Frequency doubling

Frequency doubling in a non-linear medium

Frequency doubling describes the phenomenon that when some materials are irradiated, e.g. B. with a laser with high intensity , under certain conditions radiation with twice the frequency is generated. This corresponds to halving the wavelength . For example, green light with a wavelength of 532 nm can be generated from the infrared radiation of a Nd: YAG laser ( = 1064 nm). Frequency tripling is also possible, in the example of the Nd: YAG laser then ultraviolet with = 354.7 nm is produced. ${\ displaystyle \ lambda}$ ${\ displaystyle \ lambda}$ ${\ displaystyle \ lambda}$

Frequency doubling is often abbreviated SHG ( second harmonic generation ), frequency tripling with THG ( third harmonic generation ).

Physical background

The atomic nucleus and the electron shell are shifted against each other by the action of light.

When electromagnetic radiation passes through matter , the electric field of this radiation leads to a periodic shift of the electric charges with the frequency of the radiation. These oscillating displacements in turn generate electromagnetic radiation. If the intensity of the incident light is small, the deflections of the electrical charges from the rest position are also small. They then behave like harmonic oscillators that are driven with a frequency apart from their resonance: The movement only contains the same frequency components as the excitation.

The potential that drives the dipole back to its original position has approximately the shape of a parabola only for small deflections . In the case of large deflections it deviates from this, since the nuclear charge of neighboring atoms then has an influence. This deviation is called non-linearity , since it means a non-linear relationship between deflection and back-driving force. The shape and strength of the non-linearity depends on the structure of the material through which the light shines.

The moving charge experiences an acceleration towards the zero position due to the potential. A sinusoidal course of the speed results from this only for a square potential . In the event of deviations, the charge is accelerated too slowly or too quickly in the meantime. This leads to deviations from the sinusoidal shape in the course of the speed and as a consequence to deviations in the electric field of the light emitted by the charge. In the spectrum of light this means that not only the incident frequency, but also its harmonics are contained in different strengths. Since the efficiency of the conversion decreases sharply with the degree of the harmonic, mostly only the second (SHG) or the third (THG) are of technical importance.

This generation of higher frequency light can be understood as the absorption of two or more photons and the emission of one photon. However, it is not to fluorescence . In contrast to fluorescence, the emitted light is coherent with the incident light . The mechanism is not related to the energy levels of the atoms.

Frequency doubling (SHG)

If the potential effective for the oscillating dipole is non-linear, but symmetrical to the zero position, then the velocity is equally distorted on both sides of the deflection. The resulting movement does not contain even Fourier coefficients . Therefore, only light of odd harmonics can be generated with such a potential (tripling, quintupling, ...). To generate twice the frequency, the non-linear material used must not be centrosymmetric.

The doubled light spreads in the "forward" direction like the incoming light beam: The individual phases of the forward-directed photons are in phase , so that the waves that are generated by different atoms are amplified. In other directions the waves cancel each other out .

Frequency tripling (GHG)

If the intensity of the incident light is sufficiently strong, the amplitude of the dipole oscillations is sufficient to emit light of three times the frequency. In contrast to frequency doubling, no special asymmetry in the arrangement of the atoms involved is necessary. The high intensity required and the wide range of wavelengths spanned by the tripling, however, represent technical hurdles.

Basics

The efficiency of the frequency doubling depends strongly on the strength of the field strength of the electromagnetic wave. While the polarization in linear optics only depends on the first order term , with high radiation intensities it now also depends on the other orders and in this case consists of several contributions:

{\ displaystyle {\ textbf {P}} = \ varepsilon _ {0} \ sum _ {n} \ chi ^ {(n)} {\ textbf {E}} ^ {n} = \ varepsilon _ {0} \ chi ^ {(1)} {\ textbf {E}} + \ varepsilon _ {0} \ chi ^ {(2)} {\ textbf {E}} ^ {2} + \ varepsilon _ {0} \ chi ^ {(3)} {\ textbf {E}} ^ {3} + \ dots}

where corresponds to the dielectric susceptibility . ${\ displaystyle \ chi ^ {(n)}}$

In the case of frequency doubling , the second order term ( ) of the above equation must now be considered. When a strong light wave of angular frequency propagates in the z-direction in matter, it generates a time-dependent radiation field at a given point: ${\ displaystyle n = 2}$ ${\ displaystyle \ omega}$

{\ displaystyle E (t) = E_ {0} \ sin (\ omega t)}

which induces second order polarization, and makes the above equation look like this:

{\ displaystyle | {\ textbf {P}} ^ {(2)} | = \ varepsilon _ {0} \ chi ^ {(2)} E ^ {2} = \ varepsilon _ {0} E_ {0} ^ {2} \ chi ^ {(2)} \ sin ^ {2} (\ omega t)}

With the help of the trigonometric identity : ${\ displaystyle \ sin ^ {2} (x) = {\ frac {1- \ cos (2x)} {2}}}$

{\ displaystyle | {\ textbf {P}} ^ {(2)} | = {\ frac {\ varepsilon _ {0} E_ {0} ^ {2} \ chi ^ {(2)}} {2}} - {\ frac {\ varepsilon _ {0} E_ {0} ^ {2} \ chi ^ {(2)}} {2}} \ cos (2 \ omega t)}

It is obvious that the second order polarization consists of two contributions: a constant term, corresponding to a static electric field ( optical rectification ), and a second term, which oscillates at twice the frequency . This oscillating polarization generates secondary radiation at the frequency in the non-linear medium , which is referred to as frequency doubling . ${\ displaystyle 2 \ omega}$ ${\ displaystyle 2 \ omega}$

So that the secondary radiation is also radiated when passing through the medium, the refractive index in the direction of propagation for the fundamental wave must be the same as the index for the second harmonic:

{\ displaystyle n _ {\ omega} = n_ {2 \ omega}}

If this condition is not met, the conversion still takes place in the medium, but the radiation emitted at the various points in the medium is eliminated by destructive interference . With the same refractive index, the propagation speeds of the fundamental wave and the second harmonic are the same, so that a constructive superposition takes place. This adjustment is called phase adjustment .

Since all media show dispersion , the phase matching condition cannot generally be achieved with optically isotropic materials. Therefore, the media used are mostly birefringent crystals. Three possibilities for phase matching in nonlinear optical media are known: the critical , the non-critical and the quasi-phase matching (QPM of English. Quasi phase matching ). In the critical phase matching in a birefringent material, the crystal axis with respect to the optical axis is chosen so that the refractive index of the ordinary ray of the fundamental wave and the extraordinary ray of the second harmonic match. In the case of non-critical phase matching, some media make use of the property that the refractive index of the fundamental wave and the second harmonic change differently when the temperature varies. A temperature is then found for the desired wavelength at which the phase matching condition is met. For example, this can be achieved for the conversion of light with a wavelength of 1064 nm to 532 nm with the help of an LBO crystal at a temperature of around 140 ° C. With quasi-phase matching, the ferroelectricity of materials such as lithium niobate, which is common in non-linear optics , is used. Here domains are written into the material in which the sign periodically changes ( periodic polarity ). A real phase adjustment does not take place, but the individual domains can be designed in terms of their periodicity and wavelength so that the generated partial waves of the second harmonic are constructively superimposed. ${\ displaystyle \ chi ^ {(2)}}$

Applications

With the help of frequency doubling and frequency tripling, a laser that irradiates a non-linear medium can generate higher optical frequencies than the laser itself emits. Since lasers with wavelengths in the near infrared are particularly easy to manufacture, it is often much easier to operate such a laser with frequency doubling or tripling than to construct a laser in the visible range or the near ultraviolet.

Solid-state lasers are mostly used as the radiation source, for example Nd: YAG lasers , which emit green rays after doubling the frequency, and the like. a. in green laser pointers . Frequency-doubled Nd: YAG lasers also deliver green laser beams with up to several watts of power for laser shows and in laser projectors .

The frequency doubling takes place with a non-linear medium inside the laser resonator ( intracavity SHG) or outside. The frequency doubling in the resonator offers the advantage that the intensity of the beam and thus the conversion efficiency is higher there. The disadvantage is the power and mode stability , which is difficult to achieve : due to the non-linear relationship between intensity and frequency conversion (the latter increases steeply with increasing intensity), power oscillations and competing transverse modes occur that can hardly be stabilized.

Frequency-doubled lasers emit higher-frequency radiation in a beam quality similar to that of the fundamental wave. Because of the relationship between the wavelength and the minimum focus diameter, the short-wave radiation can be more finely focused. Further, it is better absorbed by many materials, so that they are better for laser micromachining is suitable (for example, laser trimming , machining of silicon).

Doubling materials

Materials that are suitable for frequency doubling are subject to some selection criteria. In addition to the general requirement not to be symmetrical under inversion, these are particularly relevant for the technical implementation. For high output powers it is beneficial to find a substance with the highest possible coefficient . On the other hand, it must be chemically and thermally stable, i.e. it must not be destroyed by the prevailing conditions in the arrangement. It should also be noted that neither the original nor the frequency-doubled light is strongly absorbed. The choice of material therefore also depends on the laser used and its wavelength. ${\ displaystyle \ chi ^ {(2)}}$

All of these requirements are best met by crystals specially made for this purpose. Examples are lithium niobate , potassium dihydrogen phosphate , beta barium borate, and lithium triborate . However, thin films of diethylaminosulfur trifluoride , periodically polarized polymers or liquid crystals can also be used to double the frequency.

Measurement of laser pulses

To measure short laser pulses , autocorrelators are used that utilize the frequency doubling effect.

microscopy

In microscopy, frequency doubling can be used to make biological structures visible, such as collagen fibers or myosin in the striated muscles . Both structures form crystal-like lattices that are non-centrosymmetrical in the order of magnitude of the wavelength of the incident light. Frequency doubling and frequency tripling can be used with a multiphoton microscope .

Frequency doubling also occurs at surfaces and interfaces. This can be used to detect changes directly on a surface.

literature

PA Franken , AE Hill, CW Peters, G. Weinreich: Generation of Optical Harmonics . In: Physical Review Letters . tape 7 , no. 4 , August 15, 1961, p. 118–119 , doi : 10.1103 / PhysRevLett.7.118 (first observation of non-phase-adjusted frequency doubling).

Robert W. Boyd: Nonlinear Optics. 3. Edition. Academic Press, 2008, ISBN 978-0-12-369470-6 .

Individual evidence

↑ quasi-phase matching, QPM, periodic poling, nonlinear crystal, orientation-patterned GaAs In: Encyclopedia of Laser Physics and Technology.
^ Critical phase matching, angle phase matching, acceptance angle In: Encyclopedia of Laser Physics and Technology.
↑ P. Friedl, K. Wolf, UH von Andrian, G. Harms: Biological second and third harmonic generation microscopy . In: Curr Protoc Cell Biol . March 2007, Chapter 4, pp. Unit 4.15 , doi : 10.1002 / 0471143030.cb0415s34 , PMID 18228516 .

[1] quasi-phase matching, QPM, periodic poling, nonlinear crystal, orientation-patterned GaAs In: Encyclopedia of Laser Physics and Technology.

[2] Critical phase matching, angle phase matching, acceptance angle In: Encyclopedia of Laser Physics and Technology.

[pmid18228516-3] P. Friedl, K. Wolf, UH von Andrian, G. Harms: Biological second and third harmonic generation microscopy . In: Curr Protoc Cell Biol . March 2007, Chapter 4, pp. Unit 4.15 , doi : 10.1002 / 0471143030.cb0415s34 , PMID 18228516 .