The Spectrum Analyzer and the Mode Structure of a Laser

8/2/2019 The Spectrum Analyzer and the Mode Structure of a Laser

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1997 The College of Wooster

The Spectrum Analyzer and The Mode Structure of a LaserSahibzada Amir Hassana) Advisor:D. T. Jacobsb)

Physics Department, The College of Wooster, Wooster, Ohio 44691

(Written April 31 1997)

Theis experiment reveals that the output of a laser is composed of range of

frequencies, which is, theoretically contained within a Gaussiandistribution. The difference in frequency of neighbouring frequencies inthis range is v, called the mode separation; a constant given by the

equation v =c

2L

, where c is the speed of light and L is the length of

the laser cavity. The mode separation of a Melles Griot He-Ne laser, witha principal (main) emission wavelength of 543.5 nm, was experimentallydetermined (using a spectrum analyzer in conjunction with a Fabry-PerotInterferometer ) to be approximately 0.377 0.013 GHz. Using theexpression for the mode separation, the length, L, of the laser cavity wascalculated to be approximately 0.398 0.01 m. Which is in error of about1% from the value of L calculated for the published6 value of the mode

separation. 1997 The College of Wooster

INTRODUCTION

The acronym laser means lightamplification by stimulated emission ofradiation. The words 'amplification' and'stimulated emission' refer to the process of theinteraction of electromagnetic radiation withmatter, proposed by Einstein.2

Laser light1 is not an ideal monochromatic

light source; that is, the output of a laser is not aan electromagnetic radiation of a singlefrequency. Instead, as the experiment will reveal,laser light consists of discrete frequencycomponents. We can, however, determine howclose the laser light may be to a single frequencyusing a Fabry-Perot Interferometer.1

The Fabry-Perot Interferometer consists ofa pair of mirrors, with inner surfaces highlyreflective to laser light, whose separation (or theoptical cavity 1 ) can be varied. As a beam ofcollimated (coherent) light, such as a laser, entersthe interferometer it bounces back and forth

between these plates. If the original wave and thereflected wave are in phase then they reinforceeach other and optical resonance occurs. Wavesthat are in phase replicate themselves and theirelectric fields add such that the energy density ofthe resulting wave is high enough to allowtransmission through the reflecting mirrors (Fig.1).

The condition for a self replicating field(optical resonance) is that the distance betweenthe two mirrors is equal to an integral number of

half wavelengths. Therefore, for any specificseparation of the two mirrors there may exist anumber of self replicating fields, or longitudinalmodes. Conversely, only for certain frequencies isthe cavity resonant. For these longitudinalmodes, or resonant frequencies, light istransmitted through the interferometer anddetected using a photo-diode. Examining theprofile of the output would reveal the spectral

content1

of the incident laser.This experiment will present an analysis ofthe structure of the laser output from a helium-neon (He-Ne) Laser device.

Theory

The laser resonator2 (cavity) follows thesame basic principle as the Fabry-Perotinterferometer cavity. However, the laserresonator differs in functionality due to thepresence of an amplifying medium1 between themirrors and the fact that the length of the laser

cavity (or resonator) L cannot be varied.1 Moredetails may be found in the referenced texts or inany standard optics textbook.

When the incident and reflected lightwaves are self replicating or in phase1 within thelaser cavity the electric fields of both the wavesadd up due to onstructive' interference1 . Theamplitude and consequently the energy density ofthe resulting wave increases such that almost100% of the resultant wave is transmitted through


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2

the mirrors. For the laser cavity to be resonant thefollowing relationship must be satisfied;

L = m2

(1)That is the distance L must be an integral

multiple of half wavelengths. Since m can be anyinteger there may exist a number of wavelengths(or frequencies) which might satisfy therelationship. From equation (1) it follows thateach field for which the laser cavity is resonant,for a fixed separation L, may be expressed interms of the frequency of light,

vm =c

m=

c

2Lm

= mc

2L

(2)Where c is the speed of light and m is any integer.

The separation between neighboring transmittedfrequencies, the mode spacing, is given by,2

v = vm+1 vm = m + 1( )c

2L

mc

2L=

c

2L(3)

As the laser resonator cavity and theFabry-Perot cavity are identical, by applying thesame kind of formalism, we can show that theexpression for the difference between neighboringresonant frequencies set up in the Fabry-Perotcavity is,

= vm+1 vm = m +1( ) c2dmc2d =

c2d

(4)

Where d is the separation between the mirrors inthe Fabry-Perot Cavity. Since d can be varied, theFabry-Perot interferometer can accommodatedifferent integral half wavelengths at differentvalues of the separation d.

When the separation d is increasedconsiderably such that is large enough, onlyone frequency would be allowed to be transmittedand observed within the range of observablefrequencies of the interferometer thereby allowingus to analyze the frequencies contained in theincident laser light. 2

If the difference between the neighboringtransmitted frequencies (longitudinal modes) is aconstant then v (eq. 5) is equivalent to the modespacing of the laser input.It can also be shown2 that changes in d greater

than2

would then result in the same frequencies

being transmitted as observed between d = 0 and

d =2

. Thus, the range of frequencies observed,

as the mirror separation d is varied, without thepattern repeating itself is called the free spectralrange (FSR) of the interferometer, given by,

=c

2d(6)

According to Maxwell 2 there exists adistribution of velocities of the atoms in agaseous amplifying medium (He-Ne gas in ourcase) is given by a Gaussian (Maxwellian) 2 withthe most probable velocity given by;

vp =2kT

M

1/ 2

(7)

Where M is the atomic mass of the amplifying gasmedium, k is the Boltzman constant2 and T is thetemperature at which the laser transition (lasing)occurs. This causes Doppler broadening2 to occur

within the laser resonator and consequently theoutput signal consists of a band of frequenciescontained (theoretically) within a Gaussian profilerather than a single frequency (coherent) output.Therefore, the laser output can be characterized asan intensity distribution given by;3

I ( ) = I0e

c o( )0 vp

2

(8)

The full width at half maximum(FWHM)3 of thisdistribution is given by equation (9)4.

vHM =

2 ln2( )ovpc = 8ln2( )

kT

Mc2 = 2.35kT

Mc2

(9).Therefore, after examining the spectral

content of the laser, using the Fabry-PerotInterferometer, we can interpolate a Gaussian linefit (using IGOR) of the form

y = k[0] + k[1]ex k[2]

k[3]

2

through the values of theresonant frequencies and their correspondingintensities to verify the validity of theseassumptions.

If the expression given above is used for

interpolating the data then the line-fit parameterscan be compared to parameters in equation(8) togive the following correlation. k[0] should beequal to zero, since no intensity should beobserved for frequencies lying outside theGauusian distribution. The parameter k[1] shouldbe equivalent to the maximum intensity Io that

may exist within the Gaussian profile. K[2]should correspond to the central frequency vo of

the line shape and k[3] should be numerically


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3

equivalent to ovp/c, from which the FWHM canbe calculated. Once the FWHM is known, for agiven the line width and relative molecular massof the gaseous gain medium, the lasingtemperature T for the 543.5 nm He-Ne lasertransition can be determined using equation (9).Further information of the theoretical model may

be found in reference 4.

EXPERIMENT

The apparatus which consisted of a Melles Griotmodel 05 SGR 871 GreNeTM Helium-Neon laser,

a Coherent Spectrum Analyzer controller 251, aCoherent Fabry-Perot interferometer (Model 240-1-A) and a Hewlett Packard 54600 oscilloscope,was set up as shown in figure 1. The intensity ofthe transmitted frequencies from the Fabry-PerotInterferometer (measured in mVolts) weredetected using a photo diode and displayed on theHP oscilloscope. Details of the experimental

procedure may be found in ref. 4 and ref. 5.

Figure 1: Schematic Diagram of the apparatus.

The intensity of the resonance peaks (inmVolts) and their respective frequencies can beread off the Oscilloscope screen using the cursoroption (Fig. 2). The two cursors, namely V1 andT1, were used as reference (fixed) axis and thevalues of the voltages of the peaks (modes) andtheir corresponding times are measured, usingcursors V2 and T2 respectively (Fig. 2). This

method also allows us to measure the modespacing of the resonance pattern directly. It isimperative that the FSR should be greater than theline width of the laser output, so that all thepossible longitudinal modes are within thescanning range of the interferometer.

Fig 2: Identical patterns of resonant frequencies.On screen cursors are used to determine theintensities (measured in mV) and thecorresponding times. V1 and T1 used asreference axis.

The Intensities of the resonance peaks (in mV)and their respective times were collected andanalyzed using the IGOR pro software.

ANALYSIS AND INTERPRETATION

The calibration of the oscilloscope revealed that7.42 ms on the oscilloscope corresponded to 7.5GHz (FSR). Thus 1 ms on the time divisions ofthe oscilloscope corresponded to approximately1.06 0.01 GHz of the actual signal input.The pattern of the resonance peaks, observed onthe oscilloscope, was continuously varying insize. The pattern of the resonance peaks was

'skewed' in one direction and then the other,possibly, indicating thermal instability.6 Since,during the warm up process the length of thelaser resonator cavity expands the longer modecavity may be the cause of the schematic shiftin the observed resonance curves.6 Therefore,the data was not collected until the pattern wassymmetric. Thereafter, the voltage values andthe time values were collected (as shown inTable 1). Since the time values for thesuccessive resonance peaks are measured in ms,any measurement for the time must therefore bemultiplied by the scaling factor to give thecorresponding frequency of the signal input.The time base of the oscilloscope was set to 200s per/div. for better resolution of the trace.


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Therefore, the error involved in reading thevoltage values off the oscilloscope screen wasreduced.

Table 2: The mode spacing betweenneighboring resonance peaks in thetransition line shape (scaled to GHz).

The values of the mode-spacings were found tobe very consistent as shown in table 1.

Since the times corresponding to the resonancecurve was approximated by visual inspection ofthe signal trace on the oscilloscope, the readingsin table 2 may be omitting the human error that

may have been present in the data set.

Table 3: The values of the voltage of theresonance peaks and their correspondingtimes.

Plotting the data set given in table 3, thefollowing graph was obtained on IGOR pro.The Gaussian curve fit of the form

y = K[0] + K[1]ex K[2]

K[3]

2

was used tointerpolate the data points and the following

curve-fit was obtained. The curve fit shows thatthe resonance peaks are indeed contained withina Gaussian distribution. The parameter k[0] wasnot held at 0 (for the first fit) as predicted by thetheory since the background noise prevented theintensity to fall to zero on either side of theresonance pattern as reflected in the first and thelast data points.

140

120

100

80

60

40

20

Voltage(mV)

1.51.00.50.0

Time (ms)

'V (mv)'' f i t_V (mv)'

'fit_v= k[0]+k[1]*exp(-((x-k[2])/k[3])^2)W_coef={-6.553,149.39,0.88149,0.53798}

V_chisq= 2.3043; V_npnts= 6;

W_sigma={1.38,1.41,0.00281,0.00769}

Fig 3: The figure shows Voltage (mVolts) as a measure of the intensity of thefrequencies observed vs. their corresponding times (in ms). The curve used tointerpolate the data points is a Gaussian of the form: (k[0] not held equal tozero).

The parameters of the second curve-fit (holding the k[0] constant at zero) agree considerably with thefirst curve-fit.


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5

140

120

100

80

60

40

20

Voltage

(mV)

1.61.41.21.00.80.60 .40.20.0

Time (ms)

'V (mv)'Gaussian 'fit_V (mv)

fit_v= k[0]+k[1]*exp(-((x-k[2])/k[3])^2)W_coef={0,144.44,0.8817,0.50749}

V_chisq= 32.1178; V_npnts= 6;

W_sigma={0,3.01,0.00859,0.0123}

Fig 3: The figure shows Voltage (mVolts) as a measure of the intensity of the

frequencies observed vs. their corresponding times (in ms). The curve used tointerpolate the data points is a Gaussian of the form: (k[0] held equal to zero).

Using the correlation of the line fit to the Gaussian distribution for the gas medium we can calculate thefollowing,

From curve fit I:

Parameters Valuesfrom Gaussian Curve-fit.k[0] determined freely.

Theory.

K[0] = -6.6 1.4 (K[0] = -6.6 1.4)K[1] = 149.4 1.4 Io =149.4 1.4K[2] = 0.881 0.003 vo = k[2]*1.06= 0.933 0.003 GHzK[3] = 0.54 0.01 ovp/c= 0.54 0.01 *1.06

=0.57 0.01 GHz

From curve fit II:

Parameters Values from

Gaussian curve-fit.(holding k[0] =0.)

Theory

K[0] = 0 (K[0] = 0)K[1] = 144.4 3.01 Io =144.4 3.1K[2] = 0.882 0.001 vo = k[2]*1.06= 0.934 0.001

GHzK[3] = 0.51 0.01 ovp/c= 0.51 0.01 *1.06

=0.54 0.01 GHz


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The value of chi squared for the second setof line fit parameters (for Fig. 10) is 2.3 ascompared to 32.1 in the case when k[0] was fixedat zero. The value of the free parameter K[0]could be interpreted qualitatively as a verticaltranslation of the standard distribution curve. Thatis, the Gaussian line fit ought to have decreasedexponentially towards the value of the baseline

(observed on the oscilloscope) of approximately 3mV on either side of the resonance pattern.However, the parameters from the second line fitwould suggest that the respective Gaussian would

tend to -6.6 1.4 mV on either side of the datapoints. This discrepancy arises due to the fact thatthe first and the last data points in Table 1 are notresonance peaks. They were determinedarbitrarily to provide enough data for a successfulGaussian line fit. If these data points on the baseline were defined far enough from either side ofthe resonance pattern, the corresponding line fit

would yield a k[0] parameter sufficiently close tothe observed base line of 3 mV. Despite thesedifferences the parameter values for k[2] or theprincipal frequency agree remarkably for bothcurve fits.

Given that the He-Ne laser utilizes a 543.5nm laser transition6, given that the atomic mass ofneon is approximately 20 a.m.u, the lasingtemperature T is calculated as follows;( Note that the atomic mass of neon was only usedsince He metastable atoms only serve to transferenergy to the excited neon atoms, which in turnare responsible for the actual laser output).7

Since the line width value has beencalculated from the line fit parameters, therefore,we can determine the value for the lasingtemperature for the 543.5 nm transition asfollows.Using equation (9)

vHM =2 ln2( )ovp

c= 8ln2( )

kT

Mc2= 2.35

kT

Mc2vo

2 ln2( ) k[3]

2.35

=

vo

c

kT

M

2 ln2( ) (0.57 0.01)109

2.35

=

kT

M

T = 0.709 (0.57 0.01) 109 5.4 109( )2 M

k

T = 118.86 0.07 K

By inspection of equation (8) we can seethat the line width for the Gaussian line shape is

directly proportional to the resonant frequency vo .

That is, a decrease in the lasing wavelength (632.8 nm to 543.5 nm ) should result in anincrease in the corresponding frequency . Thatwould imply that the line width of the Gaussianfor the 543.5 nm transition would be relativelylarger. From the line fit parameters we can seethat the calculated line width is smaller; 0.6 GHzas compared to 1.5 GHz for the 632.8 nmtransition.5 Since the line width value is incorrect,therefore, the value for the lasing temperature T isincorrect.

Furthermore 118.86 K is below roomtemperature and it was deduced from observationthat the laser output occurred after the laser hadbeen warmed up for a couple of minutes at roomtemperature which further invalidates thecalculated value of T.

By contrast, direct measurements of themode spacing, using voltage and time cursors,

revealed that the mode spacing (calculated usingthe data in Table 4) was in fact (0.356 .002)ms* 1.06 (GHz / ms) = 0.377 0.002 GHz orapproximately 377 MHz. The published6 value isgiven to be approximately 380 MHz. Thus themeasured value is in error of about 0.8% from thepublished6 value.Furthermore, as the relationship between themode separation and the cavity length is given byequation (3) measured value ofv can be verifiedas follows.

Since v =c

2L

, substituting the values of c, the

speed of light (3x108), and the mode separationv an approximate value for the cavity length canbe calculated.That is,

L c

2v=

3108

2 377( )= 0.398 0.01m which is in

error of approximately 1% from the value of theactual cavity length (determined by a similarcalculation for the published6 v value).

CONCLUSION

The experiment was conclusive indetermining and verifying existence oflongitudinal modes in the output of a laser. Bymeasuring experimentally the separation of thespectral frequencies contained in the laseremission, we were able to verify that thesefrequencies differ by a constant of magnitude

v =c

2L

; the mode separation of the laser

output. Moreover, since the value of c, the speed


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of light, and the value of the L; length of theoptical cavity are known, the above relationshipenables us to verify the value of the modeseparation. In fact the calculated value of thelength L differed from the actual6 value by only0.8%.

On the other hand the spectral frequenciesobserved were not entirely contained within aGaussian distribution ( as seen in the dataanalysis). The resonance pattern obtained throughthe Fabry-Perot interferometer was seen to beskewed or shifted at different intervals of time (asshown in Fig. 12). This effect may be attributed tothe thermal instability 6 of the laser cavity. Thecalculation for the lasing temperature may havebeen flawed because of two possible reasons.Firstly the Doppler line width is greater than thenatural line width of the profile. Secondly, thetransition line shape is more accurately modeledby the Voigt line profile

I ( ) = Const( )e

c ' ( ) ovp

2

'( )2 + / 2( )20

d' (10)

Furthermore, one might improve theexperiment by verifying whether the laser is infact operating in the TEMoo mode by inserting a

diverging lens between the laser and theinterferometer and observing the effect on acardboard placed behind the lens. If the laser is infact operating in its TEMoo mode then diverging

lens should cause the laser output to appear asshown in figure 11. A "uniform, rotationallysymmetric, spot devoid of any internal nodes orlines signifies that the laser is" in fact operating inits TEMoo mode.1

Further study might include replacing the0.67 mW He-Ne laser with a more powerful laser,which is more thermally stable, such that morelongitudinal modes can be observed andconsequently the Gaussian shape of the spectralresonance frequencies can be determined withmore accuracy. If the Gaussian behavior of thelaser transition line shape can be established thena better estimate of the value of the lasingtemperature T can be calculated using theequation (9).

ACKNOWLEDGMENTS

1 O'shea, Donald C., Callen, W. Russell and Rhodes, T.Rhodes in Introduction to Lasers and their Applications ,(Addison Wesley, Reading, 1978). Pg. 33-92.

2Pedrotti Frank L and Pedrotti Leno S. in, Introduction toOptics, 2nd ed. (Prentice Hall, New Jersey, 1993), Pg.426-440.

3 McGrawHill Encyclopedia of Physics, Sybil, Sybil Peditor in chief.- 2nd ed. (McGraw-Hill, New York, 1993),Pg. 690.

4Garg, Sheila, Modern Optics Project #3 AssignmentLiterature, The College of Wooster, OH, 1997(unpublished).

5Physics 401, Junior Independent Study Manual , Spring1997. Pg. 30.

6Phillips, Richard A., and Robert D. Gherz, "Laser ModeStructure Experiments For Undergraduate Students," Am.J. Phys. 38 429-432 (1970).

7Melles Griot Optics Guide 4 (product catalogue),copyright 1988. Pg. 17/28 - 17/29.

8Handbook of Optics Volume I, Michael Bass editor in

chief.- 2nd ed. (McGraw-Hill, New York, 1995), Pg.11.32.

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The Spectrum Analyzer and the Mode Structure of a Laser