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Harmonic oscillator fiducials for Hermite-Gaussian laser beams
Optics and Lasers in Engineering 28 (1997) 213-228 0 1997 Elsevier Science Limited All rights reserved. Printed in Northern Ireland. 0143%8166/97/$17Nl PII: so143-8166(w)oooo3-1
ELSEVIER
Harmonic
Oscillator Fiducials for Hermite-Gaussian Laser Beams
s. L. Sverdrup Technology
Steely
Inc., 1077 Ave. C, MS-6400, Arnold AFB, TN 37389, USA
(Received 22 August 1996: accepted 5 November
1996)
ABSTRACT This paper exploits the identical functional forms of quantum mechanical harmonic oscillators and Hermite-Gaussian laser beams to investigate the probability densities of large-order modes. The classical limits of a two-dimensional harmonic oscillator provide corresponding integration limits for the photon probability densities of the laser-beam modes to determine the fraction of photons detected therein, The probabilities of detecting photons within the ‘classical limits’ of Hermite-Gaussian laser beams exhibit a power-law dependence in the limit of large-order modes and asymptotically approach unity, in agreement with the Correspondence Principle. The classical limits for large-order modes are shown to include all of the nodes and peaks for Hermite-Gaussian laser beams; Sturm’s theorem provides a direct proof that is further illustrated by examination of the concavity and convexity of Hermite functions. 0 1997 Elsevier Science Ltd.
1 INTRODUCTION There are many instances in science where different physical models have similar or identical functional forms. Scientists often exploit and glean ideas from other disciplines to better understand new areas of research, especially if the physical models exhibit similar functional forms. The harmonic oscillator is a powerful tool for explaining and understanding many similar disciplines of physics. Since exact solutions exist for the classical and quantum harmonic oscillator, it is a tool and simple model to understand basic principles of vibrational motion and normal modes. In addition, the harmonic oscillator is an excellent pedagogical system to help model and understand the basic properties of quantum mechanics, quantized radiation fields, quantum optics, and other disciplines of 213
214
S. L. Steely
physics. The harmonic oscillator rightfully deserves its place ‘on a pedestal’.’ In this paper we will exploit and use the similarity of the functional forms of quantum harmonic oscillators and Hermite-Gaussian laser beams to investigate the spot size of laser-beam modes and the fractional energy and photons incident therein. Although two slightly different definitions for Hermite polynomials exist,2-4 one can define a laser-beam spot size and show that it contains a large fraction of the energy or photon probability density for lower-order modes.‘,6 In view of the Correspondence Principle, one can also argue that the probability of finding the quantum oscillator within the classical limits asymptotically increases to unity for higher-order modes.‘**.‘,’ Since the functional forms of the quantum oscillator and laser-beam mode are similar, we should expect the probability of detecting photons within the corresponding ‘classical limits’ of Hermite-Gaussian laser-beam modes to similarly approach unity for higher-order modes. In order to extend earlier similar work,S,h.7,X Mathematics” is used to integrate the laser-beam mode probability densities for both small- and large-order modes to illustrate these principles. Sturm’s theorem is then used to provide a direct proof that the classical limits also contain all of the probability density peaks as is also illustrated by examining the concavity and convexity of Hermite functions. The harmonic oscillator’s classical limits, therefore, serve to provide a good measure of spot size for not only small-order modes, but also for large-order modes of Hermite-Gaussian laser beams. For comparison, the classical and quantum oscillator, the classical limits, the probability densities, and the Correspondence Principle are briefly reviewed in Section 2. The Hermite-Gaussian laser-beam modes are reviewed in Section 3 and compared to the quantum oscillator solutions. Section 4 provides a discussion of the results from integrating the laser-beam mode probability densities and presents a power-law dependence for higher-order laser-beam modes. Sturm’s theorem and its application in determining the extent of the peaks and zeros of the probability densities are discussed in Section 5 along with the functional concavity and convexity of Hermite functions. 2 CLASSICAL
LIMITS AND PROBABILITY
DENSITIES
Although the classical harmonic oscillator problem can be solved by the Lagrangian formulation,“’ one obtains a more direct solution using the Hamiltonian formulation to solve the equation of motion for a onedimensional problem’.“’ x(t) =x,, cos (or + 4) (1) The classical harmonic oscillators angular frequency depends simply on
Harmonic oscillator fiducials
-1
0
215
1
Oscillator Position (X/X,) Fig. 1.
Classical harmonic oscillator probability
density $7,(x).
the force constant k and its mass m, while its amplitude x,, = (2EJk)“’ is a continuous variable depending also on the total energy E,. One can solve for the speed as a function of time and subsequently the classical oscillators probability density’-“’
(2) From eqns (1) and (2) we see that the classical oscillator remains within the classical limits LX,, as illustrated in Fig. 1. After a number of tedious operations, the quantum mechanical harmonic oscillator energy levels and eigenstates are derived from the Schrodinger equation and determine the quantum mechanical probability amplitudes’.’ IL”
By introducing a dimensionless variable, the probability written in a more simplified form’*.“’
amplitude
can be
Relative to the standard weighting function w(t) = emC’, the Hermite polynomials H,,(s) are nth-degree orthogonal polynomials H,,,([)H,,(Qe-C’d[ = 6,,,,,~“~2”r2!
216
S.
L. Steely
that satisfy the differential equation3.4 y” + (2n + 1 - x2)y = 0 y(x) = e-“z’2Hn(X) The probability density I&l* = $z+” for observing monic oscillator between 5 and 5 + d,$ is
(6) the quantum
har-
while the corresponding quantum oscillator’s energy values En =(n + l/2)60 can be equated to the classical oscillator’s energy to derive the desired form for the classical limits of the quantum oscillator X0= (h/mw)“2(211 + 1)“2
(8)
The classical eqn (2) and quantum eqn (7) probability densities are plotted together in Fig. 2 for a few of the oscillator modes. As the order of the quantum oscillator mode increases, it appears that the integrated fraction of the probability to be within the classical limits increases. We also see that the classical probability density is near the average of the quantum probability densities, as is more apparent for large-order modes. The classical limits appear to increase with a corresponding increase in the mode order, in such a manner that the outer peaks of the probability densities are always contained within the classical limits. The classical and quantum probability densities are quite different yet similar in a number of ways. In particular, a position measurement of the quantum oscillator of energy E, can result in any value between --CCand +m. However, when measuring the classical oscillator’s position, only values between - -lco= - q2E,/k and xg = m will be obtained. If we consider an oscillator having a huge mass of 10” kg and oscillating at 1 rad/s with an amplitude of 1 cm, then the energy would be mw2xz/2 = 0.5 x lo-‘J. We can compare this to the energy difference between the quantum oscillator levels AE = fiw = lo-j4 J. Experimentally it would be practically impossible to detect energy differences separated by 1O-‘4J. Similarly, if one inverts the quantum energy equation to determine the mode level for this huge oscillator, we see that the mode order, II = (E/&0)-1/2 = 102’, is quite large. Because the mode order, ~1, is equal to the number of nodes in the quantum oscillator’s probability density, it would be virtually impossible to observe 102’ oscillation nodes within the 2cm classical-limit interval. We would, instead, only detect or measure the average of the quantum probability density, which is just the classical result shown previously in
Harmonic
oscillator jiducials
217
Probability Densities 0.6
n=O 0.4
0.2.
0
-6
-4
-2
0
2
4
6
0.6
n=3
/
I
I
0.4
0.2.
0
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
0.6
0
Oscillator Position (B Fig. 2.
Quantum harmonic oscillator probability densities. Dashed vertical lines represent the classical limits; thin curves correspond to the classical probability densities.
Fig. 1. For large it, the classical and quantum results become indistinguishable as expected and required by the Correspondence Principle.’ In the limit of large-order modes, this special case of the Correspondence Principle illustrates how the classical picture is indeed regained. From the Correspondence Principle and the limit of large-order modes II + =, we should expect the quantum mechanical probability densities to
218
S. L. Steely
be functionally similar to the classical harmonic oscillator probability density, which can be derived in a number of ways.“,” If we examine the quantum oscillator’s asymptotic functional form when the mode order increases to infinity, we find a rapid oscillatory behavior” cos
,
forevenn (9)
sin2
that averages out to the classical probability observed in Fig. 2.
3 HERMITE-GAUSSIAN
LASER
for odd IZ density eqn (2) as is also
BEAM MODES
We now consider an Hermite-Gaussian laser beam propagating along the z direction. The laser beam considered can have different beam waists along the x and y directions. The Hermite-Gaussian laser-beam irradiance (optical intensity) at some +z direction is obtained from a scalar-wave (Helmholtz) equation.*“.*’ The irradiance of an Hermite-Gaussian laser beam that is focused at z = 0 can be written as’,**
The beam waists w, and wYare the distances at which the lowest-order beam mode irradiance drops to ee’ times the value on the optical axis (note that many references use eP2 factors to similarly define a beam waist). The x-axis beam waist K(Z) = w,(O)(l + $)“* depends on the Rayleigh-range A
(11)
z,,_~,which is a function of the wavelength
zox=
$5g(O)
(12)
Similar results are obtained for the y-axis laser-beam waist and laser-beam parameter. Equation (10) provides the irradiance at some position in the laser beam; with proper normalization, eqn (10) can also be interpreted as the probability density to detect photons at some position in the laser beam. If we divide eqn (10) by the total power in the laser beam, then the result is
Harmonic oscillator fiducials
219
interpreted as a probability density to detect a photon at the corresponding position, assuming the probability density is proportional to the irradiance. To determine the spot size for large-order beam modes, we next consider the mean-squared value (second moment) of x and y. As an example, we look at the mean-squared value of x x X x2E(x,y,z)d-+ II u:(z)“, = -x, -x, (13) E(x,y,z)hdy II-x -7z and noting from eqn (10) that the integral is easily separable and reduces to a simpler form by integrating and canceling the y-dependent factors
with a similar result obtained for the y-direction. The integrals in the numerator and denominator of eqn (14) occur frequently in quantum mechanics in relation to the harmonic oscillator problem and are readily solved using the generating function for the Hermite po1ynomials’6.‘y
Using eqns (15) and (16) in eqn (14) we obtain the mean-squared u:(z),, = r+?(z>(m + l/2) and taking the square root of twice the mean-squared obtain the desired form for the beam waists 2UXZ),?I= W:(z),,! = wXz)(2m + 1) w.,(z),, = w,(z)@
+ 1)“2
w,(z), = wy(z)(2n + l)“*
value
(17) value, we then
(18) (19a) (19b)
Equations (19a) and (19b) define the beam waists for large-order modes and depend on the order m,n of the mode. We see that the beam waists, W.,(Z),,,, have a mode-order dependence that is identical to that of the
S. L. Steely
220
classical limits, x0, of the harmonic oscillator [eqn (8)]. Therefore, we note that eqn (10) is functionally identical to the quantum harmonic oscillator density [eqn (7)]. Similarly, the functional form of the Hermite-Gaussian laser-beam mode is identical to a two-dimensional quantum harmonic oscillator probability density. To illustrate the beam waists, two laserbeam modes are plotted in Fig. 3 along with the corresponding limits
I
m,n=l,l
I
Fig. 3.
*
!
! m,n= 3,2
Photon probability density plots and classical limits for TEM,, and TEM,, modes. (Vertical and horizontal ticks represent the classical limits w,(z),,, and w,.(z),,.)
221
Harmonic oscillator jiducials
[eqns (19a) and (19b)] that define the rectangular region and size of the laser-beam. We again see that the corresponding beam waists or ‘classical limits’ seem to increase such that the irradiance peaks are always contained therein. Since the photon probability density for an Hermite-Gaussian laser beam is similar to a two-dimensional quantum oscillator, we can anticipate that the probability of detecting photons within the corresponding classical limits of Hermite-Gaussian laser-beam modes will also asymptotically approach unity as the laser-beam mode order increases to infinity, m,n + x.
4 FRACTIONAL
POWER
AND PHOTON
PROBABILITIES
In Section 2, we saw that the classical limits seem to contain most of the probability to detect the quantum oscillator and the photons for the Hermite-Gaussian laser beams. For the large-order mode spot size to be meaningful and useful, it should contain a large portion of the power or photons of the laser beam. The probability to detect photons within the corresponding classical limits that define the spot size of the laser beam should similarly increase for large-order beams, just as in the quantum oscillator case and in agreement with the Correspondence Principle. To investigate the fraction of the power or the photon probability within the classical limits, as illustrated in Fig. 3, an integration over the classical limits is performed
where 5 =X/W,(Z) and 5 = y/w,.(z). Using eqn (16) in eqn (20) we obtain Hi,([)exp(
- E2)dt/v”I
Hf,(S)exp( - 5’)dl
1) ?,I + I ,L.,,
-
2”‘+“nm!n!
(21)
The photon probabilities [eqn (21)] were computed using Mathematics@ and are presented in Table 1 for the low-order modes. Mathematics@ was also used to compute the photon probabilities for higher-order modes.
222
S. L. Steely
TABLE 1 Probability $,1,,,of Detecting Photons within the Classical Limits (Extended of Table 1 in Carterh)
n=O
1 2 3 4 5 6 7 8 9 10
Recomputation
m=O
1
2
3
4
5
6
7
8
9
10
0.7101 0.7486 0.7626 0.7707 0.7762 0.7803 0.7836 0.7862 0.7884 0.7904 0.7920
0.7484 0.7892 0.8039 0.8124 0.8183 0.8226 0.8260 0.8288 0.8312 0.8332 0.8350
0.7626 0.8039 0.8189 0.8276 0.8335 0.8379 0.8414 0.8443 0.8467 0.8487 0.8505
0.7707 0.8124 0.8276 0.8363 0.8423 0.8468 0.8503 0.8532 0.8556 0.8577 0.8595
0.7762 0.8183 0.8335 0.8423 0.8484 0.8529 0.8564 0.8593 0.8618 0.8639 0.8657
0.7803 0.8226 0.8379 0.8468 0.8529 0.8574 0.8610 0.8639 0.8663 0.8684 0.8703
0.7836 0.8260 0.8414 0.8503 0.8564 0.8610 0.8646 0.8675 0.8700 0.8721 0.8739
0.7862 0.8288 0.8443 0.8532 0.8593 0.8639 0.8675 0.8704 0.8729 0.8750 0.8769
0.7884 0.8312 0.8467 0.8556 0.8618 0.8663 0.8700 0.8729 0.8754 0.8775 0.8793
0.7904 0.8332 0.8487 0.8577 0.8639 0.8684 0.8721 0.8750 0.8775 0.87% 0.8815
0.7920 0.8350 0.8505 0.8595 0.8657 0.8703 0.8739 0.8769 0.8793 0.8815 0.8833
Figure 4 illustrates the asymptotic behavior anticipated for the higher-order Hermite-Gaussian laser-beam modes. As the order of the laser-beam mode increases to infinity, we see that the probabilities to detect photons within the corresponding classical limits asymptotically approach unity, although somewhat slower than anticipated for large-order modes, but as expected from the quantum oscillator problem and the Correspondence Principle. In particular, we see from Table 1 and Fig. 4 that fn,,, + 1 as m,n + m.
The solid dots in Fig. 4 are computed from equation (21), while the curve is a plot from m = 10 to 1000 of a nonlinear least-squares curve fit
060 .o
200
400
600
800
1000
Laser beam order(m) Fig. 4. Plot of the probability f*,,,,, of detecting photons within the classical limits of an Hermite-Gaussian laser beam. (Dots are integrated values; the solid curve is a nonlinear least-squares curve fit using the last five integrated values (m = 600 to lOOO).)
Harmonic oscillator fiducials
223
using the five largest integrated values (m = 600, 700, 800, 900, and 1000). For modes larger than m = 10, the curve fit exhibits a very good where the constants are deterpower-law dependence, fm,, = 1 -am-“, mined to be a = 0.259729 and b = 0.329946. Using this power-law curve fit, we can estimate the fractional probabilities for even larger-order modes although obs&vation and practical utility of laser-beam modes beyond 40-50 are unlikely. One might not expect the probabilities to increase as slowly as illustrated in Fig. 4, but for a two-dimensional oscillator or an Hermite-Gaussian laser-beam mode with m = 10” we use the power-law dependence to compute jH,, = 0.999999999679, which is close to unity as one would reasonably expect (although with some reservation of accuracy).
5 PROBABILITY
DENSITY NODES, THEOREM
PEAKS,
AND STURM’S
It is not entirely obvious that all probability density peaks of the quantum oscillator or of the large-order Hermite-Gaussian laser beams are contained within the corresponding classical limits. The Hermite functions y(x) = e-X”2H,(x) determine the nodes (zeros) of both the quantum oscillator densities and the Hermite-Gaussian laser-beam intensities for all modes. The nodes of orthogonal polynomials are all real, distinct, and lie within the interior of the orthogonality region.23 Figure 5 illustrates that the nodes of the Hermite functions also determine the nodes of the
0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -6
I
I
’
I -4
-2
0
2
4
6
Oscillator Position (4) Fig. 5. Quantum oscillator (The thick curve corresponds ing normalized
Hermite
probability
density
and Hermite
function
of order n = IO.
to the probability density. The light curve is the correspondfunction. Dashed lines correspond to the classical limits.)
224
S. L. Steely
quantum oscillator and the corresponding Hermite-Gaussian laser-beam modes. The orthogonality region of Hermite polynomials extends from minus infinity to plus infinity as seen from the integral eqn (5). The Hermite polynomials determine the location of y1 nodes of interest; however, the exponential factor in the Hermite functions dominate the polynomials at large x, driving the Hermite functions to zero at plus and minus infinity resulting in a total of y1+ 2 nodes. Some method, therefore, is desired that will provide a limit to the extent of the n primary nodes of the Hermite polynomials, the Hermite functions, and consequently the nodes of the probability densities of the quantum harmonic oscillator and of the Hermite-Gaussian laser-beam modes. Sturm’s classic work on differential forms and zeros of functions is one such method for analysis of the nodes of the Hermite functions. Sturm’s theorem2’ provides a useful method to determine the limits of the nodes in many functions, especially the classical orthogonal polynomials such as the Hermite polynomials. For two consecutive zeros x0, X,, of y(x), Sturm’s theorem:*’ Let f(x) be continuous and negative in x,, (2n + l)“*, then f(x) will be negative as desired and y(x) remains either positive or negative for a given mode order. We then need only correlate this result with the corresponding classical limits for the quantum oscillator eqn (8) and the HermiteGaussian laser beam [eqns (19a) and (19b)] to see that the nodes determined by the Hermite functions are contained therein. In addition and complementary to Sturm’s method, the concavity and convexity of a function are also useful.‘” The differential eqn (6) can be rewritten as y”/y =x2 - (2n + 1)
(22)
where < 0, is concave for Y”lY1 > 0, is convex for
Ix]< (2n + 1)“’ lx]> (2n + l)“*
determines the concavity or convexity of the Hermite illustrated in Fig. 6 for orders n = 3 and yt = 4.
(23) functions
as
Harmonic oscillator fiducials
225
n=4 15 10 5 0 -5 -10
Fig. 6.
Concavity and convexity of the Hermite functions. [Thick curves denote Hermite functions: thin curves illustrate eqn (22) and eqn (23).]
the
Noting the regions of concavity and convexity [eqn (23)] for the Hermite functions as illustrated in Fig. 6, we see that the classical limits separate the concave and convex regions of the Hermite functions. One sees that the concave region lies between the classical limits while the convex regions lie outside of the classical limits. From Sturm’s theorem and for a given order, y remains nonzero within the regions of convexity and the sign of y is therefore constant, remaining either positive or negative for the Hermite functions. No local maximum or peak can occur for y (or y’) within the convex regions; a positive y would require a negative second derivative, y”, contradicting eqn (23) when 1x1> (2n + 1)“‘. A similar argument holds for any negative y in the convex regions. The classical limits always reside at inflection points and the peaks always reside within the concave region delimited by the classical limits. We can therefore assert that the beam waists for small- and large-order Hermite-Gaussian laser beams contain not only most of the laser-beam power or incident photons, but also contain all nodes and irradiance peaks, as expected from comparison with the quantum harmonic oscillator and the Correspondence Principle as diagrammatically illustrated in
S. L. Steely
226 7
.
‘)
I
Schrcdinger’s Equation
Maxwell’s Equations
c
, Wave Equation Helmholtz Equation
’
r Hermite-Gaussian Laser-Beam Mode
2-D Quantum Harmonic Oscillator
\
have Identical Functional Forms / Beam Waists
Classical Limits <
J&Y”
+ ‘From the Correspondence Principle, We Expect to Find the Oscillator Within the Classical Limits
Fig. 7.
\
, 7
’
w,(&,
w,(z),
/
# We Should Similarly Expect to Detect most Photons Within the “Classical Limits”
Flow diagram illustrating the correspondence between the quantum ... oscmator and Hermite-Gaussian laser-beam modes.
harmonic
Fig. 7. Although the probability densities are derived separately from the Schriidinger equation for the quantum harmonic oscillator and from Maxwell’s equations for the laser-beam modes, further assumptions and reductions lead to the wave equation and Helmholtz equation in both cases, hence the common link and origin of the functional similarity.
6 CONCLUDING
REMARKS
The ubiquitous harmonic oscillator is indeed a useful tool to help model physical systems and, as shown in this paper, to help clarify and better understand some aspects of the probability densities of
Harmonic oscillatorfiducials
227
Hermite-Gaussian laser beams. In particular, the probability densities for two-dimensional quantum harmonic oscillator modes are functionally identical to the photon probability densities of Hermite-Gaussian laserbeam modes. This functional similarity and the Correspondence Principle provide guidance to determine that the corresponding beam waists or ‘classical limits’ for Hermite-Gaussian laser beams define a spot size that contains a large portion of the laser beam’s incident photons and power. As computed with Mathematics, the fraction of the Hermite-Gaussian laser-beam power or photons contained within the beam waists asymptotically increases to unity exhibiting a power-law dependence as the laser-beam order increases to infinity. The classical limits and the corresponding laser-beam spot, as delimited by the beam waists, contain all nodes and probability-density peaks for both small- and large-order modes of the quantum oscillator as well as the Hermite-gaussian laser beam.
REFERENCES 1. Shankar, R., Principles of Quantum Mechanics. New York, Plenum Press, 1980, p. 195. 2. Steely, S. L., Using harmonic oscillators to determine the spot size of Hermite-Gaussian laser beams. NASA Workshop on Harmonic Oscillators, NASA Conf. Publication 3197, 1993, pp. 155-177. 3. Abramowitz, M. & Stegun, I. A., Handbook or Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Washington, DC, Government Printing Office, 1964, pp. 773-802. 4. Magnus, W., Oberhettinger, F. & Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics. New York, Springer, 1966, pp. 249-255. 5. Carter, W. H. Spot size and divergence for Hermite-Gaussian beams of any order. Appl. Opt., 19(7) (1980) 1027-1029. 6. Carter, W. H. Energy carried over the rectangular spot within a HermiteGaussian beam. Appl. Opt., 21(l) (1982) 7-9. 7. Luxon, J. T. & Parker, D. E. Practical spot size definition for single higher-order rectangular-mode laser beams. Appl. Opt., 20( 10) (1981) 17281729. 8. Luxon, J. T. & Parker, 9.
10. 11. 12.
D. E. Higher order CO* laser beam spot size and depth of focus determined. Appl. Opt., 20(11) (1981) 1933-1935. Wolfram, S., Mathematics A System for Doing Mathematics by Computer. Addison-Wesley, New York, 1991. Goldstein, H., Classical Mechanics. Addison-Wesley, Reading, MA, 1980. Condon, E. & Morse, P. M., Quantum Mechanics. McGraw-Hill, New York, 1929. Pauling, L. & Bright Wilson, E. Jr., Introduction to Quantum Mechanics. Reprinted 1965, by Dover Publications, New York, 1935.
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13. Rojansky, V., Introductory Quantum Mechanics. Prentice-Hall, Englewood Cliffs, NJ, 1938. 14. Messiah, A., Quantum Mechanics. Wiley, New York, 1958. 15. Leighton, R. B., Principles of Modern Physics. McGraw-Hill, New York, 1959. 16. Merzbacher, E., Quantum Mechanics. Wiley, New York, 1970. 17. Sakurai, J. J., Modern Quantum Mechanics. Benjamin-Cummings, Menlo Park, CA, 1985. 18. Gardiner, C. W., Quantum Noise. Springer, New York, 1991. 19. Schiff, L. I., Quantum Mechanics. McGraw-Hill, New York, 1968. 20. Kogelnik, H. & Li, T. Laser beams and Resonators. Proceedings of the IEEE, 54(10) (1966) 1312-1329.
21. Yariv, A., Introduction to Optical Electronics. Holt Rinehart and Winston, New York, 1971. 22. Tompkins, D. R. & Rodney, P. F. Six parameter Gaussian modes in free space. Phys. Rev. A, 12(2) (1975) 599-610. 23. Szego, G., Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, RI, 1975, pp. 18-21, 111-129.
ELSEVIER
Harmonic
Oscillator Fiducials for Hermite-Gaussian Laser Beams
s. L. Sverdrup Technology
Steely
Inc., 1077 Ave. C, MS-6400, Arnold AFB, TN 37389, USA
(Received 22 August 1996: accepted 5 November
1996)
ABSTRACT This paper exploits the identical functional forms of quantum mechanical harmonic oscillators and Hermite-Gaussian laser beams to investigate the probability densities of large-order modes. The classical limits of a two-dimensional harmonic oscillator provide corresponding integration limits for the photon probability densities of the laser-beam modes to determine the fraction of photons detected therein, The probabilities of detecting photons within the ‘classical limits’ of Hermite-Gaussian laser beams exhibit a power-law dependence in the limit of large-order modes and asymptotically approach unity, in agreement with the Correspondence Principle. The classical limits for large-order modes are shown to include all of the nodes and peaks for Hermite-Gaussian laser beams; Sturm’s theorem provides a direct proof that is further illustrated by examination of the concavity and convexity of Hermite functions. 0 1997 Elsevier Science Ltd.
1 INTRODUCTION There are many instances in science where different physical models have similar or identical functional forms. Scientists often exploit and glean ideas from other disciplines to better understand new areas of research, especially if the physical models exhibit similar functional forms. The harmonic oscillator is a powerful tool for explaining and understanding many similar disciplines of physics. Since exact solutions exist for the classical and quantum harmonic oscillator, it is a tool and simple model to understand basic principles of vibrational motion and normal modes. In addition, the harmonic oscillator is an excellent pedagogical system to help model and understand the basic properties of quantum mechanics, quantized radiation fields, quantum optics, and other disciplines of 213
214
S. L. Steely
physics. The harmonic oscillator rightfully deserves its place ‘on a pedestal’.’ In this paper we will exploit and use the similarity of the functional forms of quantum harmonic oscillators and Hermite-Gaussian laser beams to investigate the spot size of laser-beam modes and the fractional energy and photons incident therein. Although two slightly different definitions for Hermite polynomials exist,2-4 one can define a laser-beam spot size and show that it contains a large fraction of the energy or photon probability density for lower-order modes.‘,6 In view of the Correspondence Principle, one can also argue that the probability of finding the quantum oscillator within the classical limits asymptotically increases to unity for higher-order modes.‘**.‘,’ Since the functional forms of the quantum oscillator and laser-beam mode are similar, we should expect the probability of detecting photons within the corresponding ‘classical limits’ of Hermite-Gaussian laser-beam modes to similarly approach unity for higher-order modes. In order to extend earlier similar work,S,h.7,X Mathematics” is used to integrate the laser-beam mode probability densities for both small- and large-order modes to illustrate these principles. Sturm’s theorem is then used to provide a direct proof that the classical limits also contain all of the probability density peaks as is also illustrated by examining the concavity and convexity of Hermite functions. The harmonic oscillator’s classical limits, therefore, serve to provide a good measure of spot size for not only small-order modes, but also for large-order modes of Hermite-Gaussian laser beams. For comparison, the classical and quantum oscillator, the classical limits, the probability densities, and the Correspondence Principle are briefly reviewed in Section 2. The Hermite-Gaussian laser-beam modes are reviewed in Section 3 and compared to the quantum oscillator solutions. Section 4 provides a discussion of the results from integrating the laser-beam mode probability densities and presents a power-law dependence for higher-order laser-beam modes. Sturm’s theorem and its application in determining the extent of the peaks and zeros of the probability densities are discussed in Section 5 along with the functional concavity and convexity of Hermite functions. 2 CLASSICAL
LIMITS AND PROBABILITY
DENSITIES
Although the classical harmonic oscillator problem can be solved by the Lagrangian formulation,“’ one obtains a more direct solution using the Hamiltonian formulation to solve the equation of motion for a onedimensional problem’.“’ x(t) =x,, cos (or + 4) (1) The classical harmonic oscillators angular frequency depends simply on
Harmonic oscillator fiducials
-1
0
215
1
Oscillator Position (X/X,) Fig. 1.
Classical harmonic oscillator probability
density $7,(x).
the force constant k and its mass m, while its amplitude x,, = (2EJk)“’ is a continuous variable depending also on the total energy E,. One can solve for the speed as a function of time and subsequently the classical oscillators probability density’-“’
(2) From eqns (1) and (2) we see that the classical oscillator remains within the classical limits LX,, as illustrated in Fig. 1. After a number of tedious operations, the quantum mechanical harmonic oscillator energy levels and eigenstates are derived from the Schrodinger equation and determine the quantum mechanical probability amplitudes’.’ IL”
By introducing a dimensionless variable, the probability written in a more simplified form’*.“’
amplitude
can be
Relative to the standard weighting function w(t) = emC’, the Hermite polynomials H,,(s) are nth-degree orthogonal polynomials H,,,([)H,,(Qe-C’d[ = 6,,,,,~“~2”r2!
216
S.
L. Steely
that satisfy the differential equation3.4 y” + (2n + 1 - x2)y = 0 y(x) = e-“z’2Hn(X) The probability density I&l* = $z+” for observing monic oscillator between 5 and 5 + d,$ is
(6) the quantum
har-
while the corresponding quantum oscillator’s energy values En =(n + l/2)60 can be equated to the classical oscillator’s energy to derive the desired form for the classical limits of the quantum oscillator X0= (h/mw)“2(211 + 1)“2
(8)
The classical eqn (2) and quantum eqn (7) probability densities are plotted together in Fig. 2 for a few of the oscillator modes. As the order of the quantum oscillator mode increases, it appears that the integrated fraction of the probability to be within the classical limits increases. We also see that the classical probability density is near the average of the quantum probability densities, as is more apparent for large-order modes. The classical limits appear to increase with a corresponding increase in the mode order, in such a manner that the outer peaks of the probability densities are always contained within the classical limits. The classical and quantum probability densities are quite different yet similar in a number of ways. In particular, a position measurement of the quantum oscillator of energy E, can result in any value between --CCand +m. However, when measuring the classical oscillator’s position, only values between - -lco= - q2E,/k and xg = m will be obtained. If we consider an oscillator having a huge mass of 10” kg and oscillating at 1 rad/s with an amplitude of 1 cm, then the energy would be mw2xz/2 = 0.5 x lo-‘J. We can compare this to the energy difference between the quantum oscillator levels AE = fiw = lo-j4 J. Experimentally it would be practically impossible to detect energy differences separated by 1O-‘4J. Similarly, if one inverts the quantum energy equation to determine the mode level for this huge oscillator, we see that the mode order, II = (E/&0)-1/2 = 102’, is quite large. Because the mode order, ~1, is equal to the number of nodes in the quantum oscillator’s probability density, it would be virtually impossible to observe 102’ oscillation nodes within the 2cm classical-limit interval. We would, instead, only detect or measure the average of the quantum probability density, which is just the classical result shown previously in
Harmonic
oscillator jiducials
217
Probability Densities 0.6
n=O 0.4
0.2.
0
-6
-4
-2
0
2
4
6
0.6
n=3
/
I
I
0.4
0.2.
0
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
0.6
0
Oscillator Position (B Fig. 2.
Quantum harmonic oscillator probability densities. Dashed vertical lines represent the classical limits; thin curves correspond to the classical probability densities.
Fig. 1. For large it, the classical and quantum results become indistinguishable as expected and required by the Correspondence Principle.’ In the limit of large-order modes, this special case of the Correspondence Principle illustrates how the classical picture is indeed regained. From the Correspondence Principle and the limit of large-order modes II + =, we should expect the quantum mechanical probability densities to
218
S. L. Steely
be functionally similar to the classical harmonic oscillator probability density, which can be derived in a number of ways.“,” If we examine the quantum oscillator’s asymptotic functional form when the mode order increases to infinity, we find a rapid oscillatory behavior” cos
,
forevenn (9)
sin2
that averages out to the classical probability observed in Fig. 2.
3 HERMITE-GAUSSIAN
LASER
for odd IZ density eqn (2) as is also
BEAM MODES
We now consider an Hermite-Gaussian laser beam propagating along the z direction. The laser beam considered can have different beam waists along the x and y directions. The Hermite-Gaussian laser-beam irradiance (optical intensity) at some +z direction is obtained from a scalar-wave (Helmholtz) equation.*“.*’ The irradiance of an Hermite-Gaussian laser beam that is focused at z = 0 can be written as’,**
The beam waists w, and wYare the distances at which the lowest-order beam mode irradiance drops to ee’ times the value on the optical axis (note that many references use eP2 factors to similarly define a beam waist). The x-axis beam waist K(Z) = w,(O)(l + $)“* depends on the Rayleigh-range A
(11)
z,,_~,which is a function of the wavelength
zox=
$5g(O)
(12)
Similar results are obtained for the y-axis laser-beam waist and laser-beam parameter. Equation (10) provides the irradiance at some position in the laser beam; with proper normalization, eqn (10) can also be interpreted as the probability density to detect photons at some position in the laser beam. If we divide eqn (10) by the total power in the laser beam, then the result is
Harmonic oscillator fiducials
219
interpreted as a probability density to detect a photon at the corresponding position, assuming the probability density is proportional to the irradiance. To determine the spot size for large-order beam modes, we next consider the mean-squared value (second moment) of x and y. As an example, we look at the mean-squared value of x x X x2E(x,y,z)d-+ II u:(z)“, = -x, -x, (13) E(x,y,z)hdy II-x -7z and noting from eqn (10) that the integral is easily separable and reduces to a simpler form by integrating and canceling the y-dependent factors
with a similar result obtained for the y-direction. The integrals in the numerator and denominator of eqn (14) occur frequently in quantum mechanics in relation to the harmonic oscillator problem and are readily solved using the generating function for the Hermite po1ynomials’6.‘y
Using eqns (15) and (16) in eqn (14) we obtain the mean-squared u:(z),, = r+?(z>(m + l/2) and taking the square root of twice the mean-squared obtain the desired form for the beam waists 2UXZ),?I= W:(z),,! = wXz)(2m + 1) w.,(z),, = w,(z)@
+ 1)“2
w,(z), = wy(z)(2n + l)“*
value
(17) value, we then
(18) (19a) (19b)
Equations (19a) and (19b) define the beam waists for large-order modes and depend on the order m,n of the mode. We see that the beam waists, W.,(Z),,,, have a mode-order dependence that is identical to that of the
S. L. Steely
220
classical limits, x0, of the harmonic oscillator [eqn (8)]. Therefore, we note that eqn (10) is functionally identical to the quantum harmonic oscillator density [eqn (7)]. Similarly, the functional form of the Hermite-Gaussian laser-beam mode is identical to a two-dimensional quantum harmonic oscillator probability density. To illustrate the beam waists, two laserbeam modes are plotted in Fig. 3 along with the corresponding limits
I
m,n=l,l
I
Fig. 3.
*
!
! m,n= 3,2
Photon probability density plots and classical limits for TEM,, and TEM,, modes. (Vertical and horizontal ticks represent the classical limits w,(z),,, and w,.(z),,.)
221
Harmonic oscillator jiducials
[eqns (19a) and (19b)] that define the rectangular region and size of the laser-beam. We again see that the corresponding beam waists or ‘classical limits’ seem to increase such that the irradiance peaks are always contained therein. Since the photon probability density for an Hermite-Gaussian laser beam is similar to a two-dimensional quantum oscillator, we can anticipate that the probability of detecting photons within the corresponding classical limits of Hermite-Gaussian laser-beam modes will also asymptotically approach unity as the laser-beam mode order increases to infinity, m,n + x.
4 FRACTIONAL
POWER
AND PHOTON
PROBABILITIES
In Section 2, we saw that the classical limits seem to contain most of the probability to detect the quantum oscillator and the photons for the Hermite-Gaussian laser beams. For the large-order mode spot size to be meaningful and useful, it should contain a large portion of the power or photons of the laser beam. The probability to detect photons within the corresponding classical limits that define the spot size of the laser beam should similarly increase for large-order beams, just as in the quantum oscillator case and in agreement with the Correspondence Principle. To investigate the fraction of the power or the photon probability within the classical limits, as illustrated in Fig. 3, an integration over the classical limits is performed
where 5 =X/W,(Z) and 5 = y/w,.(z). Using eqn (16) in eqn (20) we obtain Hi,([)exp(
- E2)dt/v”I
Hf,(S)exp( - 5’)dl
1) ?,I + I ,L.,,
-
2”‘+“nm!n!
(21)
The photon probabilities [eqn (21)] were computed using Mathematics@ and are presented in Table 1 for the low-order modes. Mathematics@ was also used to compute the photon probabilities for higher-order modes.
222
S. L. Steely
TABLE 1 Probability $,1,,,of Detecting Photons within the Classical Limits (Extended of Table 1 in Carterh)
n=O
1 2 3 4 5 6 7 8 9 10
Recomputation
m=O
1
2
3
4
5
6
7
8
9
10
0.7101 0.7486 0.7626 0.7707 0.7762 0.7803 0.7836 0.7862 0.7884 0.7904 0.7920
0.7484 0.7892 0.8039 0.8124 0.8183 0.8226 0.8260 0.8288 0.8312 0.8332 0.8350
0.7626 0.8039 0.8189 0.8276 0.8335 0.8379 0.8414 0.8443 0.8467 0.8487 0.8505
0.7707 0.8124 0.8276 0.8363 0.8423 0.8468 0.8503 0.8532 0.8556 0.8577 0.8595
0.7762 0.8183 0.8335 0.8423 0.8484 0.8529 0.8564 0.8593 0.8618 0.8639 0.8657
0.7803 0.8226 0.8379 0.8468 0.8529 0.8574 0.8610 0.8639 0.8663 0.8684 0.8703
0.7836 0.8260 0.8414 0.8503 0.8564 0.8610 0.8646 0.8675 0.8700 0.8721 0.8739
0.7862 0.8288 0.8443 0.8532 0.8593 0.8639 0.8675 0.8704 0.8729 0.8750 0.8769
0.7884 0.8312 0.8467 0.8556 0.8618 0.8663 0.8700 0.8729 0.8754 0.8775 0.8793
0.7904 0.8332 0.8487 0.8577 0.8639 0.8684 0.8721 0.8750 0.8775 0.87% 0.8815
0.7920 0.8350 0.8505 0.8595 0.8657 0.8703 0.8739 0.8769 0.8793 0.8815 0.8833
Figure 4 illustrates the asymptotic behavior anticipated for the higher-order Hermite-Gaussian laser-beam modes. As the order of the laser-beam mode increases to infinity, we see that the probabilities to detect photons within the corresponding classical limits asymptotically approach unity, although somewhat slower than anticipated for large-order modes, but as expected from the quantum oscillator problem and the Correspondence Principle. In particular, we see from Table 1 and Fig. 4 that fn,,, + 1 as m,n + m.
The solid dots in Fig. 4 are computed from equation (21), while the curve is a plot from m = 10 to 1000 of a nonlinear least-squares curve fit
060 .o
200
400
600
800
1000
Laser beam order(m) Fig. 4. Plot of the probability f*,,,,, of detecting photons within the classical limits of an Hermite-Gaussian laser beam. (Dots are integrated values; the solid curve is a nonlinear least-squares curve fit using the last five integrated values (m = 600 to lOOO).)
Harmonic oscillator fiducials
223
using the five largest integrated values (m = 600, 700, 800, 900, and 1000). For modes larger than m = 10, the curve fit exhibits a very good where the constants are deterpower-law dependence, fm,, = 1 -am-“, mined to be a = 0.259729 and b = 0.329946. Using this power-law curve fit, we can estimate the fractional probabilities for even larger-order modes although obs&vation and practical utility of laser-beam modes beyond 40-50 are unlikely. One might not expect the probabilities to increase as slowly as illustrated in Fig. 4, but for a two-dimensional oscillator or an Hermite-Gaussian laser-beam mode with m = 10” we use the power-law dependence to compute jH,, = 0.999999999679, which is close to unity as one would reasonably expect (although with some reservation of accuracy).
5 PROBABILITY
DENSITY NODES, THEOREM
PEAKS,
AND STURM’S
It is not entirely obvious that all probability density peaks of the quantum oscillator or of the large-order Hermite-Gaussian laser beams are contained within the corresponding classical limits. The Hermite functions y(x) = e-X”2H,(x) determine the nodes (zeros) of both the quantum oscillator densities and the Hermite-Gaussian laser-beam intensities for all modes. The nodes of orthogonal polynomials are all real, distinct, and lie within the interior of the orthogonality region.23 Figure 5 illustrates that the nodes of the Hermite functions also determine the nodes of the
0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -6
I
I
’
I -4
-2
0
2
4
6
Oscillator Position (4) Fig. 5. Quantum oscillator (The thick curve corresponds ing normalized
Hermite
probability
density
and Hermite
function
of order n = IO.
to the probability density. The light curve is the correspondfunction. Dashed lines correspond to the classical limits.)
224
S. L. Steely
quantum oscillator and the corresponding Hermite-Gaussian laser-beam modes. The orthogonality region of Hermite polynomials extends from minus infinity to plus infinity as seen from the integral eqn (5). The Hermite polynomials determine the location of y1 nodes of interest; however, the exponential factor in the Hermite functions dominate the polynomials at large x, driving the Hermite functions to zero at plus and minus infinity resulting in a total of y1+ 2 nodes. Some method, therefore, is desired that will provide a limit to the extent of the n primary nodes of the Hermite polynomials, the Hermite functions, and consequently the nodes of the probability densities of the quantum harmonic oscillator and of the Hermite-Gaussian laser-beam modes. Sturm’s classic work on differential forms and zeros of functions is one such method for analysis of the nodes of the Hermite functions. Sturm’s theorem2’ provides a useful method to determine the limits of the nodes in many functions, especially the classical orthogonal polynomials such as the Hermite polynomials. For two consecutive zeros x0, X,, of y(x), Sturm’s theorem:*’ Let f(x) be continuous and negative in x,,
(22)
where < 0, is concave for Y”lY1 > 0, is convex for
Ix]< (2n + 1)“’ lx]> (2n + l)“*
determines the concavity or convexity of the Hermite illustrated in Fig. 6 for orders n = 3 and yt = 4.
(23) functions
as
Harmonic oscillator fiducials
225
n=4 15 10 5 0 -5 -10
Fig. 6.
Concavity and convexity of the Hermite functions. [Thick curves denote Hermite functions: thin curves illustrate eqn (22) and eqn (23).]
the
Noting the regions of concavity and convexity [eqn (23)] for the Hermite functions as illustrated in Fig. 6, we see that the classical limits separate the concave and convex regions of the Hermite functions. One sees that the concave region lies between the classical limits while the convex regions lie outside of the classical limits. From Sturm’s theorem and for a given order, y remains nonzero within the regions of convexity and the sign of y is therefore constant, remaining either positive or negative for the Hermite functions. No local maximum or peak can occur for y (or y’) within the convex regions; a positive y would require a negative second derivative, y”, contradicting eqn (23) when 1x1> (2n + 1)“‘. A similar argument holds for any negative y in the convex regions. The classical limits always reside at inflection points and the peaks always reside within the concave region delimited by the classical limits. We can therefore assert that the beam waists for small- and large-order Hermite-Gaussian laser beams contain not only most of the laser-beam power or incident photons, but also contain all nodes and irradiance peaks, as expected from comparison with the quantum harmonic oscillator and the Correspondence Principle as diagrammatically illustrated in
S. L. Steely
226 7
.
‘)
I
Schrcdinger’s Equation
Maxwell’s Equations
c
, Wave Equation Helmholtz Equation
’
r Hermite-Gaussian Laser-Beam Mode
2-D Quantum Harmonic Oscillator
\
have Identical Functional Forms / Beam Waists
Classical Limits <
J&Y”
+ ‘From the Correspondence Principle, We Expect to Find the Oscillator Within the Classical Limits
Fig. 7.
\
, 7
’
w,(&,
w,(z),
/
# We Should Similarly Expect to Detect most Photons Within the “Classical Limits”
Flow diagram illustrating the correspondence between the quantum ... oscmator and Hermite-Gaussian laser-beam modes.
harmonic
Fig. 7. Although the probability densities are derived separately from the Schriidinger equation for the quantum harmonic oscillator and from Maxwell’s equations for the laser-beam modes, further assumptions and reductions lead to the wave equation and Helmholtz equation in both cases, hence the common link and origin of the functional similarity.
6 CONCLUDING
REMARKS
The ubiquitous harmonic oscillator is indeed a useful tool to help model physical systems and, as shown in this paper, to help clarify and better understand some aspects of the probability densities of
Harmonic oscillatorfiducials
227
Hermite-Gaussian laser beams. In particular, the probability densities for two-dimensional quantum harmonic oscillator modes are functionally identical to the photon probability densities of Hermite-Gaussian laserbeam modes. This functional similarity and the Correspondence Principle provide guidance to determine that the corresponding beam waists or ‘classical limits’ for Hermite-Gaussian laser beams define a spot size that contains a large portion of the laser beam’s incident photons and power. As computed with Mathematics, the fraction of the Hermite-Gaussian laser-beam power or photons contained within the beam waists asymptotically increases to unity exhibiting a power-law dependence as the laser-beam order increases to infinity. The classical limits and the corresponding laser-beam spot, as delimited by the beam waists, contain all nodes and probability-density peaks for both small- and large-order modes of the quantum oscillator as well as the Hermite-gaussian laser beam.
REFERENCES 1. Shankar, R., Principles of Quantum Mechanics. New York, Plenum Press, 1980, p. 195. 2. Steely, S. L., Using harmonic oscillators to determine the spot size of Hermite-Gaussian laser beams. NASA Workshop on Harmonic Oscillators, NASA Conf. Publication 3197, 1993, pp. 155-177. 3. Abramowitz, M. & Stegun, I. A., Handbook or Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Washington, DC, Government Printing Office, 1964, pp. 773-802. 4. Magnus, W., Oberhettinger, F. & Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics. New York, Springer, 1966, pp. 249-255. 5. Carter, W. H. Spot size and divergence for Hermite-Gaussian beams of any order. Appl. Opt., 19(7) (1980) 1027-1029. 6. Carter, W. H. Energy carried over the rectangular spot within a HermiteGaussian beam. Appl. Opt., 21(l) (1982) 7-9. 7. Luxon, J. T. & Parker, D. E. Practical spot size definition for single higher-order rectangular-mode laser beams. Appl. Opt., 20( 10) (1981) 17281729. 8. Luxon, J. T. & Parker, 9.
10. 11. 12.
D. E. Higher order CO* laser beam spot size and depth of focus determined. Appl. Opt., 20(11) (1981) 1933-1935. Wolfram, S., Mathematics A System for Doing Mathematics by Computer. Addison-Wesley, New York, 1991. Goldstein, H., Classical Mechanics. Addison-Wesley, Reading, MA, 1980. Condon, E. & Morse, P. M., Quantum Mechanics. McGraw-Hill, New York, 1929. Pauling, L. & Bright Wilson, E. Jr., Introduction to Quantum Mechanics. Reprinted 1965, by Dover Publications, New York, 1935.
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13. Rojansky, V., Introductory Quantum Mechanics. Prentice-Hall, Englewood Cliffs, NJ, 1938. 14. Messiah, A., Quantum Mechanics. Wiley, New York, 1958. 15. Leighton, R. B., Principles of Modern Physics. McGraw-Hill, New York, 1959. 16. Merzbacher, E., Quantum Mechanics. Wiley, New York, 1970. 17. Sakurai, J. J., Modern Quantum Mechanics. Benjamin-Cummings, Menlo Park, CA, 1985. 18. Gardiner, C. W., Quantum Noise. Springer, New York, 1991. 19. Schiff, L. I., Quantum Mechanics. McGraw-Hill, New York, 1968. 20. Kogelnik, H. & Li, T. Laser beams and Resonators. Proceedings of the IEEE, 54(10) (1966) 1312-1329.
21. Yariv, A., Introduction to Optical Electronics. Holt Rinehart and Winston, New York, 1971. 22. Tompkins, D. R. & Rodney, P. F. Six parameter Gaussian modes in free space. Phys. Rev. A, 12(2) (1975) 599-610. 23. Szego, G., Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, RI, 1975, pp. 18-21, 111-129.