Ikkinchi garmonikaning hosil boʻlishi nochiziqli optikaning asosiy elementlaridan biri boʻlib, lazer nurlanishlarini hosil qilishda muhim rol oʻynaydi.
Avval kvadratik nochiziqli muhitda amplitudasi sekin oʻzgaruvchi toʻlqinlar uchun tenglamalarni keltirib chiqaramiz. Buning uchun Makswell tenglamalarini yozamiz[1]:
![{\displaystyle {\textrm {rot}}{\textbf {E}}-{\dfrac {1}{c}}{\dfrac {\partial B}{\partial t}},\ \ \ {\textrm {div}}{\textbf {B}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1260389d9edd4a6dc91aa39d44387fb515173e14)
![{\displaystyle {\text{rot}}{\textbf {H}}={\dfrac {1}{c}}{\dfrac {\partial D}{\partial t}},\ \ \ {\text{div}}{\textbf {D}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34f2c7a860cbb221b93d48966f41882f947b2633)
Bu tenglamalarning birinchisidan rotor olib uchinchi tenglama yordamida undan magnit maydonini yoʻqotamiz. Natijada elektr maydon uchun quyidagi tenglamalar sistemasini olamiz[2]:
![{\displaystyle {\begin{cases}\Delta {\textbf {E}}+{\text{grad}}\ {\text{div}}{\textbf {E}}={\dfrac {1}{c^{2}}}{\dfrac {\partial ^{2}D}{\partial t^{2}}}\\{\text{div}}{\textbf {D}}=0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6234eedf6ada7eb535cc01149f5bb9d5ea832032)
Bu yerda muhit dielektrik boʻlgani uchun magnit singdiruvchanlik
deb olindi.
Toʻlqinning elektr maydonini Furye integraliga yoyamiz:
![{\displaystyle E_{\alpha }({\textbf {r}},t)={\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }E_{\alpha }({\textbf {r}},\omega )e^{i\omega t}d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3132bda24fe6c47b907526abc33d412251ba5b0a)
Buni yuqoridagi ikkita tenglamalarga qoʻyib quyidagini hosil qilamiz:
![{\displaystyle {\text{div}}(\varepsilon (\omega ){\textbf {E}}(r,\omega ))=-4\pi {\text{div}}{\textbf {P}}^{nch}(r,\omega )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e67625508661d8012c787fceca810e135be97d09)
![{\displaystyle \Delta {\textbf {E}}(r\omega )-{\text{grad}}({\text{div}}{\textbf {E}}(r,\omega ))+{\dfrac {\omega ^{2}\varepsilon (\omega )}{c^{2}}}{\textbf {E}}(r,\omega )={\dfrac {4\pi \omega ^{2}}{c^{2}}}{\textbf {P}}^{nch}(r,\omega )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5664f16691667db9d7489aec69ce0ed105140ee)
Chastotasi
va toʻlqin vektori k
boʻlgan lazer hosil qilayotgan yorugʻlik dastasining muhitga taʼsirini koʻrib chiqamiz.
Asosiy garmonikadagi nurlanish intensivligini
pasayishini inobatga olmaslik uchun ikkinchi
garmonikaning intensivligini kichik deb faraz
qilamiz.
Qutblanish vektorining nochiziqli qismini
va elektr maydonni quyidagi koʻrinishda yozamiz:
![{\displaystyle {\textbf {E}}(z,t)={\hat {e}}{\textbf {E}}(\omega _{1})e^{i(k_{1}z-\omega _{1}t)}+{\hat {e}}{\textbf {E}}^{*}(\omega _{1})e^{-i(k_{1}z-\omega _{1}t)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07881f095c76469a53cd3ed5a375a235a9d4cc4f)
![{\displaystyle P_{\alpha }^{nch}(z,t)=E^{2}(\omega _{1})\sum \limits _{\beta ,\gamma }\chi _{\alpha \beta \gamma }{\hat {e}}_{\beta }{\hat {e}}_{\gamma }e^{2i(k_{1}z-\omega _{1}t)}+k.q.\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e919b3b1ff3c259ca0839b34f57ba5e44833e00)
Bu yerda
toʻlqinning qutblanish yoʻnalishidagi birlik vektor.
Izotrop dielektrikda toʻlqin
oʻqi boʻylab tarqalganda, elektr maydon (
) tekisligida yotadi. Maʼlumki, tushayotgan toʻlqin koʻndalang
boʻlishiga qaramasdan, muhit qutblanish vektorining koʻndalang va boʻylama tashkil etuvchilari paydo boʻladi:
![{\displaystyle {\textbf {P}}^{nch}={\hat {z}}{\textbf {P}}_{\parallel }^{nch}+{\textbf {P}}_{\perp }^{nch}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/825866b70706b4f33fae3be263444382b64439a2)
Shu sababli
chastotali ikkinchi garmonika maydonining koʻndalang va parallel tashkil etuvchilari boʻladi, yaʼni
![{\displaystyle {\textbf {E}}(z,\omega _{2})={\hat {z}}E_{\parallel }(z,\omega _{2})+E_{\perp }(z,\omega _{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/344d7ef6efdd7aa9450a5a10f6cdc9c7e578ef14)
Bularni inobatga olsak, boshlangʻich toʻlqin tenglamalari quyidagi koʻrinishni qabul qiladi:
![{\displaystyle {\dfrac {\partial ^{2}E_{\perp }}{\partial z^{2}}}+{\dfrac {\omega _{2}^{2}}{c^{2}}}\left(\varepsilon _{\perp }(\omega _{2}){\textbf {E}}_{\perp }+4\pi {\textbf {P}}_{\perp }^{nch}\right)+{\dfrac {\omega _{2}^{2}}{c^{2}}}\left(\varepsilon _{\parallel }(\omega _{2}){\textbf {E}}_{\parallel }+4\pi {\textbf {P}}_{\parallel }^{nch}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe56913e1c1c841f5c8128e37cb98513077140ee)
![{\displaystyle {\dfrac {\partial }{\partial z}}\left(\varepsilon _{\parallel }(\omega _{2})E_{\parallel }+4\pi P_{\parallel }^{nch}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32958ed3a1f92387e0fff5dfd99fa656833af7ab)
Bu yerda
,
optik oʻqqa parallel va perpendikulyar dielektrik singdiruvchanliklardir. Yuqoridagilarga asosan tenglamani parallel va perpendikulyar tashkil etuvchilarga ajratamiz:
![{\displaystyle \left({\dfrac {\partial ^{2}}{\partial z^{2}}}+k_{2}^{2}\right){\textbf {E}}_{\perp }=-{\dfrac {4\pi \omega _{2}^{2}}{c^{2}}}{\textbf {P}}_{\perp }^{(nch)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/553f658917bf62b8bd85d88633bbd0753767e72b)
![{\displaystyle \varepsilon _{\parallel }(\omega _{2})E_{\parallel }=-4\pi P_{\parallel }^{(nch)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/687ea52d72bfb7fece045bc53e5e987e526f35ae)
Yuqoridagi tenglamalarni haqiqiy va mavhum qismlarga ajratib,
va
uchun toʻrtta tenglamalar sistemasi hosil qilamiz:
![{\displaystyle {\begin{cases}{\dfrac {\partial \rho _{2}}{\partial \xi }}=-\rho _{1}^{2}\sin(2\varphi _{1}-\varphi _{2})\\{\dfrac {\partial \rho _{1}}{\partial \xi }}=\rho _{1}\rho _{2}\sin(2\varphi _{1}-\varphi _{2})\\\rho _{2}{\dfrac {\partial \varphi _{2}}{\partial \xi }}=\rho _{1}^{2}\cos(2\varphi _{1}-\varphi _{2})\\{\dfrac {\partial \varphi _{1}}{\partial \xi }}=\rho _{2}\cos(2\varphi _{1}-\varphi _{2})\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85e19078144ccd64a6150b2f79f23fd58c749c1b)
Nisbiy faza
kiritib, yuqoridagi toʻrtta tenglamani uchta tenglamaga keltirish mumkin[3]:
![{\displaystyle {\begin{cases}{\dfrac {\partial \rho _{2}}{\partial \xi }}=-\rho _{1}^{2}\sin(\psi )\\{\dfrac {\partial \rho _{1}}{\partial \xi }}=\rho _{1}\rho _{2}\sin(\psi )\\{\dfrac {\partial \psi }{\partial \xi }}=\left(2\rho _{2}-{\dfrac {\rho _{1}^{2}}{\rho _{2}}}\right)\cos \psi \\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3efb0de1b91c43cd004c502c94833db6b24bcf)
Bu tenglamalar yordamida quyidagi tenglikni yozish mumkin:
![{\displaystyle {\text{tg}}\psi {\dfrac {\partial \psi }{\partial \xi }}-{\dfrac {2}{\rho _{1}}}{\dfrac {\partial \rho _{1}}{\partial \xi }}-{\dfrac {1}{\rho _{2}}}{\partial \rho _{2}}{\partial \xi }=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a753f012de6696fd58f9a003bce2f81e38b060c1)
yoki
![{\displaystyle {\dfrac {\partial }{\partial \xi }}\left[\ln \left(\rho _{1}^{2}\rho _{2}\cos \psi \right)\right]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e70c48467388777aa045bb4a5371c4793997acf6)
Demak, bu yerda qavs ichidagi kattalik
![{\displaystyle \rho _{1}^{2}\rho _{2}\cos \psi =const}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b81a8ff03a6c415766f5180af98f2d2e94be34ed)
saqlanuvchi kattalik, yaʼni harakat integrali ekan.
Shunday qilib, hamma
lar uchun
boʻlishi mumkinligi kelib chiqadi. Muhitda toʻlqin tarqalishi davomida ikkinchi garmonikaning intensivligi oʻsib borishini inobatga olsak, bu ikkita holdan minus ishorani tanlash kerak. U holda
![{\displaystyle {\dfrac {\partial \rho _{2}}{\partial \xi }}=\rho _{1}^{2},\ \ \ {\dfrac {\partial \rho _{1}}{\partial \xi }}=-\rho _{1}\rho _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecc543c6ae714ea70b2a60fc493bf7195b0f9fb)
Bundan quyidagini topamiz:
![{\displaystyle \rho _{2}{\dfrac {\partial \rho _{2}}{\partial \xi }}+\rho _{1}{\dfrac {\partial \rho _{1}}{\partial \xi }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7df3596ee4ec9206a50bfa3886dba1f183b3fb0e)
yoki
![{\displaystyle \rho _{1}^{2}+\rho _{2}^{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e17621db19d364ba792fb3e237e8f9a309575ed2)
Bu tenglik energiyaning saqlanish qonunini anglatadi. Yuqoridagi shart yordamida sistemaning birinchisidan
ni yoʻqotib tenglamani integrallaymiz va
uchun quyidagi ifodalarni topamiz:
![{\displaystyle \rho _{2}={\text{th}}\xi ,\ \ \ \rho _{1}={\text{sech}}\xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2585821616fdb627cc3f847084efc270667a61b)
Endi boshlangʻich funksiya va oʻzgaruvchilarga qaytamiz:
![{\displaystyle E(z,\omega _{2})=iE(\omega _{1}){\text{th}}\left({\frac {z}{L_{c}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/812fed87a3b0aee958c71dc4785d5da7ed3e68ef)
![{\displaystyle E(z,\omega _{1})=E(\omega _{1}){\text{sech}}\left({\frac {z}{L_{c}}}\right),\ \ \varphi _{1}=0,\ \ \varphi _{2}={\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6646be94375c32a4123658185d640e8a2ee6aff1)
Nochiziqli optika
Dielektriklarning qutblanishi
- ↑ Marcel J. E. Golay, 1947 Rev. Sci. Instrum. 18, 347
- ↑ A. A. Abdumalikov, Elektrodinamika, Toshkent, Choʻlpon, 2011
- ↑ Marcel J. E. Golay, 1947 Rev. Sci. Instrum. 18, 347