October 19th, 2023

APPLICATION NOTE

LINEWIDTH MEASUREMENT OF DIODE LASERS
Dr. Steffen Schmidt-Eberle, Application Scientist Quantum Technologies, TOPTICA Photonics AGThe spectral purity of a laser is one of its core features. It is however non-trivial to find quantities which fully
characterize this spectral purity. In this paper we discuss two linewidth definitions which TOPTICA uses tocharacterize the frequency stability of its diode lasers. These quantities reflect the two conceptual sources of phaseor frequency noise in diode lasers.

SOURCES OF PHASE AND FREQUENCY NOISETWO DEFINITIONS OF LASER LINEWIDTH

The fundamental limit for the linewidth of any laser isgiven by the Shawlow-Townes [1] expression which
describes the impact of phase changes due tospontaneous emission on the laser linewidth. For
diode lasers this limit depends e.g. on the laser´sresonator geometry, properties of the semiconductor
and laser power. This limit manifests itself as a whitefrequency noise plateau and will therefore be called
white noise limit in the following.1 Because this noise
depends on fundamental properties of the laser it isreferred to as intrinsic noise.
Any frequency noise which adds to the intrinsic noise
is summarized as technical noise. Most generally,
these are contributions introduced from theenvironment like vibrations or acoustic noise and on
longer time scales temperature and air pressure
drifts. For this type of noise, the technical realization
concerning mechanical stability and isolation from theenvironment is key to low noise.

The difference between intrinsic and technical noise
becomes more apparent when considering the power
spectral density (PSD) of the laser´s frequency noise[2]. The PSD of the frequency noise is the absolutesquare of the Fourier transform of the timedependent frequency deviation 𝜈̃(𝑡) of the laser fromits central frequency 𝜈0 . It contains all typically
relevant spectral properties of the laser2. Figure 1
shows an example of the frequency PSD of a DL prowith narrow linewidth option at 725 nm. The
frequency PSD is largest for small Fourier
frequencies and decreases until it converges into awhite noise plateau at higher frequencies. This
plateau corresponds to the white noise limit caused
by intrinsic noise, while the noise at lower
frequencies which exceeds this plateau corresponds
to technical noise. In the following we will present twolinewidth definitions which represent the two different
types of noise.

1enhancement factor 𝛼 [4] and the white noise limit is increased bya factor of (1 + 𝛼 2 ).
2
Of course, the absolute square of the Fourier transform neglects
the phase spectrum of the signal. For most applications however,this phase spectrum is not of importance.

For semiconductor lasers this limit is further increased by acoupling between intensity and phase noise, caused by adependence of the refractive index on the carrier density in thesemiconductor [2], first explained by Charles H. Henry [3]. The
strength of this coupling is described by the linewidth

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October 19th, 2023

APPLICATION NOTE

specifications which can be found on TOPTICA´swebsite and the laser diode stock lists. 4
Second, we define the beta separation linewidth,
which is based on the beta separation line introducedby Di Domenico et al. [5]Δ𝜈𝛽 = √8 ln (2)𝐴,Fig. 1: Power spectral density (PSD) of the laser´sfrequency noise (blue curve). The black, horizontal
line indicates the white noise level caused by intrinsic
noise. The orange line is the beta separation line.
Only components of the frequency PSD which lie
above this line contribute to the beta separation
linewidth. The red line is the beta separation linewidth
as a function of the lower integration limit.

First, we define the instantaneous linewidth, whichmainly describes the intrinsic noise evident at highFourier frequencies

Δ𝜈𝑖𝑛𝑠𝑡 = 𝜋 ⋅ 𝑃𝑆𝐷𝑤ℎ𝑖𝑡𝑒 ,(1)where 𝑃𝑆𝐷𝑤ℎ𝑖𝑡𝑒 is the white noise level of the PSD inHz²/Hz. 𝑃𝑆𝐷𝑤ℎ𝑖𝑡𝑒 is obtained by fitting to the dataa power law function of the form 𝑦(𝑓) = 𝑎𝑓 𝑏 + 𝑐,
where the fitting constant 𝑐 is interpreted as
𝑃𝑆𝐷𝑤ℎ𝑖𝑡𝑒 3. Empirically we see that most spectra
are approximated well by this type of fitting
function. Δ𝜈𝑖𝑛𝑠𝑡 is the full width at half maximum(FWHM) linewidth of a hypothetical laser with afrequency PSD given by 𝑃𝑆𝐷𝜈̃ (𝑓) = 𝑃𝑆𝐷𝑤ℎ𝑖𝑡𝑒 for all𝑓. Such a laser yields a Lorentzian line shape. Thisdefinition is TOPTICA´s core definition of the laser
linewidth and is referred to in laser and diode3

The value of Δ𝜈𝑖𝑛𝑠𝑡 that we obtain in this way has to beconsidered as a typical value and not as a specification, as theexact value depends on many parameters like e.g. diode current,
temperature, wavelength, etc.
4

For some diodes and lasers yet another linewidth definition isused which will be discussed in the section “Comparison to otherlinewidth definitions”. This definition is still used for diodes for(2)where 𝐴 is the surface under the portions of 𝑃𝑆𝐷𝜈̃ (𝑓)that exceed the beta separation line [5]∞8 ln(2) 𝑓
𝐴 = ∫ 𝐻 (𝑃𝑆𝐷𝜈̃ (𝑓) −) 𝑃𝑆𝐷𝜈̃ (𝑓)𝑑𝑓,(3)𝜋2𝑓0with 𝐻(𝑥) being the Heaviside unit step function(𝐻(𝑥) = 1 if 𝑥 ≥ 0 and 𝐻(𝑥) = 0 if 𝑥 < 0) and 𝑓0 beinga lower integration limit. This quantity is a FWHM
linewidth which approximates the linewidth of theactual power spectrum of the laser´s electric field withan integration time of 𝑇0 = 1/𝑓0. For many diodes
TOPTICA can provide the beta separation linewidth
upon request for various lower integration limits. For
the example in Fig. 1 only noise below approximately
20 kHz contributes to this FWHM linewidth, while
noise above this threshold contributes to the pedestalof the line profile only.
Thus, while the instantaneous linewidth is a measurefor frequency stability at high Fourier frequencies, thebeta separation linewidth describes the frequency
stability at low Fourier frequencies. Therefore, in thecontext of laser locking, the instantaneous linewidth
is connected to the residual noise after locking
because locking bandwidths are typically limited toseveral MHz. The beta separation linewidth on theother hand indicates how well the laser can bestabilized within the locking bandwidth. It is generally
true that the quality of the locked laser depends on
the level of instability of the unlocked laser. 5 Also if
the beta separation linewidth is on the order ofseveral MHz, an actuator capable of changing thelaser frequency by several MHz is required forfrequency stabilization.

which no measurement of the frequency PSD has been performedyet.
5Of course, the linewidth of the stabilized laser depends on manyother properties of the lock setup, like the reference and locking
scheme, the bandwidth of the PID controller, etc. A detailed
introduction to laser locking can be found athttps://www.toptica.com/application-notes/phase-and-frequencylocking-of-diode-lasers/.

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APPLICATION NOTE
Another interesting quantity is the frequency at which
the frequency PSD crosses the beta separation line.
In order to achieve significant linewidth reduction afeedback loop with a bandwidth larger than thiscrossing frequency is required. TOPTICA can also
provide this number upon request.

MEASUREMENT OF THE POWER SPECTRALDENSITY
The most straightforward way to obtain the PSD of
the frequency noise is to overlap two lasers withidentical noise properties at slightly different central
frequencies and record the beat signal as a functionof time using a photodiode.6 From this signal thetime-dependent relative frequency fluctuations
between the two lasers can be obtained. The
absolute square of the Fourier transform of the timedependent frequency fluctuations is the PSD of therelative frequency noise between the two lasers. 7
However, in most cases only one laser is available.
Therefore, TOPTICA measures the PSD of thefrequency noise using a delayed self-heterodyne
detection setup which is sketched in Fig. 2. Thismethod was used to obtain the blue curve in Fig. 1.
Laser light is separated into two parts of similar
power using a non-polarizing beam splitter (NPBS).
One part is delayed using a 20 m long fiber, whichcorresponds to a delay of 100 ns. The other part isfrequency shifted by 40 MHz using an acousto-optic
modulator (AOM). Both beams are recombined using
a second NPBS. The beat signal at 40 MHz isrecorded using a photodiode (PD) driven by a low
noise power supply.

October 19th, 2023

frequency PSD of diode lasers. It is based on an
asymmetric interferometer. The optical path length
difference between the arms is 20 m. This asymmetry
is used to transform frequency fluctuations of thelaser into phase fluctuations.
Instead of evaluating the signal in the frequency
domain using a spectrum analyzer, the timedependent beat signal is recorded using anoscilloscope. From this signal the time-dependentinterferometric phase noise 𝜑(𝑡) can be calculatedusing the Hilbert transformation [6]. The absolutesquare of the Fourier transform of 𝜑(𝑡) yields thepower spectral density (PSD) of the interferometricphase noise. The relation between this quantity and
the PSD of the laser frequency noise is given by [7]4𝑃𝑆𝐷𝜑 (𝑓) = 2 sin2 (𝜋𝑓𝜏0 ) 𝑃𝑆𝐷𝜈̃ (𝑓),(4)𝑓where 𝜏0 is the relative delay between bothinterferometer arms. Thus the interferometer can beused as a laser frequency detector whose sensitivity
is largest for frequencies 𝑓 → 0 and which has zero𝑁sensitivity for frequencies fulfilling 𝑓 = , where N is
𝜏0an integer number. For a fiber length of 20 m the firstfrequency of zero sensitivity is at 𝑓 = 10 MHz.
Therefore, the interferometric phase signal allows toextract the full PSD of laser frequency noise down tofrequencies limited by the overall integration time of
the interferometric phase signal up to the firstfrequency of zero sensitivity at 10 MHz. This can be
intuitively understood by considering the fact thateven slow laser frequency drifts will causeinterferometric phase drifts if both interferometerarms have different length. This sensitivity of thephase to laser frequency drifts is more pronounced
the larger the path length difference between thearms is. This intuition coincides with the sensitivity
expression extracted from eq. (4) for frequencies 𝑓 →0 which is given by (2𝜋𝜏)2 , which shows that thesensitivity is larger the longer the relative delay
between both arms is.

Fig. 2: Optical setup for delayed self-heterodyne
detection. This setup is used to measure the6

A spectrum analyzer can be used to obtain the power spectrumof the laser field from this signal. The FWHM linewidth obtained
from the spectrum analyzer trace corresponds to twice or √2 times

the linewidth of a single laser if the spectrum is Lorentzian orGaussian, respectively.
7
The PSD of the frequency noise of a single laser is half the PSDof the relative frequency noise.

TOPTICA Photonics AG
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info@toptica.com

October 19th, 2023

APPLICATION NOTECOMPARISON TO OTHER LINEWIDTHDEFINITIONS
Like described in the section before, the moststraightforward way to obtain the laser linewidth is tooverlap two lasers with identical noise properties andrecord the beat signal using a spectrum analyzer.
The FWHM linewidth obtained from the spectrum
analyzer trace corresponds to twice or √2 times thelinewidth of a single laser if the spectrum isLorentzian or Gaussian, respectively. The linewidth
obtained using this method corresponds to the beta
separation linewidth where the lower integration limit
corresponds to the integration time of the spectrum
analyzer.
However, as in many cases there is no second
identical laser available, a large number of methods
have been developed to obtain the linewidth without
the use of a second laser. We will discuss one othervery popular way to determine the laser linewidth
which is also based on delayed self-heterodyne
detection like described above but with a much
longer fiber. The original idea was to use a fiber solong that the corresponding relative delay exceeds
the coherence time of the laser by far [8]. Thus, thetwo fields superimposed on the second NPBS are
assumed to be uncorrelated. Therefore, theinterferometric beat corresponds to a beat between
two identical but uncorrelated lasers. This method is
not applicable for most of TOPTICA´s diode lasers
because their coherence time is so large that many
kilometers of fiber would be necessary. But very longfibers have the problem that they easily collectenvironmental disturbances like vibrations andacoustics.
A more advanced method is based on fibers ofmoderate length, typically on the order of 1 km. The
method takes into account partial coherence of thetwo paths [9]. Knowing the form of the PSD of thelaser frequency noise, it is possible to directly extract
the laser linewidth by fitting to the laser power

8

The power spectrum of two partially correlated beams typicallyyields a delta peak at the AOM frequency and a sinusoidal envelop
with a periodicity given by the delay between the twointerferometer arms. The modulation depth of the sinusoidal
envelop is a measure for the coherence and thus for the linewidth

spectrum a function which is derived from the form of
the laser frequency PSD8. The drawback of thismethod is that the fit is typically not sensitive to noise
components below Fourier frequencies of 𝑓0 = 1/𝜏0[10]. For a fiber length of 1 km this corresponds to𝑓0 = 200 kHz. It is important to note that this limit
cannot be pushed to lower frequencies by increasing
the measurement time. In this sense the linewidth
obtained from this method rather corresponds to an
instantaneous linewidth as it only takes into accounthigh-frequency components. However, in most cases
it does not reproduce the instantaneous linewidth likedefined in this paper because the frequency 𝑓0 =200 kHz cannot be understood as a sharp cut-off.
Noise at frequencies below but close to 𝑓0 are
suppressed but still have a significant contribution.
Therefore, the linewidth estimation obtained by fittinga spectrum analyzer trace can be considered an
upper bound of the instantaneous linewidth describedabove. For some diodes TOPTICA specifies thisupper bound as a linewidth estimation instead of theinstantaneous linewidth. This used to be TOPTICA´sprimary definition for short term linewidths. For
reasons of consistency, therefore, in all relevant
documents, this value is denoted as “typical shortterm linewidth [5µs]” 9.

CONCLUSION
The instantaneous and beta separation linewidths
presented in this paper therefore are preferable forseveral reasons. Most importantly, they areindependent of the details of the measurement setup,e.g. the length of the fiber. Beyond that, for themethod presented in this paper, a fiber length of 20 m
is sufficient, which is less sensitive to environmental
disturbances. And last, no assumptions about theform of the laser frequency PSD are required in thefirst place.

of the laser. It is this modulation depth which is typically being
fitted and which gives rise to the laser linewidth.
9In all relevant documents, the * behind the linewidth value
denotes that this value is used instead of the instantaneous
linewidth.

TOPTICA Photonics AG
www.toptica.com
info@toptica.com

APPLICATION NOTE

October 19th, 2023

REFERENCES[1]

A. L. Schawlow and C. H. Townes, Infrared and Optical Masers, Phys. Rev. 112, 1940 (1958).

[2]

Wikipedia Article: Spectral Density, https://en.wikipedia.org/wiki/Spectral_density.

[3]

C. H. Henry, Theory of the Linewidth of Semiconductor Lasers, IEEE J. Quantum Electron. 18, 259 (1982)[4]

R. Paschotta, RP Photonics - Linewidth Enhancement Factor, https://www.rpphotonics.com/linewidth_enhancement_factor.html.

[5]G. Di Domenico, S. Schilt, and P. Thomann, Simple Approach to the Relation between Laser Frequency
Noise and Laser Line Shape, Appl. Opt. 49, 4801 (2010).

[6]

Stanford.Edu - Analytic Signals and Hilbert Transform Filters,
https://ccrma.stanford.edu/~jos/mdft/Analytic_Signals_Hilbert_Transform.html.

[7]

H. Tsuchida, Laser Frequency Modulation Noise Measurement by Recirculating Delayed Self-Heterodyne
Method, Opt. Lett. Vol. 36, Issue 5, Pp. 681-683 36, 681 (2011).

[8]

T. Okoshi, K. Kikuchi, and A. Nakayama, Novel Method for High Resolution Measurement of Laser Output
Spectrum, Electron. Lett. 16, 630 (1980).

[9]

H. Ludvigsen and E. Bødtker, New Method for Self-Homodyne Laser Linewidth Measurements with a Short
Delay Fiber, Opt. Commun. 110, 595 (1994).

[10]

L. B. Mercer, 1/f Frequency Noise Effects on Self-Heterodyne Linewidth Measurements, J. Light. Technol.
9, 485 (1991).

TOPTICA Photonics AG
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