Ziyosfera

Ikkinchi garmonika generatsiyasi

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]:

rotE−1c∂B∂t,divB=0{\displaystyle \textrm{rot} \textbf{E}-\dfrac{1}{c}\dfrac{\partial B}{\partial t},\ \ \ \textrm{div}\textbf{B}=0}
rotH=1c∂D∂t,divD=0{\displaystyle \text{rot}\textbf{H}=\dfrac{1}{c}\dfrac{\partial D}{\partial t},\ \ \ \text{div}\textbf{D}=0}

Bu tenglamalarning birinchisidan rotor olib uchinchi tenglama yordamida undan magnit maydonini yoʻqotamiz. Natijada elektr maydon uchun quyidagi tenglamalar sistemasini olamiz[2]:

{ΔE+graddivE=1c2∂2D∂t2divD=0{\displaystyle \begin{cases} \Delta\textbf{E} +\text{grad}\ \text{div}\textbf{E} = \dfrac{1}{c^2}\dfrac{\partial^2D}{\partial t^2}\\ \text{div}\textbf{D}=0 \end{cases}} Bu yerda muhit dielektrik boʻlgani uchun magnit singdiruvchanlik μ=1{\displaystyle \mu=1} deb olindi.

Toʻlqinning elektr maydonini Furye integraliga yoyamiz:

Eα(r,t)=12π∫−∞∞Eα(r,ω)eiωtdω{\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}

Buni yuqoridagi ikkita tenglamalarga qoʻyib quyidagini hosil qilamiz:

div(ε(ω)E(r,ω))=−4πdivPnch(r,ω){\displaystyle \text{div}(\varepsilon(\omega)\textbf{E}(r,\omega))=-4\pi\text{div}\textbf{P}^{nch}(r,\omega)}
ΔE(rω)−grad(divE(r,ω))+ω2ε(ω)c2E(r,ω)=4πω2c2Pnch(r,ω){\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)}

Chastotasi ω{\displaystyle \omega} 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:

E(z,t)=e^E(ω1)ei(k1z−ω1t)+e^E*(ω1)e−i(k1z−ω1t){\displaystyle \textbf{E}(z, t)=\hat{e}\textbf{E}(\omega_1)e^{i(k_1z-\omega_1t)}+\hat{e}\textbf{E}^*(\omega_1)e^{-i(k_1z-\omega_1t)}}
Pαnch(z,t)=E2(ω1)∑β,γχαβγe^βe^γe2i(k1z−ω1t)+k.q.…{\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_1z-\omega_1t)}+k.q.\ldots}

Bu yerda e^{\displaystyle \hat{e}} toʻlqinning qutblanish yoʻnalishidagi birlik vektor. Izotrop dielektrikda toʻlqin z{\displaystyle z} oʻqi boʻylab tarqalganda, elektr maydon (xy{\displaystyle xy}) tekisligida yotadi. Maʼlumki, tushayotgan toʻlqin koʻndalang boʻlishiga qaramasdan, muhit qutblanish vektorining koʻndalang va boʻylama tashkil etuvchilari paydo boʻladi:

Pnch=z^P∥nch+P⊥nch{\displaystyle \textbf{P}^{nch}=\hat{z}\textbf{P}_\parallel^{nch}+\textbf{P}_\perp^{nch}}

Shu sababli ω2=2ω1{\displaystyle \omega_2=2\omega_1} chastotali ikkinchi garmonika maydonining koʻndalang va parallel tashkil etuvchilari boʻladi, yaʼni

E(z,ω2)=z^E∥(z,ω2)+E⊥(z,ω2){\displaystyle \textbf{E}(z,\omega_2)=\hat{z}E_\parallel(z,\omega_2)+E_\perp(z,\omega_2)}

Bularni inobatga olsak, boshlangʻich toʻlqin tenglamalari quyidagi koʻrinishni qabul qiladi:

∂2E⊥∂z2+ω22c2(ε⊥(ω2)E⊥+4πP⊥nch)+ω22c2(ε∥(ω2)E∥+4πP∥nch){\displaystyle \dfrac{\partial^2E_\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)}
∂∂z(ε∥(ω2)E∥+4πP∥nch){\displaystyle \dfrac{\partial}{\partial z}\left(\varepsilon_\parallel(\omega_2)E_\parallel+4\pi P_\parallel^{nch}\right)}

Bu yerda ε∥{\displaystyle \varepsilon_\parallel}, ε⊥{\displaystyle \varepsilon_\perp} optik oʻqqa parallel va perpendikulyar dielektrik singdiruvchanliklardir. Yuqoridagilarga asosan tenglamani parallel va perpendikulyar tashkil etuvchilarga ajratamiz:

(∂2∂z2+k22)E⊥=−4πω22c2P⊥(nch){\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)}}
ε∥(ω2)E∥=−4πP∥(nch){\displaystyle \varepsilon_\parallel(\omega_2)E_\parallel=-4\pi P_\parallel^{(nch)}}

Yuqoridagi tenglamalarni haqiqiy va mavhum qismlarga ajratib, ρ1,2{\displaystyle \rho_{1,2}} va φ1,2{\displaystyle \varphi_{1,2}} uchun toʻrtta tenglamalar sistemasi hosil qilamiz:

{∂ρ2∂ξ=−ρ12sin⁡(2φ1−φ2)∂ρ1∂ξ=ρ1ρ2sin⁡(2φ1−φ2)ρ2∂φ2∂ξ=ρ12cos⁡(2φ1−φ2)∂φ1∂ξ=ρ2cos⁡(2φ1−φ2){\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}}

Nisbiy faza ψ=2φ1−φ2{\displaystyle \psi=2\varphi_1-\varphi_2} kiritib, yuqoridagi toʻrtta tenglamani uchta tenglamaga keltirish mumkin[3]:

{∂ρ2∂ξ=−ρ12sin⁡(ψ)∂ρ1∂ξ=ρ1ρ2sin⁡(ψ)∂ψ∂ξ=(2ρ2−ρ12ρ2)cos⁡ψ{\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}}

Bu tenglamalar yordamida quyidagi tenglikni yozish mumkin:

tgψ∂ψ∂ξ−2ρ1∂ρ1∂ξ−1ρ2∂ρ2∂ξ=0{\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}

yoki

∂∂ξ[ln⁡(ρ12ρ2cos⁡ψ)]=0{\displaystyle \dfrac{\partial}{\partial\xi}\left[\ln\left(\rho_1^2\rho_2\cos\psi\right)\right]=0}

Demak, bu yerda qavs ichidagi kattalik

ρ12ρ2cos⁡ψ=const{\displaystyle \rho_1^2\rho_2\cos\psi=const}

saqlanuvchi kattalik, yaʼni harakat integrali ekan.

Shunday qilib, hamma ξ{\displaystyle \xi} lar uchun ψ=±π/2{\displaystyle \psi=\pm\pi/2} boʻlishi mumkinligi kelib chiqadi. Muhitda toʻlqin tarqalishi davomida ikkinchi garmonikaning intensivligi oʻsib borishini inobatga olsak, bu ikkita holdan minus ishorani tanlash kerak. U holda

∂ρ2∂ξ=ρ12,∂ρ1∂ξ=−ρ1ρ2{\displaystyle \dfrac{\partial\rho_2}{\partial\xi}=\rho_1^2,\ \ \ \dfrac{\partial\rho_1}{\partial\xi}=-\rho_1\rho_2}

Bundan quyidagini topamiz:

ρ2∂ρ2∂ξ+ρ1∂ρ1∂ξ=0{\displaystyle \rho_2\dfrac{\partial\rho_2}{\partial\xi}+\rho_1\dfrac{\partial\rho_1}{\partial\xi}=0}

yoki

ρ12+ρ22=1{\displaystyle \rho_1^2+\rho_2^2=1}

Bu tenglik energiyaning saqlanish qonunini anglatadi. Yuqoridagi shart yordamida sistemaning birinchisidan ρ1{\displaystyle \rho_1} ni yoʻqotib tenglamani integrallaymiz va ρ1,2{\displaystyle \rho_{1,2}} uchun quyidagi ifodalarni topamiz:

ρ2=thξ,ρ1=sechξ{\displaystyle \rho_2=\text{th}\xi,\ \ \ \rho_1=\text{sech}\xi}

Endi boshlangʻich funksiya va oʻzgaruvchilarga qaytamiz:

E(z,ω2)=iE(ω1)th(zLc){\displaystyle E(z,\omega_2)=iE(\omega_1)\text{th}\left(\frac{z}{L_c}\right)}
E(z,ω1)=E(ω1)sech(zLc),φ1=0,φ2=π2{\displaystyle E(z,\omega_1)=E(\omega_1)\text{sech}\left(\frac{z}{L_c}\right),\ \ \varphi_1=0,\ \ \varphi_2=\frac{\pi}{2}}


Shuningdek qarang

Nochiziqli optika

Dielektriklarning qutblanishi


Manbalar

  1. ↑ Marcel J. E. Golay, 1947 Rev. Sci. Instrum. 18, 347
  2. ↑ A. A. Abdumalikov, Elektrodinamika, Toshkent, Choʻlpon, 2011
  3. ↑ Marcel J. E. Golay, 1947 Rev. Sci. Instrum. 18, 347 
"https://new.ziyosfera.uz/ziyo/Ikkinchi_garmonika_generatsiyasi" dan olindi