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Laser Linewidth

9006-010

 

9006-010a


Candidate lasers must be able to produce a highly stable (coherent) single longitudinal mode, and we should be able to electronically stabilize them and tune them over hundreds of GHz. As long as linewidths are less than about 1% of the data rates, no significant impairment in SNR should occur. It is relatively easy to produce lasers with short term linewidths less than 1 kHz.

The spectral width of a typical single mode laser line has a Lorentzian profile, and is given by:

 
                       
dv = ------------------
             Pt
 

where:

Rf = reflectivity of front-facet (0.90),
Rr = reflectivity of rear-facet (0.99),
Pt = laser output power (1 kW),
n = refractive index of lasing medium (~ 1.0).
 

9006-010b


Substituting the values in parentheses into the above equation for a gas laser, we find that:

 

= Hz

 

Clearly, even if high-Q mirror facets are not used, high power lasers will have much lower linewidth. This behavior is common to all types of oscillator.

 

The Lorenzian lineshape is defined by:

 
                  dv
S(f) =  ------------------------
        2.pi[(v - vo)2 + (dv/2)2]
 

where:

dv = half-power linewidth (Hz),
v = frequency offset (Hz),
vo = center frequency (Hz).
 

9006-010c


The Gaussian lineshape is defined by:

 
        exp[-(v - vo)2/dv2{2.loge(2)}]
S(f) =  -----------------------------
             [pi.dv/{2.loge(2)}]
 

Graph 9008-041 shows a linear plot of Lorentzian and Gaussian lineshapes, while Graph 9008-042 is a logarithmic plot. The responses have been individually normalized to unity and a -3 dB linewidth of 1 Hz has been assumed in each case. For these general profile characteristics we have set the center frequency vo = 0. Since the Gaussian profile is more peaky and a steeper function, its actual amplitude at zero offset is greater than for the Lorentzian for the same total power. The Lorentzian profile is typical for laser lines that are subjected to homogeneous broadening, while the Gaussian profile is typical of inhomogeneous broadening caused by a spread in energy level transition frequencies and Doppler shifts in lasing gases. We shall assume the more demanding situation of the Lorentzian profile, though in actual practice a line may be a combination of both functions.

 

9006-010d


Notice that the Lorentzian line spectral density is only 20 dB down at an offset of 5 Hz, just five times the 3 dB linewidth. At an offset of 10 Hz the spectral density has fallen to -26 dB. We need to go out to 50 Hz before the spectral density falls below -40 dB.

If this optical carrier was intensity modulated at 100 Hz, so that the first-order sidebands where equal to the amplitude of the carrier, the noise spectral density from the wings of the carrier would be 46 dB below the sideband level. Hence, the general conclusion that if the modulation rate is about 100 times the linewidth, then the noise from the noise sidebands around the carrier signal will not be above about the -40 dB level. SNRs of about 30 dB are required to produce bit error rates (BERs) of less than 10-9. Thus, a 1 Hz linewidth would not degrade a 100 bit/s digital system. Similarly, a 1 GBit/s communications link could cope with a 10 MHz linewidth. Note that after demodulation of the optical carrier, the double-sided noise spectral density will increase by 3 dB.

It should be noted that for two uncorrelated lasers with identical linewidth beating (mixing) with each other on a photodetector, the linewidth of the resulting beat frequency is 2dv .

 

9006-010e


Clearly, if we are attempting to receive very low data-rate SETI signals, high constraints are put on laser spectral linewidth at both ends of the link. For this reason, it could be argued that if very high transmitter powers are not a problem with alien technical civilizations, that there will be a desire to use a much larger bandwidth. For given constraints on transmitter power and available laser linewidths, there is an optimum data-rate or bandwidth which will maximize the CNR or BER at the receiver for baseband modulation.

It may not be desirable for the alien transmissions to consist of simple low- frequency analog baseband modulation. It would appear better to place the intelligence on the optical carrier via a subcarrier modulation. In this way, the subcarrier frequency can be chosen high enough to get away from the skirts of the laser line profile so that the recovered SNR will not be degraded. As long as the SNR is adequate to detect the subcarrier, further demodulation of the subcarrier will yield an SNR improvement. While the linewidth of the alien transmitting laser may be very narrow, it may be prudent for the aliens to employ a system with higher immunity to linewidth effects. There may be interstellar effects which will broaden the linewidth, and since they do not know anything about the linewidth of our receiving system, an approach with greater immunity to SNR degradation from finite linewidths appears desirable.

 

The Columbus Optical SETI Observatory
Copyright (c), 1990

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