propagation equation 2 E - 1/c 2 2 E/ t 2 = μ o 2 P/ t 2

32 To solve for P one needs Taylor expansion: quantum mechanics. Far from resonances, P = ε o [ χ (1) E + χ (2) E E + χ (3) E E E + ] Electric dipole approximation (terms such as B E, E. E, etc are neglected) Pulse propagation equation 2 E - 1/c 2 2 E/ t 2 = μ o 2 P/ t 2 P L = ε o [ χ (1) E]

33 Low amplitude of electric field (no nonlinearity) P = P L = ε o [ χ (1) E] Unidimensional case: Let E(z,t) = ½ [E(z)exp(iωt) + E * (z)exp(-iωt)] 2 E/ z 2-1/c 2 2 E/ t 2 = μ o ε o χ (1) 2 E/ t 2 2 E/ z 2 + [1 + χ (1) ] ω 2 /c 2 E = 0 Dielectric constant, depends on ω ε (ω) = (n + iαc/2ω) 2 n = refractive index α = loss Low loss α ~ 0 2 E/ z 2 + n 2 ω 2 /c 2 E = 0

34 Unidimensional case: 2 E/ z 2 + n 2 ω 2 /c 2 E = 0 General case (low power): 2 E/ x E/ y E/ z 2 + n 2 ω 2 /c 2 E = 0 Solution: Separate variables and get rid of x and y E (r,t) = F(x,y) A(z,t) exp [i(βz-ωt)] x

35 Solution: Boundary conditions lead to the appearance of modes F(x,y) are Bessel functions, combined as HE mn and EH mn LP 01 LP 11

36 2 E/ z 2-1/c 2 2 E/ t 2 = μ o 2 P/ t 2 E (r,t) = F(x,y) A(z,t) exp [i(βz-ωt)] x Slowly varying envelope approximation 2 E / z 2 = 2 A/ z 2 + 2iβ o A/ z + (β 2 -β o2 )A. Assume χ (2) = 0 F F F x x x

37 P = o (1) E + o (3) E.E.E Let E tot = ½ [E (z) exp (iωt) + E * (z) exp (-iωt)] E.E.E = 1/8E 3 exp 3(iωt) + 3/8EE * E exp (iωt) + 3/8E*EE* exp (-iωt) + 1/8E *3 exp3(-iωt) THG E 2 = I E.E.E = 1/8 E 3 exp (i 3ω t) + 3/8 I E exp (iωt) + cc Ignoring 3ω for now E tot The nonlinear term can be expressed as a correction to n P = o (1) E + (3/8 o (3) I.) E Refractive index depends on intensity: SPM (n 0 + n 2 I) n 2 ~ 3/8 χ (3) / n

38 Redefine n n = n o + n 2 I Join the driving term 2 P / t 2 into the field term 2 E / t 2 In the slowly varying envelope approximation A/ z + β 1 A/ t + i/2 β 2 2 A/ t 2 + αa/2 = + i γ A 2 A Redefine time origin (travel with pulse referential) i A/ z = -iαa/2 + 1/2 β 2 2 A/ T 2 - γ A 2 A γ = n 2 ω o /ca eff nonlinear coefficient of the fiber (in STF γ ~2/W km) β 1 = 1/v G β 2 = GVD parameter

39 i A/ z = -iαa/2 + 1/2 β 2 2 A/ T 2 - γ A 2 A Absorption Dispersion Nonlinearity How do we estimate the importance of these effects? (Govind rules ok!) L D = T o 2 / β 2 Dispersion length L NL = 1/γP o Nonlinear length Nonlinear Schroedinger equation (when α~0) i A/ z = 1/2 β 2 2 A/ T 2 - γ A 2 A

40 Four regimes: 1) L<<L D and L<<L NL no dispersion, no nonlinearity 2) L>L D and L<<L NL dispersion, no nonlinearity 3) L<<L D and L>L NL no dispersion, nonlinearity 4) L>L D and L>L NL dispersion, nonlinearity Case 1) L D = T o2 /β 2 large: long pulse (or low dispersion) L NL =1/γP o large: low power (or low nonlinearity) i U/ z = ± 1/2L D 2 U/ t 2 1/L NL exp(-αz) U 2 U i A/ z = -iαa/2 and thus A = A o exp(- αz)

41 Case 2) No nonlinearity (intensity or γ too low) dispersion, no nonlinearity GVD governs propagation: i U/ z = β 2 /2 2 U/ T 2 Solution The phase depends on the frequency (and propagated distance) Red and blue are phase shifted by different amounts The pulse broadens in time The spectrum remains the same

42 Case 3) No dispersion but SPM; L<<L D and L>L NL U/ z = i/l NL exp(-αz) U 2 U Typically for SMF: L NL ~1 km (at 1W power) Solution: U gains a nonlinear phase along z U(z,T) = U(0,T) exp(iφ NL (z,t)) where Φ NL (z,t) = 1/L NL U(0,T) 2 z eff The shape of the pulse does not change under SPM only U(z,T) 2 = U(0,T) 2 exp(iφ NL (z,t) exp(-iφ NL (z,t)

43 More intuitive: Pulse = cos (ω o t-kz) Instant phase = (ω o t-kz) = (ω o t-2πnz/λ) Instant frequency Φ/ t = ω o ω o -2πn 2 z/λ di/dt if n = n o if n = n o + n 2 I back ω+ δω ω ω o ω - δω front Time Time Front becomes red shifted Tail becomes blue shifted Linear chirp where pulse is most intense Positive chirp: frequency grows Square pulses only have chirp during ramping

44 Qualitatively: Why oscillations? ω Time Same frequency, different times Interference

45 Sum-up: SPM broadens spectrum of unchirped pulse New frequencies appear on front and trailing edges The pulse becomes less coherent In the absence of dispersion, no change in pulse shape in time time SPM time λ λ What happens to a chirped pulse when it propagates under a regime dominated by SPM? If the pulse is chirped to start with, the spectrum can narrow due to SPM

46 Case 4) SPM and Dispersion; L>L D and L>L NL i U/ z = ± 1/2L D 2 U/ t 2 1/L NL exp(-αz) U 2 U i U/ Z = ± 1/2 2 U/ t 2 N 2 exp(-αz) U 2 U where Z = z/l D and t = T/T o N 2 = L D /L NL = γp o T o 2 / β 2 N>>1, SPM prevails N<<1, dispersion dominates The sign of β 2 is decisive: - the effects of dispersion and SPM add to each other (normal regime) + the effects of dispersion and SPM can cancel each other (anomalous) Soliton!

47 Normal regime, β 2 positive (D negative) Pulse broadens, spectrum broadens

48 Anomalous regime, β 2 negative (D positive) Pulse and spectrum reach a steady state Pulse shape U(t) = sech (t)

49 Dispersion Demetrios Christodoulides No dispersion Soliton

50 Self-regulating mechanism: N 2 = L D /L NL = γp o T o 2 / β 2 = 1 Attenuation reduces power: SPM drops, dispersion dominates, Pulse broadens Power needed for N = 1 drops, N=1 maintained Typically: 1 ps pulse broadens to 100 ps after 5 km due to dispersion 1 ps soliton broadens to 2 ps due to attenuation

51 Cross-phase modulation Refractive index at λ 1 is affected by the presence of pulse at λ 2 P NL (ω 1 ) = χ eff ( E E 2 2 )E 1 P NL (ω 2 ) = χ eff ( E E 1 2 )E 2 P NL (2ω 1 - ω 2 ) = χ eff E 2 1 E 2 * P NL (2ω 2 - ω 1 ) = χ eff E 2 2 E 1 * Parametric mixing, four-photon mixing Second harmonic generation

52 Parametric amplification

53 SH Power (mw) SHG: Optical poling P 2ω ~ E ω E ω E rec Nd:YAG laser Fiber SH IR E-3 1E-4 1E-5 1E-6 1E Time (minutes)

54 Stimulated Raman scattering (Blillouin ) Energy is lost to vibrations (in silica peak ~440 cm -1 ) Shift from 1.06 µm to 1.12 µm, and then 1.18 µm, 1.24 µm Shift at 1.48 µm is to 1.58 µm At room temperature, most atoms are in their vibrational ground state Laser excites vibrations (Stokes) Laser de-excites vibrations extremely unlikely (no anti-stokes peak seen)

55 Intrapulse Raman scattering (soliton self-frequency shift)

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57 Ready for use (or cabling)

58 Fiber anatomy

59 Fiber cables

60 Cleaving 125 µm

61 Holder Press Holder Metal blade for scribing Push Fiber

62 Splicing Splicing station Splice protection Hand-held

63 Fiber 1 Holder Fiber 2 Holder Arc discharge

64 Connectors and collimators Alumina ferrule Fiber Fiber pigtail Grin lens Collimated light

65 Microbench Fiber components Miniature bulk optics (polarizer, isolator, filter)

66 Fiber coupler Beam splitter Beam combiner WDM coupler Polarization splitter Optical tap

67 Fiber coupler E. Pone et al, 12, 2909 (2004) E. Pone et al, Opt Exp 12, 1036 (2004)

68 Fabrication Stripped region Twisted fibers Fuse and draw

69 Microscope Step motors Fiber holder Heating element

70 Fabrication

71 How it works - Do not quote me!

72 Coupling theory Supermodes: In 1 Out 1 2 Out 2 Anti-symmetrical Symmetrical

73 Power coupling between two parallel WG 16 um Power transfer between cores along device

74 When to stop coupler fabrication 3 db WDM What is bad about these couplers?

75 Novel designs 7x1 MM combiner MM: 94% transmission 3x3 3x3 Novel designs PM Combiners Higher port count 10x1 PM 6+1x1 19x1

76 Concatenated splitters 1x8 tree splitter fabricated by concatenating 1x2 splitters Useful in passive optical networks

77 WDM coupler

78 Polarization coupler/splitter P-polarization S-polarization

79 Polarization splitter/combiner Long fiber coupler where each polarization couples to one fiber Polarization splitter/combiner Polarization splitter/combiner Extinction ratio >20 db Can combine pumps for laser

80 Mechanical polarization controllers Mickey mouse polarization controllers λ/4, λ/2, λ/4 plates

81 Fiber polarizer Typical extinction ratio: 40 db

82 Optical activity Rotation of linearly polarized light in chiral materials (e.g., sugar, quartz, spin polarized glasses) Glucose rotates light to the right Fructose rotates light to the left H R V Circular birefringence Δn = n R -n L Faraday effect: magnetic field induces optical activity L

83 Faraday rotators Material properties required: High Verdet constant (magneto-optical) Low absorption coefficient High damage threshold Low nonlinear response (prevent self-focusing, SPM) Typical materials: Terbium doped borosilicate glass, Tb gallium garnet ( µm) Yttrium Iron Garnet (YIG) ( µm) Typical isolation: >30 db

84 Faraday rotator Microbench Faraday rotator

85 Faraday rotator mirror Reflected light is rotated by 90 o in relation to input light everywhere in the fiber

86 Faraday isolator Faraday rotator Faraday rotator

87 Polarization insensitive Faraday isolator Faraday rotator Faraday rotator

88 Optical circulator Principle of operation 2 1 3

89 Fiber circulator US$ 2000 miracle

90 Arrayed waveguide gratings (AWG) Useful in Wavelength Division Multiplexing Separates (combines) channels of different wavelengths

91 AWG

92 AWG performance (datasheet)

93 Conclusions 1- Nonlinear optics in fibers is a vast and interesting research field 2- Nonlinear Scroedinger equation describes propagation 3- Many effects are dominated by the index dependence on lambda 4- Nonlinearity is given by the polarization of silica in response to E 5- When dispersion dominates, the pulse spreads in time 6- When nonlinearity (SPM) dominates, the pulse spreads spectrally 7- In the anomalous regime, dispersion and nonliearity can cancel

94 Conclusions 8- Solitons are robust and self-adjusting 9- Many processes not covered 10- Fiber components are very mature and have high performance 11- Generally, Asian components cost 1/5 to 1/10th 12- An isolator costs 1000x more than a diode

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