This page is under construction.   

Benchmark results for spheroids


In the following tables we present the normalized cross-sections tabulated for prolate and oblate spheroids having the refractive index m = 1.3, 2.5, 1.7 +0.7i, 2.5 +1.5i, 3.0 +4.0i; the semiaxis ratio a/b = 1.1, 2, 4, 10; the incidence angle alpha = 0 and 90 (TE and TM modes), and different diffraction parameter xV=2 pi rV/lambda, where rV is the radius of an equal volume sphere and lambda the wavelength of incident radiation. The results were obtained by at least two different methods, and all given digits were the same.

Extinction and scattering cross-sections

Table 1. Normalized extinction (scattering) cross-sections Cext/pi rV2 for spheroids with m = 1.3
prolate oblate
a/b xV alpha = 0 alpha = 90 (TM) alpha = 90 (TE) alpha = 0 alpha = 90 (TM) alpha = 90 (TE)
1.1 0.1 9.182080E-06 9.584544E-06 9.185689E-06 9.453529E-06 9.049059E-06 9.450043E-06
0.3 7.353049E-04 7.711279E-04 7.378481E-04 7.608544E-04 7.250718E-04 7.583891E-04
1.0 7.447337E-02 8.134987E-02 7.649733E-02 8.037420E-02 7.362975E-02 7.832033E-02
3.0 1.540311E-00 1.437064E-00 1.395298E-00 1.380468E-00 1.472662E-00 1.519971E-00
10.0 2.230447E-00 2.822134E-00 2.867127E-00 3.034794E-00 2.416342E-00 2.441388E-00
30.0 2.006623E+00 2.572447E+00
2.0 0.1 8.497336E-06 1.125232E-05 8.529378E-06 1.042893E-05 7.558188E-06 1.040419E-05
0.3 6.605367E-04 9.119126E-04 6.825033E-04 8.450193E-04 5.950242E-04 8.275387E-04
1.0 5.447619E-02 1.025111E-01 6.691507E-02 9.418041E-02 5.141485E-02 7.960716E-02
3.0 1.968352E-00 1.333052E-00 1.083852E-00 1.024408E-00 1.427102E-00 1.865927E-00
10.0 1.538734E-004.353976E-00 4.273941E-00 5.229347E-00 1.270176E-00 1.336617E-00
4.0 0.1 7.979755E-06 1.273937E-05 8.074529E-06 1.160638E-05 6.345801E-06 1.155186E-05
0.3 5.722482E-04 1.026860E-03 6.318278E-04 9.361833E-04 4.847639E-04 8.983728E-04
1.0 3.647086E-021.085455E-01 4.947725E-029.776480E-023.277807E-02 7.254568E-02
3.0 2.208284E-001.168012E-00 7.362909E-017.018565E-011.055080E-00 1.967821E-00
10.0 1.812869E+005.680530E+00 5.416727E+005.183390E-001.225550E-00 1.425348E-00
10.0 0.1 7.436218E-06 1.363190E-05 7.751983E-06 1.280422E-05 5.467201E-06 1.268771E-05
0.3 4.111163E-04 1.049295E-03 5.521820E-04 1.010622E-03 3.922034E-04 9.341944E-04
1.0 2.210801E-028.074041E-02 2.516490E-028.226629E-021.862260E-02 5.558987E-02
3.0 1.704760E-009.341732E-01 4.068989E-015.908465E-015.660661E-01 1.610737E-00
Note: The values printed by italics were obtained using only our Separation of Variables code. Since in these cases we got the same extinction and scattering cross-sections with (much) more than seven significant digits, the numbers were believed to be correct.

Table 2. Normalized extinction (scattering) cross-sections Cext/pi rV2 for spheroids with m = 1.3
prolate oblate
a/b xV alpha = 0 alpha = 90 (TM) alpha = 90 (TE) alpha = 0 alpha = 90 (TM) alpha = 90 (TE)
1.1 0.1 1.035794E-04 1.199431E-04 1.036199E-04 1.142661E-04 9.863955E-05 1.142242E-04
0.3 8.770520E-03 1.027102E-02 8.799159E-03 9.764975E-03 8.338921E-03 9.735097E-03
1.0 1.803740E-00 2.086949E-00 1.688006E-00 1.905530E-00 1.634429E-00 2.042280E-00
3.0 3.619850E-00 2.702825E-00 1.617508E-00 1.483010E-00 2.695134E-00 4.391523E-00
10.0 2.069972E-00 2.534723E-00 2.552629E-00 2.244362E-00 2.500382E-00 2.485442E-00
2.0 0.1 8.132300E-05 2.262901E-04 8.162806E-05 1.641502E-04 5.756336E-05 1.637629E-04
0.3 6.610480E-03 2.018734E-02 6.819180E-03 1.433111E-02 4.685330E-03 1.404947E-02
1.0 1.299951E-00 3.736834E-00 9.854978E-01 2.464760E-00 6.647633E-01 3.680223E-00
3.0 2.756951E-00 2.423317E-00 1.456577E-00 4.484763E-00 1.777398E-00 3.846601E-00
4.0 0.1 6.871662E-05 4.258699E-04 6.952952E-05 2.612223E-04 3.688507E-05 2.600006E-04
0.3 5.105309E-033.919277E-025.615168E-03 2.307084E-022.874841E-032.217663E-02
1.0 7.055768E-015.493155E-005.426740E-01 3.362340E-002.595873E-015.360019E-00
3.0 1.673046E-005.755844E-004.730299E-00 7.392485E-002.580128E-003.791884E-00
10.0 6.174717E-00
10.0 0.1 6.058991E-056.745045E-046.315583E-05 4.423506E-042.643817E-054.383382E-04
0.3 3.446015E-035.950350E-024.588377E-03 3.852378E-021.918230E-033.569470E-02
1.0 3.088317E-016.153316E-002.283901E-01 3.605985E-001.053865E-015.126838E-00
3.0 4.612520E-019.781859E-007.081355E-00 7.766600E-007.497780E-004.701476E-00
10.0 8.892024E-00
Note: The values printed by italics were obtained using only our Separation of Variables code. Since in these cases we got the same extinction and scattering cross-sections with (much) more than seven significant digits, the numbers were believed to be correct.

Table 3a. Normalized extinction and scattering cross-sections for prolate spheroids with m = 1.7 + 0.7i.
a/b xV Cext/pi rV (alpha=0) Csca/pi rV (alpha=0) Cext/pi rV (alpha=90; TM) Csca/pi rV (alpha=90; TM) Cext/pi rV (alpha=90; TE) Csca/pi rV (alpha=90; TE)
1.1 0.1 1.112202E-01 7.864300E-05 1.237998E-01 8.760800E-05 1.112025E-01 7.867500E-05
0.3 3.606976E-01 6.456163E-03 4.013411E-01 7.232842E-03 3.602422E-01 6.478287E-03
1.0 1.905984E-00 5.355063E-01 2.032662E-00 5.973659E-01 1.887403E-00 5.471636E-01
3.0 2.701331E-00 1.231402E-00 2.806987E-00 1.315163E-00 2.786968E-00 1.278780E-00
10.0 2.291426E-00 1.219106E-00 2.446885E-00 1.312233E-00 2.452426E-00 1.289792E-00
2.0 0.1 9.260996E-02 6.520100E-05 1.867292E-01 1.323250E-04 9.250492E-02 6.544660E-05
0.3 3.001614E-01 5.177635E-03 6.027228E-01 1.099226E-02 2.974687E-01 5.348186E-03
1.0 1.697352E-00 3.846683E-01 2.477888E-00 7.841288E-01 1.556310E-00 4.492447E-01
3.0 2.481402E-00 1.039641E-00 3.195685E-00 1.600695E-00 3.021085E-00 1.360101E-00
10.0 1.804330E-00 9.737252E-01 2.874307E-00 1.607439E-00 2.908106E-00 1.453882E-00
4.0 0.1 8.155853E-02 5.681280E-05 2.550670E-01 1.807060E-04 8.141510E-02 5.748690E-05
0.3 2.621134E-01 4.141768E-03 8.113704E-01 1.476105E-02 2.585912E-01 4.568657E-03
1.0 1.501459E-00 2.669788E-01 2.713089E-00 8.506337E-01 1.243460E-00 3.192207E-01
3.0 2.580295E-00 9.865859E-01 3.848184E-00 1.966160E-00 3.409582E-00 1.515164E-00
10.0 1.506484E-00 8.354435E-01 3.584853E-00 2.355147E-00 3.677056E-00 1.920250E-00
10.0 0.1 7.602801E-02 5.084200E-05 3.011971E-01 2.122690E-04 7.589427E-02 5.299880E-05
0.3 2.398816E-01 2.843924E-03 9.378379E-01 1.631801E-02 2.368506E-01 3.807946E-03
1.0 1.262913E-00 1.653620E-01 2.963340E-00 7.538500E-01 9.994957E-01 1.659754E-01
3.0 3.124793E-00 1.400933E-00 4.859030E-00 2.372744E-00 3.342797E-00 1.749225E-00

Table 3b. Normalized extinction and scattering cross-sections for prolate spheroids with m = 1.7 + 0.7i.
a/b xV Cext/pi rV (alpha=0) Csca/pi rV (alpha=0) Cext/pi rV (alpha=90; TM) Csca/pi rV (alpha=90; TM) Cext/pi rV (alpha=90; TE) Csca/pi rV (alpha=90; TE)
1.1 0.1 1.195146E-01 8.458200E-05 1.071869E-01 7.580000E-05 1.195327E-01 8.455100E-05
0.3 3.872282E-01 6.983265E-03 3.473992E-01 6.223923E-03 3.876931E-01 6.960843E-03
1.0 1.979097E-00 5.846022E-01 1.849906E-00 5.229885E-01 1.997317E-00 5.725476E-01
3.0 2.830111E-00 1.319935E-00 2.722550E-00 1.234025E-00 2.742786E-00 1.271537E-00
10.0 2.505115E-00 1.330005E-00 2.345806E-00 1.234293E-00 2.340025E-00 1.256973E-00
2.0 0.1 1.533476E-01 1.086960E-04 6.964599E-02 4.913110E-05 1.534705E-01 1.084390E-04
0.3 4.941085E-01 9.029149E-03 2.246184E-01 3.938983E-03 4.971749E-01 8.843852E-03
1.0 2.198062E-00 7.105949E-01 1.291186E-00 3.151001E-01 2.326983E-00 6.221156E-01
3.0 3.378955E-00 1.672422E-00 2.603296E-00 1.040019E-00 2.768031E-00 1.330862E-00
10.0 3.424418E-00 1.823962E-00 2.102060E-00 1.075898E-00 2.078338E-00 1.254424E-00
4.0 0.1 2.021662E-01 1.432990E-04 4.737106E-02 3.331120E-05 2.023435E-01 1.426270E-04
0.3 6.442415E-01 1.176912E-02 1.509486E-01 2.574738E-03 6.483922E-01 1.129677E-02
1.0 2.394211E-00 7.756880E-01 8.254487E-01 1.707659E-01 2.658211E-00 6.204606E-01
3.0 4.160786E-00 2.063017E-00 2.908122E-00 1.093205E-00 3.118426E-00 1.527032E-00
10.0 0.1 2.592012E-01 1.833420E-04 3.519341E-02 2.458170E-05 2.593550E-01 1.816740E-04
0.3 8.101747E-01 1.452893E-02 1.102168E-01 1.774836E-03 8.134503E-01 1.343542E-02
1.0 2.582792E-00 7.310081E-01 5.183328E-01 8.548927E-02 2.994641E-00 5.616921E-01
3.0 5.071606E-00 2.388579E-00 3.466482E-00 1.457912E-00 4.053009E-00 1.871199E-00


Table 4a. Normalized extinction and scattering cross-sections for prolate spheroids with m = 2.5 + 1.5i.
a/b xV Cext/pi rV (alpha=0) Csca/pi rV (alpha=0) Cext/pi rV (alpha=90; TM) Csca/pi rV (alpha=90; TM) Cext/pi rV (alpha=90; TE) Csca/pi rV (alpha=90; TE)
1.1 0.1 4.857457E-02 2.662824E-04 6.075554E-02 3.380031E-04 4.829951E-02 2.663845E-04
0.3 2.500260E-01 2.355979E-02 3.008235E-01 3.052714E-02 2.440823E-01 2.363001E-02
1.0 3.010733E-00 1.943640E-00 3.660982E-00 2.472199E-00 3.107353E-00 2.043307E-00
3.0 2.712810E-00 1.952128E-00 2.884653E-00 2.137721E-00 2.780978E-00 2.018383E-00
10.0 2.353277E-00 1.773023E-00 2.497087E-00 1.920239E-00
2.0 0.1 3.459049E-02 1.836889E-04 1.929191E-01 1.115412E-03 3.339258E-02 1.843746E-04
0.3 1.944302E-01 1.537309E-02 9.098178E-01 1.129382E-01 1.661101E-01 1.585916E-02
1.0 1.894857E-00 9.206887E-01 5.495079E-00 3.970900E-00 2.283547E-00 1.438695E-00
3.0 2.388466E-00 1.478325E-00 3.453263E-00 2.702600E-00 2.762240E-00 1.941164E-00
4.0 0.1 2.730303E-02 1.441102E-04 9.964447E-01 5.837496E-03 2.615886E-02 1.458221E-04
0.3 1.519406E-01 1.085174E-02 4.991269E-00 6.624600E-01 1.226751E-01 1.200416E-02
1.0 1.463283E-00 3.937427E-01 5.398163E-00 3.832019E-00 1.604720E-00 9.183725E-01
3.0 2.300718E-00 1.086833E-00 4.265932E-00 3.396000E-00 3.006776E-00 2.056315E-00
10.0 0.1 2.346226E-02 1.225093E-04 3.457393E-00 2.017965E-02 2.275013E-02 1.277193E-04
0.3 1.133338E-01 6.935209E-03 8.769814E-00 1.125691E-00 9.556359E-02 9.380992E-03
1.0 1.386188E-00 2.456454E-01 7.150973E-00 4.383860E-00 1.036091E-00 4.588306E-01

Table 4b. Normalized extinction and scattering cross-sections for oblate spheroids with m = 2.5 + 1.5i.
a/b xV Cext/pi rV (alpha=0) Csca/pi rV (alpha=0) Cext/pi rV (alpha=90; TM) Csca/pi rV (alpha=90; TM) Cext/pi rV (alpha=90; TE) Csca/pi rV (alpha=90; TE)
1.1 0.1 1.063091E-01 2.027530E-04 8.850409E-02 1.684017E-04 1.063972E-01 2.026790E-04
0.3 3.884586E-01 1.754138E-02 3.248624E-01 1.439400E-02 3.909319E-01 1.748734E-02
1.0 3.299242E-00 1.519356E-00 2.960052E-00 1.285217E-00 3.294445E-00 1.481333E-00
3.0 2.915360E-00 1.616238E-00 2.779851E-00 1.487376E-00 2.832633E-00 1.561297E-00
2.0 0.1 1.683208E-01 3.228280E-04 4.590543E-02 8.685830E-05 1.690525E-01 3.220660E-04
0.3 6.046926E-01 2.854920E-02 1.682741E-01 7.112129E-03 6.246817E-01 2.798368E-02
1.0 4.007697E-00 2.005398E-00 1.852325E-00 6.182319E-01 3.928646E-00 1.726163E-00
3.0 3.644115E-00 2.139153E-00 2.436512E-00 1.057701E-00 2.916411E-00 1.654727E-00
4.0 0.1 3.072162E-01 5.903690E-04 2.714797E-02 5.130230E-05 3.085216E-01 5.876060E-04
0.3 1.089947E-00 5.221910E-02 9.612426E-02 4.009507E-03 1.123093E-00 5.017062E-02
1.0 4.550598E-00 2.313767E-00 1.043029E-00 2.876447E-01 4.684036E-00 1.847488E-00
3.0 5.233787E-00 3.234486E-00 2.370631E-00 7.777997E-01 3.251025E-00 1.885156E-00
10.0 0.1 5.869667E-01 1.126115E-03 1.854663E-02 3.498488E-05 5.756589E-01 1.114894E-03
0.3 1.984941E-00 9.372724E-02 6.193207E-02 2.540922E-03 2.042807E-00 8.672525E-02
1.0 5.252690E-00 2.480982E-00 4.762698E-01 1.301479E-01 6.089094E-00 2.016472E-00
3.0 7.991256E-00 4.978289E-00 3.318379E-00 1.142783E-00 4.196434E-00 2.432267E-00


Table 5a. Normalized extinction and scattering cross-sections for prolate spheroids with m = 3 + 4i.
a/b xV Cext/pi rV (alpha=0) Csca/pi rV (alpha=0) Cext/pi rV (alpha=90; TM) Csca/pi rV (alpha=90; TM) Cext/pi rV (alpha=90; TE) Csca/pi rV (alpha=90; TE)
1.1 0.1 4.857457E-02 2.662824E-04 6.075554E-02 3.380031E-04 4.829951E-02 2.663845E-04
0.3 2.500260E-01 2.355979E-02 3.008235E-01 3.052714E-02 2.440823E-01 2.363001E-02
1.0 3.010733E-00 1.943640E-00 3.660982E-00 2.472199E-00 3.107353E-00 2.043307E-00
3.0 2.712810E-00 1.952128E-00 2.884653E-00 2.137721E-00 2.780978E-00 2.018383E-00
10.0 2.353277E-00 1.773023E-00 2.497087E-00 1.920239E-00
2.0 0.1 3.459049E-02 1.836889E-04 1.929191E-01 1.115412E-03 3.339258E-02 1.843746E-04
0.3 1.944302E-01 1.537309E-02 9.098178E-01 1.129382E-01 1.661101E-01 1.585916E-02
1.0 1.894857E-00 9.206887E-01 5.495079E-00 3.970900E-00 2.283547E-00 1.438695E-00
3.0 2.388466E-00 1.478325E-00 3.453263E-00 2.702600E-00 2.762240E-00 1.941164E-00
4.0 0.1 2.730303E-02 1.441102E-04 9.964447E-01 5.837496E-03 2.615886E-02 1.458221E-04
0.3 1.519406E-01 1.085174E-02 4.991269E-00 6.624600E-01 1.226751E-01 1.200416E-02
1.0 1.463283E-00 3.937427E-01 5.398163E-00 3.832019E-00 1.604720E-00 9.183725E-01
3.0 2.300718E-00 1.086833E-00 4.265932E-00 3.396000E-00 3.006776E-00 2.056315E-00
10.0 0.1 2.346226E-02 1.225093E-04 3.457393E-00 2.017965E-02 2.275013E-02 1.277193E-04
0.3 1.133338E-01 6.935209E-03 8.769814E-00 1.125691E-00 9.556359E-02 9.380992E-03
1.0 1.386188E-00 2.456454E-01 7.150973E-00 4.383860E-00 1.036091E-00 4.588306E-01

Table 5b. Normalized extinction and scattering cross-sections for oblate spheroids with m = 3 + 4i.
a/b xV Cext/pi rV (alpha=0) Csca/pi rV (alpha=0) Cext/pi rV (alpha=90; TM) Csca/pi rV (alpha=90; TM) Cext/pi rV (alpha=90; TE) Csca/pi rV (alpha=90; TE)
1.1 0.1 5.612657E-02 3.119622E-04 4.500012E-02 2.464553E-04 5.643647E-02 3.118502E-04
0.3 2.780221E-01 2.801834E-02 2.319978E-01 2.170902E-02 2.845873E-01 2.794121E-02
1.0 3.509895E-00 2.361293E-00 2.873149E-00 1.847522E-00 3.399752E-00 2.253059E-00
3.0 2.878324E-00 2.124266E-00 2.694723E-00 1.929101E-00 2.805930E-00 2.055328E-00
10.0 2.541936E-00 1.943805E-00
2.0 0.1 1.017897E-01 5.864570E-04 2.106561E-02 1.116025E-04 1.046258E-01 5.850970E-04
0.3 4.583023E-01 5.527781E-02 1.197910E-01 9.258967E-03 5.167953E-01 5.426552E-02
1.0 5.236749E-00 3.731015E-00 1.404715E-00 6.821323E-01 4.215200E-00 2.800509E-00
3.0 3.647681E-00 2.814034E-00 2.014876E-00 1.173930E-00 3.039072E-00 2.239774E-00
4.0 0.1 2.631899E-01 1.539965E-03 1.151162E-02 6.162460E-05 2.692782E-01 1.532804E-03
0.3 1.177409E-00 1.544476E-01 6.279030E-02 4.840135E-03 1.295292E-00 1.484126E-01
1.0 6.852812E-00 4.898007E-00 8.678566E-01 2.635941E-01 5.134813E-00 3.146706E-00
3.0 5.352621E-00 4.207347E-00 1.567910E-00 6.070681E-01 3.573088E-00 2.544887E-00
10.0 0.1 1.128218E-00 6.607310E-03 7.300898E-03 4.054916E-05 1.134808E-00 6.547190E-03
0.3 4.937262E-00 6.479088E-01 3.253295E-02 2.950166E-03 4.945121E-00 5.975757E-01
1.0 8.641034E-00 5.952626E-00 4.895687E-01 1.239376E-01 7.872300E-00 4.088165E-00
3.0 8.980463E-00 7.079498E-00 1.952602E-00 5.393900E-01 4.617530E-00 3.219906E-00

Backscattering cross-section

Table 6. Normalized backscattering cross-sections Cbk/pi rV2 for spheroids with m = 1.3
...

To the top
Created by V.I.
Last modified: 04/10/00, V.I.