# (p.268) Appendix B Laser optics

# (p.268) Appendix B Laser optics

Since lasers are our primary tools for the investigation of atoms, advances in laser optics are strictly connected with progress in atomic physics. In this appendix we present a concise summary of the main results concerning the propagation of Gaussian laser beams, the physics of optical cavities, and the basic principles of nonlinear optics.

We start with the wave equation for the electromagnetic field in free space:

where *E _{i}
* refers to a component of the electric field and c is the speed of light. We look for stationary solutions describing the propagation of an electromagnetic wave along ̂

*z*with angular frequency ω, wavenumber

*k*= ω/

*c*, and time‐independent amplitude

*u(x, y, z):*

Substituting this trial solution into eqn (B.1) one finds

Assuming a weak variation of *u _{i}
* on the coordinate

*z*(i.e. assuming that the beam propagates in a quasi‐collimated way along ̂

*z)*the component

*∂*of the Laplacian operator can be neglected

^{2}/∂z^{2}^{1}and one finds the

*paraxial wave equation*

This equation admits different classes of solutions according to the boundary conditions which set the symmetry of the problem (in the case of a laser beam, these are fixed by the geometry of the laser optical cavity).
^{2}
For a rectangular symmetry an orthonormal
(p.269)

*Hermite‐Gaussian*beams TEM

_{ mn }, in which higher‐order transverse modes are characterized by an increasing number

*m*+

*n*of nodal lines along which the electric field is zero. For cylindrical symmetry the solutions can be given in terms of

*Laguerre‐Gaussian*beams, which possess a nontrivial azimuthal phase profile and are characterized by a nonzero orbital angular momentum. Examples of low‐order Hermite‐Gaussian and Laguerre‐Gaussian modes are shown in Fig. B.1 .

Here we consider only the fundamental transverse mode, the Gaussian TEM00 mode, which is both the lowest‐order Hermite‐Gaussian mode and the lowest‐order Laguerre‐Gaussian mode. A Gaussian TEM00 beam does not present any nodal points or lines and is a very good approximation for the spatial mode of many lasers. The dependence of the electric field amplitude on the spatial coordinates can be written as:

where *r* denotes the distance from the beam axis. The following definitions for the beam radius *w(z)*, the wavefront curvature radius *R(z)*, and the *Gouy phase ζ(z)* hold:

*beam waist radius w0*, which univocally determines all the properties of the Gaussian beam, and of the

*Rayleigh length*

From eqn (B.5) we can evaluate the beam intensity I = ∈_{0}
*c|E|*
^{2}/2, which is described by a Gaussian function of the radial coordinate *r*

the beam radius *w(z)* representing the distance from the axis at which the intensity is reduced by a factor *e*
^{2}. Figure B.2a
shows a plot of the beam intensity as a function of r in the *beam waist* at *z* = 0, where the beam has the smallest radius *w*
_{0}. Out of the beam waist the beam radius *w(z)* increases with *z* according to eqn (B.6), reaching a value $\sqrt{2}{w}_{0}$ at a distance corresponding to the Rayleigh length *z _{R}.* This dependence is shown in Fig. B.2b
together with a greyscale plot of the Gaussian beam intensity as a function of

*z*and

*r*.

Far from the beam waist, for *z » z _{R}
*, the beam size

*w(z)*⋍

*w*

_{0}

*z/z*increases linearly with

_{R}*z*and the beam is characterized by a divergence angle

which increases as the beam waist size *w*
_{0} is reduced. This is a consequence of diffraction: the more the beam is focused at the beam waist, the more rapidly it spreads as it propagates out of the focus.
^{3}
(p.271)

*Optical resonators* (or *optical cavities)* are important systems in atomic and optical physics: besides constituting an essential part of the design of a laser, they provide frequency references for spectroscopy and tools for the stabilization and line‐narrowing of the laser emission spectrum as well. They also represent a physical system in which the atom‐photon interaction can be studied in regimes of strong coupling, as studied in cavity‐QED experiments (Haroche and Raimond, 2006
). The simplest example of optical cavity is provided by two partially reflective mirrors one in front of the other. This configuration takes the name of *Fabry‐Perot resonator* from the French physicists Charles Fabry and Alfred Perot who proposed it at the end of the nineteenth century.

We consider two mirrors, separated by a distance L, characterized by a field reflectivity *r _{i}
* and field transmittivity

*t*= 1,2). The reflectivity and transmittivity coefficients for the intensities are

_{i}(i*R*= |

_{i}*r*|

_{i}^{2}and

*T*=

_{i}*|t*

_{i}|^{2}respectively, with the condition

*R*+

_{i}*T*= 1 imposed by the conservation of energy. We assume that the space between the mirrors is empty, i.e. no absorption takes place and the index of refraction of the intra‐cavity medium is 1. An incident beam (a plane wave) with electric field

_{i}*E*

_{0}entering the cavity experiences an infinite sequence of partial reflections and transmissions from the two mirrors, as schematically represented in Fig. B.3 . Although the reflections are drawn at an angle in order to graphically distinguish the different beams, we assume the situation of normal incidence, in which all the beams propagate along the same direction and overlap with each other. The figure indicates the attenuation of each partially reflected and transmitted beam, including the phase

*e*acquired by the field because of its propagation over a distance

^{ikL}*L.*The (p.272) total transmitted electric field can be calculated by summing over the (infinite) partial beams leaking from the cavity in the forward direction:

where we have used the sum formula for the geometric series (which is convergent since *|r*
_{1}
*r*
_{2}
*e ^{i2kL}|* 〈 1 and, more physically, because of energy conservation). In order to simplify the presentation of the results, we consider a particular case in which the field reflectivity and transmittivity coefficients are real‐valued and the same for the two mirrors: ${r}_{1}={r}_{2}=\sqrt{R}$ and ${t}_{1}={t}_{2}=\sqrt{T}$. By taking the squared modulus of eqn (B.12) we can write the transmitted intensity as

where we have defined the *finesse* of the resonator as

and its *free spectral range* as

Looking at eqn (B.13) we observe that the transmitted intensity is maximal when the sine term at the denominator is null, which happens for radiation frequencies *υ _{n}
* that satisfy the resonance condition

*n* being an integer number. This condition corresponds to a constructive interference of the infinite number of transmitted partial waves, which are all summed coherently with the same phase.
^{4}
The transmitted spectrum of the Fabry‐Perot resonator is made up of a comb of peaks centred at equally spaced frequencies, separated by a frequency interval FSR which only depends on the *geometry* of the resonator. This makes a Fabry‐Perot resonator a simple system for the calibration of laser scans in spectroscopy, e.g. for the determination of hyperfine structures or isotope shifts. The same physics also determines the uniform spacing of the emission frequencies in an optical frequency comb (discussed in Sec. 1.4.2
).

The reflectivity of the mirrors, which enters the definition of the finesse F, determines the width of the resonances. The transmitted intensity *I _{T}
* is plotted in Fig. B.4a
as a
(p.273)

The finesse of the resonator only depends on the mirror reflectivity *R*, according to eqn (B.14), plotted in Fig. B.4b
.

Finally, we note that the intensity of light *inside* the optical cavity can be much larger than the intensity outside. Performing a similar analysis to the one carried out to derive eqn (B.12), it is possible to demonstrate that for *F »* 1 the average intra‐cavity intensity
^{5}
at resonance is enhanced by a factor on the same order as the finesse:
^{6}

In the presence of a dielectric medium inside the cavity, this enhancement of the intensity allows the investigation of strong nonlinear effects in light‐matter interaction, which we are going to discuss in the next section.

In this section we give a very brief introduction to nonlinear optics. We aim at giving only the essential information on this topic, leaving the interested reader to more specialized textbooks (Boyd, 2008 ).

Nonlinear optics requires the interaction of light with matter. As a matter of fact, the Maxwell equations form a system of partial differential equations which are linear in the electromagnetic fields:

Nonlinearities may come into play when the constitutive relations between the displacement field **D** and **E**, and the magnetic field **H** and **B** are considered:

These relations describe how the fields interact with matter by means of the macroscopic electric polarization **P** and the magnetization **M**. We consider the usual case of a dielectric non‐magnetic material, for which **M** = 0. For the sake of illustration, we assume that the medium polarization **P** has the same direction as the electric field **E** and we write it as a series expansion in the electric field amplitude:
^{7}

for weak fields only the first term is relevant: the polarization is proportional to the field and the linear electric susceptibility x defines the index of refraction of the medium. The following terms in the expansion, described by nonlinear susceptibilities *X*
^{(i≥2)}, become important only for strong fields and are responsible for nonlinear effects that we are going to describe in the following.

We first derive the wave equation for the electromagnetic field propagating in a non‐magnetic dielectric medium characterized by the nonlinear polarization in eqn (B.22) in absence of charge or current field sources *(ρ* = 0 and **J** = 0). From eqns (B.19), (B.20), and (B.22):
(p.275)

where we have separated the polarization into a linear part **P**
*
_{L}
* = ∈

_{0}χ

^{(1)}

**E**and into a nonlinear part

**P**

*NL*= ∈

_{0}χ

^{(2)}

*E*+ ∈

**E**_{0}

*χ*

^{(3)}

*E*+ … containing the remaining terms of the expansion in eqn (B.22). As in the case of linear media, the linear part of the polarization leads to a redefinition of the dielectric constant ∈ = ∈

^{2}**E**_{0}(1 + χ

^{(1)}), which is associated with an index of refraction $\eta =\sqrt{1+\mathrm{\chi}{}^{\text{(}1\text{)}}}$ and a propagation velocity $\upsilon =c/\eta =1/\sqrt{{\mu}_{0}{\epsilon}_{0}(1+\mathrm{\chi}{}^{\text{(}1\text{)}})}$ inside the medium. The left‐hand side of the above equation can be rewritten using the vectorial identity

where we have made the assumption ∇ · E ⋍ 0.
^{8}
Combining Eqs. (B.23) and (B.24) we obtain the nonlinear wave equation

which differs from the ordinary wave equation in a linear medium for the presence of the nonlinear polarization **P**
*NL* which acts as a source term. Since this term depends nonlinearly on the field, it allows the generation of radiation at frequencies which are different from the frequency of the driving field. Below we consider some relevant cases.

*Second‐harmonic and sum/difference frequency generation.* In the presence of a second‐order nonlinearity eqn (B.25) becomes

where, for the sake of illustration, we consider only the amplitude of the electric field, neglecting its polarization.

(p.276)
As a first example, we consider an electromagnetic wave with time dependence *E(t) = E*
_{0}
*cos(ωt)* entering the nonlinear medium. The right‐hand side of eqn (B.26) can be recast as

which represents a source of electromagnetic field oscillating at twice the frequency *ω* of the incident wave. This process, called *second‐harmonic generation* (SHG) allows the conversion of energy from the fundamental beam at frequency *ω* to frequency‐doubled radiation at frequency 2*ω*.

As a second example, we consider two electromagnetic waves *E*
_{1}
*(t) = E*
_{0} cos(*ω*
_{1}
*t*) and *E*
_{2}
*(t) = E*
_{0}
*cos(ω*
_{2}
*t)* with different frequencies. The right‐hand side of eqn (B.26) can now be written as

In addition to the second‐harmonic source terms oscillating at frequencies 2*ω*
_{1} and 2*ω*
_{2}, there are two additional terms at frequencies *ω*
_{1} + *ω*
_{2} and *ω*
_{1} − *ω*
_{2} arising from the nonlinear mixing of the two frequencies. These two processes are called, respectively, *sum‐frequency generation* (SFG) and *difference‐frequency generation* (DFG).

*Phase matching.* The nonlinear effects described above are visualized in a more “quantum‐optical” way in Fig. B.5
by examining the elementary processes involving photons. In SHG two photons with frequency w are annihilated and a photon with frequency *2ω* is created by the interaction with the nonlinear medium. In SFG two photons with frequencies *ω*
_{1} and *ω2* are annihilated and a photon with frequency *ω*
_{1} + *ω*
_{2} is generated. In DFG one photon with frequency *ω*
_{1} is annihilated and two photons, with frequencies *ω*
_{1} − *ω*
_{2} and *102* respectively, are generated.
^{9}
In all these elementary processes energy conservation is satisfied.

For these processes to be efficient momentum conservation has to be satisfied as well. In the case of SHG momentum conservation implies that

While this condition would be automatically satisfied in vacuum, in a dielectric medium it is more difficult to achieve, because of dispersion, which makes the index of refraction (p.277)

*ω)*dependent on the radiation frequency

*ω.*As a matter of fact, the above equation can be recast as a condition on the index of refraction:

This *phase‐matching* condition requires the fundamental wave at frequency *ω* to travel at the same speed as the generated wave at frequency 2*ω*, in order to maintain a well‐defined phase relation between them and achieve constructive interference of the radiation at frequency 2*ω* generated at different positions along the crystal.

We do not discuss the details of the methods to achieve phase matching (see Boyd (2008) for details). We just mention two possibilities. The first one is *birefringent phase matching*, in which a birefringent nonlinear crystal is used and the frequency‐doubled radiation is polarized orthogonally to the radiation at fundamental frequency. In this way, the different indices of refraction of the crystal for the two polarizations are used to achieve the condition in eqn (B.30). A second possibility is quasi *phase matching*, which can be achieved in a periodically poled crystal, i.e. a nonlinear crystal in which the sign of the *χ*(2) coefficient is spatially modulated: although eqn (B.30) is not satisfied (which means that the frequency‐doubled radiation rapidly dephase with respect to the fundamental radiation), the modulation of *χ*(2) periodically inverts the phase of the radiation at 2*ω* in such a way as to mantain, on average, a condition of constructive interference.

When (quasi‐)phase matching is achieved, nonlinear optical processes become quite efficient. With lasers powers of a few hundreds of mW and nonlinear crystals placed in optical cavities to enhance the intracavity power (see Sec. B.2 ), it is possible to achieve SHG with a conversion efficiency of ≈ 50% or even larger.

(p.278)
*Third‐order nonlinearity.* We briefly mention the effect of a third‐order nonlinearity, described by a nonlinear wave equation

If we consider an electromagnetic wave with time dependence *E(t) = E*
_{0}
*cos(ωt)*, the right‐hand side of eqn (B.31) can be written as

While the second term oscillating at 3ω is responsible for third‐harmonic generation, the first term describes a component of the nonlinear polarization oscillating at the same frequency as the driving field. Its effects can be treated as those of the linear component **P**
*
_{L}
* of the atomic polarization and lead to a modification of the index of refraction

*η*by an additional term which is proportional to ${E}_{0}^{2}$, and therefore to the field intensity

*I:*

This dependence of the refraction index on the laser intensity, known as *Kerr effect*, leads to a self‐focusing behaviour of the laser beam and its control is important in the design of high‐power laser cavities and in the operation of many pulsed mode‐locked lasers (used e.g. for the implementation of frequency combs, see Sec. 1.4.2
).

## Notes:

(
^{1}
)
Neglecting ∂ *u _{i}/∂z* over

*2ik∂u*means that the variations of

_{i}/∂z*ui*along the

*z*direction are small on the length scale set by the wavelength λ =

*2*π

*/k.*

(
^{2}
)
Note that this equation has the same mathematical structure as the Schrödinger equation for the wavefunction of a quantum‐mechanical particle moving in the *xy* plane: the time evolution is mapped onto the propagation along the ̂*z* direction.

(
^{3}
)
The angular spread of a wave diffracted by an aperture can be estimated from the uncertainty principle Δ*x*Δ*k*⊥ ~ 1, where Δ*x* is the size of the aperture and Δ*k*⊥ is the transverse wavevector component after the aperture. The divergence angle for an aperture size Δ*x = w*
_{0} can be estimated as *θ* = Δ*k*⊥/*k* ~ *1/kw*
_{0} = *λ/2*π*w*
_{0}, of the same order as the result in eqn (B.11).

(
^{4}
)
As a matter of fact, the phase shift of the field after a round trip of length *2L* in the cavity is *»φ* = *2kL* = 4πυ*L/c* = 2πυ/FSR, which is an integer multiple of *2*π at the resonance frequencies *υ _{n}
* given by eqn (B.16).

(
^{5}
)
The average refers to the fact that inside the optical cavity light propagates both forwards and backwards, which produces a standing wave. The result is given for the spatially averaged intensity.

(
^{6}
)
This intensity enhancement does not violate the conservation of energy! Energy is accumulated inside the cavity during an initial transient time τ = *FL/*π*c* after the laser is switched on (this time is given by the round‐trip time *2L/c* times the average number of round trips per photon *F*/2π*).* At steady state, the energy stored inside the optical cavity does not change and the coupled light is entirely transmitted, which is in perfect agreement with energy conservation. When the laser is switched off, the accumulated energy is then released, again in a time τ.

(
^{7}
)
Here we are deliberately oversimplifying the treatment. More generally, the medium polarization may not be parallel to the applied field and the susceptibilities are tensorial quantities. This is indeed the case for many nonlinear crystals, in which birefringence is important for achieving the phase‐matching condition, but neglecting it does not spoil the main conclusions of the elementary introduction given in this section.

(
^{8}
)
The first Maxwell equation states that V · **D** = 0 in the absence of free charges. In a homogeneous linear medium with dielectric constant ∈ one has **D** = ∈**E**, therefore also ∇ · **E** = 0. However, in a nonlinear medium, the divergence of **E** can be different from zero (also in isotropic media), but it is possible to show that in most of the cases its contribution to eqn (B.24) can be neglected (Boyd, 2008).

(
^{9}
)
As evident from Fig. B.5
, the DFG process has a slightly different nature from SHG and SFG, since one of the input photons is amplified instead of being destroyed. The DFG process can be also interpreted as an *optical parametric amplifier* (OPA) in which a higher‐frequency pump beam at *ω* is used to amplify the *signal* wave at *ω*
_{2}, generating at the same time an *idler* wave at *ω*
_{1} − *ω*
_{2}. In an *optical parametric generator* the process is not stimulated and the conversion of photons at *ω*
_{1} into pairs of photons at *ω _{s}
* and

*ω*(with

_{i}*ω*

_{s}+

*ω*

_{i}=

*ω*

_{1}

*)*occurs spontaneously without any input photon at

*ω2.*