Simple definition of "natural linewidth"
Samuel M. Goldwasser
What I have so far is as follows. My apologies if this has mangled
some things in the process, but there don't seem to be coherent (no pun...)
explanation available of natural linewidth, without a lot of hairy math.
And even finding explanations with hairy math is not easy! :)
The context for this is in describing the origin of the Lamb dip in a
HeNe laser, thus referring only to excited atoms in the following:
"Greatly simplified, the natural line width is what would be present if all
the atoms in the upper (laser) energy state were in an identical environment
(no Doppler or other broadening) and emitted their
photons spontaneously (no laser cavity). Classically, it takes the form
of a damped harmonic oscillator; quantum-mechanically, it is
determined by the uncertainty principle as the reciprocal of the
radiating (upper laser energy state) lifetime of the atom. Or, put
another way, it is the bandwidth of the damped oscillation
that would result if atoms present in the upper energy state were allowed
to decay via spontaneous emission to the lower energy state
without being replenished."
If there is anything one can say about the Lamb dip, that would also
be welcome.
I have the following to obtain a significant Lamb dip:
1. The natural linewidth should be much narrower than the gain bandwidth.
2. The center of the Doppler-broadened neon gain curve should correspond
to zero velocity (not just an average), meaning isotopically pure
gases (or at lesat neon) for the HeNe laser.
3. The gain should be fully saturated in at least part of the mode volume.
4. The laser should use a standing wave resonator (e.g., Fabry-Perot).
5. The laser should run with a single longitudinal mode.
I'd really appreciate Professor Siegman's comments on any of this!
If this is in your book, my further apologies as I don't have acces
to a copy at present.
Thanks!
--
sam | Sci.Electronics.Repair FAQ: http://www.repairfaq.org/
Repair | Main Table of Contents: http://www.repairfaq.org/REPAIR/
+Lasers | Sam's Laser FAQ: http://www.repairfaq.org/sam/lasersam.htm
| Mirror Sites: http://www.repairfaq.org/REPAIR/F_mirror.html
Important: Anything sent to the email address in the message header above is
ignored unless my full name AND either lasers or electronics is included in the
subject line. Or, you can contact me via the Feedback Form in the FAQs.
Bret Cannon
"Samuel M. Goldwasser" <s...@repairfaq.org> wrote in message
news:d4bu4g...@repairfaq.org...
The natural linewidth of a transition is determined by the lifetimes of both
the upper and lower levels. For the 633 nm HeNe transition, the radiative
lifetime of the lower level is much shorter than the lifetime of the upper
level and so dominates the natural linewidth of that transition.
Point 1. above should say that the homogenous linewidth needs to be much
narrower than the gain bandwidth. For a HeNe laser in particular,
collisions increase the homogenous linewidth from the natural linewidth to a
larger value that is proportional to pressure. At about 10 MHz/torr,
pressure broadening is significant in at typical He-Ne fill pressures.
Another source of homogeneous broadening is transit time broadening, which
can occur when the mean free path of molecules or atoms is larger than the
laser beam diameter and the average time to cross the laser beam is shorter
than the lifetime of the upper lasing level. With a beam diameter of a
fraction of a millimeter and an average velocity of several hundred meters
per second, the crossing times can be about 1 microsecond. This lifetime
broadening is probably not a big effect in HeNe lasers, but I couldn't
quickly find a value for the upper state lifetime. For a cavity that gives
a small beam diameter and operating a low enough pressures, this could be
significant.
Bret Cannon
Samuel M. Goldwasser
> The natural linewidth of a transition is determined by the lifetimes of both
> the upper and lower levels. For the 633 nm HeNe transition, the radiative
> lifetime of the lower level is much shorter than the lifetime of the upper
> level and so dominates the natural linewidth of that transition.
Does anyone have numbers? :)
>
> Point 1. above should say that the homogenous linewidth needs to be much
> narrower than the gain bandwidth. For a HeNe laser in particular,
> collisions increase the homogenous linewidth from the natural linewidth to a
> larger value that is proportional to pressure. At about 10 MHz/torr,
> pressure broadening is significant in at typical He-Ne fill pressures.
OK, so how about this for a rewording?:
1. The homogeneous line width of the lasing process must be much narrower
than the gain bandwidth of the lasing medium. The homogeneous line width
results from homogeneous broadening of the natural linewidth,
which is what would be present if all the atoms
were in an identical environment (no Doppler or other broadening)
and emitted their photons spontaneously (no laser cavity). In a gas
laser, collisions between atoms and other effects result in homogeneous
broadening, while the Doppler effect of the velocity distribution of
atoms is the dominent contribution to inhomogeneous broadening.
> Another source of homogeneous broadening is transit time broadening, which
> can occur when the mean free path of molecules or atoms is larger than the
> laser beam diameter and the average time to cross the laser beam is shorter
> than the lifetime of the upper lasing level. With a beam diameter of a
> fraction of a millimeter and an average velocity of several hundred meters
> per second, the crossing times can be about 1 microsecond. This lifetime
> broadening is probably not a big effect in HeNe lasers, but I couldn't
> quickly find a value for the upper state lifetime. For a cavity that gives
> a small beam diameter and operating a low enough pressures, this could be
> significant.
Need numbers!!!!! :)
Thanks and thanks!
Lostgallifreyan
news:d4bu4g...@repairfaq.org:
> If there is anything one can say about the Lamb dip, that would also
> be welcome.
>
Mint sauce. Works for me. Sorry. >:)
confused2
I will copy what Prof Siegman wrote about Lamb dip on page 1199 of
chapter 30: Hole Burning and Saturation Spectroscpoy. Hopefully he
won't mind and I interpret it correctly... It's very late here. At the
beginning of the chapter he starts with saturation of homogenous media
and notes that sauration at a particular frequency within the atomic
linewidth results in an reduction or shrinkage everywhere of both the
real and imaginary susceptibilty curves. He emphasises this with "a
strong saturating signal even well out in the wing of the atomic
transition will, if strong enough, saturate the entire transition
uniformily across its lineshape." So that hole burning cannot exist
in homogenous media.
In section 30.6 Inhomogenous Laser oscilation: Lambs dips
"In an early and widely read analysis of the gas lasers, Willis Lamb
predicted, and experimenter soon confirmed an unexpected aspect of
doppler-broadenened gas laser oscillation. If we tune the resonance
frequency of a single oscillating cavity mode across a doppler-
broadened gas laser transition, The curve of oscilation power output
versus cavity frequency shows a comparatively sharp and narrow dip in
oouput power when hte oscilation frequency coincides with the centre
of the doppler b raodened line." He notes that Lambdip only occurs in
standing wave cavities and "is a consequence of saturation and whole
burning effects in the doppler broadened line, caused by two
oppositely travelling waves in the cavity."
"Physical explanation
The signla field inside a standing wave cavity can be divided into two
oppositely directed travelling waves which we have referred to as +z
and -z waves. Any single atom with axial velocity v thus sees two
opposing travelling waves, for which it has equal and opposite doppler
shifts, even thought he two waves are at the same frequency. This
leads to double whole burning effects " and thus lamb dip.
"Consider a laser with an inhogenous doppler braodened tranisition,
oscillating in a single frquency standing-wave axial mode resonance,
with the frequency w of this resonance detuned from the atomic line
centre by several inhomogeneous linewidths or whole widths.... the
travelling +z wave component of the standing wave cavity fields will
interact with and burn a hole in only those atoms in the velocity
class given by v/c-w0-w/w0; while at the same time the fields in the -
z travelling wave component will burn an equal and opposite hole in
the symmetrically located velocity class at opposite value of v/c.
Whenever the cavity frequency is well away from line centre on either
side, therefore, two symmetric holes are burned, and in essence the
laser is able to extract power from two seperated set of atoms or
velocity classes in the atomic velocity distributi8on.
If, however, the cavity frequency is tuned exactly to the line centre,
both the +z and _z waves can interact the with v=0 velocity class in
the doppler distribution. This velocity class is therfore saturated
twice as heavily as either of the seperate velocity classes in the off-
resonance situaiton, because it sees two signal instead of one. But
this means that the laser need only oscillate roughly half as hard to
produce the same degree of saturation needed to reduce the gain to
equal the cavity losses. In essence the two symettric holes coelesce
into one, and the laser power is taken from the single velocity classs
v=0. In the inhomogenously broadened single frequency laser this
results in the slight, but definited dip in laser power at the line
centre known as Lamb Dip."
Prehaps the Prof. can give a definition of velocity class, as I can't
see one in the book right now... is it some differentially small range
of velocity or finite?
Hope that helps I have copied it as careful;ly as possible..
Alex
PS there must be an electronic version of his book to save carrying
the tomb around, as I have managed to view a long excerpt from it on
the net, but with pages missing here and there. Will find the url
sometime
Samuel M. Goldwasser
> Dear Sam,
This is exactly what I am interested. I hope Prof. Siegman won't mind
me further copying this into the Laser FAQ!
Thanks so much for taking the time!!! :)
AES
<8063fca8-9529-44e8...@z9g2000yqi.googlegroups.com>,
confused2 <aesb...@hotmail.com> wrote:
> Dear Sam,
>
> I will copy what Prof Siegman wrote about Lamb dip on page 1199 of
> chapter 30: Hole Burning and Saturation Spectroscpoy. Hopefully he
> won't mind and I interpret it correctly... It's very late here. At the
Don't really mind, from my point of view as author -- but seems to me
the newsgroup could get a bit clogged if people started doing this in
lots of cases, from lots of books . . .
> Prehaps the Prof. can give a definition of velocity class, as I can't
> see one in the book right now... is it some differentially small range
> of velocity or finite?
"Velocity class" is jargon for a small range of velocities (never
thought about why the term "class" was used, but that was the usual
terminology from the beginning). As usual when considering continuous
distributions, "small" is not precisely defined, but is to be taken as
meaning a range small enough that all the atoms within it act more or
less the same -- in other words, somewhat smaller than the velocity
spread that would create a doppler frequency shift larger than the
atomic linewidth of those atoms. You can start off thinking of a
discrete number of velocity classes, which you sum over; then make these
classes narrower and more numerous, until you're really integrating
rather than summing, in which case each velocity class takes on in fact
a differentially small range.
> PS there must be an electronic version of his book to save carrying
> the tomb around, as I have managed to view a long excerpt from it on
> the net, but with pages missing here and there. Will find the url
> sometime
Haven't seen that myself; maybe it's what Google has been up to.
AES
s...@repairfaq.org (Samuel M. Goldwasser) wrote:
>
> This is exactly what I am interested. I hope Prof. Siegman won't mind
> me further copying this into the Laser FAQ!
Nope.
> Thanks so much for taking the time!!! :)
Didn't respond initially myself, because busy with other things here.
--AES
Samuel M. Goldwasser
:)
BTW, while it says "standing wave cavity", wouldn't a bidirection
ring such as found in a ring laser gyro satisfy the requirement
as well if the other conditions were met? Perhaps the Lamb Dip
would disappear if the angular velocity were high enough?
AES
s...@repairfaq.org (Samuel M. Goldwasser) wrote:
> BTW, while it says "standing wave cavity", wouldn't a bidirection
> ring such as found in a ring laser gyro satisfy the requirement
> as well if the other conditions were met? Perhaps the Lamb Dip
> would disappear if the angular velocity were high enough?
No. (And this is an important point.)
A traveling wave whose frequency f1 is tuned off the center frequency f0
of a doppler broadened atomic line burns one off-center hole in the
velocity distribution, in a certain velocity class centered at a
velocity v1. An oppositely traveling wave at the same frequency f1
burns a hole in the -v1 velocity class. A standing wave at frequency f1
consists of both of these traveling waves, and burns _both_ holes
simultaneously. As f1 moves into to the line center the two holes merge
into just one hole, located at v1 = -v1 = 0. It's that behavior that is
basically responsible for the Lamb dip.
Samuel M. Goldwasser
Are you saying that a bidirectional ring cavity with both traveling waves
of similar amplitude can't have a Lamb Dip, or something else? I'm not
sure what the "No" applies to! :) And if this is the case, how are
two equal and opposite traveling waves at line center different than
a standing wave at line center?
AES
s...@repairfaq.org (Samuel M. Goldwasser) wrote:
> AES <sie...@stanford.edu> writes:
>
> > In article <7i1z16...@repairfaq.org>,
> > s...@repairfaq.org (Samuel M. Goldwasser) wrote:
> >
> > > BTW, while it says "standing wave cavity", wouldn't a bidirection
> > > ring such as found in a ring laser gyro satisfy the requirement
> > > as well if the other conditions were met? Perhaps the Lamb Dip
> > > would disappear if the angular velocity were high enough?
> >
> > No. (And this is an important point.)
> >
> > A traveling wave whose frequency f1 is tuned off the center frequency f0
> > of a doppler broadened atomic line burns one off-center hole in the
> > velocity distribution, in a certain velocity class centered at a
> > velocity v1. An oppositely traveling wave at the same frequency f1
> > burns a hole in the -v1 velocity class. A standing wave at frequency f1
> > consists of both of these traveling waves, and burns _both_ holes
> > simultaneously. As f1 moves into to the line center the two holes merge
> > into just one hole, located at v1 = -v1 = 0. It's that behavior that is
> > basically responsible for the Lamb dip.
>
> Are you saying that a bidirectional ring cavity with both traveling waves
> of similar amplitude can't have a Lamb Dip, or something else? I'm not
> sure what the "No" applies to! :) And if this is the case, how are
> two equal and opposite traveling waves at line center different than
> a standing wave at line center?
In a ring laser cavity, assuming only negligible backscattering effects
in the cavity, the CW and CCW waves are independent (that is, they exist
independently).
In an inhomogeneously broadened ring-cavity gas laser with the cavity
resonance frequency tuned off center, the CW and CCW modes draw their
energy from separate velocity classes at +v1 and -v1, and thus can (and
often do) oscillate quite independently. That's how ring laser gyros
work.
In a homogeneously broadened ring-cavity laser (e.g., YAG), the CW and
CCW modes both draw their energy from the entire line (or at least try
to), and thus compete for gain, or strongly cross-saturate each other.
As a result, the laser can generally run in one direction only, or jump
randomly between directions -- unless you add some nonreciprocal element
that favors one direction over the other.
Samuel M. Goldwasser
Or is supposed to work!
So, this would imply that there *can* be a Lamb Dip in a HeNe ring laser gyro
or similar inhomogeneously broadened laser as the (now two) cavity modes
approach line center (assuming the other required conditions are met) since
they would then be drawing on the same velocity class.
Bopefully this isn't getting totally confusing. :)
> In a homogeneously broadened ring-cavity laser (e.g., YAG), the CW and
> CCW modes both draw their energy from the entire line (or at least try
> to), and thus compete for gain, or strongly cross-saturate each other.
> As a result, the laser can generally run in one direction only, or jump
> randomly between directions -- unless you add some nonreciprocal element
> that favors one direction over the other.
No problem there since it's homogeneously broadened.
Thanks!