{"id":1229,"date":"2012-04-03T00:28:25","date_gmt":"2012-04-03T04:28:25","guid":{"rendered":"http:\/\/blogs.vassar.edu\/magnes\/?p=1229"},"modified":"2013-07-11T10:30:35","modified_gmt":"2013-07-11T14:30:35","slug":"project-plan-2","status":"publish","type":"post","link":"https:\/\/pages.vassar.edu\/magnes\/2012\/04\/03\/project-plan-2\/","title":{"rendered":"Project Plan"},"content":{"rendered":"

Second Harmonic Generation is a special case of optical mixing. It is a process by which photons from a laser beam are mixed in a nonlinear medium and the output photon has double the energy and frequency and half the wavelength. Conditions satisfy $\\omega_{1}=\\omega_{2}=\\omega$ and $\\omega_{3}=2\\omega$. Both energy and momentum conservation must be satisfied. Energy by\u00a0$\\omega_{3}=\\omega_{1}=\\omega_{2}$\u00a0and momentum by $k_{3}=k_{1}+ k_{2}$<\/p>\n

In more detail:
\n\u201cNonlinear optics is the study of phenomena that occur as a consequence of the modification of the optical properties of a material system by the presence of light\u201d. Input waves are at frequencies $\\omega_{1}$ and $\\omega_{2}$. By the nonlinear effects of incident beams (at the atomic level) each atom develops an oscillating dipole moment which contains a component at frequency $\\omega_{1}+\\omega_{2}$. Each atom radiates this frequency but there are many atoms in our medium and hence many atomic dipoles oscillating with a phase determined by the phases of the incident waves. When the relative phasing matches the waves radiated by each dipole will add constructively turning the system into a phased array of dipoles. When this happens the electric field strength of the radiation emitted will be the number of atoms times larger and hence the intensity will be the number of atoms squared.<\/h4>\n

I will assume my system to be lossless and dispersion-less for simplifying equations<\/p>\n

(1) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

2.1.19 Nonlinear optics Robert W.Boyd<\/p>\n

$\\widetilde{P}^{NL}_{n} =$ Non-Linear part of Polarization Vector<\/p>\n

$\u00a0\\widetilde{E}_{n} =$ Electric Field vector<\/p>\n

$\\epsilon^{(1)}(\\omega_{n}) =$ Frequency dependent dielectric Tensor<\/p>\n

Equation (1) is derived from Maxwell\u2019s equation and is the equation for waves in medium. It is valid for each frequency component of the field.<\/p>\n

$\\widetilde{E}_{2}(z,t) =A_{2}e^{i(k_{2}z-wt)}, \u00a0\u00a0\\widetilde{P}_{j}(z,t) =P_{j}e^{-i\\omega_{j}t},$ \u00a0 \u00a0$P_{1} =4dA_{2}A^{*}_{1}e^{i(k_{2}-k_{1})z}, P_{2} =2dA^{2}_{1}e^{i2k_{1}z}$<\/p>\n

2.2.1, 2.2.3, 2.2.4,2.2.5, 2.2.7 \u00a0Nonlinear optics Robert W.Boyd<\/p>\n

$\\widetilde{E}_{2}(z,t)$\u00a0will be my equation for the transmitted wave at frequency \u00a0propagating in the z direction,$\u00a0\\widetilde{P}_{2}(z,t)$ the nonlinear source term and $P_{2}, P_{1}$ \u00a0the amplitude of the nonlinear polarization and amplitude of incident beam respectively. I will make diagrams of incident waves hitting the medium and resulting transmitted wave.<\/p>\n

Substituting the transmitted wave equation in the wave equation and solving by hand I will find coupled amplitude equation.<\/p>\n

(2) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

2.6.11 Nonlinear optics Robert W.Boyd<\/p>\n

$\\bigtriangleup k = k_{1}+k_{1}-k_{2}$ is the wave vector mismatch<\/p>\n

2.6.12 Nonlinear optics Robert W.Boyd<\/p>\n

$A_{i} =$ amplitude of the wave<\/p>\n

The coupled amplitude equation shows how the amplitude of $\\omega_{2}$ wave varies due to it\u2019s coupling of two $\\omega_{1}$ waves. And from this we can find intensity, which is more useful.<\/p>\n

  <\/span>   <\/span>\"\begin{displaymath}<\/p>\n

(3) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

2.2.17, 2.20 Nonlinear optics Robert W.Boyd<\/p>\n

Where $\\lambda_{2}=\\frac{2\\pi c}{\\omega_{2}}$ , L is the length of the medium and d is the tensor<\/p>\n

I will try and model the effect of $\\bigtriangleup k$\u00a0of the wave vector on the efficiency and take special notice when $\\bigtriangleup k=0$ since this is the condition for perfect phase matching.\u00a0I will make an animation to vary $(\\frac{\\bigtriangleup kL}{2})$ \u00a0in the intensity equation and this will show the effects of wave vector mismatch on the efficiency of harmonic-generation.<\/p>\n

I will also attempt to model the effects of absorption.<\/p>\n

As an example I will use the nonlinear media KDP (Potassium Dihydrogen Phospate) crystal to model second harmonic generation for a laser beam at $1.06\\mu$ meters. KDP is widely used in commercial Non linear optical materials because of its electro-optic effects and it\u2019s high non-linear coefficients.<\/p>\n

REFERENCES:<\/strong><\/p>\n

Boyd.\u00a0Nonlinear Optics. New York:\u00a0 Academic Press, 1992<\/strong><\/p>\n

SHEN. The Principles of Nonlinear Optics. New York:\u00a0 Wiley-Interscience, 1984<\/strong><\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Second Harmonic Generation is a special case of optical mixing. It is a process by which photons from a laser beam are mixed in a nonlinear medium and the output photon has double the energy and frequency and half the wavelength. Conditions satisfy $\\omega_{1}=\\omega_{2}=\\omega$ and $\\omega_{3}=2\\omega$. Both energy and momentum conservation must be satisfied. Energy […]<\/p>\n","protected":false},"author":1019,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4101,29905,29902],"tags":[],"class_list":["post-1229","post","type-post","status-publish","format-standard","hentry","category-advanced-em","category-spring-2012","category-tariq"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1229","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/users\/1019"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/comments?post=1229"}],"version-history":[{"count":15,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1229\/revisions"}],"predecessor-version":[{"id":2417,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1229\/revisions\/2417"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/media?parent=1229"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/categories?post=1229"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/tags?post=1229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}