Raman scattering

From Wikipedia, the free encyclopedia

Raman scattering or the Raman effect /ˈrɑːmən/ is the inelastic scattering of a photon by molecules which are excited to higher vibrational or rotational energy levels. It was discovered in 1928 by C. V. Raman and his student K. S. Krishnan in liquids, and independently by Grigory Landsberg and Leonid Mandelstam in crystals. The effect had been predicted theoretically by Adolf Smekal in 1923.

When photons are scattered from an atom or molecule, most of them are elastically scattered (Rayleigh scattering), such that the scattered photons have the same energy (frequency and wavelength) as the incident photons. A small fraction of the scattered photons (approximately 1 in 10 million) are scattered inelastically by an excitation, with the scattered photons having a frequency and energy different from, and usually lower than, those of the incident photons. In a gas, Raman scattering can occur with a change in energy of a molecule due to a transition to another (usually higher) energy level. Chemists are primarily concerned with this "transitional" Raman effect.

History

The inelastic scattering of light was predicted by Adolf Smekal in 1923 (and in German-language literature it may be referred to as the Smekal-Raman effect). In 1922, Indian physicist C. V. Raman published his work on the "Molecular Diffraction of Light," the first of a series of investigations with his collaborators that ultimately led to his discovery (on 28 February 1928) of the radiation effect that bears his name. The Raman effect was first reported by C. V. Raman and K. S. Krishnan, and independently by Grigory Landsberg and Leonid Mandelstam, on 21 February 1928 (that is why in the former Soviet Union the priority of Raman was always disputed; thus in Russian scientific literature this effect is usually referred to as "combination scattering" or "combinatory scattering"). Raman received the Nobel Prize in 1930 for his work on the scattering of light.

In 1998 the Raman effect was designated a National Historic Chemical Landmark by the American Chemical Society in recognition of its significance as a tool for analyzing the composition of liquids, gases, and solids.

Description

Degrees of freedom

For any given chemical compound, there are a total of 3N degrees of freedom, where N is the number of atoms in the compound. This number arises from the ability of each atom in a molecule to move in three different directions (x, y, and z). When dealing with molecules, it is more common to consider the movement of the molecule as a whole. Consequently, the 3N degrees of freedom are partitioned into molecular translational, rotational, and vibrational motion. Three of the degrees of freedom correspond to translational motion of the molecule as a whole (along each of the three spatial dimensions). Similarly, three degrees of freedom correspond to rotations of the molecule about the ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$-axes. Linear molecules only have two rotations because rotations along the bond axis do not change the positions of the atoms in the molecule. The remaining degrees of freedom correspond to molecular vibrational modes. These modes include stretching and bending motions of the chemical bonds of the molecule. For a linear molecule, the number of vibrational modes is:

${\displaystyle 3N-3-2=3N-5}$

whereas for a non-linear molecule the number of vibrational modes are

${\displaystyle 3N-6}$

Molecular vibrations and infrared radiation

The frequencies of molecular vibrations range from less than 10¹² to approximately 10¹⁴ Hz. These frequencies correspond to radiation in the infrared (IR) region of the electromagnetic spectrum. At any given instant, each molecule in a sample has a certain amount of vibrational energy. However, the amount of vibrational energy that a molecule has continually changes due to collisions and other interactions with other molecules in the sample.

At room temperature, most of the molecules will be in the lowest energy state, which is known as the ground state. A few molecules will be in higher energy states, which are known as excited states. The fraction of molecules occupying a given vibrational mode at a given temperature can be calculated using the Boltzmann distribution. Performing such a calculation shows that, for relatively low temperatures (such as those used for most routine spectroscopy), most of the molecules occupy the ground vibrational state. Such a molecule can be excited to a higher vibrational mode through the direct absorption of a photon of the appropriate energy. This is the mechanism by which IR spectroscopy operates: infrared radiation is passed through the sample, and the intensity of the transmitted light is compared with that of the incident light. A reduction in intensity at a given wavelength of light indicates the absorption of energy by a vibrational transition. The energy, ${\displaystyle E}$, of a photon is

${\displaystyle E=h\nu }$,

where ${\displaystyle h}$ is Planck's constant and ${\displaystyle \nu }$ is the frequency of the radiation. Thus, the energy required for such a transition may be calculated if the frequency of the incident radiation is known.

Raman scattering

It is also possible to observe molecular vibrations by an inelastic scattering process. In inelastic (Raman) scattering, an absorbed photon is re-emitted with lower energy; the difference in energy between the incident photons and scattered photons corresponds to the energy required to excite a molecule to a higher vibrational mode.

Typically, in Raman spectroscopy high intensity laser radiation with wavelengths in either the visible or near-infrared regions of the spectrum is passed through a sample. Photons from the laser beam produce an oscillating polarization in the molecules, exciting them to a virtual energy state. The oscillating polarization of the molecule can couple with other possible polarizations of the molecule, including vibrational and electronic excitations. If the polarization in the molecule does not couple to these other possible polarizations, then it will not change the vibrational state that the molecule started in and the scattered photon will have the same energy as the original photon. This type of scattering is known as Rayleigh scattering.

When the polarization in the molecules couples to a vibrational state that is higher in energy than the state they started in, then the original photon and the scattered photon differ in energy by the amount required to vibrationally excite the molecule. In perturbation theory, the Raman effect corresponds to the absorption and subsequent emission of a photon via an intermediate quantum state of a material. The intermediate state can be either a "real", i.e. stationary state, or a virtual state.

Stokes and anti-Stokes

The different possibilities of light scattering: Rayleigh scattering (no exchange of energy: incident and scattered photons have the same energy), Stokes Raman scattering (atom or molecule absorbs energy: scattered photon has less energy than the incident photon) and anti-Stokes Raman scattering (atom or molecule loses energy: scattered photon has more energy than the incident photon)

The Raman interaction leads to two possible outcomes:

• the material absorbs energy and the emitted photon has a lower energy than the absorbed photon. This outcome is labeled Stokes Raman scattering in honor of George Stokes who showed in 1852 that fluorescence is due to light emission at longer wavelength (now known to correspond to lower energy) than the absorbed incident light;
• the material loses energy and the emitted photon has a higher energy than the absorbed photon. This outcome is labeled anti-Stokes Raman scattering.

The energy difference between the absorbed and emitted photon corresponds to the energy difference between two resonant states of the material and is independent of the absolute energy of the photon.

The spectrum of the scattered photons is termed the Raman spectrum. It shows the intensity of the scattered light as a function of its frequency difference Δν to the incident photons. The locations of corresponding Stokes and anti-Stokes peaks form a symmetric pattern around Δν=0. The frequency shifts are symmetric because they correspond to the energy difference between the same upper and lower resonant states. The intensities of the pairs of features will typically differ, though. They depend on the populations of the initial states of the material, which in turn depend on the temperature. In thermodynamic equilibrium, the lower state will be more populated than the upper state. Therefore, the rate of transitions from the more populated lower state to the upper state (Stokes transitions) will be higher than in the opposite direction (anti-Stokes transitions). Correspondingly, Stokes scattering peaks are stronger than anti-Stokes scattering peaks. Their ratio depends on the temperature, and can therefore be exploited to measure it.

Distinction from fluorescence

The Raman effect differs from the process of fluorescence in that it is a scattering process. For fluorescence, the incident light is completely absorbed, transferring the system to an excited state. After a certain resonance lifetime, the system de-excites to lower energy states via emission of photons. The result of both processes is in essence the same: A photon with a frequency different from that of the incident photon is produced and the molecule is brought to a higher or lower energy level. But the major difference is that the Raman effect can take place for any frequency of incident light. In contrast to the fluorescence effect, the Raman effect is therefore not a resonant effect. In practice, this means that a fluorescence peak is anchored at a specific frequency, whereas a Raman peak maintains a constant separation from the excitation frequency.

Selection rules

A Raman transition from one state to another is allowed only if the molecular polarizability of those states is different. For a vibration, this means that the derivative of the polarizability with respect to the normal coordinate associated to the vibration is non-zero: ${\displaystyle {\frac {\partial \alpha }{\partial Q}}\neq 0}$. In general, a normal mode is Raman active if it transforms with the same symmetry of the quadratic forms (${\displaystyle x^{2},y^{2},z^{2},xy,xz,yz}$), which can be verified from the character table of the molecule's symmetry group.
The specific selection rules state that the allowed rotational transitions are ${\displaystyle \Delta J=\pm 2}$, where ${\displaystyle J}$ is the rotational state.

The allowed vibrational transitions are ${\displaystyle \Delta \nu =\pm 1}$, where ${\displaystyle \nu }$ is the vibrational state.

Stimulated Raman scattering and Raman amplification

The Raman-scattering process as described above takes place spontaneously; i.e., in random time intervals, one of the many incoming photons is scattered by the material. This process is thus called spontaneous Raman scattering.

On the other hand, stimulated Raman scattering can take place when some Stokes photons have previously been generated by spontaneous Raman scattering (and somehow forced to remain in the material), or when deliberately injecting Stokes photons ("signal light") together with the original light ("pump light"). In that case, the total Raman-scattering rate is increased beyond that of spontaneous Raman scattering: pump photons are converted more rapidly into additional Stokes photons. The more Stokes photons are already present, the faster more of them are added. Effectively, this amplifies the Stokes light in the presence of the pump light, which is exploited in Raman amplifiers and Raman lasers.

Stimulated Raman scattering is a nonlinear-optical effect. It can be described using a third-order nonlinear susceptibility ${\displaystyle \chi ^{(3)}}$.

Need of space-coherence

Suppose that the distance between two points A and B of an exciting beam is x. Generally, as the exciting frequency is not equal to the scattered Raman frequency, the corresponding relative wavelengths λ and λ' are not equal. Thus, a phase-shift Θ = 2πx(1/λ − 1/λ') appears. For Θ = π, the scattered amplitudes are opposite, so that the Raman scattered beam remains weak.

– A crossing of the beams may limit the path x.

Several tricks may be used to get a larger amplitude:

– In an optically anisotropic crystal, a light ray may have two modes of propagation with different polarizations and different indices of refraction. If energy may be transferred between these modes by a quadrupolar (Raman) resonance, phases remain coherent along the whole path, transfer of energy may be large. It is an Optical parametric generation.

– Light may be pulsed, so that beats do not appear.

It is the Impulsive Stimulated Raman Scattering (ISRS), in which the length of the pulses must be shorter than all relevant time constants. Interference of Raman and incident lights is too short to allow beats, so that it produces a frequency shift roughly, in best conditions, inversely proportional to cube of length of pulses. In labs, femtosecond laser pulses must be used because the ISRS becomes very weak if the pulses are too long. Thus ISRS cannot be observed using nanosecond pulses making ordinary time-incoherent light.

Inverse Raman effect

The inverse Raman effect in optics (the branch of physics which deals with the properties and behavior of light) is a form of Raman scattering. It was first noted by W.J.Jones and B.P. Stoicheff.
If a material is simultaneously irradiated by intense monochromatic light of frequency ν_L (typically a laser beam) and light of a continuum of higher frequencies, among the possibilities for light scattering are:

• from the monochromatic beam at ν_L to the continuum at ν_L+ν_M (anti-Stokes Raman scattering);
• from the continuum at ν_L+ν_M to the monochromatic beam at ν_L (Stokes Raman scattering).

where ν_M is a Raman frequency of the material. The strength of these two scatterings depends (among other things) on the energy levels of the material, their occupancy, and the intensity of the continuum. In some circumstances Stokes scattering can exceed anti-Stokes scattering; in these cases the continuum (on leaving the material) is observed to have an absorption line (a dip in intensity) at ν_L+ν_M. This phenomenon is referred to as the inverse Raman effect; the application of the phenomenon is referred to as inverse Raman spectroscopy, and a record of the continuum is referred to as an inverse Raman spectrum.

In the original description of the inverse Raman effect, the authors discuss both absorption from a continuum of higher frequencies and absorption from a continuum of lower frequencies. They note that absorption from a continuum of lower frequencies will not be observed if the Raman frequency of the material is vibrational in origin and if the material is in thermal equilibrium.

Applications

Raman spectroscopy employs the Raman effect for substances analysis. The spectrum of the Raman-scattered light depends on the molecular constituents present and their state, allowing the spectrum to be used for material identification and analysis. Raman spectroscopy is used to analyze a wide range of materials, including gases, liquids, and solids. Highly complex materials such as biological organisms and human tissue can also be analyzed by Raman spectroscopy.

For solid materials, Raman scattering is used as a tool to detect high-frequency phonon and magnon excitations.

Raman lidar is used in atmospheric physics to measure the atmospheric extinction coefficient and the water vapour vertical distribution.

Stimulated Raman transitions are also widely used for manipulating a trapped ion's energy levels, and thus basis qubit states.

Raman spectroscopy can be used to determine the force constant and bond length for molecules that do not have an infrared absorption spectrum.

Raman amplification is used in optical amplifiers.

The Raman effect is also involved in producing the appearance of the blue sky (see Rayleigh Effect: 'Rayleigh scattering of molecular nitrogen and oxygen in the atmosphere includes elastic scattering as well as the inelastic contribution from rotational Raman scattering in air').

Supercontinuum generation

For high-intensity continuous wave (CW) lasers, SRS can be used to produce broad bandwidth spectra. This process can also be seen as a special case of four-wave mixing, wherein the frequencies of the two incident photons are equal and the emitted spectra are found in two bands separated from the incident light by the phonon energies. The initial Raman spectrum is built up with spontaneous emission and is amplified later on. At high pumping levels in long fibers, higher-order Raman spectra can be generated by using the Raman spectrum as a new starting point, thereby building a chain of new spectra with decreasing amplitude. The disadvantage of intrinsic noise due to the initial spontaneous process can be overcome by seeding a spectrum at the beginning, or even using a feedback loop as in a resonator to stabilize the process. Since this technology easily fits into the fast evolving fiber laser field and there is demand for transversal coherent high-intensity light sources (i.e., broadband telecommunication, imaging applications), Raman amplification and spectrum generation might be widely used in the near-future.

Nonlinear theory of semiconductor lasers

From Wikipedia, the free encyclopedia

Laser theory of Fabry-Perot (FP) semiconductor lasers proves to be nonlinear, since the gain, the refractive index and the loss coefficient are the functions of energy flux. The nonlinear theory made it possible to explain a number of experiments some of which could not even be explained (for example, natural linewidth), much less modeled, on the basis of other theoretical models; this suggests that the nonlinear theory developed is a new paradigm of the laser theory.

Equations in the gain medium

Maxwell's equations describe the field for passive medium and cannot be used in describing the field in laser and quantum amplifier. Phenomenological equations are derived for electromagnetic field in the gain medium, i.e. Maxwell's equations for the gain medium, and Poynting's theorem for these equations. Maxwell's equations in the gain medium are used to obtain equations for energy flux, and to describe nonlinear phase effect.  The equation

${\displaystyle (1)\quad rot{\vec {H}}=(\sigma -\eta ){\vec {E}}+{\partial {\vec {D}} \over \partial t},}$

defines η as a specific gain factor; σ is specific conductivity that describes incoherent losses (for example, on free electrons). Other Maxwell's equations are used unchanged.

${\displaystyle (2)\quad rot{\vec {E}}=-{\partial B \over \partial t},divD=0,divB=0,}$
${\displaystyle (3)\quad D=\epsilon '\epsilon _{0}E,B=\mu _{0}\mu H,}$

Poynting theorem follows from (1)-(3):

${\displaystyle (4)\quad \int _{V}\eta |E(\omega )|^{2}\,dv=\int _{V}\sigma |E(\omega )|^{2}\,dv+\oint _{S}(Sn)ds,}$

where S is Poynting vector; V=sz, 0 < z < L, where s is cross section (to axis z) of active laser medium.

Equations for energy flux follow from (4):

${\displaystyle (5)\quad {\frac {dI^{+}}{dz}}=(\Gamma g(I^{+})-\alpha (I^{+}))I^{+},}$
${\displaystyle (6)\quad {\frac {-dI^{-}}{dz}}=(\Gamma g(I^{-})-\alpha (I^{-}))I^{-},}$

where

${\displaystyle (7)\quad \alpha (I)=(1-\Gamma )\alpha _{out}+\Gamma \alpha _{in}+\alpha _{x}+\alpha _{2p}(I),}$

where I is the energy flux; s is sectional area of the active zone of the laser; Г is confinement factor; α_in is absorption factor in active zone; α_out is absorption factor outside active zone; α_x is losses due to incoherent scattering α_2p(I) is two-photon absorption factor, and α_2p(I)= β⋅I.

Formulas for the line shape and natural linewidth

Theory of natural linewidth in semiconductor lasers has been developed, it follows that refractive index n in FP lasers and effective refractive index n_ef in Distributed FeedBack (DFB) lasers are the functions of E:

${\displaystyle (8)\quad n=n_{0}+n_{1}(\omega )E+n_{2}(\omega )E^{2},}$
${\displaystyle (9)\quad n=n_{ef}^{0}+n_{ef}^{1}E+n_{ef}^{2}E^{2},}$

from which the formulas for the line shape in FP and in DFB lasers were derived. The formulas for the line shape are similar and have the following form:

${\displaystyle (10)\quad L(v)=2\int _{0}^{\infty }\cos(2\pi v\tau )exp(A_{pa}|\tau |-B\tau ^{2})d\tau ,}$

where ${\displaystyle v={\tfrac {\omega -\omega L}{2\pi }};\quad \omega L}$ is laser generation frequency:

${\displaystyle (11)\quad A_{pa}={\frac {D_{0}}{P}},\quad B=(D_{1}+D_{2}P^{(1/2)})^{2},}$

where D₀,D₁, D₂ have different form for FP and for DFB lasers. Let us write the natural linewidth Δν:

${\displaystyle (12)\quad \Delta v=\Delta v^{(c)}T\left(\left(mW\right){\frac {A_{pa}}{P}}+B\right)F_{N}\left({\frac {P^{(c)}}{P}},k\right),}$
where ${\displaystyle T\left(\left(mW\right){\frac {A_{pa}}{P}}+B\right)}$ is the bridge function; ${\displaystyle \Delta v^{(c)}}$ and ${\displaystyle P^{(c)}}$ are characteristic linewidth and characteristic laser power; k is characteristic parameter of laser nonlinearity; q is non-dimensional inverse power:

${\displaystyle (13)\quad F_{N}(q,k)=1/\int _{0}^{\infty }exp(-xq-x^{2}(k/q^{1/2}-1)^{2})dx}$

The theory of natural linewidth in semiconductor lasers has an independent significance. At the same time, the developed theory is an integral part of the nonlinear theory of lasers, and its concepts and the introduced characteristic parameters are used in all parts of the nonlinear theory.

Gain in a semiconductor laser

Using the density matrix equations with relaxation, the following derivations have been made: Einstein’s spectral coefficient in a semiconductor laser and, accordingly, Einstein’s coefficient; formula for the saturation effect in a semiconductor laser was derived; it was shown that the saturation effect in a semiconductor laser is small. Gain in a semiconductor laser has been derived using the density matrix equations with relaxation. It has been found that Fabry-Perot laser gain depends on energy flux, and this determines the ‘basic nonlinear effect’ in a semiconductor laser

${\displaystyle (14)\quad g=g(I)=R\Delta \omega _{e}/(1-I(\delta \Delta \omega _{e}/\delta I)/\Delta \omega _{e}),}$

where

${\displaystyle (15)\quad R=(\hbar \omega n_{g}/c)B_{21}(f_{2}-f_{1})S_{P}(\omega ).}$

where ${\displaystyle B_{21}}$ is Einstein coefficient for induced transition between the two energy levels when exposed to a narrow-band wave is written in the following form:

${\displaystyle (16)\quad B_{21}=({\hat {H}}_{21}/\hbar )^{2}/(2\Gamma _{0})}$ where ${\displaystyle \Delta \omega _{e}}$ is effective natural linewidth; ${\displaystyle I}$ is the energy flux; ${\displaystyle S_{P}(\omega )}$ is spectral density of transitions.

Necessary condition for induced radiation of the 1st kind

Necessary conditions for induced radiation of the 1st and 2nd kind have been defined in. Necessary conditions for induced radiation are determined by the requirement for the gain to be greater than zero. Necessary condition for induced radiation of the 1st kind formulated by Bernard and Duraffourg is that the population of levels located above becomes more than the population of levels located below

${\displaystyle (17)\quad f_{2}(E_{2})>f_{1}(E_{1}).}$

Necessary condition for induced radiation of the 2nd kind

The necessary condition of induced radiation of the 2nd kind formulated by Noppe is that:

${\displaystyle (18)\quad \Delta \omega _{e}/(1-I(\delta \Delta \omega _{e}/\delta I)/\Delta \omega _{e})>0.}$

Figure.1. Functions ${\displaystyle g_{1}(I)}$ and ${\displaystyle g_{2}(I)}$ versus energy flux I for two sets of characteristic parameters.

The necessary condition of induced radiation of the 2nd kind allows formulating the basic restriction of laser capacity, which has been confirmed experimentally:

${\displaystyle (19)\quad I$

where I is energy flux; I(M) is the characteristic parameter of ultimate power. Figure 1 shows the function g(I) for two sets of characteristic parameters.

Simulation of experiments

4.1. Maxwell's equations in the gain medium are used to obtain equations for energy flux.Nonlinear phase effect has been described and simulated, using the nonlinearity of refractive index. (see Fig.3).

4.2. Based on the developed theory, experimental output characteristics have been simulated: natural linewidth (see Fig.2), experimental watt - ampere characteristics (see Fig.4) and dependence of experimental output radiation line-length on the current in Fabry-Perot semiconductor injection lasers, (see Fig.3), as well as linewidth in DFB lasers (see simulation in). Created theory makes it possible to simulate the majority published experiments on the measurement of the natural linewidth in Fabry-Perot lasers and distributed feedback DFB lasers with help of two methods (using (13) and (15)). Based on the formula derived for the line shape, 12 experiments on measuring the natural linewidth in Fabry-Perot lasers (for example see Fig.2) and 15 experiments in DFB lasers have been simulated. Based on the formula derived for the natural linewidth, 15 experiments on measuring the natural linewidth in Fabry-Perot lasers and 15 experiments in DFB lasers have been simulated. The derived formula for line shape of radiation (of FP lasers and DFB lasers) is distinguished from Lorentz line formula.

4.3. Based on the developed theory, experimental output characteristics have been simulated: natural linewidth, experimental watt - ampere characteristics (see Fig.4), and dependence of experimental output radiation line-length on the current in Fabry-Perot semiconductor injection lasers.

4.4. On the basis of the nonlinear theory, recommendations have been made for the development of lasers with smaller natural linewidth and lasers with higher output power.

Figure.2. Simulating experimental curve of the natural linewidth of Fabry-Perot semiconductor lasers as functions of inverse output power Δν_e(1/P) (Ke=14) by theoretical curve Δνe(1/P)  (K_t=14).

 

Figure 3. Wavelength shift Δλ (theoretical and experimental) versus normalized current (J/Jth).

Figure 4. Experimental and theoretical output power versus current for a powerful laser.

Conclusion

Based on the solution of the density matrix equations, Einstein coefficient for induced transition has been derived; it has been shown that the saturation effect is small for semiconductor lasers. The formula of gain depending on the energy flux has been derived; it is the basic nonlinear effect in a laser. It has been stated that the main effect resulting in nonlinearity is the saturation effect. For semiconductor lasers, the saturation effect is negligible. We derived the gain g for a Fabry-Perot semiconductor laser based on the density matrix equations and expressions for the natural linewidth. Thus, the linewidth theory is an integral part of the nonlinear theory. The resulting dependence of g on the energy flux has been called the main nonlinear effect in semiconductor lasers; derivation of this relation formula is presented in. Experimental wavelength shift versus normalized current (J/Jth), and the output power versus current have been simulated for a high-power laser with a quantum well of intrinsic semiconductor. Broadening of the states density due to different effects has been taken into consideration. The nonlinear theory made it possible to explain a number of experiments some of which could not even be explained (for example, natural linewidth), much less modeled, on the basis of other theoretical models; this suggests that the nonlinear theory developed is a new paradigm of the laser theory. Due to the nonlinear theory development, recommendations can be given for creating lasers with smaller natural linewidth, and lasers with higher output power.