Nonlinear optics

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Nonlinear optics ( NLO ) is an optical branch that describes the behavior of light in nonlinear media , ie, media where dielectric polarization P responds non-linearly to the electric field E of light. Nonlinearity is usually observed only at very high light intensities (the value of the tatomic electric field, usually 10 ⁸ V/m) as provided by the laser. Above the Schwinger limit, the vacuum itself is expected to be nonlinear. optical, superposition principle no longer applies.

Nonlinear optics remain unexplored until the discovery in 1961 of the second harmonic generation by Peter Franken et al. at the University of Michigan, shortly after the first laser construction by Theodore Harold Maiman. However, some nonlinear effects were found before the development of the laser. The theoretical foundation for many nonlinear processes was first described in the monument "Nonlinear Optics" Bloembergen.

Video Nonlinear optics

Nonlinear optical process

In this process, the medium has a linear response to light, but the nature of the medium is influenced by other causes:

• The pockel effect, refractive index affected by static electric field; used in electro-optical modulator.
• Acousto-optics, refractive index affected by acoustic waves (ultrasound); used in acousto-optical modulator.
• Raman scattering, photon interaction with optical phonons.

Maps Nonlinear optics

Parametric process

The nonlinear effects fall into two distinct categories qualitatively, parametrically and non-parametric effects. A parametric non-linearity is an interaction in which the quantum state of a nonlinear material is not altered by interaction with the optical plane. As a consequence of this, the process is "instantaneous". Energy and momentum are preserved in the optical field, making the matching phase important and dependent on polarization.

Theory

Parametric and "instantaneous" (ie matter must be without loss and dispersion without via Kramers-Kronig relationship) nonlinear optical phenomena, where the optical field is not very large, can be illustrated by the Taylor series expansion of the dielectric polarization density (dipole moment) per unit volume) P ( t ) when t in the electric field E ( t ) :

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\mathbf{P}Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â(Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂtÂ\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          ?                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                         (                   ?                      (   Â 1           )                                    E                (          t        )                           ?                      (    Â 2           )                                                E                         Â 2                         (          t        )                           ?                      (   Â 3           )                                                E                        Â 3                         (          t        )                 ...        )         ,               {\ displaystyle \ mathbf {P} (t) = \ varepsilon _ {0} {\ chi ^ {(1)} \ mathbf {E} (t ) \ mathbf {E} ^ {2} (t) \ chi ^ {(3)} \ mathbf {E} ^ {3} (t) \ ldots),}  Â}

where is the coefficient? ^{( n )} is n -the vulnerability state of the media, and the presence of such a term is generally referred to as nonlinearity of order n . Note that the polarization density P ( t ) and the electric field E ( t ) are considered scalars for simplicity. Generally,? ^{( n )} is ( n 1) -the level of tensor representing properties that depend on the polarization of the parametric interactions and the symmetry (or lack of) of the material nonlinear.

The wave equation in nonlinear material

Pusat untuk mempelajari gelombang elektromagnetik adalah persamaan gelombang. Dimulai dengan persamaan Maxwell dalam ruang isotropik, yang tidak mengandung biaya gratis, dapat ditunjukkan itu

${\ displaystyle \ nabla \ times \ nabla \ times \ mathbf {E} {\ frac {n ^ {2}} {c ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} \ mathbf {E} = - {\ frac {1} {\ varepsilon _ {0} c ^ {2}}} {\ frac {\ parsial ^ {2}} {\ partial t ^ {2}}} \ mathbf {P} ^ {\ text {NL}},}  ÂÂ$

where P ^NL is the nonlinear part of the polarization density, and n is the refractive index, derived from the linear term in P .

Perhatikan bahwa seseorang biasanya dapat menggunakan identitas vektor

${\ displaystyle \ nabla \ times \ left (\ nabla \ times \ mathbf {V} \ right) = \ nabla \ left (\ nabla \ cdot \ mathbf {V} \ kanan) - \ nabla2 \ mathbf {V}} Â Â$

dan hukum Gauss (dengan asumsi tidak ada biaya gratis, ${\ displaystyle \ rho _ {\ text {free}} = 0} Â$ ),

${\ displaystyle \ nabla \ cdot \ mathbf {D} = 0,} Â Â$

untuk mendapatkan persamaan gelombang yang lebih familiar

${\ displaystyle \ nabla ^ {2} \ mathbf {E} - {\ frac {n ^ 2} {c2}} { \ frac {\ parsial2}} {\ partial t2}} \ mathbf {E} = 0.} Â Â$

Untuk media nonlinear, hukum Gauss tidak menyiratkan bahwa identitas

${\ displaystyle \ nabla \ cdot \ mathbf {E} = 0} Â Â$

benar secara umum, bahkan untuk media isotropik. Namun, bahkan ketika istilah ini tidak identik 0, itu sering diabaikan kecil dan dengan demikian dalam prakteknya biasanya diabaikan, memberi kita persamaan gelombang nonlinier standar:

${\ displaystyle \ nabla ^ {2} \ mathbf {E} - {\ frac {n ^ {2}} {c ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} \ mathbf {E} = {\ frac {1} {\ varepsilon _ {0} c ^ {2}}} {\ frac {\ partial ^ { 2}} {\ partial t ^ {2}}} \ mathbf {P} ^ {\ text {NL}}.}  ÂÂ$

Nonlinier sebagai proses pencampuran gelombang

Persamaan gelombang nonlinier adalah persamaan differential homogeneous. Solusi umum berasal dari studi persamaan diferensial biasa dan dapat diperoleh dengan menggunakan fungsi Green. Secara fisik seseorang mendapat solusi gelombang elektromagnetik yang normal ke bagian homogen dari persamaan gelombang:

${\ displaystyle \ nabla ^ {2} \ mathbf {E} - {\ frac {n ^ 2} {c2}} { \ frac {\ parsial2}} {\ partial t2}} \ mathbf {E} = 0,} Â Â$

give istilah tidak homogen

${\ displaystyle {\ frac {1} {\ varepsilon _ 0} c2}} {\ frac {\ parsial2} {\ parsial t2}} \ mathbf {P} ^ {\ text {NL}}} Â Â$

acting as a propulsion/source of electromagnetic waves. One consequence of this is a nonlinear interaction that produces energy that is mixed or combined between different frequencies, often called "wave mixing".

Secara umum, ketidaklinieran order n -th akan mengarah ke ( n   1) -mencampur campuran. Sebagai contoh, jika kita hanya mempertimbangkan non-linear urutan kedua (pencampuran tiga gelombang), maka polarisasi P mengambil bentuk

${\ displaystyle \ mathbf {P} ^ {\ text {NL}} = \ varepsilon _ {0} \ chi ^ {(2)} \ mathbf {E} ^ { 2} (t).}  ÂÂ$

Jika kita berasumsi bahwa E ( t ) terdiri dari dua komponen pada frekuensi ? ₁ dan ? ₂ , kita bisa menulis E ( t ) sebagai

${\ displaystyle \ mathbf {E} (t) = E_ {1} \ cos (\ omega _ {1} t) E_ {2} \ cos (\ omega _ {2} t),}  ÂÂ$

dan menggunakan rumus Euler untuk meng -versi que eksponensial,

${\ displaystyle \ mathbf {E} (t) = {\ frac {1} {2}} E_1 e ^ {- i \ omega _ {1} t} {\ frac {1} {2}} E2 e ^ {- i \ omega2 t} {\ text {cc}},} Â Â$

which has a frequency component at 2 ? ₁ , 2 ? ₂ , ? ₁ Ã, ? ₂ , ? ₁ Ã, -Ã, ? ₂ , and 0. This three-wave mixing process corresponds to a nonlinear effect known as second harmonic generation, frequency generation generation, frequency frequency generation and optical rectification.

Note: Parametric generation and amplification is a variation of frequency generation, where the lower frequency of one of the two generating fields is much weaker (parametric amplification) or completely absent (parametric generation). In the latter case, the fundamental quantum mechanical uncertainty in the electric field starts the process.

Phase match

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${\ displaystyle E_ {j} (\ mathbf {x}, t) = E_ {j, 0} e ^ {i (\ mathbf {k} _ j} cdot \ mathbf {x} - \ omega _ {j} t)} {\ text {cc}}} Â Â$

The above equation is known as the phase matching condition . Typically, mixing of three waves is performed in birefringent crystalline materials, where the refractive index depends on the polarization and the direction of light passing. The polarization of the plane and the orientation of the crystal is chosen such that the matching-phase conditions are met. This phase matching technique is called angle adjustment. Usually the crystal has three axes, one or two of which have refractive indexes that are different from others (s). Uniaxial crystals, for example, have a single preferred axis, called the outer axis (e), while the other two are the ordinary axes (o) (see optical crystal). There are several schemes to choose the polarization for this type of crystal. If the signal and the idler have the same polarization, it is called "phase-matching type I," and if their polarization is perpendicular, it is called "phase-II type matching". However, other conventions exist which further determine which frequencies have polarization relative to the crystal axis. These types are listed below, with the convention that the wavelength of the signal is shorter than the idling wavelength.

Most nonlinear crystals are uniaxially negative, meaning that the axis e has a smaller refractive index than the o axis. In such crystals, phase matching types-I and -II are usually the most appropriate schemes. In positive uniaxial crystals, types VII and VIII are more appropriate. Type II and III are essentially the same, except that the names of signals and idlers swap when the signal has a longer wavelength than the idler. For this reason, they are sometimes called IIA and IIB. The V-VIII number is less common than I and II and variants.

One of the undesirable effects of angular tuning is that the optical frequencies involved do not spread collectively with each other. This is due to the fact that the extraordinary waves propagating through the birefringent crystal have a Poynting vector that is not parallel to the propagation vector. This will cause the beam to run, which limits the efficiency of nonlinear optical conversion. Two other methods of matching phase avoid beam walk-off by forcing all frequencies to spread at 90 ° with respect to the optical axis of the crystal. This method is called temperature-setting and quasi-phase matching.

Temperature tuning is used when the pump (laser) polarization frequency is orthogonal to the signal and the polarization of the idler frequency. Birefringence in some crystals, especially lithium niobate is highly temperature dependent. The crystal temperature is controlled to achieve phase matching conditions.

Another method is quasi-phase-matching. In this method, the frequencies involved are not always locked in phase with each other, but the crystal axis is reversed at regular intervals, typically 15 micrometers in length. Therefore, these crystals are called polarized periodically. This results in a crystalline polarization response which will be shifted back to phase by the pump beam by reversing the nonlinear susceptibility. This allows a clean positive energy flow from the pump to the signal and idle frequencies. In this case, the crystal itself provides an additional wavevector k Ã, = Ã,2?/? (and hence the momentum) to meet the phase matching conditions. Quasi-phase-matching can be extended to the chirp grille to get more bandwidth and to form SHG pulses as done in dazzler. SHG from the pump and self-phase modulation (imitated by a second-order process) of the optical parametric signal and amplifier can be integrated monolithically.

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High-frequency mixing

Di atas berlaku untuk ${\ displaystyle \ chi ^ {(2)}} Â$ process. Ini dapat diperpanjang untuk proses di mana ${\ displaystyle \ chi ^ {(3)}} Â Â$ adalah nol, sesuatu yang umumnya benar dalam media apa pun tanpa batasan simetri apa pun; khususnya peningkatan jumlah atau perbedaan frekuensi yang sangat tinggi pencampuran gas sering digunakan untuk pembangkit cahaya Ultra Violet yang ekstrim atau "vakum". Dalam skenario umum, seperti pencampuran dalam gas encer, non-linearitas lemah dan begitu sinar cahaya difokuskan yang, tidak seperti pendekatan gelombang bidang yang digunakan di atas, memperkenalkan pergeseran phase pi pada setiap sinar, mempersulit persyaratan pencocokan phase. Mudah, perbedaan frekuensi pencampuran dengan ${\ displaystyle \ chi ^ {(3)}} Â Â$ membatalkan perleyal fokal phase ini dan sering memiliki phase ma

Source of the article : Wikipedia