From 191d81af3b7a56563b599710106aa30bc4193a2a Mon Sep 17 00:00:00 2001 From: Lorenzo Date: Tue, 14 Jun 2016 14:04:48 +0200 Subject: [PATCH] prima versione tex + nuove immagini --- relazione/FitSHG_exp.eps | 3346 +++++++++++++++++++++++ relazione/FitSHG_exp.pdf | Bin 0 -> 13883 bytes relazione/FitSHG_sim.eps | 5542 ++++++++++++++++++++++++++++++++++++++ relazione/FitSHG_sim.pdf | Bin 0 -> 29911 bytes relazione/kerr.tex | 150 ++ 5 files changed, 9038 insertions(+) create mode 100755 relazione/FitSHG_exp.eps create mode 100755 relazione/FitSHG_exp.pdf create mode 100755 relazione/FitSHG_sim.eps create mode 100755 relazione/FitSHG_sim.pdf create mode 100755 relazione/kerr.tex diff --git a/relazione/FitSHG_exp.eps b/relazione/FitSHG_exp.eps new file mode 100755 index 0000000..3528eb2 --- /dev/null +++ b/relazione/FitSHG_exp.eps @@ -0,0 +1,3346 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 0 0 772 591 +%%LanguageLevel: 2 +%%Creator: OriginLab Corporation +%%Title: 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831 0 c +closepath } bind def +end +/BuildGlyph { + exch dup /Metrics get 2 index get 0 setcharwidth +/CharProcs get exch 2 copy known not{pop/.notdef}if get exec fill}bind def +/BuildChar { +1 index /Encoding get exch get +1 index /BuildGlyph get exec +} bind def +currentdict end /MC1_Arial exch definefont pop +%%EndFont +%%BeginFont: /MC2_Arial +12 dict begin +/FontInfo 8 dict dup begin +/ItalicAngle 0 def +/UnderlinePosition -217.00 def +/UnderlineThickness 150.00 def +/StrikeoutPosition 530.00 def +/StrikeoutThickness 102.00 def +/isFixedPitch false def +end readonly def +/FontName /MC2_Arial def +/FontType 3 def +/FontMatrix [1 2048 div 0 0 1 2048 div 0 0] def +/FontBBox [-1361 -665 4096 2129] def +/Encoding 256 array def 0 1 255{Encoding exch/.notdef put}for +Encoding 32/s32 put +Encoding 40/s40 put +Encoding 41/s41 put +Encoding 46/s46 put +Encoding 73/s73 put +Encoding 97/s97 put +Encoding 101/s101 put +Encoding 105/s105 put +Encoding 110/s110 put +Encoding 115/s115 put +Encoding 116/s116 put +Encoding 117/s117 put +Encoding 224/s224 put +/BoundingBoxes 14 dict def +BoundingBoxes begin +/s32 [0 -434 569 1854] def +/s40 [0 -434 682 1854] def +/s41 [0 -434 682 1854] def +/s46 [0 -434 569 1854] def +/s73 [0 -434 569 1854] def +/s97 [0 -434 1139 1854] def +/s101 [0 -434 1139 1854] def +/s105 [0 -434 455 1854] def +/s110 [0 -434 1139 1854] def +/s115 [0 -434 1024 1854] def +/s116 [0 -434 569 1854] def +/s117 [0 -434 1139 1854] def +/s224 [0 -434 1139 1854] def +end +/Metrics 14 dict def + Metrics begin +/s32 569 def +/s40 682 def +/s41 682 def +/s46 569 def +/s73 569 def +/s97 1139 def +/s101 1139 def +/s105 455 def +/s110 1139 def +/s115 1024 def +/s116 569 def +/s117 1139 def +/s224 1139 def +end +/CharProcs 14 dict def +CharProcs begin +/.notdef {} def +/s40 { +479 -431 m +379.67 -305.67 295.67 -159 227 9 c +158.33 177 124 351 124 531 c +124 689.67 149.67 841.67 201 987 c +261 1155.67 353.67 1323.67 479 1491 c +608 1491 479 1491 608 1491 c +527.33 1352.33 474 1253.33 448 1194 c +407.33 1102 375.33 1006 352 906 c +323.33 781.33 309 656 309 530 c +309 209.33 408.67 -111 608 -431 c +479 -431 608 -431 479 -431 c +closepath } bind def +/s41 { +253 -431 m +124 -431 253 -431 124 -431 c +323.33 -111 423 209.33 423 530 c +423 655.33 408.67 779.67 380 903 c +357.33 1003 325.67 1099 285 1191 c +259 1251 205.33 1351 124 1491 c +253 1491 124 1491 253 1491 c +378.33 1323.67 471 1155.67 531 987 c +582.33 841.67 608 689.67 608 531 c +608 351 573.5 177 504.5 9 c +435.5 -159 351.67 -305.67 253 -431 c +closepath } bind def +/s46 { +186 0 m +186 205 186 0 186 205 c +391 205 186 205 391 205 c +391 0 391 205 391 0 c +186 0 391 0 186 0 c +closepath } bind def +/s73 { +191 0 m +191 1466 191 0 191 1466 c +385 1466 191 1466 385 1466 c +385 0 385 1466 385 0 c +191 0 385 0 191 0 c +closepath } bind def +/s97 { +828 131 m +761.33 74.33 697.17 34.33 635.5 11 c +573.83 -12.33 507.67 -24 437 -24 c +320.33 -24 230.67 4.5 168 61.5 c +105.33 118.5 74 191.33 74 280 c +74 332 85.83 379.5 109.5 422.5 c +133.17 465.5 164.17 500 202.5 526 c +240.83 552 284 571.67 332 585 c +367.33 594.33 420.67 603.33 492 612 c +637.33 629.33 744.33 650 813 674 c +813.67 698.67 814 714.33 814 721 c +814 794.33 797 846 763 876 c +717 916.67 648.67 937 558 937 c +473.33 937 410.83 922.17 370.5 892.5 c +330.17 862.83 300.33 810.33 281 735 c +105 759 281 735 105 759 c +121 834.33 147.33 895.17 184 941.5 c +220.67 987.83 273.67 1023.5 343 1048.5 c +412.33 1073.5 492.67 1086 584 1086 c +674.67 1086 748.33 1075.33 805 1054 c +861.67 1032.67 903.33 1005.83 930 973.5 c +956.67 941.17 975.33 900.33 986 851 c +992 820.33 995 765 995 685 c +995 445 995 685 995 445 c +995 277.67 998.83 171.83 1006.5 127.5 c +1014.17 83.17 1029.33 40.67 1052 0 c +864 0 1052 0 864 0 c +845.33 37.33 833.33 81 828 131 c +closepath 813 533 m +747.67 506.33 649.67 483.67 519 465 c +445 454.33 392.67 442.33 362 429 c +331.33 415.67 307.67 396.17 291 370.5 c +274.33 344.83 266 316.33 266 285 c +266 237 284.17 197 320.5 165 c +356.83 133 410 117 480 117 c +549.33 117 611 132.17 665 162.5 c +719 192.83 758.67 234.33 784 287 c +803.33 327.67 813 387.67 813 467 c +813 533 813 467 813 533 c +closepath } bind def +/s101 { +862 342 m +1048 319 862 342 1048 319 c +1018.67 210.33 964.33 126 885 66 c +805.67 6 704.33 -24 581 -24 c +425.67 -24 302.5 23.83 211.5 119.5 c +120.5 215.17 75 349.33 75 522 c +75 700.67 121 839.33 213 938 c +305 1036.67 424.33 1086 571 1086 c +713 1086 829 1037.67 919 941 c +1009 844.33 1054 708.33 1054 533 c +1054 522.33 1053.67 506.33 1053 485 c +261 485 1053 485 261 485 c +267.67 368.33 300.67 279 360 217 c +419.33 155 493.33 124 582 124 c +648 124 704.33 141.33 751 176 c +797.67 210.67 834.67 266 862 342 c +closepath 271 633 m +864 633 271 633 864 633 c +856 722.33 833.33 789.33 796 834 c +738.67 903.33 664.33 938 573 938 c +490.33 938 420.83 910.33 364.5 855 c +308.17 799.67 277 725.67 271 633 c +closepath } bind def +/s105 { +136 1259 m +136 1466 136 1259 136 1466 c +316 1466 136 1466 316 1466 c +316 1259 316 1466 316 1259 c +136 1259 316 1259 136 1259 c +closepath 136 0 m +136 1062 136 0 136 1062 c +316 1062 136 1062 316 1062 c +316 0 316 1062 316 0 c +136 0 316 0 136 0 c +closepath } bind def +/s110 { +135 0 m +135 1062 135 0 135 1062 c +297 1062 135 1062 297 1062 c +297 911 297 1062 297 911 c +375 1027.67 487.67 1086 635 1086 c +699 1086 757.83 1074.5 811.5 1051.5 c +865.17 1028.5 905.33 998.33 932 961 c +958.67 923.67 977.33 879.33 988 828 c +994.67 794.67 998 736.33 998 653 c +998 0 998 653 998 0 c +818 0 998 0 818 0 c +818 646 818 0 818 646 c +818 719.33 811 774.17 797 810.5 c +783 846.83 758.17 875.83 722.5 897.5 c +686.83 919.17 645 930 597 930 c +520.33 930 454.17 905.67 398.5 857 c +342.83 808.33 315 716 315 580 c +315 0 315 580 315 0 c +135 0 315 0 135 0 c +closepath } bind def +/s115 { +63 317 m +241 345 63 317 241 345 c +251 273.67 278.83 219 324.5 181 c +370.17 143 434 124 516 124 c +598.67 124 660 140.83 700 174.5 c +740 208.17 760 247.67 760 293 c +760 333.67 742.33 365.67 707 389 c +682.33 405 621 425.33 523 450 c +391 483.33 299.5 512.17 248.5 536.5 c +197.5 560.83 158.83 594.5 132.5 637.5 c +106.17 680.5 93 728 93 780 c +93 827.33 103.83 871.17 125.5 911.5 c +147.17 951.83 176.67 985.33 214 1012 c +242 1032.67 280.17 1050.17 328.5 1064.5 c +376.83 1078.83 428.67 1086 484 1086 c +567.33 1086 640.5 1074 703.5 1050 c +766.5 1026 813 993.5 843 952.5 c +873 911.5 893.67 856.67 905 788 c +729 764 905 788 729 764 c +721 818.67 697.83 861.33 659.5 892 c +621.17 922.67 567 938 497 938 c +414.33 938 355.33 924.33 320 897 c +284.67 869.67 267 837.67 267 801 c +267 777.67 274.33 756.67 289 738 c +303.67 718.67 326.67 702.67 358 690 c +376 683.33 429 668 517 644 c +644.33 610 733.17 582.17 783.5 560.5 c +833.83 538.83 873.33 507.33 902 466 c +930.67 424.67 945 373.33 945 312 c +945 252 927.5 195.5 892.5 142.5 c +857.5 89.5 807 48.5 741 19.5 c +675 -9.5 600.33 -24 517 -24 c +379 -24 273.83 4.67 201.5 62 c +129.17 119.33 83 204.33 63 317 c +closepath } bind def +/s116 { +528 161 m +554 2 528 161 554 2 c +503.33 -8.67 458 -14 418 -14 c +352.67 -14 302 -3.67 266 17 c +230 37.67 204.67 64.83 190 98.5 c +175.33 132.17 168 203 168 311 c +168 922 168 311 168 922 c +36 922 168 922 36 922 c +36 1062 36 922 36 1062 c +168 1062 36 1062 168 1062 c +168 1325 168 1062 168 1325 c +347 1433 168 1325 347 1433 c +347 1062 347 1433 347 1062 c +528 1062 347 1062 528 1062 c +528 922 528 1062 528 922 c +347 922 528 922 347 922 c +347 301 347 922 347 301 c +347 249.67 350.17 216.67 356.5 202 c +362.83 187.33 373.17 175.67 387.5 167 c +401.83 158.33 422.33 154 449 154 c +469 154 495.33 156.33 528 161 c +closepath } bind def +/s117 { +831 0 m +831 156 831 0 831 156 c +748.33 36 636 -24 494 -24 c +431.33 -24 372.83 -12 318.5 12 c +264.17 36 223.83 66.17 197.5 102.5 c +171.17 138.83 152.67 183.33 142 236 c +134.67 271.33 131 327.33 131 404 c +131 1062 131 404 131 1062 c +311 1062 131 1062 311 1062 c +311 473 311 1062 311 473 c +311 379 314.67 315.67 322 283 c +333.33 235.67 357.33 198.5 394 171.5 c +430.67 144.5 476 131 530 131 c +584 131 634.67 144.83 682 172.5 c +729.33 200.17 762.83 237.83 782.5 285.5 c +802.17 333.17 812 402.33 812 493 c +812 1062 812 493 812 1062 c +992 1062 812 1062 992 1062 c +992 0 992 1062 992 0 c +831 0 992 0 831 0 c +closepath } bind def +/s224 { +828 131 m +761.33 74.33 697.17 34.33 635.5 11 c +573.83 -12.33 507.67 -24 437 -24 c +320.33 -24 230.67 4.5 168 61.5 c +105.33 118.5 74 191.33 74 280 c +74 332 85.83 379.5 109.5 422.5 c +133.17 465.5 164.17 500 202.5 526 c +240.83 552 284 571.67 332 585 c +367.33 594.33 420.67 603.33 492 612 c +637.33 629.33 744.33 650 813 674 c +813.67 698.67 814 714.33 814 721 c +814 794.33 797 846 763 876 c +717 916.67 648.67 937 558 937 c +473.33 937 410.83 922.17 370.5 892.5 c +330.17 862.83 300.33 810.33 281 735 c +105 759 281 735 105 759 c +121 834.33 147.33 895.17 184 941.5 c +220.67 987.83 273.67 1023.5 343 1048.5 c +412.33 1073.5 492.67 1086 584 1086 c +674.67 1086 748.33 1075.33 805 1054 c +861.67 1032.67 903.33 1005.83 930 973.5 c +956.67 941.17 975.33 900.33 986 851 c +992 820.33 995 765 995 685 c +995 445 995 685 995 445 c +995 277.67 998.83 171.83 1006.5 127.5 c +1014.17 83.17 1029.33 40.67 1052 0 c +864 0 1052 0 864 0 c +845.33 37.33 833.33 81 828 131 c +closepath 813 533 m +747.67 506.33 649.67 483.67 519 465 c +445 454.33 392.67 442.33 362 429 c +331.33 415.67 307.67 396.17 291 370.5 c +274.33 344.83 266 316.33 266 285 c +266 237 284.17 197 320.5 165 c +356.83 133 410 117 480 117 c +549.33 117 611 132.17 665 162.5 c +719 192.83 758.67 234.33 784 287 c +803.33 327.67 813 387.67 813 467 c +813 533 813 467 813 533 c +closepath 715 1194 m +570 1194 715 1194 570 1194 c +339 1474 570 1194 339 1474 c +580 1474 339 1474 580 1474 c +715 1194 580 1474 715 1194 c +closepath } bind def +end +/BuildGlyph { + exch dup /Metrics get 2 index get 0 setcharwidth +/CharProcs get exch 2 copy known not{pop/.notdef}if get exec fill}bind def +/BuildChar { +1 index /Encoding get exch get +1 index /BuildGlyph get exec +} bind def +currentdict end /MC2_Arial exch definefont pop +%%EndFont +%%BeginFont: /MC3_Symbol +12 dict begin +/FontInfo 8 dict dup begin +/ItalicAngle 0 def +/UnderlinePosition -223.00 def +/UnderlineThickness 100.00 def +/StrikeoutPosition 530.00 def +/StrikeoutThickness 102.00 def +/isFixedPitch false def +end readonly def +/FontName /MC3_Symbol def +/FontType 3 def +/FontMatrix [1 2048 div 0 0 1 2048 div 0 0] def +/FontBBox [-1 -450 2279 2059] def +/Encoding 256 array def 0 1 255{Encoding exch/.notdef put}for +Encoding 32/s32 put +/BoundingBoxes 4 dict def +BoundingBoxes begin +/s32 [0 -450 512 2059] def +/s61472 [0 -450 0 2059] def +/s61556 [0 -450 0 2059] def +end +/Metrics 4 dict def + Metrics begin +/s32 512 def +/s61472 0 def +/s61556 0 def +end +/CharProcs 4 dict def +CharProcs begin +/.notdef {} def +/s61556 { +813 916 m +813 769 813 916 813 769 c +501 769 813 769 501 769 c +478.33 549.67 467 399.33 467 318 c +467 253.33 479.17 206.5 503.5 177.5 c +527.83 148.5 558.33 134 595 134 c +622.33 134 646 143.33 666 162 c +686 180.67 700.67 212.67 710 258 c +745 258 710 258 745 258 c +733.67 148.67 709.33 73.17 672 31.5 c +634.67 -10.17 591 -31 541 -31 c +488.33 -31 444.33 -7.17 409 40.5 c +373.67 88.17 356 173.33 356 296 c +356 392.67 368 550.33 392 769 c +298 769 392 769 298 769 c +244 769 201.67 759.17 171 739.5 c +140.33 719.83 110 682.67 80 628 c +43 628 80 628 43 628 c +91 756 134 836 172 868 c +210 900 264 916 334 916 c +813 916 334 916 813 916 c +closepath } bind def +end +/BuildGlyph { + exch dup /Metrics get 2 index get 0 setcharwidth +/CharProcs get exch 2 copy known not{pop/.notdef}if get exec fill}bind def +/BuildChar { +1 index /Encoding get exch get +1 index /BuildGlyph get exec +} bind def +currentdict end /MC3_Symbol exch definefont pop +%%EndFont +/clipproc{clippath{[{/m cvx}{/l cvx}{/c cvx}{/cp cvx}pathforall}stopped +{cleartomark[pathbbox exch 3 i sub exch 2 i sub/rectpath cvx}if]np}bd +/iclmtx mtx cmtx def/iclpath clipproc xd/clinit{pathproc initclip np iclpath cl np exec}bd +/clmove{pathproc 3 1 roll clippath pathproc 2 i neg 2 i neg tr initclip np exec cl tr np exec}bd +/clrect[gs np iclpath{pathbbox}stopped{0 0 771.85 590.88 }if gr]def +/cloper{mtx cmtx iclmtx smtx exch exec smtx}bd +/cland{{exch cvx exec}cloper}bd +/clcopy{{exch clinit cvx exec}cloper}bd +/cldiff{{exch pathbbox 4 -1 roll clrect 0 get 2 copy gt{exch}if pop 4 1 roll + 3 -1 roll clrect 1 get 2 copy gt{exch}if pop 3 1 roll + 2 -1 roll clrect 2 get 2 copy lt{exch}if pop 2 1 roll + 1 -1 roll clrect 3 get 2 copy lt{exch}if pop 1 1 roll + exch 3 i sub exch 2 i sub rectpath cvx exec}cloper}bd +/icl{{clinit}cloper}bd +0.438407 0.437939 sc + +/initmtx mtx cmtx def +initmtx smtx +[1.0000 0.0000 0.0000 1.0000 0.0000 0.0000 ] concat + [0.547441 -0 0 -0.548032 0 1349.23] concat +gs +575 2052 m +2770 2052 l +7 sw +1 sj +1 scap +st +746 2052 m +746 2069 l +st +1089 2052 m +1089 2069 l +st +1432 2052 m +1432 2069 l +st +1775 2052 m +1775 2069 l +st +2118 2052 m +2118 2069 l +st +2461 2052 m +2461 2069 l +st +575 2052 m +575 2086 l +st +918 2052 m +918 2086 l +st +1261 2052 m +1261 2086 l +st +1604 2052 m +1604 2086 l +st +1947 2052 m +1947 2086 l +st +2290 2052 m +2290 2086 l +st +2633 2052 m +2633 2086 l +st +575 2052 m +575 285 l +st +575 2052 m +558 2052 l +st +575 1758 m +558 1758 l +st +575 1463 m +558 1463 l +st +575 1169 m +558 1169 l +st +575 874 m +558 874 l +st +575 580 m +558 580 l +st +575 285 m +558 285 l +st +575 1905 m +541 1905 l +st +575 1610 m +541 1610 l +st +575 1316 m +541 1316 l +st +575 1021 m +541 1021 l +st +575 727 m +541 727 l +st +575 432 m +541 432 l +st +gs +gr +gs +gr +gs +1 -1 sc +/MC1_Arial 75 self 521 -2169 m +(-60)[24 41 41 ] xs +1 -1 sc +gr +gs +gr +gs +gr +gs +1 -1 sc +/MC1_Arial 75 self 864 -2169 m +(-40)[24 41 41 ] xs +1 -1 sc +gr +gs +gr +gs +gr +gs +1 -1 sc +/MC1_Arial 75 self 1207 -2169 m +(-20)[24 41 41 ] xs 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+gs +gr +gs +gr +gs +gr +gs +gr +gs +gr +gs +gr +gs +gr +gs +gr +gs +gr +gs +gr +gs +pathproc 575 285 2196 1768 np rectpath +/eocl cland +np exec +569 1900 8 8 rectfill +541 432 m +1 sw +569 1900 8 8 rectstroke +578 1900 8 8 rectfill +578 1900 8 8 rectstroke +587 1900 8 8 rectfill +587 1900 8 8 rectstroke +596 1900 8 8 rectfill +596 1900 8 8 rectstroke +605 1900 8 8 rectfill +605 1900 8 8 rectstroke +614 1900 8 8 rectfill +614 1900 8 8 rectstroke +623 1900 8 8 rectfill +623 1900 8 8 rectstroke +632 1900 8 8 rectfill +632 1900 8 8 rectstroke +641 1900 8 8 rectfill +641 1900 8 8 rectstroke +650 1900 8 8 rectfill +650 1900 8 8 rectstroke +659 1900 8 8 rectfill +659 1900 8 8 rectstroke +668 1900 8 8 rectfill +668 1900 8 8 rectstroke +678 1900 8 8 rectfill +678 1900 8 8 rectstroke +687 1900 8 8 rectfill +687 1900 8 8 rectstroke +696 1900 8 8 rectfill +696 1900 8 8 rectstroke +705 1900 8 8 rectfill +705 1900 8 8 rectstroke +714 1900 8 8 rectfill +714 1900 8 8 rectstroke +723 1900 8 8 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+%\graphicspath{immagini/} + +%-------------------------------------------% +%Intestazioni delle pagine +\lfoot{} +\cfoot{\thepage} +\rfoot{} +\lhead{} +\chead{} +\rhead{\small{\textbf{\nouppercase{\leftmark}}}} +\renewcommand{\headrulewidth}{0.2pt} + +%-------------------------------------------% +%Dimensioni della pagina +\linespread{1.5} +\setlength{\textheight}{23cm} +\setlength{\textwidth}{15cm} +\setlength{\headheight}{12pt} +\setlength{\voffset}{-1cm} +\setlength{\hoffset}{-0.5cm} +%\setlength{}{} +%\setlength{}{} +%\setlength{}{} + +%-------------------------------------------% +%Applico il nuovo stile alla pagina +\pagestyle{fancy} + + + +%\frontmatter + %detuning + \title{Effetto ottico Kerr (OKE) per lo studio dei modi vibrazionali in liquidi molecolari.} + \author{Chilleri Gabriele \\ Zolfanelli Lorenzo} + \date{08/06/2016} + + \begin{document} +\maketitle + + \cleardoublepage + \begin{flushright} + \null\vspace{\stretch {1}} + \emph{Physics would be dull and life most unfulfilling if all physical phenomena around us were linear. Fortunately, we are living in a nonlinear world.\\(Shen)\break} + \vspace{\stretch{2}}\null + \end{flushright} + \cleardoublepage +%\tableofcontents + +%\mainmatter +\chapter*{Introduzione} +\chaptermark{Introduzione} +Considerando la densità di corrente $\textbf{J}$ in approsimazione di dipolo, si può ricavare l'espressione della polarizzazione di un mezzo dall'equazione di continuità e dalle equazioni di Maxwell. Essa si esprime come +\begin{equation} +\left[ \nabla\times \nabla\times + \dfrac{1}{c^2} \dfrac{\partial ^2}{\partial t^2} \right] \textbf{E}(\textbf{r},t) = -\dfrac{4\pi}{c^2} \dfrac{\partial ^2}{\partial t^2} \textbf{P}(\textbf{r},t). +\end{equation} +Può essere utile sviluppare la soluzione dell'equazione differenziale in potenze del campo elettrico, +$$ \textbf{P}(\textbf{r},t)=\textbf{P}^{(1)}(\textbf{r},t) +\textbf{P}^{(2)}(\textbf{r},t) + \textbf{P}^{(3)}(\textbf{r},t)+...\, ,$$ e quindi scrivere i contributi tensoriali come +\begin{eqnarray} +P_i^{(1)}(\textbf{r},t) &=& \int \chi_{ij}^{(1)}(\textbf{r}-\textbf{r}', t - t') \,\textbf{E}_j(\textbf{r}',t') \nonumber \\ +P_i^{(2)}(\textbf{r},t)&=& \int \chi_{ijk}^{(2)}(\textbf{r}-\textbf{r}',\textbf{r}-\textbf{r}'',t - t',t-t'') \,\textbf{E}_j(\textbf{r}',t')\textbf{E}_k(\textbf{r}'',t'') \nonumber \\ +P_i^{(3)}(\textbf{r},t)&=& \int \chi_{ijkl}^{(3)}(\textbf{r}-\textbf{r}',\textbf{r}-\textbf{r}'',\textbf{r}-\textbf{r}''', t - t',t-t'',t-t''') \,\textbf{E}_j(\textbf{r}',t')\textbf{E}_k(\textbf{r}'',t'')\textbf{E}_l(\textbf{r}''',t''') \nonumber. +\end{eqnarray} +Un campo elettrico è capace di modificare le proprietà dielettriche di un mezzo. Infatti il tensore suscettività $\chi_{ij..n}^{(n)}$ è la variabile che descrive propriamente l'interazione radiazione-materia. Inoltre le equazioni di Maxwell forniscono la connessione tra il tensore dielettrico $\epsilon_{ij..n} +$ e la suscettività nel sistema C.G.S.,$\epsilon_{ij..n} = 1 + 4\pi\chi_{ij..n}$, quindi direttamente anche con l'indice di rifrazione $\eta$, dalla relazione $\eta= \sqrt{\epsilon}$, cioè con le proprietà ottiche di un mezzo. +I termini del primo termino descrivono l'ottica lineare, mentre i termini successivi quella non lineare. Essi sono messi in evidenza per campi molto intensi come quelli generati dai laser impulsati. I tensori suscettività dipendono fortemente dalle proprietà di simmetria. Il contributo del secondo ordine è nullo per mezzi isotropi e centro-simmetrici, come per alcuni liquidi e il vetro, mentre quello del terzo ordine è sempre presente per ogni simmetria. +Uno degli effetti legati al termine del secondo ordine è la \emph{generazione di seconda armonica}, SHG. Questo effetto è dovuto all'interazione, ad esempio in un cristallo BBO, tra due pacchetti con diverso vettore d'onda $\textbf{k}_1 \neq \textbf{k}_2$, che si combinano generando un terzo campo con vettore d'onda $\textbf{k}=\textbf{k}_1+\textbf{k}_2$ e con frequenza $\omega=\omega_1+\omega_2$. +Quando i due campi sono forniti dallo stesso laser, considerando solo il caso in cui la frequenza del laser non è risonante con le transizioni elettroniche del mezzo, e trascurando gli effetti non locali, è possibile ottenere la duplicazione della frequenza della radiazione incidente. + +Il termine del terzo ordine della suscettività è invece responsabile dell'\emph{effetto Kerr ottico}, OKE. L'applicazione di un campo elettrostatico produce su un mezzo isotropo, come un liquido o un gas, un'anisotropia ottica transiente. Considerando effetti locali e non risonanti si parla di \emph{birifrangenza indotta}, altrimenti si parla di \emph{dicroismo indotto}. Per mezzo di un secondo impulso, opportunamente polarizzato, può essere misurata l'evoluzione temporale dell'eccitazione ottica dello stato fuori equilibrio del materiale. Da questa misura si osserva la dinamica di rilassamento della birifranenza indotta, dalla quale si può ottenere informazioni sui modi vibrazionali intermolecolari. + +\chapter{Apparato strumentale} +I processi dinamici riguardanti i liquidi molecolari lavorano su intervalli temporali molto veloci, dai femtosecondi fino ai nanosecondi. Per investigare su tali processi è indispensabile lavorare con laser capaci di produrre impulsi molto corti. Tali laser sono definiti \emph{impulsati}. Al contrario del regime continuo, dove il campo elettromagnetico generato deve essere monocromatico, quindi di durata molto lunga e coerente, nel regime pulsato il campo è rappresentabile come somma dei campi dei modi longitudinali, i quali vanno a costituire un inviluppo gaussiano. +$$ +\textbf{E}(t)= \sum_{n=-k}^{+k} A_n\, e^{i\left[(\omega_L+n\delta \omega )t+ \varphi_n \right]} +$$ +$A_n$ rappresenta la distribuzione gaussiana, $ \omega_L$ il valore della pulsazione in cui il campo assume il massimo valore dell'intensità dell'inviluppo, $n$ il numero dei modi, $\delta\omega$ la distanza tra due modi e $\varphi_n$ la fase dell'$n$-esimo modo. Per il nostro esperimento abbiamo utilizzato un laser con mezzo attivo Ti:sapphire che ci assicura il principio di \emph{mode-locking}, per cui tutti i modi hanno una relazione di fase costante e che le ampiezze dei modi seguono una funzione gasussiana. La durata temporale minima degli impulsi è data dalla condizione di \emph{transform limited}, la quale definisce costante il prodotto $ \Delta \tau_L \Delta \omega_L= k = cost$, che nel caso gaussiano è $ k = 0.441.$ Data la lunghezza d'onda spettrale, questa relazione ci permette di stimare la durata minima del segnale laser. Il laser con mezzo attivo Ti:sapphire è pompato dalla seconda armonica di un laser in continua $\mathrm{Nd:YVO_4}$, il quale consente l'inversione di popolazione nel secondo mezzo attivo e la generazione di impulsi. Tale sistema laser produce una dispersione positiva dovuta al mezzo attivo, quindi risulta necessario aggiungere un sistema di compensazione composto da due \emph{chirped mirror} ed un \emph{compressore a prismi} che consentirà di ridurre la dispersione anche per gli altri sistemi ottici dell'apparato strumentale. In figura \ref{spettrolaser} si osserva lo spettro dell'impulso laser confrontato con una funzione gausssiano predeterminato. Confrontando l'impulso con la gaussiana si può calcolare da essa la durata minima nelle condizioni di transform limited che risulta di 14 fs. +\begin{figure}[h] +\includegraphics[scale=0.5]{SpettroLaser} +\caption{Impulso laser in ingresso nell'apparato sperimentale (nero) confrontato con una funzione gaussiana (rosso).} +\label{spettrolaser} +\end{figure} +Una lamina $\dfrac{\lambda}{2}$ è inserita nel setup sperimentale per assicurare che l'impulso laser sia ben polarizzato verticalmente o orizzontalmente. Un beam-splitter permette di dividere il fascio in due: un fascio \emph{pump} (pompa) e un fascio \emph{probe} (sonda). Il fascio pompa permette di generare la birifrangenza nel campione, mentre il fascio sonda serve a misurare tali modifiche delle proprietà ottiche. +\begin{figure}[h] +\includegraphics[width = \textwidth]{setup} +\caption{Apparato sperimentale.} +\label{setup} +\end{figure} +In figura \ref{setup} si osserva che la procedura di misura pump-probe si manifesta ritardando il fascio di sonda tramite un sistema di specchi montato su una slitta mobile. Variando la separazione temporale tra pump e probe è possibile indagare l'evoluzione nel tempo della birifrangenza indotta dal pump. Quest'ultimo attraversa un chopper, cioè un disco in rotazione che introduce una modulazione dell'intensità del fascio. Esso è usato in combinazione con un amplificatore lock-in, il quale demodulando il segnale, riduce i contributi di rumore. Si parla di tecnica \emph{omodina} quando si misura direttamente il campo segnale con un rivelatore. Nel nostro caso si compie una misura in \emph{eterodina}. Tale tecnica necessita l'utilizzo di un campo locale che viene generato per mezzo di una lamina ${\lambda}/{4}$, cioè con polarizzazione circolare. Un sistema di lenti permette di focalizzare il fascio di pompa e di sonda nella stessa regione del campione. Definiamo campo locale la porzione di fascio la cui polarizzazione non viene alterata dall'interazione con il campione. Mentre la porzione che interagisce contribuirà alla generazione del campo segnale. Le componenti x e y del campo locale sono interessate da uno sfasamento di ${\pi}/{2}$, mentre quelle del campo segnale avranno un ulteriore sfasamento dovuto alla birifrangenza. Il percorso delle componenti ortogonali viene separato da un cubo polarizzatore e inviato a un rivelatore a due fotodiodi, il quale lavora in amplificazione differenziale. L'intensità data dal modulo quadro della somma dei due campi determina la presenza di un termine, detto \emph{eterodino}, che ha segno opposto per le due polarizzazioni; l'amplificatore differenziale consente di eliminare gli altri contributi. +$$ S \propto |E_S|^2 + |E_{loc}|^2 + 2\, Re \, E_S \cdot E_{loc}^* $$ +Quindi in uscita dal sistema di rivelazione avremo il segnale +$$ S_{ete} = \dfrac{S_+ - S_-}{2} $$ +La tecnica di misura in eterodina favorisce l'eliminazione dei contributi di fase ed un aumento in intensità del segnale, in quanto presenta un andamento lineare rispetto al campo segnale, al contrario del caso omodino che è quadratico. Tale aspetto facilita il rivelamento di campi deboli come quello prodotto dall'effetto Kerr. + +\chapter{Misura della durata dell'impulso tramite generazione di seconda armonica} +Un metodo per ricavare la durata dell'impulso laser è sfruttare il fenomeno della generazione di seconda armonica. Come scritto in precedenza, tale fenomeno si ottiene dall'interazione di due impulsi in un mezzo anisotropo, quindi con il termine del secondo ordine della polarizzazione diverso da zero. Per procedere a tale tecnica abbiamo sostituito il campione liquido con un cristallo BBO?? in una disposizione ottica come in figura \ref{shg}. +\begin{figure} +\includegraphics[width = \textwidth]{shg} +\caption{Generazione seconda armonica per la misura temporale dell'impulso laser.} +\label{shg} +\end{figure} La generazione di seconda armonica avviene quando i due impulsi giungono sul cristallo nello stesso istante. Con un diaframma posto dopo il campione si interrompono i fasci con frequenza $\omega$ e si lascia propagare il fascio di seconda armonica $2\omega$. Si misura l'intensità del fascio in funzione del ritardo $\tau$ tra i due impulsi, ottenuto tramite la slitta mobile. L'andamento del segnale in funzione di $\tau$ è dato dalla funzione di autocorrelazione +$$ S(\tau) = \int dt\,\left[\,| E(t)+ E(t-\tau)|^2\,\right]^2 .$$ +\begin{figure} +\includegraphics[width = \textwidth]{SHGcorr} +\caption{Andamento della funzione di autocorrelazione priva di filtro (in nero),in cui si osservano le frange di interferenza e l'andamento con filtro (in rosso) per eliminare i termini costanti. Si può osservare che la funzione di autocorrelazione segue un andamento gaussiano.} +\label{SHGcorr} +\end{figure} +Calcolando numericamente la funzione di autocorrelazione si ottiene l'andamento del segnale. Trascurando i contributi costanti del segnale, l'andamento calcolato risolto tramite una simulazione MATLAB segue una funzione gaussiana. Per mezzo della relazione $\Delta \tau_L= \dfrac{\Delta \tau_S}{\sqrt{2}}$ si ricava la durata dell'impulso laser simulato +$$\Delta \tau_{L_{sim}}=14.15\,\mathrm{fs},$$ +invece, per quanto riguarda il dato sperimentale si ricava una durata temporale dell'impulso pari a +$$\Delta \tau_{L_{exp}}=15.14\,\mathrm{fs}.$$ +\begin{figure} +\centering +\subfigure[]{\includegraphics[width = 0.48\textwidth]{FitSHG_sim}} +\subfigure[]{\includegraphics[width = 0.48\textwidth]{FitSHG_exp}} +\caption{Fit gaussiani dell'impulso laser simulato (a) e dell'impulso laser sperimentale (b).} +\label{impulsi} +\end{figure} + + + + + + + + + + + + + +\end{document} \ No newline at end of file