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Anomalies in the field of a gaussian beam near focus
Volume
7, number
3
ANOMALIES
OPTICS
March
COMMUNICATIONS
IN THE FIELD OF A GAUSSIAN
1973
BEAM NEAR FOCUS
William H. CARTER Naval Research Laboratory, Received
Washington, D.C. 20390,
7 December
USA
1972
A theoretical analysis of the electromagnetic field of a focused gaussian beam, like that produced by some lasers, indicates anomalies in the field distribution over the focal region. The transverse component of the electric field is shown graphically in this region using data obtained through the numerical integration of an expression that is exact according to Maxwell’s equations. These data agree with the standard paraxial theories used to describe gaussian beams only asymtotically, in the limit of zero beam divergence about focus. As the divergence is increased, so that the beam is brought more sharply to focus, the gaussian beam evolves toward a dipole field from which all evanescent plane waves have been removed. Gaussian beams observed in nature should show some evidence of the associated anomalies, even if the beam is almost collimated.
The field of a gaussian beam, which is similar to the light beam produced by some lasers, has been studied extensively by the use of mathematical models which employ a paraxial approximation [ 11. More recently, a more precise description was found which indicates that gaussian beams should exhibit anomalies that were not previously recognized [2]. One such anomaly, a complicated spatial variation in the field near focus, is discussed in this paper. In an earlier paper, the electromagnetic field of a gaussian beam in free space was described through the use of a modal expansion into an angular spectrum of plane waves [2]. Expressions involving surface integrals were obtained for the rectangular components of the electric and magnetic fields that are exact according to Maxwell’s equations. Relations describing the field over the region away from focus were obtained by carrying out the necessary integrations asymptotically in the limit of large kd, where k is the propagation constant, and d is the distance from the focus to the field point. However, similar relations describing the field in the focal region were determined only after making the conventional paraxial approximation. To obtain a more exact description of the field near focus, the component of the electric field transverse to the direction of propagation has now been evaluated through the use of numerical integration. In formulating
model, it is assumed that the beam propagates along the positive z axis and is focused in the z = 0 plane. We restrict the analysis to a common type of plane-polarized gaussian beam for which the transverse component of the electric field is symmetric in amplitude about the z axis. For this case, the surface integral - cf., ref. [2], eqs. (1) and (7) ~ can be reduced to the Hankel transform, this
E,(r,z)=~
j
exp(-k202p2/2)
0
x exp [ik( 1-P~)“~z]
J 0 (kpr)pdp,
(1)
where r = (x2 +~*)l’~ , u is a constant which characterizes the divergence of the beam as it passes through focus, et is a constant which determines the maximum amplitude of the field, and the eeiwt time dependence of this monochromatic field has been suppressed. The integral appearing in eq. (1) was evaluated using the trapezoidal rule to sum the kernel over 200, equally spaced points within the range 0 G p S 1. A difficulty is encountered asp approaches the upper limit due to an increasingly fast variation of the kernel. This difficulty is overcome by the use of a suitable transformation of the variable of integration as discussed previously [3] The field amplitude, calculated in this manner, is shown here through the use of maps of constant magni211
Volume
7, number
3
OPTICS
COMMUNICATIONS
tude and phase. In figs. 1 and 2 the amplitude of the transverse electric field is shown for o/A = 2.0, where X is the wavelength *a **. The transverse electric field, as given by eq. (l), is clearly conjugate symmetric with respect to the z = 0 plane and symmetric about the z axis so that the data given in these figures are sufficient to visualize the field in three dimensions over the region about the focal point. The transverse electric field for a beam with o/h = 2.0, as shown in figs. 1 and 2, appears to agree with the usual theories for a gaussian beam. The magnitude of the beam decays with increasing r as an approximate gaussian function, and the beam decays slowly away from focus with increasing 1.~1.The wave fronts, as shown in fig. 2, are plane in the focal plane but become sperical as Iz 1increases. The beam that is represented in figs. 1 and 2 and appears to agree with the usual theories, is almost collimated The divergence of a gaussian beam from focus can be defined, in following manner, as a function of the parameter a/X. If the radius of the beam d(z) is defined as the value of r with fixed z for which the magnitude of the transverse electric field component decays monotonically from the maximum value at r = 0 to l/e% of that value, then according to the usual paraxial beam theories the parameter u is numerically equal to the minimum beam radius d(0) which occurs at focus. However, as we shall prove shortly, the actual minimum radius d(O) is always larger than u, and d(O) only approaches u closely as u/h becomes large. Thus u does not rigorously have this meaning. But if the beam boundary is defined in the usual manner as the surface generated
* These amplitude
data were renormalized before plotting to simplify comparison. The loci of constant amplitude in figs. 1, 3, 5, and 7 indicate values given by the formula Et = 1.0 exp[ -0.25(N-l)] , for integer values of N. The value of N for any contour line in these figures can be determined by starting with N = I nearest the origin and counting contours along any path of monotonically decreasing amplitude to N = 30 near the nulls. The amplitude values for some of the contours are indicated on the figures for convenience. ** The contours in figs. 2, 4, 6, and 8 represent loci of constant phase in the field. The phase interval between the narrow contour lines is 30’ and between the wide lines is 360° (i.e., it corresponds to an interval of one wavelength). The abrupt jumps in the wide contour lines are computational artifacts and not a property of the field.
212
March
1973
by the locus of all beam radii about the z axis, then away from focus the beam boundary becomes asymptotic to a right, circular cone with the apex at the origin and making a half angle, B = tan-’ (h/2nu),
(2)
with the z axis [2]. Thus, o/h always specifies the divergence of the beam about focus according to eq. (2). Using this formula, the beam described by the data in figs. 1 and 2 diverges by only 0 = 4.56” from focus and is therefore, almost collimated. As u/h is made smaller, the divergence of the beam becomes greater, and the field near focus becomes increasingly anomalous. In figs. 3 and 4 the transverse electric field near focus is shown for u/h = 1.O. In the figures, this field is clearly not in agreement with the usual theories for a gaussian beam. The beam divergence is increased only to u = 9.12” according to eq. (2) but already sharp anomalies appear. The magnitude of the transverse electric field vanishes along an annulus encircling the beam axis for a value of constant z = + 3.5 h on each side of the focal plane. Along these annuli, the phase of the field varies rapidly and in such a manner that a wavefront appears to terminate. As o/X is decreased to 0.5 so that the beam divergence becomes 0 = 17.8”, the data in figs. 5 and 6 show that more annuli of zero field magnitude appear near focus and the distribution is even less like that given by the paraxial theory. For u/X = 0.1 (0 = 57X3”), the tangential electric components of the field cease to resemble a gaussian beam but resemble, instead, the field of a scalar dipole from which all evanescent plane waves have been removed. This modified scalar dipole field, appearing in figs. 7 and 8, has been studied previously (cf., ref. [3]). The corresponding transverse component of the electric field (i.e., Et 1 z axis) for a magnetic dipole [4] oscillating in the X_Yplane has this amplitude dependence if all evanescent plane waves are removed. From eq. (1) we find by direct integration that the field in the z = 0 plane is given by Et(r, 0) - (k2et/277)J,(kr)/kr,
as u/X -+ 0.
(3)
From the figures, we observe that the annuli of zero magnitude for the transverse electric field of the gaussian
Volume
7, number
OPTICS
3
March
COMMUNICATIONS
1973
OX
IA
2x X I 3x
4x
/
_.
-.
^I
^\
I,
Fig. 1. The magnitude of the transverse electric field of a gaussian beam with o/h = 2.0, which is focused at the origin. The contours represent loci of constant magnitude in the (Y = 0) plane. The magnitude intervals between contours are noteequal but decrease ex, p. 21 ponentially from the peak value of the field at the origin. The format of this figure is described in footnote
OA
I I
4X
L
L
Fig. 2. The phase of the transverse electric field of a gaussian beam with o/h = 2.0 corresponding to the magnitudes shown in fig. 1. The contours represent loci of constant phase in the Cy = 0) plane. The phase interval between the narrow contours is 30°, $2d be, p. 212. tween the wide contours is 360’ (i.e., it corresponds to one wavelength). The format of this figure is described in footnote
213
Volume
5x
7, number
OX
Fig. 3. The magnitude
3
IX
OPTICS
3x
2x
of the transverse
electric
COMMUNICATIONS
4A
March
6X
5x
field with a/h = 1.0. The format
of this figure is identical
IOA
9x
8X
7A
1973
to that of fig. 1.
ox-
IX -
2x
-
X I 3x -
4x
5x-
-
5X
01
6X
7x
8X
91
IOX
z-
Fig. 4. The phase of the transverse is identical to that of fig. 2.
214
electric
field with o/h = 1.0, corresponding
to the magnitude
in fig. 3. The format
of this figure
Volume
7, number
3
OPTICS COMMUNICATIONS
March
1973
OX
IX
-2
X
I 31
4x
5x Ok
IA
21
3A
5A
6X
71
9A
ZFig. 5. The magnitude
of the transverse
Fig. 6. The phase of the transverse is identical to that of fig. 2.
electric
electric
field with o/h = 0.5. The format
field with u/h = 0.5, corresponding
of this figure is identical
to the magnitude
to that of fig. 1
in fig. 5. The format
of this figure
21.5
Volume
7, number
‘7. The magnitude
3
OPTICS
of the transverse
Fig. 8. The phase of the transverse is identical to that of fig. 2. 216
electric
electric
COMMUNICATIONS
field with o/h = 0.1. The format
field with u/h = 0.1
, corresponding
March
of this figure is identical
to the magnitude
to that of fig. 1.
in fig. 7. The format
of this figure
1973
Volume 7, number 3
March 1973
OPTICS COMMUNICATIONS
beam evolve with decreasing a/h into the axis crossing zeros in eq. (3) and that the phase changes so as to keep the field in the z = 0 plane real. Thus the anomalies which appear in the gaussian field are characteristic of this dipole field. It is interesting to note that the modified scalar dipole field, appearing in figs. 7 and 8, is very closely related to the image of a point source formed by a perfect lens system [5]. By comparison of fig. 7 with figs. 1, 3, and 5, we observe that the minimum beam radius (at focus) is greater than u for o/h = 0.1, but approaches it closely as o/X increases. This is in agreement with the general rule, cited previously, that E,(r,O) = ( l/eX)Et(O,O) for some r > u but that this value of r approaches u asymptotically as u/h becomes large. To prove this rule, we need only recognize that EJr, o) according to eq. (1) is the two-dimensional Fourier transform of a cylindrically symmetric gaussian function which is truncated by the finite upper limit of integration, and then recall the effect this truncation has on the standard deviation of E&r,O) according to elementary Fourier theory. It should be emphasized that this finite upper limit of integration, which appears in eq. (1) and gives rise mathematically to the anomalies in the figure, does not result from truncation of the beam due to a physical aperture. Indeed, in formulating eq. (1) it was assumed that the beam propagates in free space with no matter save its sources which are continuously disturbed over the plane z = zs, zs < 0. This finite upper limit is, instead, an intrinsic property of Maxwell’s equations in free space which has been discussed in an earlier paper [2] To properly describe the electromagnetic fields associated with a gaussian beam over the focal region, the other components in addition to the transverse electric field are generally required. However, for sufficiently large u/X (i.e., u/h % l/271 so that 0 Q 45”), the approximate expressions derived previously - cf., ref. [2], eq. (10) - show that the longitudinal components of the electric and magnetic fields over the region inside the beam (i.e., where r 2 a) are much smaller than the transverse components and can be neglected. These expressions also show that the transverse component of the magnetic field is approximately proportional in amplitude to the transverse component of the electric field over this region. Thus, in figs. 1 through 6 where u/h % 1/2n, the data shown is sufficient to characterize the electromagnetic field. As u/X approaches zero, however, the behavior of the
lognitudinal electric and the magnetic components of the field must be properly taken into account. These components are not symmetrical about the z axis for small u/h, but are distributed in a much more complicated manner. From the data in figs. 7 and 8 it is evident that the field is somehow related to that of a magnetic dipole. This relationship can be clarified by comparing the fields through their angular spectra. If the electromagnetic field for this beam is represented by an expansion of the transverse electric component (which is the component parallel to the xy plane) into an angular spectrum of plane waves, then it is uniquely specified by the plane wave amplitudes A@,q), where p and 4 are direction cosines describing the direction of propagation for a particular plane wave relative to the x and y axes respectively - cf., ref. [2] , eqs. (1) and (7). In the limit as u/X approaches zero these amplitudes are all equal to the constants A($, q) - p,/h2, if p2 + q2 < 1, (4) A(g,q)=i
,ifp2+q2>1.
In the same manner, the electromagnetic field for a magnetic dipole at the origin with a moment M polarized in the xy plane is uniquely specified by the plane wave amplitudes A’@, q) = -(ik3/2n)lMl,
for all p and 4.
(5)
If we compare eq. (4) with eq. (5) and take due account of a requirement that these two fields be similarly polarized, then it is immediately evident that the gaussian beam described here evolves with decreasing u/h into a magnetic dipole field from which all evanescent plane waves (i.e., those for which p2 + q2 > 1) have been removed. It should be mentioned that a different, closely related, gaussian beam may be defined through an angular spectrum expansion of the transverse magnetic field component rather than the transverse electric field component as given here in eq. (1). For sufficiently large u/X these two types of beams are nearly identical. However, as o/X is made smaller it is the transverse magnetic component rather than the transverse electric component that now remains symmetric about the z axis and obeys eq. (1). The characteristics of the electric and magnetic fields are exchanged in the same manner as in 217
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OPTICS COMMUNICATIONS
multipole theory. This second type of beam evolves with decreasing o/h into an electric dipole field from which all evanescent plane waves have been removed. Thus, we will name it an electric beam, as opposed to the beam treated here which we will name a magnetic beam. This beam duality is interesting, per se, and is currently under investigation. From this analysis it appears that the ideal gaussian beam as described by the conventional theories (cf., ref. [l] ) exists only as an asymptotic approximation in the limit of large o/h, and that a gaussian beam found in nature, with finite o/h, should exhibit the anomalies described here. These anomalies are seen to arise due to the partial evolution of the beam toward a dipole field. The anomalies should be observed experimentally as a modification in the intensitv distribution at the focus of a laser beam. The focal spot will not be a perfect gaussian, will have a slightly larger diameter, and will have a system of concentric encircling rings somewhat like an Airy disk but of weaker intensity. This is not related to the modification in the focal spot which oc-
218
March
1973
curs if the beam is truncated by a lens aperture but is somewhat similar in appearance. Thus, care must be taken to avoid truncation effects if the, more fundamental, effects described here are to be observed. The effect: due to these basic anomalies will be weak for well collimated beams but should become observable if the beam divergence about focus is made sufficiently large. The author would like to thank Dr. D.L. Weinberg for his assistance in proof reading the manuscript, and Professor E. Wolf for several helpful comments.
References [ 1) [2] [ 31 [4]
H. Kogelnik and T. Li, Proc. IEEE 54 (1966) 1312. W.H. Carter, J. Opt. Sot. Am., to be published. W.H. Carter, Opt. Commun. 2 (1970) 142. J.D. Jackson, Classical electrodynamics (Wiley, New York,
1967‘)n -__‘I r- 771. -‘1!5] M. Born and E. Wolf, Principles of optics, mon Press, New York, 1970) p. 435.
4th Ed. (Perga
7, number
3
ANOMALIES
OPTICS
March
COMMUNICATIONS
IN THE FIELD OF A GAUSSIAN
1973
BEAM NEAR FOCUS
William H. CARTER Naval Research Laboratory, Received
Washington, D.C. 20390,
7 December
USA
1972
A theoretical analysis of the electromagnetic field of a focused gaussian beam, like that produced by some lasers, indicates anomalies in the field distribution over the focal region. The transverse component of the electric field is shown graphically in this region using data obtained through the numerical integration of an expression that is exact according to Maxwell’s equations. These data agree with the standard paraxial theories used to describe gaussian beams only asymtotically, in the limit of zero beam divergence about focus. As the divergence is increased, so that the beam is brought more sharply to focus, the gaussian beam evolves toward a dipole field from which all evanescent plane waves have been removed. Gaussian beams observed in nature should show some evidence of the associated anomalies, even if the beam is almost collimated.
The field of a gaussian beam, which is similar to the light beam produced by some lasers, has been studied extensively by the use of mathematical models which employ a paraxial approximation [ 11. More recently, a more precise description was found which indicates that gaussian beams should exhibit anomalies that were not previously recognized [2]. One such anomaly, a complicated spatial variation in the field near focus, is discussed in this paper. In an earlier paper, the electromagnetic field of a gaussian beam in free space was described through the use of a modal expansion into an angular spectrum of plane waves [2]. Expressions involving surface integrals were obtained for the rectangular components of the electric and magnetic fields that are exact according to Maxwell’s equations. Relations describing the field over the region away from focus were obtained by carrying out the necessary integrations asymptotically in the limit of large kd, where k is the propagation constant, and d is the distance from the focus to the field point. However, similar relations describing the field in the focal region were determined only after making the conventional paraxial approximation. To obtain a more exact description of the field near focus, the component of the electric field transverse to the direction of propagation has now been evaluated through the use of numerical integration. In formulating
model, it is assumed that the beam propagates along the positive z axis and is focused in the z = 0 plane. We restrict the analysis to a common type of plane-polarized gaussian beam for which the transverse component of the electric field is symmetric in amplitude about the z axis. For this case, the surface integral - cf., ref. [2], eqs. (1) and (7) ~ can be reduced to the Hankel transform, this
E,(r,z)=~
j
exp(-k202p2/2)
0
x exp [ik( 1-P~)“~z]
J 0 (kpr)pdp,
(1)
where r = (x2 +~*)l’~ , u is a constant which characterizes the divergence of the beam as it passes through focus, et is a constant which determines the maximum amplitude of the field, and the eeiwt time dependence of this monochromatic field has been suppressed. The integral appearing in eq. (1) was evaluated using the trapezoidal rule to sum the kernel over 200, equally spaced points within the range 0 G p S 1. A difficulty is encountered asp approaches the upper limit due to an increasingly fast variation of the kernel. This difficulty is overcome by the use of a suitable transformation of the variable of integration as discussed previously [3] The field amplitude, calculated in this manner, is shown here through the use of maps of constant magni211
Volume
7, number
3
OPTICS
COMMUNICATIONS
tude and phase. In figs. 1 and 2 the amplitude of the transverse electric field is shown for o/A = 2.0, where X is the wavelength *a **. The transverse electric field, as given by eq. (l), is clearly conjugate symmetric with respect to the z = 0 plane and symmetric about the z axis so that the data given in these figures are sufficient to visualize the field in three dimensions over the region about the focal point. The transverse electric field for a beam with o/h = 2.0, as shown in figs. 1 and 2, appears to agree with the usual theories for a gaussian beam. The magnitude of the beam decays with increasing r as an approximate gaussian function, and the beam decays slowly away from focus with increasing 1.~1.The wave fronts, as shown in fig. 2, are plane in the focal plane but become sperical as Iz 1increases. The beam that is represented in figs. 1 and 2 and appears to agree with the usual theories, is almost collimated The divergence of a gaussian beam from focus can be defined, in following manner, as a function of the parameter a/X. If the radius of the beam d(z) is defined as the value of r with fixed z for which the magnitude of the transverse electric field component decays monotonically from the maximum value at r = 0 to l/e% of that value, then according to the usual paraxial beam theories the parameter u is numerically equal to the minimum beam radius d(0) which occurs at focus. However, as we shall prove shortly, the actual minimum radius d(O) is always larger than u, and d(O) only approaches u closely as u/h becomes large. Thus u does not rigorously have this meaning. But if the beam boundary is defined in the usual manner as the surface generated
* These amplitude
data were renormalized before plotting to simplify comparison. The loci of constant amplitude in figs. 1, 3, 5, and 7 indicate values given by the formula Et = 1.0 exp[ -0.25(N-l)] , for integer values of N. The value of N for any contour line in these figures can be determined by starting with N = I nearest the origin and counting contours along any path of monotonically decreasing amplitude to N = 30 near the nulls. The amplitude values for some of the contours are indicated on the figures for convenience. ** The contours in figs. 2, 4, 6, and 8 represent loci of constant phase in the field. The phase interval between the narrow contour lines is 30’ and between the wide lines is 360° (i.e., it corresponds to an interval of one wavelength). The abrupt jumps in the wide contour lines are computational artifacts and not a property of the field.
212
March
1973
by the locus of all beam radii about the z axis, then away from focus the beam boundary becomes asymptotic to a right, circular cone with the apex at the origin and making a half angle, B = tan-’ (h/2nu),
(2)
with the z axis [2]. Thus, o/h always specifies the divergence of the beam about focus according to eq. (2). Using this formula, the beam described by the data in figs. 1 and 2 diverges by only 0 = 4.56” from focus and is therefore, almost collimated. As u/h is made smaller, the divergence of the beam becomes greater, and the field near focus becomes increasingly anomalous. In figs. 3 and 4 the transverse electric field near focus is shown for u/h = 1.O. In the figures, this field is clearly not in agreement with the usual theories for a gaussian beam. The beam divergence is increased only to u = 9.12” according to eq. (2) but already sharp anomalies appear. The magnitude of the transverse electric field vanishes along an annulus encircling the beam axis for a value of constant z = + 3.5 h on each side of the focal plane. Along these annuli, the phase of the field varies rapidly and in such a manner that a wavefront appears to terminate. As o/X is decreased to 0.5 so that the beam divergence becomes 0 = 17.8”, the data in figs. 5 and 6 show that more annuli of zero field magnitude appear near focus and the distribution is even less like that given by the paraxial theory. For u/X = 0.1 (0 = 57X3”), the tangential electric components of the field cease to resemble a gaussian beam but resemble, instead, the field of a scalar dipole from which all evanescent plane waves have been removed. This modified scalar dipole field, appearing in figs. 7 and 8, has been studied previously (cf., ref. [3]). The corresponding transverse component of the electric field (i.e., Et 1 z axis) for a magnetic dipole [4] oscillating in the X_Yplane has this amplitude dependence if all evanescent plane waves are removed. From eq. (1) we find by direct integration that the field in the z = 0 plane is given by Et(r, 0) - (k2et/277)J,(kr)/kr,
as u/X -+ 0.
(3)
From the figures, we observe that the annuli of zero magnitude for the transverse electric field of the gaussian
Volume
7, number
OPTICS
3
March
COMMUNICATIONS
1973
OX
IA
2x X I 3x
4x
/
_.
-.
^I
^\
I,
Fig. 1. The magnitude of the transverse electric field of a gaussian beam with o/h = 2.0, which is focused at the origin. The contours represent loci of constant magnitude in the (Y = 0) plane. The magnitude intervals between contours are noteequal but decrease ex, p. 21 ponentially from the peak value of the field at the origin. The format of this figure is described in footnote
OA
I I
4X
L
L
Fig. 2. The phase of the transverse electric field of a gaussian beam with o/h = 2.0 corresponding to the magnitudes shown in fig. 1. The contours represent loci of constant phase in the Cy = 0) plane. The phase interval between the narrow contours is 30°, $2d be, p. 212. tween the wide contours is 360’ (i.e., it corresponds to one wavelength). The format of this figure is described in footnote
213
Volume
5x
7, number
OX
Fig. 3. The magnitude
3
IX
OPTICS
3x
2x
of the transverse
electric
COMMUNICATIONS
4A
March
6X
5x
field with a/h = 1.0. The format
of this figure is identical
IOA
9x
8X
7A
1973
to that of fig. 1.
ox-
IX -
2x
-
X I 3x -
4x
5x-
-
5X
01
6X
7x
8X
91
IOX
z-
Fig. 4. The phase of the transverse is identical to that of fig. 2.
214
electric
field with o/h = 1.0, corresponding
to the magnitude
in fig. 3. The format
of this figure
Volume
7, number
3
OPTICS COMMUNICATIONS
March
1973
OX
IX
-2
X
I 31
4x
5x Ok
IA
21
3A
5A
6X
71
9A
ZFig. 5. The magnitude
of the transverse
Fig. 6. The phase of the transverse is identical to that of fig. 2.
electric
electric
field with o/h = 0.5. The format
field with u/h = 0.5, corresponding
of this figure is identical
to the magnitude
to that of fig. 1
in fig. 5. The format
of this figure
21.5
Volume
7, number
‘7. The magnitude
3
OPTICS
of the transverse
Fig. 8. The phase of the transverse is identical to that of fig. 2. 216
electric
electric
COMMUNICATIONS
field with o/h = 0.1. The format
field with u/h = 0.1
, corresponding
March
of this figure is identical
to the magnitude
to that of fig. 1.
in fig. 7. The format
of this figure
1973
Volume 7, number 3
March 1973
OPTICS COMMUNICATIONS
beam evolve with decreasing a/h into the axis crossing zeros in eq. (3) and that the phase changes so as to keep the field in the z = 0 plane real. Thus the anomalies which appear in the gaussian field are characteristic of this dipole field. It is interesting to note that the modified scalar dipole field, appearing in figs. 7 and 8, is very closely related to the image of a point source formed by a perfect lens system [5]. By comparison of fig. 7 with figs. 1, 3, and 5, we observe that the minimum beam radius (at focus) is greater than u for o/h = 0.1, but approaches it closely as o/X increases. This is in agreement with the general rule, cited previously, that E,(r,O) = ( l/eX)Et(O,O) for some r > u but that this value of r approaches u asymptotically as u/h becomes large. To prove this rule, we need only recognize that EJr, o) according to eq. (1) is the two-dimensional Fourier transform of a cylindrically symmetric gaussian function which is truncated by the finite upper limit of integration, and then recall the effect this truncation has on the standard deviation of E&r,O) according to elementary Fourier theory. It should be emphasized that this finite upper limit of integration, which appears in eq. (1) and gives rise mathematically to the anomalies in the figure, does not result from truncation of the beam due to a physical aperture. Indeed, in formulating eq. (1) it was assumed that the beam propagates in free space with no matter save its sources which are continuously disturbed over the plane z = zs, zs < 0. This finite upper limit is, instead, an intrinsic property of Maxwell’s equations in free space which has been discussed in an earlier paper [2] To properly describe the electromagnetic fields associated with a gaussian beam over the focal region, the other components in addition to the transverse electric field are generally required. However, for sufficiently large u/X (i.e., u/h % l/271 so that 0 Q 45”), the approximate expressions derived previously - cf., ref. [2], eq. (10) - show that the longitudinal components of the electric and magnetic fields over the region inside the beam (i.e., where r 2 a) are much smaller than the transverse components and can be neglected. These expressions also show that the transverse component of the magnetic field is approximately proportional in amplitude to the transverse component of the electric field over this region. Thus, in figs. 1 through 6 where u/h % 1/2n, the data shown is sufficient to characterize the electromagnetic field. As u/X approaches zero, however, the behavior of the
lognitudinal electric and the magnetic components of the field must be properly taken into account. These components are not symmetrical about the z axis for small u/h, but are distributed in a much more complicated manner. From the data in figs. 7 and 8 it is evident that the field is somehow related to that of a magnetic dipole. This relationship can be clarified by comparing the fields through their angular spectra. If the electromagnetic field for this beam is represented by an expansion of the transverse electric component (which is the component parallel to the xy plane) into an angular spectrum of plane waves, then it is uniquely specified by the plane wave amplitudes A@,q), where p and 4 are direction cosines describing the direction of propagation for a particular plane wave relative to the x and y axes respectively - cf., ref. [2] , eqs. (1) and (7). In the limit as u/X approaches zero these amplitudes are all equal to the constants A($, q) - p,/h2, if p2 + q2 < 1, (4) A(g,q)=i
,ifp2+q2>1.
In the same manner, the electromagnetic field for a magnetic dipole at the origin with a moment M polarized in the xy plane is uniquely specified by the plane wave amplitudes A’@, q) = -(ik3/2n)lMl,
for all p and 4.
(5)
If we compare eq. (4) with eq. (5) and take due account of a requirement that these two fields be similarly polarized, then it is immediately evident that the gaussian beam described here evolves with decreasing u/h into a magnetic dipole field from which all evanescent plane waves (i.e., those for which p2 + q2 > 1) have been removed. It should be mentioned that a different, closely related, gaussian beam may be defined through an angular spectrum expansion of the transverse magnetic field component rather than the transverse electric field component as given here in eq. (1). For sufficiently large u/X these two types of beams are nearly identical. However, as o/X is made smaller it is the transverse magnetic component rather than the transverse electric component that now remains symmetric about the z axis and obeys eq. (1). The characteristics of the electric and magnetic fields are exchanged in the same manner as in 217
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OPTICS COMMUNICATIONS
multipole theory. This second type of beam evolves with decreasing o/h into an electric dipole field from which all evanescent plane waves have been removed. Thus, we will name it an electric beam, as opposed to the beam treated here which we will name a magnetic beam. This beam duality is interesting, per se, and is currently under investigation. From this analysis it appears that the ideal gaussian beam as described by the conventional theories (cf., ref. [l] ) exists only as an asymptotic approximation in the limit of large o/h, and that a gaussian beam found in nature, with finite o/h, should exhibit the anomalies described here. These anomalies are seen to arise due to the partial evolution of the beam toward a dipole field. The anomalies should be observed experimentally as a modification in the intensitv distribution at the focus of a laser beam. The focal spot will not be a perfect gaussian, will have a slightly larger diameter, and will have a system of concentric encircling rings somewhat like an Airy disk but of weaker intensity. This is not related to the modification in the focal spot which oc-
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curs if the beam is truncated by a lens aperture but is somewhat similar in appearance. Thus, care must be taken to avoid truncation effects if the, more fundamental, effects described here are to be observed. The effect: due to these basic anomalies will be weak for well collimated beams but should become observable if the beam divergence about focus is made sufficiently large. The author would like to thank Dr. D.L. Weinberg for his assistance in proof reading the manuscript, and Professor E. Wolf for several helpful comments.
References [ 1) [2] [ 31 [4]
H. Kogelnik and T. Li, Proc. IEEE 54 (1966) 1312. W.H. Carter, J. Opt. Sot. Am., to be published. W.H. Carter, Opt. Commun. 2 (1970) 142. J.D. Jackson, Classical electrodynamics (Wiley, New York,
1967‘)n -__‘I r- 771. -‘1!5] M. Born and E. Wolf, Principles of optics, mon Press, New York, 1970) p. 435.
4th Ed. (Perga