Yorug'lik Dispersiyasi




Optikada va umuman to’ lqin tarqalishida dispersiya to’ lqinning faza tezligi uning chastotasiga bog’ liq bo’ lgan hodisadir; 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 yorug’ lik va boshqa elektromagnit to’ lqinlarni tasvirlash uchun ishlatilgan bo’ lsa-da, dispersiya xuddi shu ma’ noda tovush va seysmik to’ lqinlar holatida akustik dispersiya va tortishish to’ lqinlarida (okean) har qanday to’ lqin harakati uchun qo’ llanilishi mumkin. to’ lqinlar). Optikada dispersiya uzatish liniyalari bo’ ylab telekommunikatsiya signallarining (masalan, koaksiyal kabeldagi mikroto’ lqinlar ) yoki optik tolali yorug’ lik impulslarining xususiyatidir.

Optikada dispersiyaning muhim va tanish oqibatlaridan biri dispers prizma tomonidan ishlab chiqarilgan spektrda va linzalarning xromatik aberratsiyasida ko’ rinib turganidek, yorug’ likning ranglarining sinish burchagining o’ zgarishidir. Xromatik aberatsiya asosan bekor qilingan aralash akromatik linzalarning dizayni uning Abbe raqami V bilan berilgan shisha dispersiyasi miqdorini qo’ llaydi, bu erda pastroq Abbe raqamlari ko’ rinadigan spektrda katta dispersiyaga mos keladi. Telekommunikatsiya kabi ba’ zi ilovalarda to’ lqinning mutlaq fazasi ko’ pincha muhim emas, faqat to’ lqin paketlari yoki "impulslar" ning tarqalishi; Bunday holda, odamni faqat guruh tezligining chastota bilan o’ zgarishi qiziqtiradi, bu guruh tezligi dispersiyasi deb ataladi.

Barcha umumiy uzatish vositalari, shuningdek, chastota funktsiyasi sifatida zaiflashuvda (uzatish uzunligiga normallashtirilgan) farqlanadi, bu esa zaiflashuvning buzilishiga olib keladi; Bu dispersiya emas, garchi ba’ zida bir-biriga yaqin bo’ lgan 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 yorug’ likning turli to’ lqin uzunliklari (turli ranglar ) komponentlariga fazoviy ajralishiga olib keladi. Biroq, dispersiya boshqa ko’ plab holatlarda ham ta’ sir qiladi: masalan, guruh-tezlik dispersiyasi impulslarning optik tolalarda tarqalishiga olib keladi, uzoq masofalarda signallarni yomonlashtiradi; shuningdek, guruh-tezlik dispersiyasi va 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 indeksining o’  zgarishi tushuniladi. Shu bilan birga, 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 nol dispersiyali to’  lqin uzunligini hosil qiladi.

Optikada material dispersiyasi




Materialning dispersiyasi optik ilovalarda istalgan yoki kiruvchi ta’ sir bo’ lishi mumkin. Shisha prizmalarda yorug’ likning tarqalishi spektrometrlar va spektroradyometrlarni qurish uchun ishlatiladi. Biroq, linzalarda dispersiya xromatik aberatsiyaga olib keladi, bu mikroskoplar, teleskoplar va fotografik ob’ ektlardagi tasvirlarni buzishi mumkin bo’ lgan kiruvchi ta’ sir.




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Bu erda c- 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 = n ( f ), yoki muqobil ravishda, to’ lqin uzunligi n ga nisbatan = n ( l ). Materialning sinishi indeksining to’ lqin uzunligiga bog’ liqligi odatda uning Abbe raqami yoki uning koeffitsientlari bilan Koshi yoki 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, yo’ q bo’ lib ketish koeffitsienti deb ataladi) tomonidan tasvirlangan materialning yutilishi bilan bog’ liq. Xususan, magnit bo’ lmagan materiallar uchun ( m = m 0 ), Kramers-Kronig munosabatlarida paydo bo’ ladigan sezuvchanlik ch - elektr sezuvchanlik ch  = n  − 1.

Optikada dispersiyaning eng ko’ p ko’ rilgan oqibati oq yorug’ likning prizma orqali rang spektriga ajralishidir. Snel qonunidan ko’ rinib turibdiki, prizmadagi yorug’ likning 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 :




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{\displaystyle {\frac {dn}{d\lambda }}<0.}


Bunday holda, muhit normal dispersiyaga ega deyiladi . To’ lqin uzunligi ortishi bilan indeks ortib borsa (bu odatda ultrabinafsha), muhit anomal dispersiyaga ega deyiladi.

Bunday materialning havo yoki vakuum bilan interfeysida (indeks ~ 1) Snell qonuni normalga burchak ostida tushgan yorug’ lik yoyi burchak ostida sinishi haqida bashorat qiladi.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




To’ lqin uzunligi bo’ yicha faza tezligining o’ zgarishini oddiygina tavsiflashdan tashqari, ko’ plab ilovalarda dispersiyaning yanada jiddiy oqibati 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 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 erda ō - radian chastotasi ō. = 2 pf . Holbuki, faza tezligining bir ifodasi v  = ō / k, guruh tezligi lotin yordamida ifodalanishi mumkin: v  = dō / dk . Yoki faza tezligi bo’ yicha v ,





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{\displaystyle v_{\text{g}}={\frac {v_{\text{p}}}{1-{\dfrac {\omega }{v_{\text{p}}}}{\dfrac {dv_{\text{p}}}{d\omega }}}}.}


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 chastotasiga nisbatan o’ zaro hosilasi sifatida aniqlanadi, bu esa guruh tezligi dispersiyasiga olib keladi. = d k / dō .

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 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




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{\displaystyle D={\frac {1}{c}}\,{\frac {dn}{d\lambda }}}


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




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{\displaystyle D=-{\frac {2\pi c}{\lambda ^{2}}}\,{\frac {d^{2}k}{d\omega ^{2}}}.}


Ba’ zi mualliflarning fikriga ko’ ra, muhitning ma’ lum vakuum to’ lqin uzunligi l uchun normal dispersiya / anomal dispersiyaga ega deyiladi, agar l da hisoblangan sinishi indeksining ikkinchi hosilasi musbat/manfiy yoki ekvivalenti D ( l) bo’ lsa. ) 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). silika 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 ) qabul qilib bo’ lmaydigan darajada kuchaytiradi. Yana bir mumkin bo’ lgan variant - salbiy dispersiya rejimida soliton impulslaridan foydalanish, bu optik impulsning shakli bo’ lib, u o’ z shaklini saqlab qolish uchun 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-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.

Qisqa impulslar ishlab chiqaradigan lazerlarda dispersiyani nazorat qilish ham muhimdir . 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 prizma joylashtirilishi mumkin . Dispersiv effektlarni hosil qilish uchun, difraksion panjaralardan ham foydalanish mumkin ; bular ko’ pincha yuqori quvvatli lazer kuchaytirgich tizimlarida qo’ llaniladi. So’ nggi paytlarda prizma va panjaralarga alternativa ishlab chiqildi: 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 o’ tkazgichlar geometriyasi tufayli (faqat moddiy tarkibidan ko’ ra) juda dispersdir. 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 chastotasi ō ( b ) bo’ lgan to’ lqin o’ tkazgich rejimi uchun tarqalish doimiysi b (shundayki z tarqalish yo’ nalishidagi elektromagnit maydonlar e i ga proportsional tebranadi), guruh-tezlik dispersiyasi parametri D. sifatida belgilanadi




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{\displaystyle 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 }},}


qayerda λ=2πc/ω - vakuum to’ lqin uzunligi va v=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 ps /( nm ⋅ km ) da qayd etilgan masofani bosib o’ tgan birlik tarmoqli kengligi Δt uchun Δλ ning (asimptotik) vaqtinchalik impuls tarqalishi.

Ko’ p rejimli optik tolalar holatida modal dispersiya deb ataladigan narsa ham pulsning kengayishiga olib keladi . Hatto bitta rejimli tolalarda ham impulsning kengayishi 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 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, ultraqisqa impuls yoki jirkanch impuls yoki 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. Bu atamalar oddiygina ma’ lum bir chastota atrofida muhit yoki to’ lqin o’ tkazgichning dispersiya munosabatining b ( ō ) Teylor seriyali kengayishidir. Ularning ta’ sirini to’ lqin shaklining Furye o’ zgarishlarini raqamli baholash, asta- sekin o’ zgaruvchan konvertning yuqori darajali taxminiy integratsiyasi orqali, 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 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.

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








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{\displaystyle {\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}}}









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{\displaystyle {\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}}}

To’ lqinli qurilma uchun dispersiya munosabatlari



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{\displaystyle k\mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}}n\mathrm {(} \omega \mathrm {)} }

va faza



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{\displaystyle \varphi \mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}}{\it {OP}}\mathrm {(} \omega \mathrm {)} }

quyidagicha ifodalanishi mumkin:















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)


)

 






{\displaystyle {\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}}}

,














p






ω


p





φ

(

ω

)

=


1
c



(

p







p

1






ω


p

1







O
P



(

ω

)

+
ω







p






ω


p







O
P



(

ω

)


)





(
1
)


{\displaystyle {\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)}


Har qanday differentsiallanuvchi funktsiyaning hosilalari



f

(

ω


|


λ

)



{\displaystyle f\mathrm {(} \omega \mathrm {|} \lambda \mathrm {)} }

To’ lqin uzunligi , yoki chastota bo’ shlig’ i Lah o'zgarishi orqali quyidagicha aniqlanadi :













p






ω


p





f

(

ω

)

=






(





1


)



p





(


λ


2

π
c



)



p





m
=

0



p





A



(

p
,
m

)



λ


m









m






λ


m





f

(

λ

)








{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}f\mathrm {(} \lambda \mathrm {)} }\end{array}}}





,


{\displaystyle ,}
















p






λ


p





f

(

λ

)

=






(





1


)



p





(


ω


2

π
c



)



p





m
=

0



p





A



(

p
,
m

)



ω


m









m






ω


m





f

(

ω

)






(
2
)


{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\omega }^{m}{\frac {{\partial }^{m}}{\partial {\omega }^{m}}}f\mathrm {(} \omega \mathrm {)} }\end{array}}(2)}


Transformatsiyaning matritsa elementlari Lah koeffitsientlari:





A



(

p
,
m

)

=



p

!




(

p



m

)


!

m

!








(

p




1
)
!




(

m




1
)
!






{\displaystyle {\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)!} }}}


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















p






ω


p





G
D
D

(

ω

)

=






(





1


)



p





(


λ


2

π
c



)



p





m
=

0



p





A



(

p
,
m

)



λ


m









m






λ


m





G
D
D

(

λ

)








{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}GDD\mathrm {(} \lambda \mathrm {)} }\end{array}}}


Sinishi ko’ rsatkichi uchun ifodalangan (2) tenglamani almashtirish



n


{\displaystyle n}

yoki optik yo’ l



O
P


{\displaystyle OP}

(1) tenglamada dispersiya tartiblari uchun yopiq shakldagi ifodalar paydo bo’ ladi. Umuman olganda,




p

t
h




{\displaystyle p^{th}}

tartibli dispersiya POD - ikkinchi manfiy tartibli lager tipidagi transformatsiya:




P
O
D
=




d

p


φ
(
ω
)


d

ω

p





=
(

1

)

p


(


λ

2
π
c




)

(
p

1
)





m
=
0


p




B
(
p
,
m
)


(
λ

)

m






d

m


O
P
(
λ
)


d

λ

m







{\displaystyle 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}}}}





,


{\displaystyle ,}






P
O
D
=




d

p


k
(
ω
)


d

ω

p





=
(

1

)

p


(


λ

2
π
c




)

(
p

1
)





m
=
0


p




B
(
p
,
m
)


(
λ

)

m






d

m


n
(
λ
)


d

λ

m







{\displaystyle 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}}}}


O’ zgartirishlarning matritsa elementlari minus 2 tartibli belgisiz lager koeffitsientlari bo’ lib, ular quyidagicha berilgan






B



(

p
,
m

)

=



p

!




(

p



m

)


!

m

!








(

p




2
)
!




(

m




2
)
!






{\displaystyle {\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)!} }}}


To'lqin vektori uchun aniq yozilgan birinchi o'n dispersiya tartibi:









G
D


=





ω



k

(

ω

)

=



1

c



(

n

(

ω

)

+
ω




n

(

ω

)




ω




)

=



1

c



(

n

(

λ

)


λ




n

(

λ

)




λ




)

=

v

g
r






1









{\displaystyle {\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_{gr}^{\mathrm {-} \mathrm {1} }\end{array}}}


Guruh sinishi ko'rsatkichi




n

g




{\displaystyle n_{g}}

quyidagicha aniqlanadi:




n

g


=
c

v

g
r






1





{\displaystyle n_{g}=cv_{gr}^{\mathrm {-} \mathrm {1} }}

.










G
D
D


=







2






ω



2






k

(

ω

)

=



1

c



(


2





n

(

ω

)




ω



+
ω








2


n

(

ω

)






ω



2







)

=



1

c



(


λ


2

π
c



)


(



λ



2











2


n

(

λ

)






λ



2







)







{\displaystyle {\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}}}











T
O
D


=







3






ω



3






k

(

ω

)

=



1

c



(


3









2


n

(

ω

)






ω



2






+
ω








3


n

(

ω

)






ω



3







)

=






1

c





(


λ


2

π
c



)




2





(



3



λ



2











2


n

(

λ

)






λ



2






+


λ



3











3


n

(

λ

)






λ



3








)








{\displaystyle {\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}}}












F
O
D


=







4






ω



4






k

(

ω

)

=



1

c



(


4









3


n

(

ω

)






ω



3






+
ω








4


n

(

ω

)






ω



4







)

=



1

c





(


λ


2

π
c



)




3





(



12



λ



2











2


n

(

λ

)






λ



2






+

8



λ



3











3


n

(

λ

)






λ



3






+


λ



4











4


n

(

λ

)






λ



4








)








{\displaystyle {\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}}}












F
i
O
D


=







5






ω



5






k

(

ω

)

=



1

c



(


5









4


n

(

ω

)






ω



4






+
ω








5


n

(

ω

)






ω



5







)

=






1

c





(


λ


2

π
c



)




4





(



60



λ



2











2


n

(

λ

)






λ



2






+

60



λ



3











3


n

(

λ

)






λ



3






+

15



λ



4











4


n

(

λ

)






λ



4






+


λ



5











5


n

(

λ

)






λ



5








)








{\displaystyle {\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}}}












S
i
O
D


=







6






ω



6






k

(

ω

)

=



1

c



(


6









5


n

(

ω

)






ω



5






+
ω








6


n

(

ω

)






ω



6







)

=



1

c





(


λ


2

π
c



)




5





(



360



λ



2











2


n

(

λ

)






λ



2






+

480



λ



3











3


n

(

λ

)






λ



3






+

180



λ



4











4


n

(

λ

)






λ



4






+

24



λ



5











5


n

(

λ

)






λ



5






+


λ



6











6


n

(

λ

)






λ



6








)








{\displaystyle {\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}}}












S
e
O
D


=







7






ω



7






k

(

ω

)

=



1

c



(


7









6


n

(

ω

)





ω



6





+
ω








7


n

(

ω

)





ω



7






)

=






1

c





(


λ


2

π
c



)




6





(



2520



λ



2











2


n

(

λ

)






λ



2






+

4200



λ



3











3


n

(

λ

)






λ



3






+

2100



λ



4











4


n

(

λ

)






λ



4






+

420



λ



5











5


n

(

λ

)






λ



5






+

35



λ



6











6


n

(

λ

)






λ



6






+


λ



7











7


n

(

λ

)






λ



7








)








{\displaystyle {\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}}}












E
O
D


=







8






ω



8






k

(

ω

)

=



1

c



(


8









7


n

(

ω

)





ω



7





+
ω








8


n

(

ω

)






ω



8







)

=



1

c





(


λ


2

π
c



)




7





(



20160



λ



2











2


n

(

λ

)






λ



2






+

40320



λ



3











3


n

(

λ

)






λ



3






+

25200



λ



4











4


n

(

λ

)






λ



4






+

6720



λ



5











5


n

(

λ

)






λ



5






+

840



λ



6











6


n

(

λ

)






λ



6






+




+

48



λ



7











7


n

(

λ

)






λ



7






+


λ



8











8


n

(

λ

)






λ



8








)








{\displaystyle {\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}}}












N
O
D


=







9






ω



9






k

(

ω

)

=



1

c



(


9









8


n

(

ω

)






ω



8






+
ω








9


n

(

ω

)






ω



9







)

=






1

c





(


λ


2

π
c



)




8





(



181440



λ



2











2


n

(

λ

)






λ



2






+

423360



λ



3











3


n

(

λ

)






λ



3






+

317520



λ



4











4


n

(

λ

)






λ



4






+

105840



λ



5











5


n

(

λ

)






λ



5






+

17640



λ



6











6


n

(

λ

)






λ



6






+




+

1512



λ



7











7


n

(

λ

)






λ



7






+

63



λ



8











8


n

(

λ

)






λ



8






+


λ



9











9


n

(

λ

)






λ



9








)








{\displaystyle {\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}}}












T
e
O
D


=







10






ω



10






k

(

ω

)

=



1

c



(


10









9


n

(

ω

)






ω



9






+
ω








10


n

(

ω

)






ω



10







)

=



1

c





(


λ


2

π
c



)




9





(



1814400



λ



2











2


n

(

λ

)






λ



2






+

4838400



λ



3











3


n

(

λ

)






λ



3






+

4233600



λ



4











4


n

(

λ

)






λ



4






+

1693440



λ



5











5


n

(

λ

)






λ



5






+




+

352800



λ



6











6


n

(

λ

)






λ



6






+

40320



λ



7











7


n

(

λ

)






λ



7






+

2520



λ



8











8


n

(

λ

)






λ



8






+

80



λ



9











9


n

(

λ

)






λ



9






+


λ



10











10


n

(

λ

)






λ



10








)








{\displaystyle {\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}}}


Aniq, bosqich uchun yozilgan



φ


{\displaystyle \varphi }

, birinchi on dispersiya tartiblari to’ lqin uzunligi funksiyasi sifatida Lah o’ zgarishlari (tenglama (2)) yordamida quyidagicha ifodalanishi mumkin:













p






ω


p





f

(

ω

)

=






(





1


)



p





(


λ


2

π
c



)



p





m
=

0



p





A



(

p
,
m

)



λ


m









m






λ


m





f

(

λ

)








{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}f\mathrm {(} \lambda \mathrm {)} }\end{array}}}





,


{\displaystyle ,}
















p






λ


p





f

(

λ

)

=






(





1


)



p





(


ω


2

π
c



)



p





m
=

0



p





A



(

p
,
m

)



ω


m









m






ω


m





f

(

ω

)








{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\omega }^{m}{\frac {{\partial }^{m}}{\partial {\omega }^{m}}}f\mathrm {(} \omega \mathrm {)} }\end{array}}}













φ

(

ω

)




ω



=




(




2

π
c



ω



2





)





φ

(

ω

)




λ



=




(




λ



2





2

π
c



)





φ

(

λ

)




λ









{\displaystyle {\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}}}

















2


φ

(

ω

)






ω



2






=





ω




(




φ

(

ω

)




ω



)

=



(


λ


2

π
c



)




2




(


2

λ




φ

(

λ

)




λ



+


λ



2











2


φ

(

λ

)






λ



2







)







{\displaystyle {\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}}}

















3


φ

(

ω

)






ω



3






=






(


λ


2

π
c



)




3




(


6

λ




φ

(

λ

)




λ



+

6



λ



2











2


φ

(

λ

)






λ



2






+


λ



3











3


φ

(

λ

)






λ



3







)







{\displaystyle {\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}}}

















4


φ

(

ω

)






ω



4






=



(


λ


2

π
c



)




4





(



24

λ




φ

(

λ

)




λ



+

36



λ



2











2


φ

(

λ

)






λ



2






+

12



λ



3











3


φ

(

λ

)






λ



3






+


λ



4











4


φ

(

λ

)






λ



4








)








{\displaystyle {\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}}}


















5



φ

(

ω

)






ω



5






=






(


λ


2

π
c



)




5





(



120

λ




φ

(

λ

)




λ



+

240



λ



2











2


φ

(

λ

)






λ



2






+

120



λ



3











3


φ

(

λ

)






λ



3






+

20



λ



4











4


φ

(

λ

)






λ



4






+


λ



5











5


φ

(

λ

)






λ



5








)








{\displaystyle {\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}}}

















6


φ

(

ω

)






ω



6






=



(


λ


2

π
c



)




6





(



720

λ




φ

(

λ

)




λ



+

1800



λ



2











2


φ

(

λ

)






λ



2






+

1200



λ



3











3


φ

(

λ

)






λ



3






+

300



λ



4











4


φ

(

λ

)






λ



4






+

30



λ



5











5


φ

(

λ

)






λ



5







 
+



λ



6











6


φ

(

λ

)






λ



6








)








{\displaystyle {\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}}}

















7


φ

(

ω

)






ω



7






=






(


λ


2

π
c



)




7





(



5040

λ




φ

(

λ

)




λ



+

15120



λ



2











2


φ

(

λ

)






λ



2






+

12600



λ



3











3


φ

(

λ

)






λ



3






+

4200



λ



4











4


φ

(

λ

)






λ



4






+

630



λ



5











5


φ

(

λ

)






λ



5






+

42



λ



6











6


φ

(

λ

)






λ



6






+


λ



7











7


φ

(

λ

)






λ



7








)








{\displaystyle {\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}}}

















8


φ

(

ω

)






ω



8






=



(


λ


2

π
c



)




8





(



40320

λ




φ

(

λ

)




λ



+

141120



λ



2











2


φ

(

λ

)






λ



2






+

141120



λ



3











3


φ

(

λ

)






λ



3






+

58800



λ



4











4


φ

(

λ

)






λ



4






+

11760



λ



5











5


φ

(

λ

)






λ



5






+

1176



λ



6











6


φ

(

λ

)






λ



6






+

56



λ



7











7


φ

(

λ

)






λ



7






+




+


λ



8









8


φ

(

λ

)






λ



8








)








{\displaystyle {\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}}}

















9


φ

(

ω

)






ω



9






=






(


λ


2

π
c



)




9





(



362880

λ




φ

(

λ

)




λ



+

1451520



λ



2











2


φ

(

λ

)






λ



2






+

1693440



λ



3











3


φ

(

λ

)






λ



3






+

846720



λ



4











4


φ

(

λ

)






λ



4






+

211680



λ



5











5


φ

(

λ

)






λ



5






+

28224



λ



6











6


φ

(

λ

)






λ



6






+




+

2016



λ



7











7


φ

(

λ

)






λ



7






+

72



λ



8











8


φ

(

λ

)






λ



8






+


λ



9










9



φ

(

λ

)






λ



9








)








{\displaystyle {\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}}}

















10


φ

(

ω

)






ω



10






=



(


λ


2

π
c



)




10





(



3628800

λ




φ

(

λ

)




λ



+

16329600



λ



2











2


φ

(

λ

)






λ



2






+

21772800



λ



3











3


φ

(

λ

)






λ



3






+

12700800



λ



4











4


φ

(

λ

)






λ



4






+

3810240



λ



5











5


φ

(

λ

)






λ



5






+

635040



λ



6











6


φ

(

λ

)






λ



6






+




+

60480



λ



7











7


φ

(

λ

)






λ



7






+

3240



λ



8











8


φ

(

λ

)






λ



8






+

90



λ



9











9


φ

(

λ

)






λ



9






+


λ



10











10


φ

(

λ

)






λ



10








)








{\displaystyle {\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}}}


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 anizotrop muhit uchun elektr va elektr joy almashish maydoni o’ rtasidagi fazoviy munosabat konvolyutsiya sifatida ifodalanishi mumkin:





D

i


(
t
,
r
)
=

E

i


(
t
,
r
)
+



0







f

i
k


(
τ
;
r
,

r


)

E

k


(
t

τ
,

r


)

d

V



d
τ
,


{\displaystyle 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 ,}


yadro qaerda




f

i
k




{\displaystyle f_{ik}}

dielektrik javobdir (sezuvchanlik); uning indekslari muhitning anizotropiyasini hisobga olish uchun uni umuman tenzorga aylantiradi. Ko’ pgina makroskopik holatlarda fazoviy dispersiya ahamiyatsiz bo’ lib, o’ zgaruvchanlik shkalasi.




E

k


(
t

τ
,

r


)


{\displaystyle E_{k}(t-\tau ,r')}

atom o’ lchamlaridan ancha katta, chunki dielektrik yadrosi makroskopik masofalarda nobud bo’ ladi. Shunga qaramay, bu, ayniqsa, metallar, elektrolitlar va plazmalar kabi o’ tkazuvchi vositalarda ahamiyatsiz bo’ lmagan makroskopik ta’ sirga olib kelishi mumkin. Fazoviy dispersiya, shuningdek , optik faollik va Doppler kengayishida, shuningdek, metamateriallar nazariyasida rol o’ ynaydi.

Gemologiyada




Gemologiyaning texnik terminologiyasida dispersiya B va G nuqtalarida materialning sinishi indeksidagi farqdir (686,7). nm va 430.8 nm) yoki C va F (656.3 nm va 486.1 nm) Fraungofer to’ lqin uzunliklari va 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.

Tasvirda



Fotografik va mikroskopik linzalarda dispersiya 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, akromatlardan, turli dispersiyadagi ko’ zoynakli ko’ p elementli linzalardan foydalanish. Ular shunday tuzilganki, turli qismlarning xromatik aberratsiyalari bekor qilinadi.

Pulsar emissiyasi



Pulsarlar - bu millisekundlardan soniyalargacha bo’ lgan juda muntazam oraliqlarda 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 muhitning ionlashgan komponenti, asosan guruh tezligini chastotaga bog’ liq bo’ lgan erkin elektronlar tufayli yuzaga keladi. ν chastotada qo’ shilgan qo’ shimcha kechikish




t
=

k

DM




(


DM

ν

2




)

,


{\displaystyle t=k_{\text{DM}}\cdot \left({\frac {\text{DM}}{\nu ^{2}}}\right),}


dispersiya konstantasi k bilan berilgan





k

DM


=



e

2



2
π

m

e


c




4.149
 


GHz


2





pc



1





cm


3




ms

,


{\displaystyle 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}},}


va dispersiya o’ lchovi (DM) erkin elektronlarning ustun zichligi ( jami elektron tarkibi ) – ya’ ni fotonning pulsardan Yergacha boradigan yo’ l bo’ ylab integrallashgan elektronlarning soni zichligi  – va tomonidan beriladi





DM

=



0


d



n

e



d
l


{\displaystyle {\text{DM}}=\int _{0}^{d}n_{e}\,dl}


kub santimetr uchun parsek birliklari bilan (1 dona/sm = 30,857 ×10 m ).

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 ν va past chastotali ν komponenti orasidagi kechikish D t bo’ ladi.




Δ
t
=

k

DM




DM



(



1

ν

lo


2







1

ν

hi


2





)

.


{\displaystyle \Delta t=k_{\text{DM}}\cdot {\text{DM}}\cdot \left({\frac {1}{\nu _{\text{lo}}^{2}}}-{\frac {1}{\nu _{\text{hi}}^{2}}}\right).}


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.

Shuningdek qarang



 

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