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OPTICS BY THE NUMBERS L’Ottica Attraverso i Numeri Michael Scalora U.S. Army Research, Development, and Engineering Center Redstone Arsenal, Alabama, 35898-5000 & Universita' di Roma "La Sapienza" Dipartimento di Energetica Rome, April-May 2004 BPM:Propagation in Planar Waveguides Retarded Coordinate trasformation: time dependence, Raman scattering, self-phase modulation in PCFs Study the transmissive properties of guided modes. 1.5 5 m air core Index of Refraction 1.4 14 m 1.3 Propagation into the page 1.2 1.1 1.0 -15 -10 -5 0 5 10 15 Transverse Coordinate (m) fig.(4) n E 4 Pnl E 2 2 2 c t c t 2 2 2 2 2 Pnl E ( z, x, t ) E ( z, x, t )e i ( kz t ) (3) 2 E E c.c. 2 2 2 2 2 E E n ( x ) E 2 i n ( x ) E 2 2 2 E 2 2ik 2 k 2 n ( x) E 2 2 z z c t c t c 2 2 4 (3) 2 2 2 2i E E c t t 2 2 2 2 E E n ( x ) E 2 i n ( x) E 2 E 2 2ik 2 2 z z c t c2 t 2 2 2 2 4 (3) 2 2 k 2 n ( x) E 2 2 2i E E c c t t Assuming steady state conditions… n ( x) n E i 2 E i F n0 2 2 0 (3) 0 2 4 0 E i E E in n0 in 2 2 n ( x ) n E i 2 4 (3) 0 2 0 0 E i E i E E F n0 in n0 in 4 n0 0 F in z / 0 Fresnel Number k n0 c F F small Wave front does not distort: Plane Wave propagation Diffraction is very important 2 2 (3) n ( x ) n E i 2 4 0 2 0 0 E i E i E E F n0 in n0 in This equation is of the form: E HE Where: 2 2 n ( x ) n i 2 4 (3) 0 2 0 0 H i i E D V F n0 in n0 in E ( , x) e H ( ', x ) ' 0 E (0, x) e Using the split-step BPM algorithm H (0, x ) E (0, x) eV (0, x ) / 2e D eV (0, x ) / 2 E (0, x) Example: Incident angle is 5 degrees 1.50 air guide ~ 5m 0.8 1.00 0.4 b=1m a=1.4m 0.75 -10 Intensity glass; n=1.42 glass; n=1.42 1.25 1.2 air glass; n=1.42 glass; n=1.42 Index of Refraction air -8 -6 -4 -2 0 2 4 6 8 0 10 Transverse Coordinate (m) Assume 3=0 The cross section along x renders the problem one-dimensional in nature x 1.5 Transverse Index Profile 1.4 1.3 1.2 1.1 1.0 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 Transverse Position (microns) 4 5 6 7 8 Transmissive properties in the linear (low intensity) regime For two different fibers. We set 3=0 1.0 Normalized Transmittance 14-micron core 5-micron core 0.8 0.6 0.4 0.2 0 0.5 0.6 0.7 0.8 0.9 (m) 1.0 1.1 1.2 Field bouncing back and forth from structure's walls Inpout Field Profile 1.5 1.25 1.0 1.00 0.5 Intensity Index of Refraction 1.50 0.75 -10 -8 -6 -4 -2 0 2 4 Transverse Coordinate (m) 6 8 0 10 1.0 0.8 00. .7 5 0.6 0.4 1.0.1 1 Field tuning corresponds to High transmission state. 0.7 1.00.8 0.2 0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.0 1.1 (m) 300 1.1 1.0 8 00..7 0.7 1.00 .8 1.1 0.51.0 1.1 0.8 200 Direction of propagation 1.0 0.8 z(m) 1.0 0.8 8 0. 400 14-micron core 5-micron core Normalized Transmittance 500 0.5 00.8 .7 1.1 1.0 100 10.0.8 1.10.7 1.1 0 -12 -10 -8 -6 -4 -2 -0 x (m) 2 4 6 8 10 12 Same as previous figure. 1.0 Normalized Transmittance 14-micron core 5-micron core 500 0.8 0.6 0.4 0.2 0 0.5 400 0.6 0.7 0.8 0.9 1.0 1.1 1.2 (m) z ( m) 300 0.1 0.1 200 0.1 0.1 1 0. 0.2 0. 2 0.2 0.1 0.1 1 0. 0 -12 -10 -8 -6 -4 0.1 -2 -0 x ( m) 0.2 2 0.1 100 4 0.2 6 2 0. .1 0 8 10 12 Same as previous figure. 1.0 0.8 1.0.1 1 x x / 0 0.0125 z / 0 0.025 0.7 1.00.8 1.0 1.1 N 200000 1.1 1.0 8 00..7 N x 4096 1.00 .8 1.1 0.51.0 1.1 0.8 200 5-mm guide 1.0 0.8 z(m) 300 00. .7 5 8 0. 400 For the example discussed: 0.7 500 00.8 .7 1.1 1.0 0.5 ~ 8 minutes on this laptop 100 3.2GHz, 1Gbts RAM 10.0.8 1.10.7 1.1 0 -12 -10 -8 -6 -4 -2 -0 x (m) 2 4 6 8 10 12 n ( x) n E i 2 E i F n0 2 2 0 (3) 0 2 4 0 E i E E in n0 in If (3) is non-zero, the refractive index is a function of the local intensity. Solutions are obtained using the same algorithm but with a nonlinear potential. Optical Switch 1.5 5 m Index of Refraction 1.4 air core 1.3 14 m 1.2 1.1 1.0 -15 -10 -5 0 5 10 15 Transverse Coordinate (m) fig.(4) (3) 1.0 non-zero linear transmittance normalized transmittance 0.8 0.6 0.4 0.2 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 scaled frequency (1/ where is in microns) The band shifts because the location and the width of each gap depends on the exact values of n2 and n1, and on their local difference. Normalized Transmittance 1.0 0.8 0.6 0.4 0.2 0 0.65 0.70 0.75 0.80 0.85 0.90 (m) 0.95 1.00 1.05 1.10 fig.(5a) Optical Switch 1.0 on Transmittance 0.8 0.6 0.4 Nonlinear Transmittance 0.2 Linear Transmittance off 0 0.715 0.720 0.725 (m) 0.730 0.735 0.740 fig.(5b) 1.0 on Core Energy 0.8 0.6 0.4 0.2 off 0 0 1000 2000 3000 4000 5000 Longitudinal Position (m) fig.(6) 2 2 2 2 E E n ( x ) E 2 i n ( x) E 2 E 2 2ik 2 2 z z c t c2 t 2 2 2 2 4 (3) 2 2 k 2 n ( x) E 2 2 2i E E c c t t z t z/v v c / n0 1 z z z v 2 2 1 2 2 2 2 2 2 z v v Retarded coordinate Transformation t t t 2 2 2 2 t N.B.:An implicit and important assumption we have made is that one can go to a retarded coordinate provided the grating is shallow so that a group velocity can be defined unumbiuosly and uniquely. 2 2 2 2 E E n ( x ) E 2 i n ( x) E 2 E 2 2ik 2 2 z z c t c2 t 2 2 2 2 4 (3) 2 2 k 2 n ( x) E 2 2 2i E E c c t t z t z/v v c / n0 In other words, the effect of the grating on the group velocity is scaled away into an effective group velocity v. It is obvious that care should be excercised at every step when reaching conclusions, in order to properly account for both material index and modal dispersion, if the index discontinuity is large. 2 2 1 2 n0 2 E 2 2 2 E 2i n0 E v c c v 2 2 2 n2 2 2i n 2 2 2 E 2 E k 2 n E c c c 2 4 (3) 2 2 2 2i E E c 2 Symplifying and Dropping all Higher order Derivatives… 2 2 4 (3) 2 2 2 2 E 2i n0 E k 2 n E 2 2i E E c c c i i 2 F x 2 2 2 n ( x ) n 0 n0 4 i n0 2 2 2 * (3) 4 (3) 0 2 i n0 in 0 in 2 * * * 2 * 2 2 2 2 (3) n ( x ) n i 2 4 0 2 0 0 H i i E F n0 in n0 in 4 2 (3) 0 * * i 2 n0 in Now we look at the linear regime, by injecting a beam inside the guide from the left and then from the right. On-Axis Intensity as Beam Propagates Down the Guide. Beam is Guided. Output Field Profile in the case Light is Guided. On-Axis Intensity as Beam Propagates Down the Guide. Beam is Tuned to a Minimum of Transmission, and is Not Guided, and energy Quickly Dissipates Away. Output Field Profile in the case Light is Not Guided. 1.0 0.8 Input Spectrum ON-AXIS 0.6 0.4 0.2 0 0.80 2n2 I max in L c I 1013 W/cm2 n2 510-19 cm2/W L 8 cm 100 fs 0.82 0.84 0.86 Output Spectrum 0.88 0.90 0.92 0.94 0.96 0.98 1.00 /0 Propagating from left to right the pulse is tuned on the red curve, igniting self-phase modulation, and the spectral shifts indicated on the graph. A good portion of the input energy is transmitted. Spectra are to scale. Fig. 4 Linear transmittance for 2 slightly different guides 1.0 Transmittance 0.8 0.6 0.4 0.2 Output spectrum Input spectrum 0 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 scaled frequency (=1/ where is in m) Propagation from right to left does not induce nonlinearities because the light quickly dissipates. The pulse is tuned with respect to the blue curve. Spectra are to scale. Fig. 4 Initial pulse profile Final profile Spectrum of the pulse as it propagates. Note splitting. 1000 Initial profile power spectrum 800 600 400 200 0 0.65 0.70 0.75 scaled frequency (1/) 0.80 Self-phase modulation A process whereby new frequencies (or wavelengths) are generated such that: Example: input 100fs pulse at 800nm is broadened by ~30nm 2n2 I max in L c I max Intensity, arb. units 1,0 0,5 0,0 400 500 600 700 , nm 800 900 I t Stimulated Raman Scattering p p stokes anti stokes E p 2 i Ep * i Q E e QEs A 2 F x EA i 2 EA i QE e P F x 2 ES i 2 ES * Q EP 2 F x Q Q ES* EP QE A EP* e i The simplest case stokes E p 2 i Ep QEs 2 F x ES i 2 ES * Q EP 2 F x Q Q ES* EP Raman Soliton: A sudden relative phase shift between the pump and the Stokes at the input field generates a “phase wave”, or soliton, a temporary repletion of the pump at the expense of the Stokes intensity The simplest case stokes E p 2 i Ep QEs 2 F x ES i 2 ES * Q EP 2 F x Q Q ES* EP 0 Es Es (0, ) 0 Es 0 0 The Input Stokes field undergoes a -phase shift The gain changes sign temporarily, For times of order 1/; The soliton is the phase wave Intensity at cell output 2500 INTENSITY 2000 The Pump signal is temporarily repleted 1500 The Stokes minimum is referred to as a Dark Soliton 1000 PUMP INTENSITY STOKES INTENSITY 500 0 0 0.02 0.04 TIME 0.06 0.08 PUMP FIELD z,0,0 z,=L,0 z=0 PUMP FIELD STOKES FIELD TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT PUMP FIELD TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT STOKES FIELD ON-AXIS INTENSITY AT CELL OUTPUT 3000 INTENSITY F= F=20 2000 1000 0 0 0.02 0.04 0.06 0.08 TIME The onset of diffraction causes the soliton to decay… …almost as expected. Except that… ON-AXIS INTENSITY AT CELL OUTPUT STOKES INTENSITY 5000 F= F=20 4000 3000 2000 1000 0 0 0.02 0.04 0.06 0.08 TIME … the Stokes field undergoes significant replenishement on its axis, as a result of nonlinear self focusing ON-AXIS INTENSITY PROFILE PUMP FIELD ON-AXIS INTENSITY PROFILE STOKES FIELD TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT PUMP FIELD TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT STOKES FIELD Poisson Spot like effect Examples: single slit 0.8 intensity 0.6 0.4 0.2 0 -40 -30 -20 -10 0 10 transverse coordinate 20 30 40 Direction of Propagation