Understanding Coherence in Laser Light Emission

[eoghan]

Hi! I hope this is the right forum. I know how laser works (population inversion, electronic transitions etc...), but I can't understand why the light emitted is coherent

 

[eoghan]

please help! Why laser light is coherent?

 

[Maxim Zh]

There are few modes in the laser resonator (one mode in the best ones) and we can sonsider it as a small set of oscillators. The radiation of such system has large coherence length.

In the spiral of an incandescent lamp almost each atom is an independent oscillator. That's why it has short coherence length.

 

[eoghan]

Thank you!

 

[conway]

People tend to understate the importance of population inversion in explaining how a laser works. I've seen this in a few previous discussions. If you have a normal, thermal distribution of excited atoms in the cavity, the light is not coherent.

 

[tiny-tim]

stimulated emission

I know … population inversion …, but I can't understand why the light emitted is coherent

Hi eoghan!

As you know, …

In physics, specifically statistical mechanics, a population inversion occurs when a system (such as a group of atoms or molecules) exists in state with more members in an excited state than in lower energy states. The concept is of fundamental importance in laser science because the production of a population inversion is a necessary step in the workings of a laser.
…
As described above, a population inversion is required for laser operation, but cannot be achieved in our theoretical group of atoms with two energy-levels when they are in thermal equilibrium.
…
To achieve non-equilibrium conditions, an indirect method of populating the excited state must be used.​

(I got that from http://en.wikipedia.org/wiki/Population_inversion" )

But coherence requires more, it requires an external electromagnetic field (ie a photon) which effectively converts the atom into a dipole oscillator resonating in phase (and direction) with the field.

As a result, any electron produced from that atom will also be in phase.

See http://en.wikipedia.org/wiki/Stimulated_emission" …

When an electron decays without external influence it is said to be due to "spontaneous emission." The phase associated with the photon that is emitted is random.
…
An external electromagnetic field will affect an atom's state - changing the quantum mechanical variables mentioned above. Specifically, the atom will act like a small electric dipole which will oscillate with the external field. One of the consequences of this oscillation is that it encourages electrons to decay to the lowest energy state. When this happens due to the presence of the electromagnetic field from a photon, a photon is released in the same phase and direction as the "stimulating" photon …

 

[Cthugha]

First of all one should note that there are two concepts of coherence: The classical concept as defined via coherence time and the quantum optics concept of higher order coherence, where the n-fold joint detection rate of the light needs to factorize into the n single detection rates in order to be nth order coherent.

People tend to understate the importance of population inversion in explaining how a laser works. I've seen this in a few previous discussions. If you have a normal, thermal distribution of excited atoms in the cavity, the light is not coherent.

A laser needs inversion in order to prevent stimulated absorption, but it is not necessarily a precondition for a buildup of coherence - for example coherent outcoupling from condensates or the so-called lasers without inversion, where reabsorption is forbidden by means of destructive interference can produce coherent light, too. Normal lasers, however, of course need inversion.

As described above, a population inversion is required for laser operation, but cannot be achieved in our theoretical group of atoms with two energy-levels when they are in thermal equilibrium.
…
To achieve non-equilibrium conditions, an indirect method of populating the excited state must be used.

But coherence requires more, it requires an external electromagnetic field (ie a photon) which effectively converts the atom into a dipole oscillator resonating in phase (and direction) with the field.

As a result, any electron produced from that atom will also be in phase.

In fact this is not a case of more requirements, but rather less requirements. The in-phase emission as seen in stimulated emission defines coherence time. As an easy picture one can imagine that the incoming photon and the stimulated-emission photon become indistinguishable and you can not tell, which was the incoming photon and which is the emitted photon. Repeat this over several stimulated emission events and you will have an extremely large uncertainty about when one single photon inside your cavity was emitted - it could have been the first photon inside, but it could also have been emitted milliseconds later. This uncertainty timespan basically is your coherence time. Unfortunately, you can also get long coherence times by just taking some sunlight and passing it through an extremely narrow bandwidth filter. If it becomes spectrally just as narrow as your lasing mode, it will also have the same coherence time.
However, the other features (are the emitters in thermal equilibrium or out of equilibrium) will also show some effects. Basically the underlying photon number distribution will change. Thermal light sources follow a Bose-Einstein distribution, while coherent light source follow the distribution of statistically independent events and are therefore Poisson-distributed. However, these statistical effects will of course only show up inside the coherence time.

This is why the in-phase emission is not a good measure of whether light is coherent. All kinds of light will have some coherence time. It might be in the fs range (sunlight) or in the second range for lasers or number-squeezed light. Drawing a line somewhere and saying "light is coherent if it has a minimal coherence time of XYZ" would be rather arbitrary and not satisfying.

 

[tiny-tim]

Hi Cthugha!

In fact this is not a case of more requirements, but rather less requirements.

Not following you.

The in-phase emission as seen in stimulated emission defines coherence time.

What's coherence time?

What does it have to do with why the emitted photon is coherent?

In particular …

All kinds of light will have some coherence time. It might be in the fs range (sunlight) or in the second range for lasers or number-squeezed light.

… are you saying that sunlight is coherent?

 

[jambaugh]

As I've come to understand it...
The essential feature of the laser is the process of stimulated emission which is a phenomenon dependent on the photons' Bose-Einstein statistics. Just as fermions satisfy the exclusion principle due to their statistics, bosons follow a sort of "inclusion principle" preferring to manifest in in a common mode and that is the source for the stimulated emission effect.

 

[Maxim Zh]

… are you saying that sunlight is coherent?

There is no absolutely incoherent light.

The concepts of temporal and spatial coherence ratio are more correct.
There are also concepts of coherence time and length which are derived from correlation functions.

The coherence time for sunlight is approximately 10^-15 s

See also:
O. Svelto, Principles of Lasers, section 7.5

 

[Cthugha]

What's coherence time?

Ok, let us imagine you have completely monochromatic light. If you know its phase at some position now, you will be able to predict its phase at the same position in every possible instant of time. For real light sources, the phase will not be completely fixed, but you will lose the ability to predict the phase due to some random processes occurring on some timescale. The timescale over which the phase information is lost, is the coherence time. Basically this is what you measure with a Michelson interferometer.

What does it have to do with why the emitted photon is coherent?

In particular …

… are you saying that sunlight is coherent?

Yes, I was just stressing that according to this classical definition all kinds of light are coherent - just on different timescales. Therefore there is a second quantum optical criterion of coherence. This is what Roy Glauber got his Nobel prize for.

The essential feature of the laser is the process of stimulated emission which is a phenomenon dependent on the photons' Bose-Einstein statistics. Just as fermions satisfy the exclusion principle due to their statistics, bosons follow a sort of "inclusion principle" preferring to manifest in in a common mode and that is the source for the stimulated emission effect.

This is a rather difficult topic as the term statistics is used in several ways and the meaning of preferring to manifest in a common mode can be easily misunderstood. While photons follow bosonic statistics, the photon number does NOT follow a Bose-Einstein distribution in the case of a laser. In fact it is the suppression of the STATISTICAL (this emphasis is important) bosonic tendency to manifest in a common mode called Hanbury-Brown-Twiss effect or photon bunching, which marks the threshold between spontaneous emission and lasing although that seems counterintuitive as there indeed is a macroscopic population of bosons present in some state.

 

[tiny-tim]

If you know its phase …

Isn't this begging the question … surely an incoherent beam of photons doesn't have "a" phase?

To put it another way, aren't you talking about the coherence of an individual photon … but this thread is about coherence between two photons?

 

[Cthugha]

Isn't this begging the question … surely an incoherent beam of photons doesn't have "a" phase?

To put it another way, aren't you talking about the coherence of an individual photon … but this thread is about coherence between two photons?

First, this is awkward terminology. The underlying fields have a phase, but photons do not. Therefore it is always a bit of messy terminology to talk about coherence between photons. Accordingly also the concepts of "coherence of a single photon" and "coherence between different photons" are not defined - or at least not in the way you think.

Theoretically a completely incoherent state is a state with a fixed photon number and (as connected via phase-photon number uncertainty) completely undefined phase. Having such a light source would be the dream of every experimentalist in the realm of quantum optics as it would allow to prepare Fock states on demand. However, Fock states - eigenstates of the photon number operator - are the only states having this signature, while every other light source - from sunlight (although it is often called incoherent in a wrong fashion) to lasers does not have this property. If you do not have stimulated emission as the emission process, this does not mean that the light is incoherent. Stimulated emission usually leads to an extremely narrow linewidth because the emission process favors one single mode. Coherence time and spectral line width are inversely proportional to each other (narrow linewidth leads to long coherence time) and accordingly stimulated emission is sometimes considered as building up coherence. However, you can just as well put sunlight through a narrow filter and get the same spectral linewidth and coherence time.

 

[DrDu]

I thought that a Fock state has still full first order coherence, but no higher one.

 

[Cthugha]

If I remember correctly, a state composed as a superposition of two Fock states can show oscillatory first order coherence, but a theoretically ideal single Fock state necessarily has a well defined photon number and undefined phase and therefore vanishing first order coherence time.

 

[DrDu]

The two point correlation function is <E^*(x_2)E(x_1)>. I don't see that it vanishes for a Fock state (one mode), as E propto a, the boson anihilator, so that it becomes proportional to the particle number operator. Even the higher order Fock states <E^*(x_2)E(x_1)>(up to order n for a n-particle state) are non-zero in general. However, they do not factorize into Prod_j e^(*)(x_2j)e(x_1j) where e is a function, not an operator and j is the order of the correlation function.

 

[Cthugha]

Well, where should the coherence come from?

You can express the first order correlation function as
$$G^{(1)}=\langle E_1^*(r_1,t_1) E_2(r_2,t_2)\rangle=\langle E_1^*(r_1,t_1)\rangle \langle E_2(r_2,t_2)\rangle +\langle \delta E_1^*(r_1,t_1) \delta E_2(r_2,t_2) \rangle$$

The two terms describing the averages of the electrical field vanish because the electric field operator is proportional to $$a^{\dagger}-a$$, which leads to terms proportional to <n|n-1> and <n|n+1>, which are all 0 because of the orthogonality of the Fock states.

The third term is trivially nonzero for $$r_1=r_2, t_1=t_2$$.
In all other cases the fields at the respective times need to be correlated to give nonvanishing results. As the phase of photon number states is completely random, this cannot be the case for different times. It might be possible for different positions at the same time for a symmetric emitter. However, if that was the case, it would mean the photon carries basically no net momentum and accordingly you cannot have a free and isolated Fock state, but one, which is still entangled with its emitter so that the whole system conserves momentum.

In real systems you will see some short coherence times for (almost) Fock states, which are maybe in the fs range I suppose, but that is more or less a signature of the impossibility to create true isolated Fock states.

 

[DrDu]

Usually in the definition of the correlation functions one splits E into positive and negative frequency parts E=E^(+)+E^(-), which are hermitian adjoints. Then E^(-) depends only on the a and E^(+) only on a^+. The expression for the correlation function is normal ordered <E^(+)(x_2)E^(-)(x_1)> so that it has also a nonvanishing expectation value in a Fock state. A characteristic of Fock states is the so called "anti-bunching" which obviously is a correlation of the detection times of the photons. So a Fock state cannot be void of correlations.

 

[Cthugha]

Usually in the definition of the correlation functions one splits E into positive and negative frequency parts E=E^(+)+E^(-), which are hermitian adjoints. Then E^(-) depends only on the a and E^(+) only on a^+. The expression for the correlation function is normal ordered <E^(+)(x_2)E^(-)(x_1)> so that it has also a nonvanishing expectation value in a Fock state. A characteristic of Fock states is the so called "anti-bunching" which obviously is a correlation of the detection times of the photons. So a Fock state cannot be void of correlations.

Antibunching is seen in the 4-point field correlation (or second-order intensity correlation) function. The normal ordering ensures that the simple fact that the detection of a photon changes the light field is considered. Therefore the nonclassical correlations occur - the photon is gone.

The first order correlation function is just the classical correlation fuction of the fields. As there are no processes involved, which destroy a photon, it cannot show the nonclassical correlations seen in $$g^{(2)}$$. In fact no nonclassical features can ever be seen in $$g^{(1)}$$. See for example the Mandel/Wolf for more details on why that is so.

 

[DrDu]

I fully agree with what you say in post #19. However, the expression you use in post #17 is not the correct one, once you consider nonclassical light. Then you are forced to use the more general normal ordered version. Then you find that also a Fock state is fully coherent in first order.
What distinguishes a laser is that it is fully coherent in any order in the sense that the correlation function can be factored into equal factors e and e^*. A Fock state is fully coherent only in first order, although all correlation functions up to order n (the particle number) are different from 0.
I just freshed up my knowledge from "U. Titulaer and R. Glauber, Density operators for Coherent Fields", Phys Rev, Vol 145, (1966), pp. 1041". A nice read is also the Nobel lecture of Roy Glauber: http://nobelprize.org/nobel_prizes/physics/laureates/2005/glauber-lecture.html

Btw, I just noticed that in my post #18, the "+" and "-" signs on the operators E^(+) and E^(-) are reversed.

 

[conway]

I don't want to interrupt what seems like a very intelligent discussion about the nature of coherence, but I can't help feeling that we are ignoring its relevance to the operation of a laser. So I am going to propose a partial explanation.

I believe that the phase of an atomic oscillator is essentially random with respect to the phase of the ambient field. This random phase can be resolved into two components: the purely in-phase and the quadrature component. The quadrature component is what people commonly call "spontaneous emmision" and the in-phase component is the stimulated emmission (or absorption, according to its sign).

The rate of energy output of the quadrature component is independent of the ambient field. However the power output of the in-phase component, for reasons which can be explained via classical electromagnetics, is actually proportional to the ambient field. So in a very strong ambient field, in-phase power is sucked out of (or pumped into) the atom at such a high rate that the transition is essentially over and done before there has been enough time for significant power to be emitted in quadrature.

It is also true classically that the in-phase component will tend to re-inforce (or subtract from) the ambient field in terms of phase and direction.

This partial explanation does not explain the sign of the stimulated transition, but it does explain the phase/direction coherence.

 

[Cthugha]

What distinguishes a laser is that it is fully coherent in any order in the sense that the correlation function can be factored into equal factors e and e^*. A Fock state is fully coherent only in first order, although all correlation functions up to order n (the particle number) are different from 0.

Yes, indeed. A Fock state will be fully first order coherent in the sense that the correlation function factorizes (btw, if you have a look at the expression without normal-ordering it does in fact factorize, too, exactly because there are no field correlations). However this thread started from discussing lasers and (for the sake of easier understanding at least my arguments were based on) coherence time/length in the sense of seeing interference in interferometric experiments. Although the correlation function factorizes you will not see the interference pattern you would see using laser light. I agree that the definition by Glauber is the more fundamental one - but it is not exactly easy to understand and demonstrate all its meanings, especially not when discussing lasers in an online forum. I mean - understanding what the factorized terms mean is easy for a laser, where you get that $$\epsilon$$ will just be the expectation value of $$E^+$$, but it is not that trivial for a Fock state. But maybe we should really transfer this discussion into a different thread, where it is more suitable.

This partial explanation does not explain the sign of the stimulated transition, but it does explain the phase/direction coherence.

It should be sufficient as an easy picture as long as one keeps in mind that it should not be possible explaining more difficult features like superradiance or the Purcell effect with it.

 

[DrDu]

Ok, agreed. I would be really interested in your argument why a Fock state will yield a different interference pattern than a Glauber state. You are right that we should discuss this probably in a new thread.

 

[ZapperZ]

Can we get back to the OP's question and not hijack this thread into something else?

Zz.