MODELS FOR FITTING REFRACTIVE INDEX n vs. λ

Transcription

1 MODELS FOR FITTING REFRACTIVE INDEX vs. Preferred equatios are highlighted i GREEN. They have bee idetified as the most commo equatios used for iterpolatio i several optical desig programs or i books dealig with optical properties of materials. There are may equatios that have bee used to represet the idex of refractio of optical materials as a fuctio of wavelegth. The earliest is the Cauchy equatio, which dates back to oth Jekis & White ad Ditchbur show the followig geeral form: C ( ) = A 4 C = A CAUCHY ( ) 4 Refs: Smith, Agewadte used by TFCalc Aother group cotais the CONRADY equatios. These simple models are ofte used to fit sparse data. The first appeared i Corady s book Applied Optics ad Optical Desig (p. 659). CONRADY1 ( ) C Refs: Smith, Agewadte =A 3.5 used by OSLO, Opticad, TracePro, ZEMAX C =A CONRADY ( ) 3.5 There is a family of equatios based upo work by HELMHOLTZ. Jekis & White show: ( ) = A o m A i i= 1 ( i ) i / ( i ) These equatios are based o physical priciples ad ca be very complex. They are rarely see today. 1

2 The HARTMANN equatio has the geeral form: ( ) where the expoet D assumes values betwee 1 ad. = A ( C ) D HARTMANN 1 ( ) = A used by TFCalc C HARTMANN ( ) = A ( ) ( C ) used by TFCalc Hartma equatios also appear i the followig refs: HARTMANN3a ( ) = A Ditchbur C HARTMANN3b = HARTMANN4 ( ) ( ) 1. A ( ) 1. = A D ( C ) ( E ) Lueberg Smith ad Agewadte

3 HERZERGER devised a useful equatio for fittig over wide wavelegth rages. It comes i may guises. The first is a 4-term Herzberger equatio. The costat 0.08 represets a short wavelegth correspodig to the high frequecy absorptio edge, which is valid for a wide class of materials. From this poit we may add or subtract terms as eeded. Five-term ad six-term Herzberger equatios are quite commo. It is also possible to chage the costat 0.08 ito variables K ad L, ad the umber is occasioally see. C D =A ( -0.08) ( -0.08) Smith ad Agewadt use here HERZRGRX ( ) D E =A C ( -0.08) ( -0.08) Laiki; used for LiF ad Irtra E F =A C D ( -0.08) ( -0.08) Opticad D E F =A C ( -0.08) ( -0.08) ( -0.08) F G =A C D E ( -0.08) ( -0.08) HERZRGR3X ( ) 4 HERZRGR4X ( ) 4 6 HERZRGR3X3 ( ) 4 HERZRGR5X ( ) E F =A C D ( -J) ( -K) HERZGRJK ( ) 4 6 J ad/or K could be or a variable 3

4 SCHOTT Glasswerke devised a useful polyomial equatio based o a Lauret series that has several variats. The simplest has 5-terms. etter results are ofte achieved by addig more terms. A 6-term Schott equatio, is most commo. A 7-term Schott equatio is occasioally see. Schott Glass abadoed these formulae i 199, ad switched to a Sellmeier represetatio. These are still i widespread use elsewhere. C D E =A SCHOTTX3 ( ) 4 6 C D E F =A C D E F G Ref Agewadte, called =A glasherstellar C D E F G H =A Exteded i Zemax D E F =A C 4 6 Laiki 3 = polyomial D E F G Ref Agewadte, called =A C glasherstellar D E F G H Ref Agewadte, called =A C glasherstellar E F G H =A C D Exteded i Zemax F G H J K =A C D E Exteded Schott i TracePro SCHOTTX4 ( ) SCHOTTX5 ( ) SCHOTTX6 ( ) SCHOTT3X3 ( ) 4 SCHOTT3X4 ( ) 4 SCHOTT3X5 ( ) 4 SCHOTT4X4 ( ) 4 6 SCHOTT5X5 ( ) Ref Smith as Old Schott used by ZEMAX, OSLO, Optalix, TFCalc, TracePro, Opticad. Laiki 0 4

5 The geeral SELLMEIER equatio looks like: ( ) A i 1 i i = where i is a wavelegth correspodig to a absorptio edge. The simplest is a sigle-term with oe wavelegth costat. The costat i the deomiator represet a absorptio edge wavelegths, usually i the ultraviolet. Higher-order Sellmeier equatios add more terms; as may as four are commoly used ad five have bee see. The two-term Sellmeier equatio adds a ifrared wavelegth costat D. Higher order terms are refiemets to allow for soft edges. SELL1T A ( ) 1 = SELLT A C ( ) 1 = D ZEMAX, OSLO ad TFCalc SELL3T Sellmeier 1 i ZEMAX; also i OSLO, A C E ( ) 1 = Optalix, TracePro, Sellmeier 3 i TFCalc, D F Opticad, Schott glass catalog, Laiki 1 SELL4T A C E G ( ) 1 = D F H Sellmeier 3 i TracePro SELL5T A C E G J ( ) 1 = D F H K Sellmeier 5 i Zemax CAUTION Some authors use a liear term (e.g., A) i the deomiator of the Sellmeier terms, ad others use a squared term (e.g., A ). The term A carries the dimesio of wavelegth ad represets the wavelegth of a resoace. Occasioally, egative sigs are see betwee the Sellmeier terms. Neither are fudametally flawed, but the sigs ad magitudes must be carefully chose as seeds for most Leveberg-Marquardt o-liear curve-fittig routies. 5

6 The most obvious ehacemet to the basic Sellmeier equatio is to add a costat term to replace the 1. EQN NAME SELL1TA ( ) SELLTA ( ) SELL3TA ( ) EQUATION = A C D = A C E D F = A C E G D F H = A C E G J D F H K = A C E G J M D F H K N = A C E G J M P D F H K N Q = A C E G J M P R SELL4TA ( ) SELL5TA ( ) SELL6TA ( ) SELL7TA ( ) NOTES Sellmeier 1 i TFCalc Sellmeier 4 i ZEMAX ad TracePro; Sellmeier i TFCalc; Ghosh (C=C, E=E ) KCl i HoOII Kr i HoOII KI i HoOII AcL i HoOII Other modificatios o the basic ad ehaced Sellmeier equatios are also foud. These are hybrid equatios, combiatios of the Sellmeier equatio with added terms characteristic of the Schott or Herzberger equatios. Several examples are show below: HoO1 ( ) C = A D Hadbook of Optics 1 i ZEMAX, TracePro HoO C Hadbook of Optics i ZEMAX & ( ) = A D TracePro; Sellmeier i TFCalc 6

7 SELLMOD1 D ( ) = A C E D = A C E SELLMOD 4 D ( ) = A C E D = A C E SELLMOD3 A D ( ) = C E ADP, KDP SELLMOD4 C D F ( ) = A E G C D F = A E G Kr SELLMOD5 C E ( ) = A D F KNbO 3 i HoOII SELLMOD6 D Sellmeier i ZEMAX, TracePro ( ) = A C E ad Opticad SELLMOD7 4 D E ( ) = A C 6 F D E = A C 6 F SELLMOD8 ( ) 4 D F = A C E G SELLMOD9 C D E ( ) = A 4 6 F SELLMOD1A ( ) SELLMODA ( ) 4 SELLMOD4A ( ) SELLMOD7A ( ) 4 7

8 = D A C E KINGSLAKE ( ) KINGSLAKE1 ( ) D = A C E D F = A C E G KINGSLAKE ( ) MISC01 ( ) MISC0 ( ) MISC03 ( ) MISC04 ( ) MISC05 ( ) = A C C = A D C = A D 4 D E = A C F G / ( F) / D = A 1 / C E 1/ Geeral form Smith calls this Kettler-Drude, others call it Helmholzz- Drude. used by Li for ZS & ZSe TiO i HoOII HoOII(Ag 3 AsS 3 ) HoO CuCl i HoOII ausch ad Lomb JOSA 54, 115 NOTE: The EQN NAME colum idetifies a.eqn equatio file used i PSI-Plot. I use this program for o-liear regressio curve fittig usig the Leveberg-Marquardt algorithm. 7/07/ jmp 8