Yorugʻlik dispersiyasi sahifasining manbasini koʻrish


[[Fayl:Light_dispersion_conceptual_waves.gif|thumb| Dispersiv prizmada material dispersiyasi ( [[Toʻlqin uzunligi|to'lqin uzunligiga]] bog'liq sinishi indeksi ) turli xil ranglarning turli burchaklarda [[Refraksiya|sinishiga]] olib keladi va oq yorug'likni [[Spektr|spektrga]] bo'ladi.]]
[[Fayl:Light_dispersion_of_a_compact_fluorescent_lamp_seen_through_an_Amici_direct-vision_prism_PNr°0114.jpg|thumb| Amici prizmasi orqali ko'rilgan ixcham lyuminestsent chiroq]]
[[Optika|Optikada]] va umuman [[Toʻlqinlar|to’ lqin tarqalishida]] '''dispersiya''' [[Toʻlqinlar|to’ lqinning]] [[faza tezligi]] uning chastotasiga bog’ liq bo’ lgan hodisadir;<ref>{{Kitob manbasi |last=Born |first=Max |authorlink=Max Born |title=[[Principles of Optics]] |publisher=[[Cambridge University Press]] |date=October 1999 |location=Cambridge |pages=[https://archive.org/details/principlesofopti0006born/page/n43 14]–24 |isbn=0-521-64222-1}}</ref> baʼzan '''xromatik dispersiya''' atamasi, xususan, optikaga xoslik uchun ishlatiladi. Ushbu umumiy xususiyatga ega bo’ lgan vositani '''dispersiv vosita''' (ko’ plik ''dispersiv vosita'' ) deb atash mumkin.

Ushbu atama optika sohasida [[Yorugʻlik|yorug’ lik]] va boshqa [[Elektromagnit nurlanish|elektromagnit to’ lqinlarni]] tasvirlash uchun ishlatilgan bo’ lsa-da, dispersiya xuddi shu ma’ noda tovush va seysmik to’ lqinlar holatida [[akustik dispersiya]] va [[Gravitatsiya to'lqini|tortishish to’lqinlarida]] (okean) har qanday to’ lqin harakati uchun qo’ llanilishi mumkin. to’ lqinlar). Optikada dispersiya [[Elektr uzatish liniyasi|uzatish liniyalari]] bo’ ylab telekommunikatsiya signallarining (masalan, [[Koaksiyal kabel|koaksiyal kabeldagi]] [[Mikroto'lqinli pechlar|mikroto’ lqinlar]] ) yoki [[Optik tola|optik tolali]] yorug’ lik [[Puls (signalni qayta ishlash)|impulslarining]] xususiyatidir.

Optikada dispersiyaning muhim va tanish oqibatlaridan biri dispers [[Dispersion prizmalar|prizma]] tomonidan ishlab chiqarilgan spektrda va linzalarning [[Xromatik aberatsiya|xromatik aberratsiyasida]] ko’ rinib turganidek, yorug’ likning<ref>[http://www.proximion.com/about-dispersion Dispersion Compensation]. Retrieved 25-08-2015.</ref> ranglarining [[Refraksiya|sinish]] burchagining o’ zgarishidir. Xromatik aberatsiya asosan bekor qilingan aralash [[Axromat|akromatik linzalarning]] dizayni uning [[Abbe raqami]] ''V'' bilan berilgan shisha dispersiyasi miqdorini qo’ llaydi, bu yerda ''pastroq'' Abbe raqamlari [[Ko'rinadigan spektr|ko’ rinadigan spektrda]] ''katta'' dispersiyaga mos keladi. Telekommunikatsiya kabi ba’ zi ilovalarda to’ lqinning mutlaq fazasi ko’ pincha muhim emas, faqat [[To'lqin paket|to’ lqin paketlari]] yoki "impulslar" ning tarqalishi; Bunday holda, odamni faqat [[Guruh tezligi|guruh tezligining]] chastota bilan o’ zgarishi qiziqtiradi, bu [[Yorugʻlik dispersiyasi|guruh tezligi dispersiyasi]] deb ataladi.

Barcha umumiy [[Transmissiya vositalari|uzatish vositalari,]] shuningdek, chastota funktsiyasi sifatida [[Zaiflash|zaiflashuvda]] (uzatish uzunligiga normallashtirilgan) farqlanadi, bu esa [[Zaiflashning buzilishi|zaiflashuvning buzilishiga]] olib keladi; Bu dispersiya emas, garchi ba’ zida bir-biriga yaqin bo’ lgan [[Empedans moslashuvi|empedans chegaralarida]] aks ettirish (masalan, kabeldagi qiyshiq segmentlar) signalning buzilishiga olib kelishi mumkin, bu esa signal o’ tkazuvchanligi bo’ ylab kuzatilgan nomuvofiq o’ tish vaqtini yanada kuchaytiradi.

== Misollar ==
Dispersiyaning eng tanish misoli, ehtimol, [[kamalak]] bo’ lib, unda dispersiya oq [[Yorugʻlik|yorug’ likning]] turli [[Toʻlqin uzunligi|to’ lqin uzunliklari]] (turli [[Rang|ranglar]] ) komponentlariga fazoviy ajralishiga olib keladi. Biroq, dispersiya boshqa ko’ plab holatlarda ham ta’ sir qiladi: masalan, [[guruh-tezlik dispersiyasi]] [[Puls (signalni qayta ishlash)|impulslarning]] [[Optik tola|optik tolalarda]] tarqalishiga olib keladi, uzoq masofalarda signallarni yomonlashtiradi; shuningdek, guruh-tezlik dispersiyasi va [[Nochiziqli|chiziqli bo’ lmagan]] effektlar o’ rtasidagi bekor qilish [[soliton]] to’ lqinlariga olib keladi.

== Materiallar va to'lqin o'tkazgich dispersiyasi ==
Ko’  pincha xromatik dispersiya deganda materialning ommaviy dispersiyasi, ya’  ni optik chastota bilan [[Sinishi indeksi|sinishi indeksining]] o’  zgarishi tushuniladi. Shu bilan birga, [[To'lqin yo'riqnomasi|to’  lqin o’  tkazgichda]] ''to’  lqin o’  tkazgich dispersiyasi'' hodisasi ham mavjud bo’  lib, bu holda strukturadagi to’  lqinning [[faza tezligi]] uning chastotasiga shunchaki strukturaning geometriyasiga bog’  liq. Umuman olganda, "to’  lqin yo’  nalishi" dispersiyasi, to’  lqinlar ma’  lum bir mintaqada chegaralangan bo’  lishidan qat’  i nazar, har qanday bir hil bo’  lmagan struktura (masalan, [[fotonik kristall]] ) orqali tarqaladigan to’  lqinlar uchun sodir bo’  lishi mumkin.  To’  lqin o’  tkazgichda dispersiyaning ''har ikkala'' turi odatda mavjud bo’  ladi, garchi ular qat’  iy qo’  shimcha bo’  lmasa ham.  Masalan, optik tolali materiallarda material va to’  lqin uzatuvchi dispersiya bir-birini samarali ravishda bekor qilishi mumkin, bu esa tezkor [[optik tolali aloqa]] uchun muhim bo’  lgan [[Nolinchi dispersiya to'lqin uzunligi|nol dispersiyali to’  lqin uzunligini]] hosil qiladi.

== Optikada material dispersiyasi ==
[[Fayl:Mplwp_dispersion_curves.svg|thumb| Turli ko'zoynaklar uchun sinishi indeksining vakuum to'lqin uzunligiga o'zgarishi. Ko'rinadigan yorug'likning to'lqin uzunliklari kulrang rangga bo'yalgan.]]
[[File:Spidergraph_Dispersion.GIF|thumb| Tanlangan shisha komponentlarini qo'shishning ma'lum bir asosiy oynaning o'rtacha dispersiyasiga ta'siri ( ''l'' uchun amal qiladi ''n'' <sub>F</sub>&nbsp;=&nbsp;486&nbsp;nm (ko'k), ''n'' <sub>C</sub> ''l'' uchun amal qiladi&nbsp;=&nbsp;656&nbsp;nm (qizil))<ref>[http://glassproperties.com/dispersion/ Calculation of the Mean Dispersion of Glasses].</ref>]]
Materialning dispersiyasi optik ilovalarda istalgan yoki kiruvchi ta’ sir bo’ lishi mumkin. Shisha prizmalarda yorug’ likning tarqalishi [[Spektrometr|spektrometrlar]] va [[Spektroradiometr|spektroradyometrlarni]] qurish uchun ishlatiladi. Biroq, linzalarda dispersiya [[Xromatik aberatsiya|xromatik aberatsiyaga]] olib keladi, bu mikroskoplar, teleskoplar va fotografik ob’ ektlardagi tasvirlarni buzishi mumkin bo’ lgan kiruvchi ta’ sir.

: <math>v = \frac{c}{n},</math>

Bu yerda ''c''- [[Yorugʻlik tezligi|yorug'likning vakuumdagi tezligi]], ''n''- muhitning sindirish ko’ rsatkichi .

Umuman olganda, sindirish ko’ rsatkichi yorug’ likning ''f'' chastotasining ba’ zi bir funktsiyasidir, shuning uchun ''n''&nbsp;=&nbsp;''n'' ( ''f'' ), yoki muqobil ravishda, to’ lqin uzunligi ''n'' ga nisbatan&nbsp;=&nbsp;''n'' ( ''l'' ). Materialning sinishi indeksining to’ lqin uzunligiga bog’ liqligi odatda uning [[Abbe raqami]] yoki uning koeffitsientlari bilan [[Koshi tenglamasi|Koshi]] yoki [[Sellmeyer tenglamasi|Sellmeyer tenglamalari]] kabi empirik formulada aniqlanadi.

[[Kramers-Kronig munosabatlari]] tufayli sinishi indeksining haqiqiy qismining to’ lqin uzunligiga bog’ liqligi sinishi ko’ rsatkichining xayoliy qismi (shuningdek, [[Sinishi indeksi|yo’ q bo’ lib ketish koeffitsienti]] deb ataladi) tomonidan tasvirlangan materialning [[Absorbsiya (elektromagnit nurlanish)|yutilishi]] bilan bog’ liq. Xususan, magnit bo’ lmagan materiallar uchun ( [[O'tkazuvchanlik (elektromagnitizm)|''m'']]&nbsp;=&nbsp;[[Magnit doimiy|''m'' <sub>0</sub>]] ), Kramers-Kronig munosabatlarida paydo bo’ ladigan [[Chiziqli javob funksiyasi|sezuvchanlik]] ''ch'' - [[Elektr sezgirligi|elektr sezuvchanlik]] ''ch'' <sub>e</sub>&nbsp;=&nbsp;''n'' <sup>2</sup>&nbsp;−&nbsp;1.

Optikada dispersiyaning eng ko’ p ko’ rilgan oqibati [[Elektromagnit spektr|oq yorug’ likning]] [[Uchburchak prizma (optika)|prizma]] orqali [[Optik spektr|rang spektriga]] ajralishidir. [[Snell qonuni|Snel qonunidan]] ko’ rinib turibdiki, prizmadagi yorug’ likning [[Refraksiya|sinish]] burchagi prizma materialining sindirish ko’ rsatkichiga bog’ liq. Ushbu sinishi ko’ rsatkichi to’ lqin uzunligiga qarab o’ zgarganligi sababli, yorug’ likning sinishi burchagi ham to’ lqin uzunligiga qarab o’ zgaradi, bu ''burchak dispersiyasi'' deb nomlanuvchi ranglarning burchak bo’ linishiga olib keladi.

Ko’ rinadigan yorug’ lik uchun ko’ pgina shaffof materiallarning (masalan, havo, ko’ zoynak) sinishi ko’ rsatkichlari ''n'' to’ lqin uzunligi ortishi bilan kamayadi ''l'' :

: <math>1 < n(\lambda_\text{red}) < n(\lambda_\text{yellow}) < n(\lambda_\text{blue}),</math>

yoki umuman,

: <math>\frac{dn}{d \lambda} < 0.</math>

Bunday holda, muhit ''normal dispersiyaga'' ega deyiladi . To’ lqin uzunligi ortishi bilan indeks ortib borsa (bu odatda ultrabinafsha<ref>Born, M. and Wolf, E. (1980) "[[Principles of Optics]]", 6th&nbsp;ed., p.&nbsp;93. Pergamon Press.</ref>), muhit ''anomal dispersiyaga'' ega deyiladi.

Bunday materialning havo yoki vakuum bilan interfeysida (indeks ~ 1) Snell qonuni [[Normal (geometriya)|normalga]] burchak ostida tushgan yorug’ lik yoyi burchak ostida sinishi haqida bashorat qiladi.{{Sfrac|sin ''θ''|''n''}} ). Shunday qilib, yuqori sinishi indeksiga ega bo’ lgan ko’ k yorug’ lik qizil nurga qaraganda kuchliroq egiladi, natijada taniqli [[kamalak]] naqshlari paydo bo'ladi.

== Guruh-tezlik dispersiyasi ==
[[Fayl:Optical_dispersion_dynamics.gif|thumb| Gipotetik dispersiv muhitda qisqa pulsning vaqt evolyutsiyasi ( ''k''&nbsp;=&nbsp;''ō'' <sup>2</sup> ) uzunroq to'lqin uzunlikdagi komponentlar qisqa to'lqin uzunlikdagi komponentlarga (ijobiy GVD) nisbatan tezroq harakatlanishini ko'rsatadi, buning natijasida chiyillash va pulsning kengayishi kuzatiladi.]]
To’ lqin uzunligi bo’ yicha faza tezligining o’ zgarishini oddiygina tavsiflashdan tashqari, ko’ plab ilovalarda dispersiyaning yanada jiddiy oqibati [[Guruh-tezlik dispersiyasi|guruh tezligi dispersiyasi]] (GVD) deb ataladi. Faza tezligi ''v'' ''v'' = ''c'' / ''n'' sifatida belgilangan bo’ lsa-da, bu faqat bitta chastota komponentini tavsiflaydi. Turli chastota komponentlari birlashtirilganda, masalan, signal yoki impulsni ko’ rib chiqayotganda, ko’ pincha [[Guruh tezligi|guruh tezligiga]] ko’ proq qiziqish bildiriladi, bu impuls yoki to’ lqin (modulyatsiya) ustiga qo’ yilgan ma’ lumotlarning tarqalish tezligini tavsiflaydi. Qo’ shilgan animatsiyada to’ lqinning o’ zi (to’ q sariq-jigarrang) guruh tezligiga mos keladigan ''konvertning'' tezligidan (qora) ancha tezroq faza tezligida harakatlanishini ko’ rish mumkin. Bu impuls, masalan, aloqa signali bo’ lishi mumkin va uning ma’ lumotlari faqat guruh tezligida tarqaladi, garchi u tezroq tezlikda (faza tezligi) oldinga siljuvchi to’ lqinlardan iborat bo’ lsa ham.

Guruh tezligini sinishi indeksi egri chizig’ idan ''n'' ( ''ō'' ) yoki to’ g’ ridan-to’ g’ ri ''k'' = ''ōn'' / ''c'' to’ lqin raqamidan hisoblash mumkin, bu yerda ''ō'' - radian chastotasi ''ō.''&nbsp;=&nbsp;2 ''pf'' . Holbuki, faza tezligining bir ifodasi ''v'' <sub>p</sub>&nbsp;=&nbsp;''ō'' / ''k'', guruh tezligi [[Differensial|lotin]] yordamida ifodalanishi mumkin: ''v'' <sub>g</sub>&nbsp;=&nbsp;''dō'' / ''dk'' . Yoki faza tezligi bo’ yicha ''v'' <sub>p</sub> ,

: <math>v_\text{g} = \frac{v_\text{p}}{1 - \dfrac{\omega}{v_\text{p}} \dfrac{dv_\text{p}}{d\omega}}.</math>

Dispersiya mavjud bo’ lganda, nafaqat guruh tezligi faza tezligiga teng emas, balki odatda uning o’ zi to’ lqin uzunligiga qarab o’ zgaradi. Bu guruh-tezlik dispersiyasi deb nomlanadi va yorug’ likning qisqa pulsini kengayishiga olib keladi, chunki impuls ichidagi turli chastotali komponentlar turli tezliklarda harakatlanadi. Guruh tezligi dispersiyasi guruh tezligining [[Burchak chastota|burchak chastotasiga]] nisbatan ''o’ zaro'' hosilasi sifatida aniqlanadi, bu esa ''guruh tezligi dispersiyasiga'' olib keladi.&nbsp;=&nbsp;''d'' <sup>2</sup> ''k'' / ''dō'' <sup>2</sup> .

Agar yorug’ lik impulsi ijobiy guruh-tezlik dispersiyasiga ega bo’ lgan material orqali tarqaladigan bo’ lsa, u holda qisqa to’ lqin uzunlikdagi komponentlar uzunroq to’ lqin uzunlikdagi komponentlarga qaraganda sekinroq harakat qiladi. Shunday qilib, puls vaqt o’ tishi bilan chastotasi ortib boruvchi ''musbat [[Chirp|jiringlaydi]]'' yoki ''ko’ tariladi'' . Boshqa tomondan, agar impuls manfiy guruh-tezlik dispersiyasiga ega bo’ lgan material bo’ ylab o’ tsa, qisqa to’ lqin uzunlikdagi komponentlar uzunroqlariga qaraganda tezroq harakat qiladi va impuls vaqt o’ tishi bilan chastotasi pasayib, ''salbiy jiringlaydi'' yoki ''pastga tushadi'' .

Akustik sohadagi manfiy chiyillash signalining kundalik misoli , yaqinlashib kelayotgan poyezdning payvandlangan yo’ lda deformatsiyalarga urilishidir . Poyezdning o’ zi keltirib chiqaradigan tovush impulsiv bo’ lib, metall yo’ llarda havoga qaraganda ancha tez tarqaladi, shuning uchun poyezd kelishidan oldin uni yaxshi eshitish mumkin. Biroq, uzoqdan bu impulslarni keltirib chiqaradigan sifatida eshitilmaydi, lekin trekning tebranish rejimlarining murakkabligidan kelib chiqadigan aks sado o’ rtasida o’ ziga xos tushuvchi chiyillashga olib keladi. Guruh tezligi dispersiyasini eshitish mumkin, chunki tovushlar hajmi hayratlanarli darajada uzoq vaqt, bir necha soniyagacha eshitiladi.

''Guruh-tezlik dispersiyasi parametri''

: <math>D = \frac{1}{c} \, \frac{d n}{d \lambda}</math>

ko’ pincha salbiy omil orqali ''D'' ga proportsional bo’ lgan GVD miqdorini aniqlash uchun ishlatiladi:

: <math>D = -\frac{2\pi c}{\lambda^2} \, \frac{d^2k}{d\omega^2}.</math>

Ba’ zi mualliflarning fikriga ko’ ra,<ref>Saleh, B. E. A. and Teich, M. C. ''Fundamentals of Photonics'' (2nd ed.), Wiley, 2007.</ref> muhitning ma’ lum vakuum to’ lqin uzunligi ''l'' <sub>0</sub> uchun ''normal dispersiya'' / ''anomal dispersiyaga'' ega deyiladi, agar ''l'' <sub>0</sub> da hisoblangan sinishi indeksining ''ikkinchi'' hosilasi musbat/manfiy yoki ekvivalenti ''D'' ( ''l)'' bo’ lsa. <sub>0</sub> ) salbiy/musbat. Ushbu ta’ rif guruh-tezlik dispersiyasiga tegishli va oldingi bo’ limda berilgan ta’ rif bilan adashtirmaslik kerak. Ikkala ta’ rif umuman bir-biriga mos kelmaydi, shuning uchun o’ quvchi kontekstni tushunishi kerak.

== Dispersiyani nazorat qilish ==
GVD natijasi, salbiy yoki ijobiy bo’ ladimi, oxir-oqibat pulsning vaqtincha tarqalishidir. Bu optik tolaga asoslangan optik aloqa tizimlarida dispersiyani boshqarishni juda muhim qiladi, chunki dispersiya juda yuqori bo’ lsa, bit oqimini ifodalovchi impulslar guruhi vaqt o’ tishi bilan tarqaladi va birlashadi, bu bit oqimini tushunarsiz qiladi. Bu signal qayta tiklanmasdan yuborilishi mumkin bo’ lgan tola uzunligini cheklaydi. Bu muammoning mumkin bo’ lgan javoblaridan biri signallarni optik tolaga GVD nolga teng bo’ lgan to’ lqin uzunligida yuborishdir (masalan, taxminan 1,3-1,5).&nbsp;[[Kremniy dioksid|silika]] [[tolalari|tolalarida]] mikron ), shuning uchun bu to’ lqin uzunligidagi impulslar dispersiyadan minimal tarqaladi. Biroq, amalda bu yondashuv hal qilishdan ko’ ra ko’ proq muammolarni keltirib chiqaradi, chunki nol GVD boshqa nochiziqli ta’ sirlarni (masalan [[To'rt to'lqinli aralashtirish|, to’ rt to’ lqinli aralashtirish]] ) qabul qilib bo’ lmaydigan darajada kuchaytiradi. Yana bir mumkin bo’ lgan variant - salbiy dispersiya rejimida [[Soliton (optika)|soliton]] impulslaridan foydalanish, bu optik impulsning shakli bo’ lib, u o’ z shaklini saqlab qolish uchun [[Chiziqsiz optika|chiziqli bo’ lmagan optik]] effektdan foydalanadi. Solitonlarning amaliy muammosi borki, ular chiziqli bo’ lmagan effekt to’ g’ ri kuchga ega bo’ lishi uchun impulsda ma’ lum quvvat darajasini saqlab turishni talab qiladi. Buning o’ rniga, hozirda amalda qo’ llaniladigan yechim dispersiya ta’ sirini bekor qilish uchun odatda tolani qarama-qarshi dispersiyaning boshqa tolasi bilan moslashtirish orqali dispersiya kompensatsiyasini amalga oshirishdir; Bunday kompensatsiya oxir-oqibat [[O'z fazali modulyatsiya|o’ z-o’ zidan fazali modulyatsiya]] kabi chiziqli bo’ lmagan effektlar bilan cheklanadi, bu dispersiya bilan o’ zaro ta’ sir qiladi va uni bekor qilishni juda qiyinlashtiradi.

[[Ultra qisqa puls|Qisqa impulslar]] ishlab chiqaradigan [[Lazer|lazerlarda]] dispersiyani nazorat qilish ham muhimdir . [[Lazer konstruktsiyasi|Optik rezonatorning]] umumiy dispersiyasi lazer tomonidan chiqarilgan impulslarning, davomiyligini aniqlashda asosiy omil hisoblanadi . Lazer muhitining odatda ijobiy dispersiyasini muvozanatlash uchun ishlatilishi mumkin bo’ lgan aniq salbiy dispersiyani hosil qilish uchun bir juft [[Dispersion prizmalar|prizma]] joylashtirilishi mumkin . Dispersiv effektlarni hosil qilish uchun, [[Difraksion panjara|difraksion panjaralardan]] ham foydalanish mumkin ; bular ko’ pincha yuqori quvvatli lazer kuchaytirgich tizimlarida qo’ llaniladi. So’ nggi paytlarda prizma va panjaralarga alternativa ishlab chiqildi: [[Shiqillagan oyna|chiyillashli nometall]] . Ushbu dielektrik nometall turli xil to’ lqin uzunliklari turli xil penetratsion uzunliklarga ega bo’  lishi va shuning uchun turli guruh kechikishlari bilan qoplangan . Qoplama qatlamlari aniq salbiy dispersiyaga erishish uchun moslashtirilishi mumkin .

== To'lqin o'tkazgichlarda ==
[[To'lqin yo'riqnomasi|To’ lqin o’ tkazgichlar]] geometriyasi tufayli (faqat moddiy tarkibidan ko’ ra) juda dispersdir. [[Optik tola|Optik tolalar]] zamonaviy telekommunikatsiya tizimlarida keng qo’ llaniladigan optik chastotalar (yorug’ lik) uchun bir xil to’ lqin qo’ llanmasidir. Ma’ lumotni bitta tolada tashish tezligi boshqa hodisalar orasida xromatik dispersiya tufayli pulsning kengayishi bilan cheklangan.

Umuman olganda, [[Burchak chastota|burchak chastotasi]] ''ō'' ( ''b'' ) bo’ lgan to’ lqin o’ tkazgich rejimi uchun [[Tarqalish konstantasi|tarqalish doimiysi]] ''b'' (shundayki ''z'' tarqalish yo’ nalishidagi elektromagnit maydonlar ''e'' ''<sup>i</sup>'' <sup>( ''bz'' - ''ōt'' )</sup> ga proportsional tebranadi), guruh-tezlik dispersiyasi parametri ''D.'' sifatida belgilanadi<ref>Ramaswami, Rajiv and Sivarajan, Kumar N. (1998) ''Optical Networks: A Practical Perspective''. Academic Press: London.</ref>

: <math>D = -\frac{2\pi c}{\lambda^2} \frac{d^2 \beta}{d\omega^2} = \frac{2\pi c}{v_g^2 \lambda^2} \frac{dv_g}{d\omega},</math>

qayerda ''λ''=2π''c''/''ω'' - vakuum to’ lqin uzunligi va ''v''<sub>g</sub>=''dω'' /''dβ'' - guruh tezligi. Ushbu formula avvalgi bobdagi bir hil muhit uchun umumlashtiriladi va to’ lqin uzatuvchi dispersiyani ham, material dispersiyasini ham o’ z ichiga oladi. Dispersiyani bu tarzda belgilashning sababi shundaki, |''D''| - optik tolalar uchun odatda [[Pikosekund|ps]] /( [[Nanometr|nm]] ⋅ [[Kilometr|km]] ) da qayd etilgan masofani bosib o’ tgan birlik tarmoqli kengligi Δ''t'' uchun Δ''λ'' ning (asimptotik) vaqtinchalik impuls tarqalishi.

[[Ko'p rejimli optik tolali|Ko’ p rejimli optik tolalar]] holatida [[Modali dispersiya|modal dispersiya]] deb ataladigan narsa ham pulsning kengayishiga olib keladi . Hatto [[Yagona rejimli tola|bitta rejimli tolalarda]] ham impulsning kengayishi [[Polarizatsiya rejimi dispersiyasi|polarizatsiya rejimining tarqalishi]] natijasida yuzaga kelishi mumkin ( chunki ikkita polarizatsiya rejimi mavjud ) . Bular xromatik dispersiyaga misol ''emas'', chunki ular tarqaladigan impulslarning to’ lqin uzunligi yoki [[Spektral chiziq kengligi|tarmoqli kengligiga]] bog’ liq emas.

== Keng tarmoqli kengligi bo'yicha yuqori tartibli dispersiya ==
Agar bitta to’ lqin to’ plamida keng chastota diapazoni (keng tarmoqli kengligi) mavjud bo’ lsa, masalan, [[Ultra qisqa puls|ultraqisqa impuls]] yoki [[Chirp|jirkanch]] impuls yoki [[Spektrning tarqalishi|tarqalish spektrini]] uzatishning boshqa shakllarida, dispersiyani doimiy qiymatga yaqinlashtirish to’ g’ ri bo’ lmasligi mumkin. butun tarmoqli kengligi va impuls tarqalishi kabi effektlarni hisoblash uchun murakkabroq hisoblar talab qilinadi.

Xususan, yuqorida aniqlangan ''D'' dispersiya parametri guruh tezligining faqat bitta hosilasidan olinadi. Yuqori hosilalar ''yuqori tartibli dispersiya'' deb nomlanadi.<ref>[http://www.rp-photonics.com/chromatic_dispersion.html Chromatic Dispersion], ''Encyclopedia of Laser Physics and Technology'' (Wiley, 2008).</ref><ref>{{Cite journal|last=Mai|first1=Wending|last2=Campbell|first2=Sawyer D.|last3=Whiting|first3=Eric B.|last4=Kang|first4=Lei|last5=Werner|first5=Pingjuan L.|last6=Chen|first6=Yifan|author7-link=Douglas Werner|last7=Werner|first7=Douglas H.|date=2020-10-01|title=Prismatic discontinuous Galerkin time domain method with an integrated generalized dispersion model for efficient optical metasurface analysis|journal=[[Optical Materials Express]]|language=EN|volume=10|issue=10|pages=2542–2559|doi=10.1364/OME.399414|bibcode=2020OMExp..10.2542M|issn=2159-3930}}</ref> Bu atamalar oddiygina ma’ lum bir chastota atrofida muhit yoki to’ lqin o’ tkazgichning [[Yorugʻlik dispersiyasining elementar klassik nazariyasi|dispersiya munosabatining]] ''b'' ( ''ō'' ) [[Teylor seriyasi|Teylor seriyali]] kengayishidir. Ularning ta’ sirini to’ lqin shaklining [[Furye konvertatsiyasi|Furye o’ zgarishlarini]] raqamli baholash, asta- [[Sekin-asta o'zgaruvchan konvertning taxminiyligi|sekin o’ zgaruvchan konvertning]] yuqori darajali taxminiy integratsiyasi orqali, [[Ajratilgan bosqichli usul|bo’ lingan bosqichli usul]] bilan (teylor seriyasidan ko’ ra aniq dispersiya munosabatidan foydalanish mumkin) yoki to’ g’ ridan-to’ g’ ri hisoblash orqali hisoblash mumkin. taxminiy konvert tenglamasi emas, balki to’ liq [[Maksvell tenglamalari|Maksvell tenglamalarini]] simulyatsiya qilish.

== Yuqori dispersiya tartiblarining umumlashtirilgan formulasi - Lah-Laguerre optikasi ==
Teylor koeffitsientlari orqali xromatik dispersiyani bezovta qiluvchi tarzda tavsiflash bir nechta turli tizimlardan dispersiyani muvozanatlash kerak bo’ lgan optimallashtirish muammolari uchun foydalidir. Misol uchun, chirp puls lazer kuchaytirgichlarida optik shikastlanmaslik uchun impulslar birinchi navbatda nosilka tomonidan o’ z vaqtida cho’ ziladi. Keyin kuchaytirish jarayonida impulslar muqarrar ravishda materiallardan o’ tadigan chiziqli va chiziqli bo’ lmagan fazalarni to’ playdi. Va nihoyat, impulslar har xil turdagi kompressorlarda siqiladi. Yig’ ilgan bosqichda qolgan yuqori buyurtmalarni bekor qilish uchun odatda individual buyurtmalar o’ lchanadi va muvozanatlanadi. Biroq, bir xil tizimlar uchun bunday bezovta qiluvchi tavsif ko’ pincha kerak emas (ya’ ni, to’ lqin o’ tkazgichlarda tarqalish). Dispersiya tartiblari Lah-Lager tipidagi transformatsiyalar ko’ rinishida hisoblash uchun qulay tarzda umumlashtirildi.<ref>{{Cite journal|last=Popmintchev|first1=Dimitar|last2=Wang|first2=Siyang|last3=Xiaoshi|first3=Zhang|last4=Stoev|first4=Ventzislav|last5=Popmintchev|first5=Tenio|date=2022-10-24|title=Analytical Lah-Laguerre optical formalism for perturbative chromatic dispersion|journal=[[Optics Express]]|language=EN|volume=30|issue=22|pages=40779–40808|year=2022|doi=10.1364/OE.457139|pmid=36299007|bibcode=2022OExpr..3040779P}}</ref><ref>{{ArXiv manbasi|last=Popmintchev|first1=Dimitar|last2=Wang|first2=Siyang|last3=Xiaoshi|first3=Zhang|last4=Stoev|first4=Ventzislav|last5=Popmintchev|first5=Tenio|date=2020-08-30|title=Theory of the Chromatic Dispersion, Revisited|class=physics.optics|language=EN|eprint=2011.00066}}</ref>

Dispersiya tartiblari faza yoki to’ lqin vektorining , Teylor kengayishi bilan belgilanadi .

<math> \begin{array}{c}\varphi \mathrm{(}\omega\mathrm{)} = \varphi\left.\ \right|_{\omega_{0}} + \left. \ \frac{\partial\varphi}{\partial\omega} \right|_{\omega_{0}}\left(\omega - \omega_{0} \right) + \frac{1}{2}\left. \ \frac{\partial^{{2}}\varphi}{\partial\omega^{2}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{2}\ + \ldots + \frac{1}{p!}\left. \ \frac{\partial^{{p}}\varphi}{\partial\omega^{p}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{p} + \ldots \end{array} </math>

<math> \begin{array}{c}k\mathrm{(}\omega\mathrm{)} = k\left.\ \right|_{\omega_{0}} + \left. \ \frac{\partial k}{\partial\omega} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right) + \frac{1}{2}\left. \ \frac{\partial^{{2}}k}{\partial\omega^{2}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{2}\ + \ldots + \frac{1}{p!}\left. \ \frac{\partial^{{p}}k}{\partial\omega^{p}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{p} + \ldots \end{array}</math>To’ lqinli qurilma uchun dispersiya munosabatlari <math>k \mathrm{(}\omega\mathrm{)} = \frac{\omega}{c}n \mathrm{(}\omega\mathrm{)}</math> va faza <math>\varphi \mathrm{(}\omega\mathrm{)} = \frac{\omega}{c}{\it OP} \mathrm{(}\omega\mathrm{)}</math> quyidagicha ifodalanishi mumkin:


<math> \begin{array}{c}\frac{ {\partial }^{{p}}}{\partial {\omega }^p}k \mathrm{(}\omega \mathrm{)}=\frac{1}{c}\left(p\frac{ {\partial }^{p-1}}{\partial {\omega }^{p-1}}n \mathrm{(}\omega \mathrm{)}+\omega \frac{ {\partial }^{{p}}}{\partial {\omega }^p}n \mathrm{(}\omega \mathrm{)}\right)\ \end{array}</math>, <math>\begin{array}{c}\frac{ {\partial }^{{p}}}{\partial {\omega }^p}\varphi  \mathrm{(}\omega \mathrm{)} = \frac{1}{c}\left(p\frac{ {\partial }^{p-1}}{\partial {\omega }^{p-1}}{\it OP}  \mathrm{(}\omega \mathrm{)}+\omega \frac{ {\partial }^{{p}}}{\partial {\omega }^p}{\it OP}  \mathrm{(}\omega \mathrm{)}\right)  \end{array} (1)</math>

Har qanday differentsiallanuvchi funktsiyaning hosilalari <math>f\mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}</math> To’ lqin uzunligi , yoki chastota bo’ shlig’ i Lah o'zgarishi orqali quyidagicha aniqlanadi :

<math xmlns="http://www.w3.org/1998/Math/MathML">
	
\begin{array}{l} \frac{ {\partial {p}}}{\partial {\omega }^p}f  \mathrm{(}\omega \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\lambda }^m\frac{ {\partial }^{{m}}}{\partial {\lambda }^m}f \mathrm{(}\lambda \mathrm{)}}\end{array}  
	</math><math>,</math>  <math>
	 \begin{array}{c} 
	
\frac{ {\partial }^{{p}}}{\partial {\lambda }^p}f \mathrm{(}\lambda \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\omega }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\omega }^m\frac{ {\partial }^{{m}}}{\partial {\omega }^m}f  \mathrm{(}\omega \mathrm{)}}\end{array}  
(2)
</math>

Transformatsiyaning matritsa elementlari Lah koeffitsientlari: <math>\mathcal{A}\mathrm{(}p,m\mathrm{)} = \frac{p\mathrm{!}}{\left(p\mathrm{-}m\right)\mathrm{!}m\mathrm{!}}\frac{\mathrm{(}p\mathrm{-}\mathrm{1)!}}{\mathrm{(}m\mathrm{-}\mathrm{1)!}}</math>

GDD uchun yozilgan yuqoridagi ifoda to’ lqin uzunligi GGD bo’ lgan doimiy nol yuqori tartiblarga ega bo’ lishini bildiradi. GDD tomonidan baholangan yuqori buyurtmalar:

<math>\begin{array}{c} \frac{ {\partial }^{{p}}}{\partial {\omega }^p}GDD  \mathrm{(}\omega \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\lambda }^m\frac{ {\partial }^{{m}}}{\partial {\lambda }^m}GDD \mathrm{(}\lambda \mathrm{)}}
	
 \end{array}</math>

Sinishi ko’ rsatkichi uchun ifodalangan (2) tenglamani almashtirish <math>n</math> yoki optik yo’ l <math>OP</math> (1) tenglamada dispersiya tartiblari uchun yopiq shakldagi ifodalar paydo bo’ ladi. Umuman olganda, <math>p^{th}</math> tartibli dispersiya POD - ikkinchi manfiy tartibli lager tipidagi transformatsiya:

<math>POD = \frac{d^p \varphi(\omega) }{d\omega^p}=(-1)^p(\frac{\lambda}{2\pi c})^{(p-1)}\sum_{m=0}^{p}\mathcal{B(p,m)} (\lambda)^m\frac{d^m OP(\lambda) }{d\lambda^m} </math> <math>,</math>  
<math>POD = \frac{d^p k(\omega) }{d\omega^p}=(-1)^p(\frac{\lambda}{2\pi c})^{(p-1)}\sum_{m=0}^{p}\mathcal{B(p,m)} (\lambda)^m\frac{d^m n(\lambda) }{d\lambda^m} </math>
[[Fayl:Figure1_Wiki_FS-01.png|thumb|370x370px| Eritilgan kremniyning birinchi o'nta dispersiya tartibi]]
O’ zgartirishlarning  matritsa  elementlari minus 2 tartibli belgisiz lager koeffitsientlari bo’ lib, ular quyidagicha berilgan 

<math> \mathcal{B}\mathrm{(}p,m\mathrm{)} = \frac{p\mathrm{!}}{\left(p\mathrm{-}m\right)\mathrm{!}m\mathrm{!}}\frac{\mathrm{(}p\mathrm{-}\mathrm{2)!}}{\mathrm{(}m\mathrm{-}\mathrm{2)!}}</math>

To'lqin vektori uchun aniq yozilgan birinchi o'n dispersiya tartibi: <math>
\begin{array}{l}\boldsymbol{{\it GD}} = \frac{\partial }{\partial \omega }k \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(n \mathrm{(}\omega \mathrm{)}+\omega \frac{\partial n \mathrm{(}\omega \mathrm{)}}{\partial \omega }\right) = \frac{\mathrm{1}}{c}\left(n \mathrm{(}\lambda \mathrm{)}-\lambda \frac{\partial n \mathrm{(}\lambda \mathrm{)}}{\partial\lambda }\right) = v^{\mathrm{-}\mathrm{1}}_{gr}\end{array} 
	
</math>

Guruh sinishi ko'rsatkichi <math>n_g</math> quyidagicha aniqlanadi: <math>n_g  =  cv^{\mathrm{-}\mathrm{1}}_{gr}</math> .

<math>
	
\begin{array}{l}\boldsymbol{{\it GDD}} = \frac{ {\partial }^{{2}}}{\partial {\omega }^{\mathrm{2}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{2}\frac{\partial n\mathrm{(}\omega \mathrm{)}}{\partial \omega }+\omega \frac{ {\partial }^{{2}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}}\right)  =  \frac{\mathrm{1}}{c}\left(\frac{\lambda }{\mathrm{2}\pi c}\right)\left({\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}\right) \end{array} 
	
</math>
<math>
	
\begin{array}{l}\boldsymbol{{\it TOD}} = \frac{ {\partial }^{{3}}}{\partial {\omega }^{\mathrm{3}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{3}\frac{ {\partial }^{{2}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}}+\omega \frac{ {\partial }^{{3}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}\right)    = {-} \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{2}}\Bigl(\mathrm{3}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}\Bigr) \end{array} 
	
</math> 
<math>
	
\begin{array}{l}\boldsymbol{{\it FOD}} = \frac{ {\partial }^{{4}}}{\partial {\omega }^{\mathrm{4}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{4}\frac{ {\partial }^{{3}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}+\omega \frac{ {\partial }^{{4}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}\right)    = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{3}}\Bigl(\mathrm{12}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}    +\mathrm{8}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}\Bigr) \end{array} 
	
</math>   
<math>
	
\begin{array}{l}\boldsymbol{{\it FiOD}} = \frac{ {\partial }^{{5}}}{\partial {\omega }^{\mathrm{5}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{5}\frac{ {\partial }^{{4}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}+\omega \frac{ {\partial }^{{5}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}\right)=  {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{4}} \Bigl(\mathrm{60}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{60}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{15}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\Bigr) \end{array} 
	
</math>  
<math>
	
\begin{array}{l}\boldsymbol{{\it SiOD}} = \frac{ {\partial }^{{6}}}{\partial {\omega }^{\mathrm{6}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{6}\frac{ {\partial }^{{5}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}+\omega \frac{ {\partial }^{{6}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{6}}}\right) =   \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{5}}\Bigl(\mathrm{360}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +\mathrm{480}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{180}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{24}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}\Bigr) \end{array} 
	
</math>   
<math>    
	
\begin{array}{l}\boldsymbol{{\it SeOD}} = \frac{ {\partial }^{{7}}}{\partial {\omega }^{\mathrm{7}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{7}\frac{ {\partial }^{{6}}n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{6}}}+\omega \frac{ {\partial }^{{7}}n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{7}}}\right)    =   {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{6}} \Bigl(\mathrm{2520}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{4200}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{2100}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{420}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+   \mathrm{35}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}\Bigr) \end{array} 
	
</math>   
<math>    
	
\begin{array}{l}\boldsymbol{{\it EOD}} = \frac{ {\partial }^{{8}}}{\partial {\omega }^{\mathrm{8}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{8}\frac{ {\partial }^{{7}}n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{7}}}+\omega \frac{ {\partial }^{{8}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}\right) =    \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{7}}\Bigl(\mathrm{20160}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +\mathrm{40320}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{25200}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{6720}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+ \mathrm{840}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}} +\\+\mathrm{48}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}\Bigr) \end{array} 
	
</math>  
<math>    
	
\begin{array}{l}\boldsymbol{{\it NOD}} = \frac{ {\partial }^{{9}}}{\partial {\omega }^{\mathrm{9}}}k \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{9}\frac{ {\partial }^{{8}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}+\omega \frac{ {\partial }^{{9}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}\right) =   {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{8}}\Bigl(\mathrm{181440}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{423360}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{317520}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{105840}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+  \mathrm{17640}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{1512}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{63}{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+{\lambda }^{\mathrm{9}}\frac{ {\partial }^{{9}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}\Bigr) \end{array} 
	
</math>        
<math>    
	
\begin{array}{l}\boldsymbol{{\it TeOD}} = \frac{ {\partial }^{{10}}}{\partial {\omega }^{\mathrm{10}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{10}\frac{ {\partial }^{{9}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}+\omega \frac{ {\partial }^{{10}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{10}}}\right) =   \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{9}}\Bigl(\mathrm{1814400}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{4838400}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{4233600}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+  {1693440}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\\+\mathrm{352800}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{40320}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{2520}{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+\mathrm{80}{\lambda }^{\mathrm{9}}\frac{ {\partial }^{{9}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}+  {\lambda }^{\mathrm{10}}\frac{ {\partial }^{{10}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{10}}}\Bigr) \end{array} 
	
</math>


Aniq, bosqich uchun yozilgan <math>\varphi</math>, birinchi on dispersiya tartiblari to’ lqin uzunligi funksiyasi sifatida Lah o’ zgarishlari (tenglama (2)) yordamida quyidagicha ifodalanishi mumkin:

<math xmlns="http://www.w3.org/1998/Math/MathML">
	
\begin{array}{l} \frac{ {\partial {p}}}{\partial {\omega }^p}f  \mathrm{(}\omega \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\lambda }^m\frac{ {\partial }^{{m}}}{\partial {\lambda }^m}f \mathrm{(}\lambda \mathrm{)}}\end{array}  
</math> <math>,</math>  <math>
	\begin{array}{c} 
	
\frac{ {\partial }^{{p}}}{\partial {\lambda }^p}f \mathrm{(}\lambda \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\omega }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\omega }^m\frac{ {\partial }^{{m}}}{\partial {\omega }^m}f  \mathrm{(}\omega \mathrm{)}}\end{array}  
	
	
</math>


<math>
	
\begin{array}{l}\frac{\partial \varphi\mathrm{(}\omega \mathrm{)}}{\partial \omega }= {-}\left(\frac{\mathrm{2}\pi c}{{\omega }^{\mathrm{2}}}\right)\frac{\partial \varphi \mathrm{(}\omega \mathrm{)}}{\partial \lambda } = {-}\left(\frac{ {\lambda }^{\mathrm{2}}}{\mathrm{2}\pi c}\right)\frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }\end{array}
	
</math>

<math>
	
\begin{array}{l}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}} = \frac{\partial }{\partial \omega }\left(\frac{\partial \varphi \mathrm{(}\omega \mathrm{)}}{\partial \omega }\right) = {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{2}}\left(\mathrm{2}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}\right) \end{array}
	
</math>

<math>
	
\begin{array}{l}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}= {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{3}}\left(\mathrm{6}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{6}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}\right) \end{array}
	
 </math>

<math>
	
\begin{array}{l}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}= {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{4}}\Bigl(\mathrm{24}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{36}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{12}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}
	
+{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}\Bigr) \end{array}
	
</math>
<math>
	
\begin{array}{l}\frac{ {\partial
	
}^{\mathrm{5}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}= {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{5}}\Bigl(\mathrm{120}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{240}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}  
	
+\mathrm{120}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{20}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\Bigr) \end{array}
	
</math>
<math>
	
\begin{array}{l}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{6}}}= {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{6}}\Bigl(\mathrm{720}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{1800}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{1200}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{300}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{30}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\mathrm{\ +}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}\Bigr) \end{array}
	
</math>
<math>
	
\begin{array}{l}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{7}}}= {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{7}} \Bigl(\mathrm{5040}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{15120}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{12600}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{4200}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{630}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{42}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}} \Bigr) \end{array} 
	
</math>
<math>
	
\begin{array}{l}\frac{ {\partial }^{{8}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}= {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{8}}\Bigl(\mathrm{40320}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{141120}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{141120}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{58800}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{11760}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{1176}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{56}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\\ +{\lambda }^{\mathrm{8}}\frac{\partial^{{8}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial{\lambda }^{\mathrm{8}}}\Bigr) \end{array} 
	
</math>
<math>
	
\begin{array}{l}\frac{ {\partial }^{{9}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}= {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{9}}\Bigl(\mathrm{362880}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{1451520}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{1693440}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{846720}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{211680}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{28224}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{2016}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{72}{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+{\lambda }^{\mathrm{9}}\frac{\partial ^{\mathrm{9}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}\Bigr) \end{array}
	
</math>
<math>
	
\begin{array}{l}\frac{ {\partial }^{{10}}\varphi \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{10}}}= {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{10}}\Bigl(\mathrm{3628800}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{16329600}{\lambda }^{\mathrm{2}}\frac{ {\partial }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{21772800}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{12700800}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{3810240}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{635040}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{60480}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}} +\mathrm{3240}{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+\mathrm{90}{\lambda }^{\mathrm{9}}\frac{ {\partial }^{{9}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}+{\lambda }^{\mathrm{10}}\frac{ {\partial }^{{10}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{10}}}\Bigr) \end{array}
	
</math>

== Fazoviy dispersiya ==
Elektromagnitika va optikada ''dispersiya'' atamasi odatda yuqorida aytib o’ tilgan vaqtinchalik yoki chastota dispersiyasini anglatadi. Fazoviy dispersiya muhitning kosmosga mahalliy bo’ lmagan javobini bildiradi; Buni o’ tkazuvchanlikning to’ lqin vektorga bog’ liqligi deb qayta ta’ riflash mumkin. Namunaviy [[Anizotropiya|anizotrop]] muhit uchun [[Elektr maydon|elektr]] va [[Elektr almashinish maydoni|elektr joy almashish maydoni]] o’ rtasidagi fazoviy munosabat [[konvolyutsiya]] sifatida ifodalanishi mumkin:<ref name="landau2">{{Kitob manbasi |first=L. D. |last=Landau |archivedate=Lev Pitaevskii |year=1984 |title=Electrodynamics of Continuous Media |edition=2nd |publisher=[[Butterworth-Heinemann]] |isbn=978-0-7506-2634-7}}</ref>

: <math>D_i(t, r) = E_i(t, r) + \int_0^\infty \int f_{ik}(\tau; r, r') E_k(t - \tau, r') \,dV'\,d\tau,</math>

[[Integral konvertatsiya|yadro]] qaerda <math>f_{ik}</math> dielektrik javobdir (sezuvchanlik); uning indekslari muhitning anizotropiyasini hisobga olish uchun uni umuman [[Tensor|tenzorga]] aylantiradi. Ko’ pgina makroskopik holatlarda fazoviy dispersiya ahamiyatsiz bo’ lib, o’ zgaruvchanlik shkalasi. <math>E_k(t - \tau, r')</math> atom o’ lchamlaridan ancha katta, chunki dielektrik yadrosi makroskopik masofalarda nobud bo’ ladi. Shunga qaramay, bu, ayniqsa, [[metallar]], [[elektrolitlar]] va [[Plazma|plazmalar]] kabi o’ tkazuvchi vositalarda ahamiyatsiz bo’ lmagan makroskopik ta’ sirga olib kelishi mumkin. Fazoviy dispersiya, shuningdek [[Optik aylanish|, optik faollik]] va [[Doppler kengayishi|Doppler kengayishida]]<ref name="landau3">{{Kitob manbasi |first=L. D. |last=Landau |archivedate=Lev Pitaevskii |year=1984 |title=Electrodynamics of Continuous Media |edition=2nd |publisher=[[Butterworth-Heinemann]] |isbn=978-0-7506-2634-7}}</ref>, shuningdek, [[Metamaterial|metamateriallar]] nazariyasida rol o’ ynaydi.<ref>{{Cite journal|last=Demetriadou|first1=A.|last2=Pendry|first2=J. B.|authorlink2=John Pendry|date=1 July 2008|title=Taming spatial dispersion in wire metamaterial|journal=[[Journal of Physics: Condensed Matter]]|volume=20|issue=29|pages=295222|doi=10.1088/0953-8984/20/29/295222|bibcode=2008JPCM...20C5222D}}</ref>

== Gemologiyada ==
{| class="wikitable sortable collapsible collapsed" style="float: right; text-align: center;"
|+Dispersion values of minerals<ref name="b1">{{Kitob manbasi |last=Schumann, Walter |title=Gemstones of the World: Newly Revised & Expanded Fourth Edition |url=https://books.google.com/books?id=V9PqVxpxeiEC&pg=PA42 |accessdate=31 December 2011 |year=2009 |publisher=Sterling Publishing Company, Inc. |isbn=978-1-4027-6829-3 |pages=41–2}}</ref>
!Name
!B–G
!C–F
|-
|Cinnabar (HgS)
|0.40
|—
|-
|Synth. [[Rutil|rutile]]
|0.330
|0.190
|-
|[[Rutil|Rutile]] (TiO<sub>2</sub>)
|0.280
|0.120–0.180
|-
|[[Anataz|Anatase]] (TiO<sub>2</sub>)
|0.213–0.259
|—
|-
|[[Vulfenit|Wulfenite]]
|0.203
|0.133
|-
|[[Vanadinit|Vanadinite]]
|0.202
|—
|-
|Fabulite
|0.190
|0.109
|-
|[[Sfalerit|Sphalerite]] (ZnS)
|0.156
|0.088
|-
|[[Oltingugurt|Sulfur]] (S)
|0.155
|—
|-
|Stibiotantalite
|0.146
|—
|-
|[[Gyotit|Goethite]] (FeO(OH))
|0.14
|—
|-
|[[Brukit|Brookite]] (TiO<sub>2</sub>)
|0.131
|0.12–1.80
|-
|[[Sinkit|Zincite]] (ZnO)
|0.127
|—
|-
|Linobate
|0.13
|0.075
|-
|Synthetic moissanite (SiC)
|0.104
|—
|-
|[[Kassiterit|Cassiterite]] (SnO<sub>2</sub>)
|0.071
|0.035
|-
|Zirconia (ZrO<sub>2</sub>)
|0.060
|0.035
|-
|Powellite (CaMoO<sub>4</sub>)
|0.058
|—
|-
|[[Andradit|Andradite]]
|0.057
|—
|-
|Demantoid
|0.057
|0.034
|-
|[[Serussit|Cerussite]]
|0.055
|0.033–0.050
|-
|[[Titanit|Titanite]]
|0.051
|0.019–0.038
|-
|Benitoite
|0.046
|0.026
|-
|[[Anglezit|Anglesite]]
|0.044
|0.025
|-
|[[Olmos|Diamond]] (C)
|0.044
|0.025
|-
|Flint glass
|0.041
|—
|-
|[[Sirkon|Hyacinth]]
|0.039
|—
|-
|Jargoon
|0.039
|—
|-
|Starlite
|0.039
|—
|-
|[[Sirkon|Zircon]] (ZrSiO<sub>4</sub>)
|0.039
|0.022
|-
|GGG
|0.038
|0.022
|-
|Scheelite
|0.038
|0.026
|-
|[[Ashirit|Dioptase]]
|0.036
|0.021
|-
|[[Whe Vinay wellite]]
|0.034
|—
|-
|[[Alebastr|Alabaster]]
|0.033
|—
|-
|[[Gips|Gypsum]]
|0.033
|0.008
|-
|[[Epidot|Epidote]]
|0.03
|0.012–0.027
|-
|[[Achroite]]
|0.017
|—
|-
|[[Kordiyerit|Cordierite]]
|0.017
|0.009
|-
|[[Danburit|Danburite]]
|0.017
|0.009
|-
|[[Turmalin|Dravite]]
|0.017
|—
|-
|Elbaite
|0.017
|—
|-
|Herderite
|0.017
|0.008–0.009
|-
|Hiddenite
|0.017
|0.010
|-
|Indicolite
|0.017
|—
|-
|Liddicoatite
|0.017
|—
|-
|[[Spodumen|Kunzite]]
|0.017
|0.010
|-
|[[Rubellit|Rubellite]]
|0.017
|0.008–0.009
|-
|[[Turmalin|Schorl]]
|0.017
|—
|-
|[[Skapolit|Scapolite]]
|0.017
|—
|-
|[[Spodumen|Spodumene]]
|0.017
|0.010
|-
|[[Turmalin|Tourmaline]]
|0.017
|0.009–0.011
|-
|[[Verdelite]]
|0.017
|—
|-
|[[Andaluzit|Andalusite]]
|0.016
|0.009
|-
|[[Barit|Baryte]] (BaSO<sub>4</sub>)
|0.016
|0.009
|-
|[[Evklaz|Euclase]]
|0.016
|0.009
|-
|Alexandrite
|0.015
|0.011
|-
|Chrysoberyl
|0.015
|0.011
|-
|Hambergite
|0.015
|0.009–0.010
|-
|[[Fenakit|Phenakite]]
|0.01
|0.009
|-
|[[Rodoxrozit|Rhodochrosite]]
|0.015
|0.010–0.020
|-
|[[Sillimanit|Sillimanite]]
|0.015
|0.009–0.012
|-
|[[Smitsonit|Smithsonite]]
|0.014–0.031
|0.008–0.017
|-
|Amblygonite
|0.014–0.015
|0.008
|-
|[[Akvamarin|Aquamarine]]
|0.014
|0.009–0.013
|-
|[[Berill|Beryl]]
|0.014
|0.009–0.013
|-
|Brazilianite
|0.014
|0.008
|-
|[[Selestin|Celestine]]
|0.014
|0.008
|-
|Goshenite
|0.014
|—
|-
|[[Berill|Heliodor]]
|0.014
|0.009–0.013
|-
|Morganite
|0.014
|0.009–0.013
|-
|Pyroxmangite
|0.015
|—
|-
|Synth. scheelite
|0.015
|—
|-
|Dolomite
|0.013
|—
|-
|[[Magnezit|Magnesite]] (MgCO<sub>3</sub>)
|0.012
|—
|-
|Synth. [[Zumrad|emerald]]
|0.012
|—
|-
|Synth. alexandrite
|0.011
|—
|-
|Synth. [[Sapfir|sapphire]] (Al<sub>2</sub>O<sub>3</sub>)
|0.011
|—
|-
|Phosphophyllite
|0.010–0.011
|—
|-
|Enstatite
|0.010
|—
|-
|[[Anortit|Anorthite]]
|0.009–0.010
|—
|-
|[[Aktinolit|Actinolite]]
|0.009
|—
|-
|Jeremejevite
|0.009
|—
|-
|[[Nefelin|Nepheline]]
|0.008–0.009
|—
|-
|[[Apofillit|Apophyllite]]
|0.008
|—
|-
|Hauyne
|0.008
|—
|-
|[[Natrolit|Natrolite]]
|0.008
|—
|-
|Synth. quartz (SiO<sub>2</sub>)
|0.008
|—
|-
|[[Aragonit|Aragonite]]
|0.007–0.012
|—
|-
|Augelite
|0.007
|—
|-
|Tanzanite
|0.030
|0.011
|-
|Thulite
|0.03
|0.011
|-
|Zoisite
|0.03
|—
|-
|YAG
|0.028
|0.015
|-
|[[Almandin|Almandine]]
|0.027
|0.013–0.016
|-
|[[Grossulyar|Hessonite]]
|0.027
|0.013–0.015
|-
|[[Spessartin|Spessartine]]
|0.027
|0.015
|-
|[[Uvarovit|Uvarovite]]
|0.027
|0.014–0.021
|-
|[[Villemit|Willemite]]
|0.027
|—
|-
|Pleonaste
|0.026
|—
|-
|Rhodolite
|0.026
|—
|-
|[[Boratsit|Boracite]]
|0.024
|0.012
|-
|[[Kriolit|Cryolite]]
|0.024
|—
|-
|[[Stavrolit|Staurolite]]
|0.023
|0.012–0.013
|-
|[[Pirop|Pyrope]]
|0.022
|0.013–0.016
|-
|[[Diaspor|Diaspore]]
|0.02
|—
|-
|[[Grossulyar|Grossular]]
|0.020
|0.012
|-
|[[Gemimorfit|Hemimorphite]]
|0.020
|0.013
|-
|[[Kianit|Kyanite]]
|0.020
|0.011
|-
|[[Zabarjad|Peridot]]
|0.020
|0.012–0.013
|-
|[[Laʼl|Spinel]]
|0.020
|0.011
|-
|Vesuvianite
|0.019–0.025
|0.014
|-
|Clinozoisite
|0.019
|0.011–0.014
|-
|[[Labrador (mineral)|Labradorite]]
|0.019
|0.010
|-
|[[Aksinit|Axinite]]
|0.018–0.020
|0.011
|-
|Ekanite
|0.018
|0.012
|-
|Kornerupine
|0.018
|0.010
|-
|[[Korund|Corundum]] (Al<sub>2</sub>O<sub>3</sub>)
|0.018
|0.011
|-
|Rhodizite
|0.018
|—
|-
|[[Yoqut|Ruby]] (Al<sub>2</sub>O<sub>3</sub>)
|0.018
|0.011
|-
|[[Sapfir|Sapphire]] (Al<sub>2</sub>O<sub>3</sub>)
|0.018
|0.011
|-
|Sinhalite
|0.018
|0.010
|-
|[[Sodalit|Sodalite]]
|0.018
|0.009
|-
|Synth. [[Korund|corundum]]
|0.018
|0.011
|-
|[[Diopsid|Diopside]]
|0.018–0.020
|0.01
|-
|[[Zumrad|Emerald]]
|0.014
|0.009–0.013
|-
|[[Topaz]]
|0.014
|0.008
|-
|[[Ametist|Amethyst]] (SiO<sub>2</sub>)
|0.013
|0.008
|-
|[[Angidrit|Anhydrite]]
|0.013
|—
|-
|[[Apatit|Apatite]]
|0.013
|0.010
|-
|[[Apatit|Apatite]]
|0.013
|0.008
|-
|[[Avantyurin|Aventurine]]
|0.013
|0.008
|-
|[[Kvars|Citrine]]
|0.013
|0.008
|-
|Morion
|0.013
|—
|-
|Prasiolite
|0.013
|0.008
|-
|[[Kvars|Quartz]] (SiO<sub>2</sub>)
|0.013
|0.008
|-
|Smoky quartz (SiO<sub>2</sub>)
|0.013
|0.008
|-
|Rose quartz (SiO<sub>2</sub>)
|0.013
|0.008
|-
|[[Albit|Albite]]
|0.012
|—
|-
|[[Bytownite]]
|0.012
|—
|-
|[[Dala shpatlari|Feldspar]]
|0.012
|0.008
|-
|[[Oytosh|Moonstone]]
|0.012
|0.008
|-
|[[Ortoklaz|Orthoclase]]
|0.012
|0.008
|-
|Pollucite
|0.012
|0.007
|-
|[[Sanidin|Sanidine]]
|0.012
|—
|-
|[[Quyosh toshi|Sunstone]]
|0.012
|—
|-
|Beryllonite
|0.010
|0.007
|-
|Cancrinite
|0.010
|0.008–0.009
|-
|[[Leysit|Leucite]]
|0.010
|0.008
|-
|[[Obsidian]]
|0.010
|—
|-
|[[Stronsianit|Strontianite]]
|0.008–0.028
|—
|-
|[[Kalsit|Calcite]] (CaCO<sub>3</sub>)
|0.008–0.017
|0.013–0.014
|-
|[[Flyuorit|Fluorite]] (CaF<sub>2</sub>)
|0.007
|0.004
|-
|[[Gematit|Hematite]]
|0.500
|—
|-
|Synthetic [[Kassiterit|cassiterite]] (SnO<sub>2</sub>)
|0.041
|—
|-
|Gahnite
|0.019–0.021
|—
|-
|[[Datolit|Datolite]]
|0.016
|—
|-
|[[Tremolit|Tremolite]]
|0.006–0.007
|—
|}
[[Gemmologiya|Gemologiyaning]] [[Jargon|texnik terminologiyasida]] ''dispersiya'' B va G nuqtalarida materialning sinishi indeksidagi farqdir (686,7).&nbsp;[[Nanometr|nm]] va 430.8&nbsp;nm) yoki C va F (656.3&nbsp;nm va 486.1&nbsp;nm) [[Fraungofer chiziqlari|Fraungofer to’ lqin uzunliklari]] va [[Javohirlar|qimmatbaho]] toshdan kesilgan prizmaning "olov" darajasini ko’ rsatish uchun mo’ ljallangan. Yong’ in - bu gemologlar tomonidan qimmatbaho toshning tarqalish xususiyatini yoki uning etishmasligini tasvirlash uchun ishlatiladigan so’ zlashuv atamasi. Dispersiya moddiy xususiyatdir. Berilgan qimmatbaho tosh tomonidan ko’ rsatilgan olov miqdori qimmatbaho toshning burchak burchaklari, jilo sifati, yorug’ lik muhiti, materialning sinishi indeksi, rangning to’ yinganligi va tomoshabinning marvaridga nisbatan yo’ nalishiga bog’ liq.<ref name="b12">{{Kitob manbasi |last=Schumann, Walter |title=Gemstones of the World: Newly Revised & Expanded Fourth Edition |url=https://books.google.com/books?id=V9PqVxpxeiEC&pg=PA42 |accessdate=31 December 2011 |year=2009 |publisher=Sterling Publishing Company, Inc. |isbn=978-1-4027-6829-3 |pages=41–2}}</ref><ref>{{Veb manbasi |url=http://www.gemsociety.org/article/gemstone-dispersion/ |sarlavha=What is Gemstone Dispersion? |ish=GemSociety.org |nashriyot=International Gem Society |qaralgan sana=2015-03-09}}</ref>

== Tasvirda ==
Fotografik va mikroskopik linzalarda dispersiya [[Xromatik aberatsiya|xromatik aberatsiyaga]] olib keladi, bu esa tasvirdagi turli ranglarning to’ g’ ri bir - biriga mos kelmasligiga olib keladi. Bunga qarshi turish uchun turli xil texnikalar ishlab chiqilgan, masalan, [[Axromat|akromatlardan]], turli dispersiyadagi ko’ zoynakli ko’ p elementli linzalardan foydalanish. Ular shunday tuzilganki, turli qismlarning xromatik aberratsiyalari bekor qilinadi.

== Pulsar emissiyasi ==
[[Pulsar|Pulsarlar]] - bu millisekundlardan soniyalargacha bo’ lgan juda muntazam oraliqlarda [[Puls|pulslar]] chiqaradigan neytron yulduzlari. Astronomlarning fikriga ko’ ra, impulslar bir vaqtning o’ zida keng chastotalarda chiqariladi. Biroq, Yerda kuzatilganidek, yuqori radiochastotalarda chiqarilgan har bir impulsning tarkibiy qismlari past chastotalarda chiqarilgandan oldin keladi. Bu dispersiya [[Yulduzlararo muhit|yulduzlararo muhitning]] ionlashgan komponenti, asosan guruh tezligini chastotaga bog’ liq bo’ lgan erkin elektronlar tufayli yuzaga keladi. ''{{Mvar|ν}}'' chastotada qo’ shilgan qo’ shimcha kechikish

: <math>t = k_\text{DM} \cdot \left(\frac{\text{DM}}{\nu^2}\right),</math>

dispersiya konstantasi ''k''<sub>DM</sub> bilan berilgan

: <math>k_\text{DM} = \frac{e^2}{2 \pi m_\text{e} c} \approx 4.149~\text{GHz}^2\,\text{pc}^{-1}\,\text{cm}^3\,\text{ms},</math>

va '''dispersiya o’ lchovi''' (DM) erkin elektronlarning ustun zichligi ( [[Umumiy elektron tarkibi|jami elektron tarkibi]] ){{Snd}}ya’ ni fotonning pulsardan Yergacha boradigan yo’ l bo’ ylab ''integrallashgan'' elektronlarning [[Raqamlar zichligi|soni zichligi]] <sub>.</sub>{{Snd}}va tomonidan beriladi

: <math>\text{DM} = \int_0^d n_e\,dl</math>

kub santimetr uchun [[parsek]] birliklari bilan (1 dona/sm<sup>3</sup> = 30,857 {{E|21}}  m <sup>−2</sup>).<ref>Lorimer, D. R., and Kramer, M., ''Handbook of Pulsar Astronomy'', vol.&nbsp;4 of Cambridge Observing Handbooks for Research Astronomers ([[Cambridge University Press]], Cambridge, U.K.; New York, U.S.A, 2005), 1st edition.</ref>

Odatda astronomik kuzatishlar uchun bu kechikishni to’ g’ ridan-to’ g’ ri o’ lchash mumkin emas, chunki emissiya vaqti noma’ lum. Ikki xil chastotada kelish vaqtlaridagi farqni o’ lchash ''mumkin'' . Pulsning yuqori chastotali ''{{Mvar|ν}}'' <sub>hi</sub> va past chastotali ''{{Mvar|ν}}'' <sub>lo</sub> komponenti orasidagi kechikish D ''t'' bo’ ladi.

: <math>\Delta t = k_\text{DM} \cdot \text{DM} \cdot \left( \frac{1}{\nu_\text{lo}^2} - \frac{1}{\nu_\text{hi}^2} \right).</math>

Yuqoridagi tenglamani D ''t'' ko’ rinishida qayta yozish bir necha chastotalarda impulsning kelish vaqtini o’ lchash orqali DM ni aniqlash imkonini beradi. Bu, o’ z navbatida, yulduzlararo muhitni o’ rganish uchun ishlatilishi mumkin, shuningdek, turli chastotalarda pulsarlarni kuzatishni birlashtirishga imkon beradi.

== Yana qarang ==
 {{div col|colwidth=22em}}
* [[Abbe number]]
* [[Calculation of glass properties]] incl. dispersion
* [[Cauchy's equation]]
* [[Dispersion relation]]
* [[Fast radio burst]] (astronomy)
* [[Fluctuation theorem]]
* [[Green–Kubo relations]]
* [[Group delay]]
* [[Intramodal dispersion]]
* [[Kramers–Kronig relations]]
* [[Linear response function]]
* [[Multiple-prism dispersion theory]]
* [[Sellmeier equation]]
* [[Ultrashort pulse]]
* [[Virtually imaged phased array]]
{{div col end}}

== Manbalar ==
{{Manbalar}}

== Havolalar ==

* [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page Dispersiv Wiki] {{Webarxiv|url=https://web.archive.org/web/20170723091053/http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page |date=2017-07-23 }} - dispersiyaning matematik jihatlarini muhokama qilish.
* [http://www.rp-photonics.com/dispersion.html Dispersiya] - Lazer fizikasi va texnologiyasi entsiklopediyasi
* QED tomonidan [http://qed.wikina.org/dispersion/ optik dispersiyani namoyish qiluvchi animatsiyalar]
* [http://webdemo.inue.uni-stuttgart.de/webdemos/02_lectures/uebertragungstechnik_2/chromatic_dispersion/ Xromatik dispersiya uchun interaktiv veb-demo.] Shtutgart universiteti Telekommunikatsiyalar instituti
[[Turkum:Optika]]