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Spatial parametric characterization of general polychromatic light beams
cm .-__ @
ELSEVIER
1 October
1997
OPTICS COMMUNICATIONS Optics Communications
142 (1997) 135-145
Full length article
Spatial parametric characterization of general polychromatic beams
light
Qing Cao, Ximing Deng National Laboratory
on High Power Laser and Physics, Shanghai
Institute
800-21 I, Shanghai
Received 30 October
ofOptics
201800,
1996; revised 6 February
and Fine Mechanics,
Academia
Sinica, P.O. Box
China
1997; accepted
18 April 1997
Abstract The ABCD law and the M’ factor are generalized for the propagation of general polychromatic light beams through first-order optical systems by introducing a generalized complex radius of curvature Q, an average wavelength x and a characterization width Fc. The real part and the imaginary part of the parameter l/Q are related to the effective radius of curvature and the characterization width W, respectively. The characterization width YFc and the generalized M2 factor are shown to be appropriate for describing the propagation capability and the beam quality of a general polychromatic light beam respectively. Moreover, the relation M2 2 1 is deduced, a novel approach for interpreting the physical meaning of the effective radius of curvature is presented, and several special examples such as the polychromatic diffraction-limited light beams are analysed. 0 1997 Elsevier Science B.V.
1. Introduction In the past few years, the intensity-moments theory and the M’ factor have been revealed to be very useful to describe the propagation law and the beam quality of stationary light beams [ 1- 121. This theory was generalized for quasi-monochromatic pulsed light beams using the tensorial method three years ago [ 131. However, those results obtained in Ref. [13] are only valid for pulsed light beams which fulfilled the quasi-monochromatic condition Iv - v,] x v, (where v, is the center frequency or the carrier frequency), and are not valid for general polychromatic pulsed light beams, especially not valid for ultrabroadband pulsed light beams, because the tensorial method is established on the basis of the space-time analogy [14,15] and the generalized spatio-temporal diffraction integral [16] which only apply to quasi-monochromatic pulsed light beams. On the other hand, the research on ultrashort pulse lasers has made notable progress since the appearance of the novel ultrabroadband solid-state laser material. At present, sub-l 0-fs ultrashort pulse lasers can be obtained 0030-4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SOO30-40 18(97)00222-8
using Ti:sapphire laser material in many laboratories 1171. It is apparent that the spatio-temporal tensorial method is invalid for describing the propagation law of these sub-lofs pulse lasers, because, in these cases, the quasi-monochromatic condition 1v - v,l +z v, is not satisfied. Moreover, the spatio-temporal tensorial method is also invalid for describing the propagation of polychromatic light beams emitted from the so-called multi-wavelength laser devices which can oscillate at several different frequencies at the same time, such as the polychromatic light beams emitted from the so-called “white-light” laser devices. The existence of sub-lo-fs ultrashort pulsed laser beams, multiwavelength laser beams and other polychromatic light beams which do not fulfil the quasi-monochromatic condition shows that it is very necessary to generalize the intensity moment theory and the M2 factor for general polychromatic light beams, especially for those ultrabroadband (pulsed and CW) light beams. Recently, in Ref. [ 181 an attempt has been made to extend the stationary intensity-moments method to general polychromatic pulsed light beams, and to define the overall squared beam width and the overall squared far-field diver-
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Optics Communications
gence. However. as we shall prove, in Ref. [18] the wrong conclusion was drawn that “for general pulsed beams, the paraxial approach does not ensure that such (i.e. ABCD) propagation laws are fulfilled”, because the definition of the overall squared far-field divergence used there is not appropriate. In this paper, we shall more accurately analyse the propagation characteristics of general polychromatic light beams through first-order optical systems which do not include any dispersive element and thoroughly generalize the intensity moment theory and the M2 factor for the general polychromatic light beams using the Fourier transform method. The paper is organized as follows: in Section 2, we generalize the ABCD law for the propagation of general polychromatic light beams through first-order optical systems which do not have any dispersive element by introducing the generalized complex radius of curvature Q; in Section 3, we define the characterization width Wc and generalize the M2 factor for general polychromatic light beams, and interpret their physical meaning; in Section 4, we present a new method for interpreting the effective radius of curvature; in Section 5, we analyse several special examples such as polychromatic diffraction-limited light beams; and in Section 6, we conclude this paper. For simplicity, in this paper, we shall limit our analysis to the situation of one-dimensional, (x, z,) propagation.
2. ABCD propagation light beams
law of general
polychromatic
Let us consider a general polychromatic pulsed light beam which is represented by a real function 4( x,z,t> at a transverse plane which is perpendicular to the z-axis [ 191. In the sequel, we shall consider the z dependence, but we shall for simplicity drop it from the notations. For this pulsed light beam, we can introduce an area energy density P(x) P(s)=/‘
--P
4’(x,t)dt,
(1)
where +2(x,f) is the instantaneous light intensity. Obviously, the area energy density P(x), which represents the amount of energy which crosses a unit area in the transverse plane in the whole light pulse, is a measurable quantity. In order to simplify the following demonstrations, we assume that the z-axis is the propagation axis of the “center” of the pulsed light beam, namely _ x=
squared beam width (i.e. the beam radius width), in terms of the area energy density, as ~2~4
Z x2P( x) dx, / -r
(3)
where it is assumed that the total energy in the whole pulse is unity, namely lx P(x)dx=j/l --cc
--r
+‘(x,t)dxdt=
1.
(4)
It is well known that a general pulsed beam $(x,l) may be expressed as a superposition of monochromatic waves of different frequency, namely [19] 4(x,f)
= IX p(x,v)exp(-i2rvt)dv. -X
(5)
Since $(x,t) is real, cp(x,- V) = cp*(x,u). It is important to point out that, in this paper, the discrete frequency spectral polychromatic (or monochromatic) light beams are included in the general polychromatic pulsed light beams, because, from the point of view of the Fourier transform theory, the case of discrete frequency spectral light beams can be regarded as a special case of continuous frequency spectra1 light beams. That is to say, the results deduced from the polychromatic pulsed light beams in this paper are also valid for the discrete frequency spectra1 light beams. In terms of the Parseval’s theorem in the Fourier transform theory, the area energy density P(x) can be rewritten as P(x)
=/=
-cc
Substituting
/~(x,v)~‘dv=2/nil~(x,i)j2d~.
this result into Eq. (3), one can obtain
W2 = 8/DzjX x21’p(x,u)12 Pm
dvdx.
(6)
The propagation property of any monochromatic component cp(x,v) through a first-order optical system which does not include any dispersive element is described by Huygens’s integral in the Kirchhoff-Fresnel approximation [201
i7rv
Xexp
- E(A”: L
-2x,x,+Dx;)
dx,, I
(7)
cc / --x
142 (19971 135- 145
xP(x)dx=O
(2)
in any transverse plane. Then it is reasonable to define the
where CL is the velocity of light in free space, and the ABCD parameters are the ray’s matrix elements of the first-order optical system located between the transverse
137
Q. Cao, X. Deng/Optics Communications 142 (1997) 135-145 plane 1 and the transverse plane 2. From Eq. (7), one can obtain (see Appendix A.1) W,2=A2Wf+2ABV,+B21J,,
(8)
V, = ACW,2 + (AD + BC)V,
+ BDU,,
(9)
lJ2=C2W,2+2DCV,+D2U,,
polychromatic light beam through a first-order optical system which does not have any dispersive element. Let us now investigate the physical meaning of the parameters V and U from their propagation properties in free space. In free space, V(z) and U(z) satisfy the following relations
(10) 1 aw’(z)
where
V(z)=27:
(11)
dW(z) U(z)
(12) where cp,
2av -i~L(x,v)
= ‘p,(x,v)exp
[
L
1
and the cp(x, V)
(13)
From Eqs. (8)-(10) and the fundamental condition AD - BC = 1, one can derive a propagation invariant WC (see Appendix A.2) P-~=w~u2-v~=w~u,-v~.
1
_-
V
--
iW, w2’
v W2’
(16)
Here R could be regarded as an extension of the effective radius of curvature for a monochromatic light beam [2,3]. In Section 4, we shall present a new approach for interpreting the effective radius of curvature. It is easy to prove that the generalized complex radius of curvature Q obeys the same ABCD propagation law as that of a nonideal stationary beam does [2] (see Appendix A.3). That is to say
AQ, +B Q2 = ~ CQ,+D’
Eq. (17) is just the generalized
= 1=O2 ’
const.
(19)
Eq. (18) means that V( I> is related to the first derivative of W 2( z> and represents the variation rate of W 2( z) along the propagation axis z. Eq. (19) means that U is the squared divergence angle (i.e. the squared half-angle) in the far field. In order to make the physical meaning of the parameter U clearer, let us express the parameter U in terms of the complex temporal-frequency and spatialfrequency amplitude +(f,,u) of the polychromatic light beams +( x,l). +(f,, V) is the spatial Fourier transform of cp(x, v), namely
In the temporal-frequency main, U is expressed as U(z)
(15)
The expression of Q for a general polychromatic light beam is the same as that for a stationary light beam which was introduced by BClanger [2] except that the factor AM2/n in the latter is replaced by the new parameter Wc. By the way, the physical meaning of WC will be interpreted in Section 3. The real part of 1/Q is related to the effective radius of curvature for a general polychromatic light beam, namely 1 _=R
limz,~~
(‘4)
Employing Eqs. (8)-(10) and the relation (14) among them, one can define a generalized complex radius of curvature Q for a general polychromatic light beam,
e-W2
=
(18)
(17) ABCD law for a general
= 4/11
(AfJ21+(f,,v)12 --ra
and spatial-frequency
df, dv.
do-
(21)
Eq. (21) more apparently shows that the parameter II is the weighted average value of the squared divergence angle in the far field, because, in the paraxial approximation, Af, and 1I,/I(~,,v>I’ represent the angular direction and the angular intensity of the corresponding monochromatic plane wave component exp[ i2r(f, x - vt)] respectively. It is necessary to do a clear comparison between the intensity-moments defined in this section and those defined in Ref. [ 181, because, in Ref. [ 181, a very similar approach was considered, but a completely contrary conclusion that the ABCD law cannot be extended to general polychromatic light beams was drawn. As we shall prove in Appendix B, for a general polychromatic light beam, the relations between the results obtained in this section and those corresponding results obtained in Ref. [ 181 can be simply stated as follows: the squared beam width defined in this section is equivalent to that defined in Ref. [ 181; the squared divergence angle given in this section is not equivalent to that defined in Ref. [18], only in the quasimonochromatic case, they are approximately equivalent; Ref. [18] failed to extend the ABCD law to a general polychromatic pulsed beam, because the definition of the overall squared far-field divergence used there is not ap-
I.18
Q. Cao, X. Deng/Optics
Communications
propriate; in Ref. [ 181, the mathematical reason which cannot lead to the appropriate definition of the squared divergence is that a kind of Fourier transform whose transform coefficient is just related to the angular frequency o is used there. As pointed out by Ref. [12], the intensity-moments methods which were used in Refs. [ l- 111to describe the propagation characterization of a nonideal monochromatic (or quasi-monochromatic) light beam can be divided into two families. The first is based on the coherent field theory, such as in Refs. [l-6], and the second is based on the Wigner distribution function of a partially coherent field, such as in Refs. [8-l I]. The equivalence between these two kinds of definitions was also proved in Ref. [ 121. It is apparent that the intensity-moments method used in this section is the natural extension to general polychromatic beams of the first kind of intensity-moments method. As a logical extension of the conclusion on the equivalence of the two kinds of intensity-moments methods in Ref. [12], there should be two families of intensity-moments methods which can be used to describe the spatial propagation law of a general polychromatic beam through a first-order optical system. Here we have presented the extension to general polychromatic beams of the first kind of intensity-moments method. But what is the extension to polychromatic beams of the second kind of intensity-moments method? Fortunately, in the last decade, the spatial-temporal Wigner function has been established to describe the propagation of a general ultrashort pulsed beam [21,22]. So we guess it is possible that the second kind of intensity-moments method for a general polychromatic light beam could be established on the spatial-temporal Wigner function, and an equivalent ABCD law for a polychromatic beam could also be derived from this kind of intensity-moments method.
142 (1997) 135-145
In terms of the characterization width WC, the Rayleigh distance Z,, according to its original definition, can be expressed as za = w,‘/wc.
It can be proved that the characterization width Wc has the following properties: Wc directly describes the propagation (or focusing) capability of a polychromatic light beam because it is the product of the minimum beam width and the divergence angle; Wc completely takes the place of the factor AM’/a in the intensity moments theory of the stationary light beams; YYc is directly related to many important parameters, such as the complex radius of curvature Q and the Rayleigh distance Z,; Wc reflects that the frequency spectral distribution of a polychromatic light beam can affect its propagation capability, and also implies that, just as the fact shows, the propagation capability of a monochromatic wave with short wavelength is greater than that of a monochromatic wave with long wavelength if they have the same complex amplitude distribution at the original transverse plane. These properties show that “rc is a very important characterization parameter which can be used to describe the propagation of a general polychromatic light beam through a first-order optical system. By employing Eqs. (6), (12) and the Schwarz inequality, one can prove that (see Appendix C)
in any transverse plane, where P(V) is the frequency spectral density which represents the amount of energy included in a unit frequency width and is given by P(v)=/~
3. The characterization M2 factor
width 27,
and the generalized
(23)
-3c
]rp(x,v)]2dx=/m
q;(x,v)dx. --cc
(25)
Let us now introduce the concept of the average wavelength x in the frequency spectral domain. For a general polychromatic light beam, it is reasonable to accurately define the weighted average wavelength x as
Let us now investigate the physical meaning of the propagation invariant Yc through first-order optical systems. In free space, one can prove that vc=w,e,
(22)
where W, and 0 are the minimum beam width in the near field and the divergence angle in the far field, respectively. Eq. (22) shows that WC is the product of the minimum beam width and the divergence angle, and has the same dimension as the beam width. For these two reasons, we name the propagation invariant Wc the characterization width of a polychromatic light beam. It is apparent that the characterization width Wc directly describes the propagation (or focusing) capability of a general polychromatic light beam.
It is apparent that P(v) is a propagation invariant through an ABCD system because the ABCD system is considered as a dispersive-free system. Introducing this result in Eq. (261, one can find that x is also a propagation invariant through an ABCD system. In terms of the average wavelength x, Eq. (24) can be rewritten as w2u
2 (X/T+
Especially, reduces to PXc 2X/a
(27) at the beam waist plane,
this inequality
(28)
Q. Cao, X. Deng/Optics
Communications
Eq. (28) shows that the low limit of the characterization width Wc is restricted by the average wavelength x. In terms of the characterization width WC, the average wavelength x and Eq. (281, we define the generalized M* factor for a general polychromatic light beam as M2 = awe-X.
(29)
Obviously, this definition can ensure that M* 2 1. It is necessary to emphasize that the generalized M2 factor can only be obtained after the parameters WC and 1 are obtained because the generalized M2 factor is defined on the basis of the characterization width Wc and the average wavelength x. In the special case of a quasi-monochromatic light beam whose central frequency is v,, P(Y) can be approximately expressed as P(v)
=;6(v,),
(30)
where 6 is the Dirac 6 function. One can show that the M* factor defined by (29) can reduce to the M * factor of a stationary light beam which was introduced by Siegman [I] provided that the quasi-monochromatic condition (30) is fulfilled. This correspondence relation shows that the M* factor of a polychromatic light beam is consistent with that of a stationary light beam, and still keeps the physical meaning of “the times of the diffraction-limit”. That is to say, the generalized M2 factor can be used to describe the beam quality of a general polychromatic light beam. Especially, the polychromatic light beams whose generalized M 2 factors are 1 are diffraction-limited.
4. A new approach for interpreting of curvature
+Hexp(-i2nvt)+c.c.,
+ c.c.,
--&E.
(31) (32)
in a transverse plane which is perpendicular to the z-axis, where E and H are the time-independent parts of +and + respectively. In free-space propagation, the relation between E and H can be deduced from the Maxwell’s equations [23,24],
(33)
By employing flow density (S)
Eqs. (31)-(331, the time average energy can be expressed as
(S)a&[Ex(VxE’)]+c.c.
(34)
Let us now limit our analysis to the situation of one-dimensional propagation of a monochromatic scalar light beam cp. In order to be more consistent with the assumption of one-dimensional propagation which implies that (Jcp/ay) = 0, we assume that the polarization direction of the scalar light beam is the y direction, namely [24] E=e,cp,
(35)
where eY is the unit vector in the y direction. Obviously, in this case, the condition V. E = 0 can lead to the assumption (+/Jy) = 0 which corresponds to the situation of one-dimensional propagation. Substituting Eq. (35) into Eq. (34), one can obtain
(S)=d
aL dL e,- ax + e,az
i
1 9
(36)
where e, and e, are the unit vectors in the x and z directions respectively, and cp, and L are the real amplitude and the path length of the monochromatic scalar light field cp, respectively. Eq. (36) shows that the time average energy flow density for a monochromatic scalar light beam cp whose polarization direction is the y direction is parallel to the vector e,(JL/ax) + e,(dL/dz). In the paraxial approximation, one can obtain the following results.
the effective radius
The effective radius of curvature for a nonideal stationary light beam was introduced by BClanger [2] and was reasonably explained from several different aspects of beam propagation by Siegman [3]. This concept has been generalized for a general polychromatic light beam in Section 3 of this paper. In this section, we shall present a new and more direct approach for interpreting the physical meaning of the effective radius of curvature. Let us first consider a monochromatic vector light beam with frequency V. The time-dependent electric vector 4and the time-dependent magnetic vector &are expressed as + E exp( --i2r~t)
H=
139
142 (1997) 135-145
(37)
(38) From Eqs. (36)-(38), one can find that, at the x =x point, the included angle U between the z-axis and the actual energy flow direction is
GE. Let us now use some imaginary spherical wavefronts with different radii of curvature to approach the actual wavefront of the paraxial scalar light field cp so as to find the best-fit effective spherical wavefront for the actual wavefront L in the z = z plane. It is apparent that, at the x=x point, the included angle between the z-axis and the the energy flow direction of the imaginary scalar light field (~a which has the same amplitude cp, and the imaginary spherical wavefront with radius R is given by 6a =x/R.
(40)
At the x=x point, the deviation between the actual energy flow direction and the energy flow direction of the
Q. Cao, X. Deng/Optics Communications 142 (1997) 135-145
I10
imaginary light field (~a with spherical wavefront be characterized by (40)‘=
(H-
0J.
R can
which correspond to the different monochromatic wave components cp(n, V) of a general polychromatic light beam.
(41)
It is reasonable to define the total weighted deviation between the actual wavefront and the imaginary spherical wavefront R as
5. Some special examples 5.1. Discrete frequency
spectral light beams
A general discrete frequency expressed as 4(x,t)
= i
spectral light beam can be
Ccn(Pn(x,vn)exp(-i2Tv,t)
I (46)
(42) since
the first moment
(40)=/?,(0-
~9a)(pa2dx=O.
Eq. (42) shows that the total deviation m is a quadratic function of the variance l/R. From Eq. (421, one can find that the total deviation (ae)’ has the minimum when the variance l/R satisfies the condition
value
cc
1 -= R
/ _mx(aL/ax)cc,?dx
where cp,( x, v,) is the normalized complex amplitude which corresponds to frequency v,, and lc,/’ is the fractional power content of the individual field cp,(x, v,,). In this case, the normalization condition becomes Xnlc,,/’ = 1. For these multi-wavelength light beams, one can easily prove that w’ = cIc,12w;, n
(47)
iJ=
J&J2Q1,, n
(48)
x =
~Ic,12A,,
(49)
(43)
m .r*cp,zdx / -CC
Eq. (43) is just the definition of the effective radius of curvature for a nonideal stationary light beam [2,3]. From the above analysis, one can obtain the important conclusion that the spherical wavefront with the effective radius of curvature R given by (43) is the best-fit effective spherical wavefront for the actual wavefront, because, in this case, the total deviation between the spherical wavefront and the actual wavefront has the minimum value. For a general polychromatic light beam, in the z = z plane, the total deviation between the imaginary spherical wavefront and the actual wavefronts which correspond to the wavefronts L(x,v> of the different monochromatic wave components cp(x,u) can be reasonably defined as
(44) From Eq. (44), one can prove that the total deviation m has the minimum satisfies the condition
,
+c.c.
[ n
value when the variance
l/R
n
where W,’ and ZJ,, are the individual squared beam width and the individual squared divergence angle which correspond to frequency vn respectively, and are given by d x, and WY2= 41: r x’l~,(x,v,)l~ u,, = r A,,~~,~~cp,( x,v,)/~?xl* dx, respectively. The individual characterization width W,.., and the individual propagation factor M,‘, according to their definition, can be expressed as P-& = w”;
u*,
(50)
M,2 = -~~cfl A” ’
(51)
where W,‘, is the minimum value of W,‘(Z). Unfortunately, in general, the relation between WC and wc.,, and the relation between M2 and M,’ are both complicated. However, in some special circumstances, as we shall prove below, the two relations can become simple. The propagation of the individual squared beam width W,’ in free space is governed by the quadratic law, w,‘(z)=w,‘+u,(z-z,,)*,
(45) This result shows that the spherical wavefront with the effective radius of curvature R given by (45) is the best-fit effective spherical wavefront for the actual wavefronts
(52)
where z,,, is the axial position of the individual beam waist plane. Employing Eq. (52), one can obtain the minimum beam width W,, (~lcYz.0K)’ W,2 =
~Ic,12(w,20+ u,zi) n
+
U
(53)
141
Q. Cao, X. Deng / Optics Communications 142 (1997) 135-145 Then, ZYc and M* can be obtained from the relations YYz = W:U and M2 = rr’3Yc.Jx. Especially, when the quasi-monochromatic condition IV,,- v,l -K v,, v, is satisfied, the above results reduce to the corresponding results obtained in Ref. [5] where the propagation properties of quasi-monochromatic multitransverse-mode laser beams are analysed. Let us now consider discrete frequency spectral light beams which satisfy the conditions z,,, = z,,, and W,,‘,/U, = const. By taking these two conditions into account, one can obtain wc = Clc,12~cn, n
(54)
CIc,12A,M,2 lVf2=
(55)
“C,c.,2hn
n Especially, when the quasi-monochromatic condition \A, - A,I < &,A,, is fulfilled, Eqs. (54), (55) reduce to and M* = Cnlc,12M~. It is easy to wc = x&,\c,,1*M~/7r prove that an incoherent mixture of one-dimensional Hermite-Gaussian modes emitted from a spherical resonator with Brewster windows satisfies the above conditions, and its Wc and M 2 factor, just as we expect, reduce to Wc = XC,lc,1*(2n + l)/rr and M* = C,lc,12(2n + I), respectively.
5.2. Polychrornatic diffraction-limited
light beams
It can be proved that, according to the condition which corresponds to the case that the equal-sign can be used in relation (24), the polychromatic diffraction-limited light beams, whose generalized M2 factors are 1, need to satisfy the following conditions (see Appendix C),
(57) at their beam waist planes, where C,, is an arbitrary positive constant, q,(x,v> and L(x,v) are the real amplitude and the path length of the monochromatic wave component cp(x,v> respectively. Eq. (56) shows that the path length L(x,v) of any monochromatic wave component cp(x,v) of the diffraction-limited light beams is independent of the transverse coordinate x at the beam waist plane. That is to say, cp(x,v) has uniform phase at the beam waist plane. Eq. (57) implies that the real amplitude ~,. By taking the two properties into account, at the beam
waist plane, one can express the general diffraction-limited light beams as
polychromatic
where vc is the characterization frequency, u,‘v,/v is the beam waist width of the Gaussian monochromatic wave component cp(x,v), and 1A(v is the frequency spectral density, namely IA(v)* = P(v). It is necessary to point out that A(v) can be any function of frequency v which satisfies /“%I A( v)12 dv = 1. It is easy to prove that the parameters W,, Cl and Wc of the polychromatic diffraction-limited light beams are Waz = o&,%/C,, U = C,x/~*u&, and Wc = x/rr respectively. A discrete frequency spectral diffraction-limited light beam can be expressed as
+(x,t>=
1 Ccn 4&
Hn
X
(
exp !I0
X2 -i2Tv,t
-F
n0
+c.c. i
1
(59)
at its beam waist plane, where W,,f, is the individual beam waist width which corresponds to frequency v,, and is given by Wni = u&,/v,,. It is apparent that, in the special case of a monochromatic stationary light beam whose frequency is v,, the polychromatic diffraction-limited light beams, as we expect, reduce to the well-known Gaussian beams. From Eq. (581, one can find that the conditions for polychromatic diffraction-limited light beams can be reexpressed as follows: (i) any monochromatic wave component cp(x,v> is a Gaussian beam; (ii) at the beam waist plane, the phase of any monochromatic wave component cp(x,v) is uniform, that is to say, all the Gaussian monochromatic wave components cp(x,v) have the same beam waist planes; (iii) any Gaussian monochromatic wave component p( x,v) satisfies the relation v W,‘< v) = const, where W,(v) is the beam waist width of the Gaussian monochromatic wave component q( X, v). Fortunately, using matrix optics, one can prove that the polychromatic light beams made up of a superposition of the different fundamental transverse modes (which correspond to different frequencies) of a spherical mirror resonator, can fulfil the three conditions. This property implies that the polychromatic light beams emitted from an ultrabroadband gain medium laser device with a spherical mirror resonator are diffraction-limited provided that only the fundamental transverse modes can oscillate in the laser device. In order to understand the polychromatic diffraction-
I-11
Q. Cao, X. Deng / Optics Communications 142 (1997) 135-145
limited light beams more clearly, let us consider a frequency-independent Gaussian beam which has the form
(60) at its beam waist plane. where on is the beam waist width which corresponds to any monochromatic wave component cp(x,v). One can easily prove that the M* factor of a polychromatic light beam of this kind is always greater than I. That is to say, this kind of polychromatic light beams are not yet diffraction-limited light beams though any of their monochromatic wave component cp(x,v) has a Gaussian distribution along the transverse coordinate at the beam waist plane. 5.3. Polychromotic light beams whose spatial field distributions are independent of frequency at their beam waist plunes In order to understand the influences on the characterization width Wc and the generalized M* factor of the frequency spectral density P(v) more clearly, let us investigate the propagation of polychromatic light beams whose field distributions have the form cp(x,v> =A(v)u(x)
(6’)
at their beam waist planes, where for simplicity A(v) and u(x) are both assumed to satisfy the condition /“,IA(v)1*dv= 1 and l”,lu(x)l’dx= I. For this kind of polychromatic light beams, the average squared wavelength p and the second moment of the spatial-frequency distribution z can be defined as h2= ?C;r;P(v)dv,
(62)
f,’ = 1:” f$/G)i’df,, -x
(63)
where $(f,) VU>
In terms of 2 and 2,
rr-
M4=
(W2
I I 1+-
(q2
M4
O’
In summary, we have thoroughly generalized the ABCD propagation law and the M* factor for general polychromatic light beams in terms of the Fourier transform method, we have shown that the characterization width Vc and the M* factor are appropriate for describing the propagation capability and the beam quality of a general polychromatic light beam respectively, we presented a new approach for interpreting the physical meaning of the effective radius of curvature, and analysed several special examples such as polychromatic diffraction-limited light beams and multiwavelength laser beams. The results obtained in this paper can be used to study the spatial propagation characteristics of general polychromatic light beams through first-order optical systems.
Acknowledgements This research was supported by the National Hi-Tech (863-416) Foundation of China. The authors thank the reviewer for suggesting a clear comparison between the results obtained in this paper and those obtained in Ref.
(64) Appendix A. Derivations of some equations in Section 2
one can obtain
A* w,‘= ,M$,
6. Conclusions
[181.
is the Fourier transform of u(x), namely
= 1X u(x)exp(-i2rf,x)dx. Pm
The factors 2 and (AA>2/(h)2 reflect the influences on the characterization width ?Yc and the generalized M’ factor of the frequency spectral distribution respectively, and show that W, increases with increasing 2 and M2 increases with increasing (A A)'/m*. That is to say, for a polychromatic light beam of this kind, when its spatial field distribution U(X) at its beam waist plane has been determined, the greater A’, the weaker the propagation capability; and the greater (A A)*/m*, the bigger the M2 factor.
(65)
A.1. Deriution
(66)
For the monochromatic [2] has obtained
where (AA)’ =? - (a*, Mt = 4Wtz and Wz = 4/Y,x*lu(x)l dx. Because (AA)* is always greater than zero, the generalized M* factor for a light beam of this kind is always greater than M,‘. Obviously, the relation M,j = 4W0’z is just the expression of the M2 factor of monochromatic stationary light beams [I].
of equations (8)~(101 wave component
w;(v)=A~w~(~)+~ABL~,(v)+B*u,(v), L.*(v) =ACw;(
cp(x, v 1, Ref.
(A.1)
v) + (AD + BC)L.,( v) + BDu,( v), (‘4.2)
u2( v) = C*w:( v) + 2DCc,(
v) + D’u,( v),
(A.3)
133
Q. Cao, X. Deng/Optics Communications 142 (19971 13.5-145 where
A.3. Derivation of equation (17) From Eq. (l4), the parameter
i+J( rJ> = 4/= x%p(x,v)12dx, -z
U, can be expressed
as
V,2_tW2 u, =
(A.9)
w,2
Employing v,
Eqs. (8), (9) and (A.9), one can obtain
= (CW,2 + DV,)(
AW,2 + BV,) + BDSF; , (A.10)
W,’
( AW,2 + BV,)* + B’%f;
In the above expressions, the wavelength A used in Ref. [2] has been replaced by C,/u. Multiplying Eqs. (A.l)-(A.3) by a constant 2, and then integrating them, one can immediately obtain
WC -=
~=2w;(v)du=A2jc~2~;(v)d~+2ABjaxZu,(v)dv
In terms of expression (l5), the relation AD - BC = 1 and Eqs. (A.lO), (A.1 1), one can prove that
+B’
W,’
C+
X2+)dv, /0
(A.4)
VCW, ( A W,2 + BV, )’+ B’T2
(A.1 I) ’
D/Q,
A + B/Q, CW? + DV, - iD‘Zfc
jam2C2(V)du=ACjom2i+)dv
= AW,‘+BV,-iBYc +(AD+BC)im2il,(v)dv
=
(CWf
+ DV,)(
AWf
+ BV,) + BEYZ
- iFcW,
(AW,2 + BV,)* + B’;yTc’ + BD[2u,(v)dv,
(A.5)
V, =_--=_ W2’
c2u2(v)dv=C2c2Wj(r)dy+2DCja=2,:,(r)dY
iYYc
1
W,’
Qz’
(A.12)
It is apparent that Eq. (A.12) is an equivalent sion of Eq. (17). +D2[2U,(v)dy. Obviously,
Eqs. (A.4)-(A.6)
expres-
(A.6)
are just Eqs. (8)-(10).
Appendix B. Comparison between the results obtained in Section 2 and those obtained in Ref. US1
A.2. Derivation of equation (14) From Eqs. (8)-(lo), W;U2 =A’C*WP
Let us start from the original definition transform [25]
one can obtain
+ 4ABCDV,2 + B2D21Jf
H(f,)
= jm h(x)exp(-iTf,x)dx, TI
+ 2AC( AD + BC)W,2V,
h(x) =
+( ~*~~+~*c*)wfCi,
(B.‘)
&j: ff(fx)exp(i~fxx)dfx, 3c
+2BD(AD+BC)V,U,, v,~=A~c*w~+(AD+Bc)~v~+
(‘4.7) B~D*U,
+2AC(AD+BC)W,2V,+2ABCDW,2lJ, + 2BD( AD + BC)V,U,.
(A.81
Then, from Eqs. (A.7), (A.8), Eq. (14) can be easily derived, namely
w,'u,- v; = (AD - BC)‘W,~U, = w,‘u,
P.2)
where h(x) can be an arbitrary function provided that its integral /?,lh(x)(‘dx is finite, and r is the Fourier transform coefficient. For the Fourier transform with transform coefficient r, the Parseval’s theorem (a special case of theorem IV in Ref. [25]) and the Fourier transform of the differential operator can be expressed as /I
--P
IH(f_J*
df, = T/i
Ih(x)I* zc
dx,
03.3)
- (AD - BC)‘V, f,H(f,)=
-v,‘.
In the above derivation procedure, = 1 has been employed.
of the Fourier
the relation
AD - BC
-j:zG$$exp(-irf;*)dx.
(E.4)
Let us now investigate the relation between the overall intensity-moments defined in Section 2 and those defined
13‘4
Q. Cm, X. Deng/Optics
Communications
in Ref. [ 181. For simplicity, in the following, we shall ignore the difference 2~ between the angular frequency w and the linear frequency v, because it is very apparent that this small difference does not have any influence on the problem concerned here. In terms of the linear frequency v, the overall squared beam width (x2> and the overall squared far-field divergence ( u2> defined in Ref. [I 81 can be expressed as
142 (1997) 135-145
Substituting Eqs. (B. 13) and (B.14) into Eqs. (B. IO) and (B.6), G, and (u2> can be expressed as zz
G,=2C,
1
+(x,v)l’dnix, mccv
z jj0
jmjx v-31ap(x,v)/h12dvdx
c;
O -=
(“2)=G
I
z x’lF(x,v)/2dxdv 1 F” jj --z
(_r2)=
2
=-
(u’)
//0
(B.16) ~~(F(x,v)/~dvdx,
F 0 lI0
p=
= 6_i
P.9
u21G(U,v)12dUdv
U21G(u,v)12dvdu, --cc
(B.6)
where F(x,v)
=_/= 4(x,t)exp(-i2nvt)dt, --a
P.7)
G(u,v)
= jm F(x,v)exp --r
03.8)
F,, =
;DDIF(x,v)I’dxdv
z 2mr jj0
z
__v+(x,v)12dvdx
rx
= -$ j ja
!I‘
(B.15)
IF(x,v)12dvdx,
(B.9)
--5
From Eqs. (B.16) and Eq. (12), one can find that the squared divergence angle defined in Ref. [18] is completely different from that given in Section 2. As claimed by Ref. 1181, the squared divergence angle defined there does not obey the ABCD law. That is to say, the squared divergence angle defined in Ref. [ 181 is not appropriate to be used to extend the ABCD law to polychromatic pulsed beams. From Eqs. (B.8), (B.15) and (B.l6), one can deduce that the mathematical reason which leads to the unsuitable definition of the divergence angle in Ref. [ 181 is that the influence of the Fourier transform coefficient r= 25-v/C, which is just related to the frequency has not been taken into account. It can be easily proved that, for a quasi-monochromatic pulsed beam such as a SCLQ field [18], U = 4(u2>, namely, in this special case, the divergence angles defined in Section 2 and defined in Ref. [18] are approximately equivalent. This property is just consistent with the conclusion that the overall spatial second-moments of paraxial SCLQ fields follow the ABCD law [18].
G, = j j= IG(u,v)12dudv -Ti = 2 * jj0
= IG(u,v)l’dv
Comparing F(x,v)
du.
(B.lO) Appendix
-rx
C. Derivation of inequality
(24)
Eq. (B.7) with Eq. (.5), one can deduce that
= cp(x,-
By employing F, = 1 and
v) = cp*(x,v).
(B.11)
this relation, one can easily prove that
(B.12)
w2=4(,?),
where W* is given by Eq. (6). Expression (B.12) shows that the squared beam width defined in Section 2 is equivalent to that defined in Ref. 1181. Employing Eqs. (B.3), (B.4) and (B.l I), one can obtain
jrIG(u,v)12du=~ jrbp(x,v)12dx, --P
--co
j:XlaG(u,v)i2du=
From Eq. (I 3), one can prove that
2
(B.13)
2
I
%4(x,v) 2 ax ’ [
(C.1)
where the equal-sign can be used only when c?L(x,v)/ax = 0 in the whole transverse plane. Substituting inequality (C.1) into Eq. (12), the following result can be derived,
j;&lFi’dx. (B.14)
dvdx.
(C.2)
Q. Cao, X. Deng / Optics Communications
In terms of the Schwarz inequality,
one can obtain
CC.31 where we have assumed that x&x,v) 00, and we used the relation
vanishes for x +
-P(v)
(-9 In expression (C.31, the equal-sign can be used only when -C,xv~.Jx,v) = +a(x,v)/dx, where C, is an arbitrary positive constant. The reason why C, is not a negative constant but a positive constant is that the integral lY,&x,v)dx is finite. By employing expressions (C.21, (C.3) and Eq. (6), one can immediately obtain inequality (24). In expression (241, when the conditions only aL(x,v)/ax = 0 and - C,xvcp,(x,v) = &$x,v)/~x are both satisfied, the equal-sign can be used.
References [ll A.E. Siegman, Proc. Sot. Photo-Opt. Instrum. Eng. 1224 (1990) 2.
142 (1997) 135-145
145
[2] P.A. B&anger, Optics Lett. 16 (1991) 196. [3] A.E. Siegman, IEEE J. Quantum Electron. 27 (1991) 1146. [4] M. Morin, P. Bernard, P. Galarneall, Optics Lett. 19 (19941 1379. [5] Y. Champagne, J. Opt. Sot. Am. A 12 (19951 1707. ii H. Weber, Opt. Quantum Electron. 24 (1992) S1027. 171 H. Weber, Opt. Quantum Electron. 24 (1992) S861. 181M.J. Bastiaans, J. Opt. Sot. Am. A 3 (19861 1227. 191 M.J. Bastiaans, Optik 88 (1991) 163. 1101R. Simon, N. Mukunda, E.C.G. Sudarshan, Optics Comm. 65 (1988) 322. 1111R. Martinez-Herrero, P.M. Mejias, M. Sanchez, J.L.H. Neira, Opt. Quantum Electron. 24 (1992) S1021. 1121 R. Martinez-Herrero, P.M. Mejias, H. Weber, Opt. Quantum Electron. 25 (1993) 423. 1131Q. Lin, S. Wang, J. Alda, E. Bernabeu, Optics Lett. 18 (1993) 669. 1141A.E. Siegman, Lasers (University Science, Mill Valley, CA, 1986) Ch. 9. 1151 S.P. Dijaili, A. Dienes, J.S. Smith, IEEE J. Quantum Electron 26 (1990) 1158. [I61 A.G. Kostenbauder, IEEE J. Quantum Electron, 26 (1990) 1148. 1171 See for example, J. Zhou, G. Taft, C.P. Huang, M.M. Mumane, H. Kapteyn, I.P. Christov, Optics Len. 19 (1994) 1149. Optics Len. 20 (1995) 1181P.M. Mejias, R. Martinez-Herrero. 660. 1191M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975) Ch. 10. 1201S.A. Collins, J. Opt. Sot. Am. 60 (1970) 1168. 1211J. Paye, A. Migus, J. Opt. Sot. Am. B 12 (1995) 1480. 1221B.O. Maier, N.G. Preobrazhenskii, Opt. Spectr. (USSR) 66 (1989) 643. 1231J.D. Jackson, Classical electrodynamics (Wiley, New York, 1975) Ch. 6. 1241M. Lax, W.H. Louisell, W.B. McKnight, Phys. Rev. A 11 (19751 1365. 1251 See for example, E. Lalor, J. Opt. Sot. Am. 58 (1968) 1235.
ELSEVIER
1 October
1997
OPTICS COMMUNICATIONS Optics Communications
142 (1997) 135-145
Full length article
Spatial parametric characterization of general polychromatic beams
light
Qing Cao, Ximing Deng National Laboratory
on High Power Laser and Physics, Shanghai
Institute
800-21 I, Shanghai
Received 30 October
ofOptics
201800,
1996; revised 6 February
and Fine Mechanics,
Academia
Sinica, P.O. Box
China
1997; accepted
18 April 1997
Abstract The ABCD law and the M’ factor are generalized for the propagation of general polychromatic light beams through first-order optical systems by introducing a generalized complex radius of curvature Q, an average wavelength x and a characterization width Fc. The real part and the imaginary part of the parameter l/Q are related to the effective radius of curvature and the characterization width W, respectively. The characterization width YFc and the generalized M2 factor are shown to be appropriate for describing the propagation capability and the beam quality of a general polychromatic light beam respectively. Moreover, the relation M2 2 1 is deduced, a novel approach for interpreting the physical meaning of the effective radius of curvature is presented, and several special examples such as the polychromatic diffraction-limited light beams are analysed. 0 1997 Elsevier Science B.V.
1. Introduction In the past few years, the intensity-moments theory and the M’ factor have been revealed to be very useful to describe the propagation law and the beam quality of stationary light beams [ 1- 121. This theory was generalized for quasi-monochromatic pulsed light beams using the tensorial method three years ago [ 131. However, those results obtained in Ref. [13] are only valid for pulsed light beams which fulfilled the quasi-monochromatic condition Iv - v,] x v, (where v, is the center frequency or the carrier frequency), and are not valid for general polychromatic pulsed light beams, especially not valid for ultrabroadband pulsed light beams, because the tensorial method is established on the basis of the space-time analogy [14,15] and the generalized spatio-temporal diffraction integral [16] which only apply to quasi-monochromatic pulsed light beams. On the other hand, the research on ultrashort pulse lasers has made notable progress since the appearance of the novel ultrabroadband solid-state laser material. At present, sub-l 0-fs ultrashort pulse lasers can be obtained 0030-4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SOO30-40 18(97)00222-8
using Ti:sapphire laser material in many laboratories 1171. It is apparent that the spatio-temporal tensorial method is invalid for describing the propagation law of these sub-lofs pulse lasers, because, in these cases, the quasi-monochromatic condition 1v - v,l +z v, is not satisfied. Moreover, the spatio-temporal tensorial method is also invalid for describing the propagation of polychromatic light beams emitted from the so-called multi-wavelength laser devices which can oscillate at several different frequencies at the same time, such as the polychromatic light beams emitted from the so-called “white-light” laser devices. The existence of sub-lo-fs ultrashort pulsed laser beams, multiwavelength laser beams and other polychromatic light beams which do not fulfil the quasi-monochromatic condition shows that it is very necessary to generalize the intensity moment theory and the M2 factor for general polychromatic light beams, especially for those ultrabroadband (pulsed and CW) light beams. Recently, in Ref. [ 181 an attempt has been made to extend the stationary intensity-moments method to general polychromatic pulsed light beams, and to define the overall squared beam width and the overall squared far-field diver-
136
Q. Go, X. Deng/
Optics Communications
gence. However. as we shall prove, in Ref. [18] the wrong conclusion was drawn that “for general pulsed beams, the paraxial approach does not ensure that such (i.e. ABCD) propagation laws are fulfilled”, because the definition of the overall squared far-field divergence used there is not appropriate. In this paper, we shall more accurately analyse the propagation characteristics of general polychromatic light beams through first-order optical systems which do not include any dispersive element and thoroughly generalize the intensity moment theory and the M2 factor for the general polychromatic light beams using the Fourier transform method. The paper is organized as follows: in Section 2, we generalize the ABCD law for the propagation of general polychromatic light beams through first-order optical systems which do not have any dispersive element by introducing the generalized complex radius of curvature Q; in Section 3, we define the characterization width Wc and generalize the M2 factor for general polychromatic light beams, and interpret their physical meaning; in Section 4, we present a new method for interpreting the effective radius of curvature; in Section 5, we analyse several special examples such as polychromatic diffraction-limited light beams; and in Section 6, we conclude this paper. For simplicity, in this paper, we shall limit our analysis to the situation of one-dimensional, (x, z,) propagation.
2. ABCD propagation light beams
law of general
polychromatic
Let us consider a general polychromatic pulsed light beam which is represented by a real function 4( x,z,t> at a transverse plane which is perpendicular to the z-axis [ 191. In the sequel, we shall consider the z dependence, but we shall for simplicity drop it from the notations. For this pulsed light beam, we can introduce an area energy density P(x) P(s)=/‘
--P
4’(x,t)dt,
(1)
where +2(x,f) is the instantaneous light intensity. Obviously, the area energy density P(x), which represents the amount of energy which crosses a unit area in the transverse plane in the whole light pulse, is a measurable quantity. In order to simplify the following demonstrations, we assume that the z-axis is the propagation axis of the “center” of the pulsed light beam, namely _ x=
squared beam width (i.e. the beam radius width), in terms of the area energy density, as ~2~4
Z x2P( x) dx, / -r
(3)
where it is assumed that the total energy in the whole pulse is unity, namely lx P(x)dx=j/l --cc
--r
+‘(x,t)dxdt=
1.
(4)
It is well known that a general pulsed beam $(x,l) may be expressed as a superposition of monochromatic waves of different frequency, namely [19] 4(x,f)
= IX p(x,v)exp(-i2rvt)dv. -X
(5)
Since $(x,t) is real, cp(x,- V) = cp*(x,u). It is important to point out that, in this paper, the discrete frequency spectral polychromatic (or monochromatic) light beams are included in the general polychromatic pulsed light beams, because, from the point of view of the Fourier transform theory, the case of discrete frequency spectral light beams can be regarded as a special case of continuous frequency spectra1 light beams. That is to say, the results deduced from the polychromatic pulsed light beams in this paper are also valid for the discrete frequency spectra1 light beams. In terms of the Parseval’s theorem in the Fourier transform theory, the area energy density P(x) can be rewritten as P(x)
=/=
-cc
Substituting
/~(x,v)~‘dv=2/nil~(x,i)j2d~.
this result into Eq. (3), one can obtain
W2 = 8/DzjX x21’p(x,u)12 Pm
dvdx.
(6)
The propagation property of any monochromatic component cp(x,v) through a first-order optical system which does not include any dispersive element is described by Huygens’s integral in the Kirchhoff-Fresnel approximation [201
i7rv
Xexp
- E(A”: L
-2x,x,+Dx;)
dx,, I
(7)
cc / --x
142 (19971 135- 145
xP(x)dx=O
(2)
in any transverse plane. Then it is reasonable to define the
where CL is the velocity of light in free space, and the ABCD parameters are the ray’s matrix elements of the first-order optical system located between the transverse
137
Q. Cao, X. Deng/Optics Communications 142 (1997) 135-145 plane 1 and the transverse plane 2. From Eq. (7), one can obtain (see Appendix A.1) W,2=A2Wf+2ABV,+B21J,,
(8)
V, = ACW,2 + (AD + BC)V,
+ BDU,,
(9)
lJ2=C2W,2+2DCV,+D2U,,
polychromatic light beam through a first-order optical system which does not have any dispersive element. Let us now investigate the physical meaning of the parameters V and U from their propagation properties in free space. In free space, V(z) and U(z) satisfy the following relations
(10) 1 aw’(z)
where
V(z)=27:
(11)
dW(z) U(z)
(12) where cp,
2av -i~L(x,v)
= ‘p,(x,v)exp
[
L
1
and the cp(x, V)
(13)
From Eqs. (8)-(10) and the fundamental condition AD - BC = 1, one can derive a propagation invariant WC (see Appendix A.2) P-~=w~u2-v~=w~u,-v~.
1
_-
V
--
iW, w2’
v W2’
(16)
Here R could be regarded as an extension of the effective radius of curvature for a monochromatic light beam [2,3]. In Section 4, we shall present a new approach for interpreting the effective radius of curvature. It is easy to prove that the generalized complex radius of curvature Q obeys the same ABCD propagation law as that of a nonideal stationary beam does [2] (see Appendix A.3). That is to say
AQ, +B Q2 = ~ CQ,+D’
Eq. (17) is just the generalized
= 1=O2 ’
const.
(19)
Eq. (18) means that V( I> is related to the first derivative of W 2( z> and represents the variation rate of W 2( z) along the propagation axis z. Eq. (19) means that U is the squared divergence angle (i.e. the squared half-angle) in the far field. In order to make the physical meaning of the parameter U clearer, let us express the parameter U in terms of the complex temporal-frequency and spatialfrequency amplitude +(f,,u) of the polychromatic light beams +( x,l). +(f,, V) is the spatial Fourier transform of cp(x, v), namely
In the temporal-frequency main, U is expressed as U(z)
(15)
The expression of Q for a general polychromatic light beam is the same as that for a stationary light beam which was introduced by BClanger [2] except that the factor AM2/n in the latter is replaced by the new parameter Wc. By the way, the physical meaning of WC will be interpreted in Section 3. The real part of 1/Q is related to the effective radius of curvature for a general polychromatic light beam, namely 1 _=R
limz,~~
(‘4)
Employing Eqs. (8)-(10) and the relation (14) among them, one can define a generalized complex radius of curvature Q for a general polychromatic light beam,
e-W2
=
(18)
(17) ABCD law for a general
= 4/11
(AfJ21+(f,,v)12 --ra
and spatial-frequency
df, dv.
do-
(21)
Eq. (21) more apparently shows that the parameter II is the weighted average value of the squared divergence angle in the far field, because, in the paraxial approximation, Af, and 1I,/I(~,,v>I’ represent the angular direction and the angular intensity of the corresponding monochromatic plane wave component exp[ i2r(f, x - vt)] respectively. It is necessary to do a clear comparison between the intensity-moments defined in this section and those defined in Ref. [ 181, because, in Ref. [ 181, a very similar approach was considered, but a completely contrary conclusion that the ABCD law cannot be extended to general polychromatic light beams was drawn. As we shall prove in Appendix B, for a general polychromatic light beam, the relations between the results obtained in this section and those corresponding results obtained in Ref. [ 181 can be simply stated as follows: the squared beam width defined in this section is equivalent to that defined in Ref. [ 181; the squared divergence angle given in this section is not equivalent to that defined in Ref. [18], only in the quasimonochromatic case, they are approximately equivalent; Ref. [18] failed to extend the ABCD law to a general polychromatic pulsed beam, because the definition of the overall squared far-field divergence used there is not ap-
I.18
Q. Cao, X. Deng/Optics
Communications
propriate; in Ref. [ 181, the mathematical reason which cannot lead to the appropriate definition of the squared divergence is that a kind of Fourier transform whose transform coefficient is just related to the angular frequency o is used there. As pointed out by Ref. [12], the intensity-moments methods which were used in Refs. [ l- 111to describe the propagation characterization of a nonideal monochromatic (or quasi-monochromatic) light beam can be divided into two families. The first is based on the coherent field theory, such as in Refs. [l-6], and the second is based on the Wigner distribution function of a partially coherent field, such as in Refs. [8-l I]. The equivalence between these two kinds of definitions was also proved in Ref. [ 121. It is apparent that the intensity-moments method used in this section is the natural extension to general polychromatic beams of the first kind of intensity-moments method. As a logical extension of the conclusion on the equivalence of the two kinds of intensity-moments methods in Ref. [12], there should be two families of intensity-moments methods which can be used to describe the spatial propagation law of a general polychromatic beam through a first-order optical system. Here we have presented the extension to general polychromatic beams of the first kind of intensity-moments method. But what is the extension to polychromatic beams of the second kind of intensity-moments method? Fortunately, in the last decade, the spatial-temporal Wigner function has been established to describe the propagation of a general ultrashort pulsed beam [21,22]. So we guess it is possible that the second kind of intensity-moments method for a general polychromatic light beam could be established on the spatial-temporal Wigner function, and an equivalent ABCD law for a polychromatic beam could also be derived from this kind of intensity-moments method.
142 (1997) 135-145
In terms of the characterization width WC, the Rayleigh distance Z,, according to its original definition, can be expressed as za = w,‘/wc.
It can be proved that the characterization width Wc has the following properties: Wc directly describes the propagation (or focusing) capability of a polychromatic light beam because it is the product of the minimum beam width and the divergence angle; Wc completely takes the place of the factor AM’/a in the intensity moments theory of the stationary light beams; YYc is directly related to many important parameters, such as the complex radius of curvature Q and the Rayleigh distance Z,; Wc reflects that the frequency spectral distribution of a polychromatic light beam can affect its propagation capability, and also implies that, just as the fact shows, the propagation capability of a monochromatic wave with short wavelength is greater than that of a monochromatic wave with long wavelength if they have the same complex amplitude distribution at the original transverse plane. These properties show that “rc is a very important characterization parameter which can be used to describe the propagation of a general polychromatic light beam through a first-order optical system. By employing Eqs. (6), (12) and the Schwarz inequality, one can prove that (see Appendix C)
in any transverse plane, where P(V) is the frequency spectral density which represents the amount of energy included in a unit frequency width and is given by P(v)=/~
3. The characterization M2 factor
width 27,
and the generalized
(23)
-3c
]rp(x,v)]2dx=/m
q;(x,v)dx. --cc
(25)
Let us now introduce the concept of the average wavelength x in the frequency spectral domain. For a general polychromatic light beam, it is reasonable to accurately define the weighted average wavelength x as
Let us now investigate the physical meaning of the propagation invariant Yc through first-order optical systems. In free space, one can prove that vc=w,e,
(22)
where W, and 0 are the minimum beam width in the near field and the divergence angle in the far field, respectively. Eq. (22) shows that WC is the product of the minimum beam width and the divergence angle, and has the same dimension as the beam width. For these two reasons, we name the propagation invariant Wc the characterization width of a polychromatic light beam. It is apparent that the characterization width Wc directly describes the propagation (or focusing) capability of a general polychromatic light beam.
It is apparent that P(v) is a propagation invariant through an ABCD system because the ABCD system is considered as a dispersive-free system. Introducing this result in Eq. (261, one can find that x is also a propagation invariant through an ABCD system. In terms of the average wavelength x, Eq. (24) can be rewritten as w2u
2 (X/T+
Especially, reduces to PXc 2X/a
(27) at the beam waist plane,
this inequality
(28)
Q. Cao, X. Deng/Optics
Communications
Eq. (28) shows that the low limit of the characterization width Wc is restricted by the average wavelength x. In terms of the characterization width WC, the average wavelength x and Eq. (281, we define the generalized M* factor for a general polychromatic light beam as M2 = awe-X.
(29)
Obviously, this definition can ensure that M* 2 1. It is necessary to emphasize that the generalized M2 factor can only be obtained after the parameters WC and 1 are obtained because the generalized M2 factor is defined on the basis of the characterization width Wc and the average wavelength x. In the special case of a quasi-monochromatic light beam whose central frequency is v,, P(Y) can be approximately expressed as P(v)
=;6(v,),
(30)
where 6 is the Dirac 6 function. One can show that the M* factor defined by (29) can reduce to the M * factor of a stationary light beam which was introduced by Siegman [I] provided that the quasi-monochromatic condition (30) is fulfilled. This correspondence relation shows that the M* factor of a polychromatic light beam is consistent with that of a stationary light beam, and still keeps the physical meaning of “the times of the diffraction-limit”. That is to say, the generalized M2 factor can be used to describe the beam quality of a general polychromatic light beam. Especially, the polychromatic light beams whose generalized M 2 factors are 1 are diffraction-limited.
4. A new approach for interpreting of curvature
+Hexp(-i2nvt)+c.c.,
+ c.c.,
--&E.
(31) (32)
in a transverse plane which is perpendicular to the z-axis, where E and H are the time-independent parts of +and + respectively. In free-space propagation, the relation between E and H can be deduced from the Maxwell’s equations [23,24],
(33)
By employing flow density (S)
Eqs. (31)-(331, the time average energy can be expressed as
(S)a&[Ex(VxE’)]+c.c.
(34)
Let us now limit our analysis to the situation of one-dimensional propagation of a monochromatic scalar light beam cp. In order to be more consistent with the assumption of one-dimensional propagation which implies that (Jcp/ay) = 0, we assume that the polarization direction of the scalar light beam is the y direction, namely [24] E=e,cp,
(35)
where eY is the unit vector in the y direction. Obviously, in this case, the condition V. E = 0 can lead to the assumption (+/Jy) = 0 which corresponds to the situation of one-dimensional propagation. Substituting Eq. (35) into Eq. (34), one can obtain
(S)=d
aL dL e,- ax + e,az
i
1 9
(36)
where e, and e, are the unit vectors in the x and z directions respectively, and cp, and L are the real amplitude and the path length of the monochromatic scalar light field cp, respectively. Eq. (36) shows that the time average energy flow density for a monochromatic scalar light beam cp whose polarization direction is the y direction is parallel to the vector e,(JL/ax) + e,(dL/dz). In the paraxial approximation, one can obtain the following results.
the effective radius
The effective radius of curvature for a nonideal stationary light beam was introduced by BClanger [2] and was reasonably explained from several different aspects of beam propagation by Siegman [3]. This concept has been generalized for a general polychromatic light beam in Section 3 of this paper. In this section, we shall present a new and more direct approach for interpreting the physical meaning of the effective radius of curvature. Let us first consider a monochromatic vector light beam with frequency V. The time-dependent electric vector 4and the time-dependent magnetic vector &are expressed as + E exp( --i2r~t)
H=
139
142 (1997) 135-145
(37)
(38) From Eqs. (36)-(38), one can find that, at the x =x point, the included angle U between the z-axis and the actual energy flow direction is
GE. Let us now use some imaginary spherical wavefronts with different radii of curvature to approach the actual wavefront of the paraxial scalar light field cp so as to find the best-fit effective spherical wavefront for the actual wavefront L in the z = z plane. It is apparent that, at the x=x point, the included angle between the z-axis and the the energy flow direction of the imaginary scalar light field (~a which has the same amplitude cp, and the imaginary spherical wavefront with radius R is given by 6a =x/R.
(40)
At the x=x point, the deviation between the actual energy flow direction and the energy flow direction of the
Q. Cao, X. Deng/Optics Communications 142 (1997) 135-145
I10
imaginary light field (~a with spherical wavefront be characterized by (40)‘=
(H-
0J.
R can
which correspond to the different monochromatic wave components cp(n, V) of a general polychromatic light beam.
(41)
It is reasonable to define the total weighted deviation between the actual wavefront and the imaginary spherical wavefront R as
5. Some special examples 5.1. Discrete frequency
spectral light beams
A general discrete frequency expressed as 4(x,t)
= i
spectral light beam can be
Ccn(Pn(x,vn)exp(-i2Tv,t)
I (46)
(42) since
the first moment
(40)=/?,(0-
~9a)(pa2dx=O.
Eq. (42) shows that the total deviation m is a quadratic function of the variance l/R. From Eq. (421, one can find that the total deviation (ae)’ has the minimum when the variance l/R satisfies the condition
value
cc
1 -= R
/ _mx(aL/ax)cc,?dx
where cp,( x, v,) is the normalized complex amplitude which corresponds to frequency v,, and lc,/’ is the fractional power content of the individual field cp,(x, v,,). In this case, the normalization condition becomes Xnlc,,/’ = 1. For these multi-wavelength light beams, one can easily prove that w’ = cIc,12w;, n
(47)
iJ=
J&J2Q1,, n
(48)
x =
~Ic,12A,,
(49)
(43)
m .r*cp,zdx / -CC
Eq. (43) is just the definition of the effective radius of curvature for a nonideal stationary light beam [2,3]. From the above analysis, one can obtain the important conclusion that the spherical wavefront with the effective radius of curvature R given by (43) is the best-fit effective spherical wavefront for the actual wavefront, because, in this case, the total deviation between the spherical wavefront and the actual wavefront has the minimum value. For a general polychromatic light beam, in the z = z plane, the total deviation between the imaginary spherical wavefront and the actual wavefronts which correspond to the wavefronts L(x,v> of the different monochromatic wave components cp(x,u) can be reasonably defined as
(44) From Eq. (44), one can prove that the total deviation m has the minimum satisfies the condition
,
+c.c.
[ n
value when the variance
l/R
n
where W,’ and ZJ,, are the individual squared beam width and the individual squared divergence angle which correspond to frequency vn respectively, and are given by d x, and WY2= 41: r x’l~,(x,v,)l~ u,, = r A,,~~,~~cp,( x,v,)/~?xl* dx, respectively. The individual characterization width W,.., and the individual propagation factor M,‘, according to their definition, can be expressed as P-& = w”;
u*,
(50)
M,2 = -~~cfl A” ’
(51)
where W,‘, is the minimum value of W,‘(Z). Unfortunately, in general, the relation between WC and wc.,, and the relation between M2 and M,’ are both complicated. However, in some special circumstances, as we shall prove below, the two relations can become simple. The propagation of the individual squared beam width W,’ in free space is governed by the quadratic law, w,‘(z)=w,‘+u,(z-z,,)*,
(45) This result shows that the spherical wavefront with the effective radius of curvature R given by (45) is the best-fit effective spherical wavefront for the actual wavefronts
(52)
where z,,, is the axial position of the individual beam waist plane. Employing Eq. (52), one can obtain the minimum beam width W,, (~lcYz.0K)’ W,2 =
~Ic,12(w,20+ u,zi) n
+
U
(53)
141
Q. Cao, X. Deng / Optics Communications 142 (1997) 135-145 Then, ZYc and M* can be obtained from the relations YYz = W:U and M2 = rr’3Yc.Jx. Especially, when the quasi-monochromatic condition IV,,- v,l -K v,, v, is satisfied, the above results reduce to the corresponding results obtained in Ref. [5] where the propagation properties of quasi-monochromatic multitransverse-mode laser beams are analysed. Let us now consider discrete frequency spectral light beams which satisfy the conditions z,,, = z,,, and W,,‘,/U, = const. By taking these two conditions into account, one can obtain wc = Clc,12~cn, n
(54)
CIc,12A,M,2 lVf2=
(55)
“C,c.,2hn
n Especially, when the quasi-monochromatic condition \A, - A,I < &,A,, is fulfilled, Eqs. (54), (55) reduce to and M* = Cnlc,12M~. It is easy to wc = x&,\c,,1*M~/7r prove that an incoherent mixture of one-dimensional Hermite-Gaussian modes emitted from a spherical resonator with Brewster windows satisfies the above conditions, and its Wc and M 2 factor, just as we expect, reduce to Wc = XC,lc,1*(2n + l)/rr and M* = C,lc,12(2n + I), respectively.
5.2. Polychrornatic diffraction-limited
light beams
It can be proved that, according to the condition which corresponds to the case that the equal-sign can be used in relation (24), the polychromatic diffraction-limited light beams, whose generalized M2 factors are 1, need to satisfy the following conditions (see Appendix C),
(57) at their beam waist planes, where C,, is an arbitrary positive constant, q,(x,v> and L(x,v) are the real amplitude and the path length of the monochromatic wave component cp(x,v> respectively. Eq. (56) shows that the path length L(x,v) of any monochromatic wave component cp(x,v) of the diffraction-limited light beams is independent of the transverse coordinate x at the beam waist plane. That is to say, cp(x,v) has uniform phase at the beam waist plane. Eq. (57) implies that the real amplitude ~,
waist plane, one can express the general diffraction-limited light beams as
polychromatic
where vc is the characterization frequency, u,‘v,/v is the beam waist width of the Gaussian monochromatic wave component cp(x,v), and 1A(v is the frequency spectral density, namely IA(v)* = P(v). It is necessary to point out that A(v) can be any function of frequency v which satisfies /“%I A( v)12 dv = 1. It is easy to prove that the parameters W,, Cl and Wc of the polychromatic diffraction-limited light beams are Waz = o&,%/C,, U = C,x/~*u&, and Wc = x/rr respectively. A discrete frequency spectral diffraction-limited light beam can be expressed as
+(x,t>=
1 Ccn 4&
Hn
X
(
exp !I0
X2 -i2Tv,t
-F
n0
+c.c. i
1
(59)
at its beam waist plane, where W,,f, is the individual beam waist width which corresponds to frequency v,, and is given by Wni = u&,/v,,. It is apparent that, in the special case of a monochromatic stationary light beam whose frequency is v,, the polychromatic diffraction-limited light beams, as we expect, reduce to the well-known Gaussian beams. From Eq. (581, one can find that the conditions for polychromatic diffraction-limited light beams can be reexpressed as follows: (i) any monochromatic wave component cp(x,v> is a Gaussian beam; (ii) at the beam waist plane, the phase of any monochromatic wave component cp(x,v) is uniform, that is to say, all the Gaussian monochromatic wave components cp(x,v) have the same beam waist planes; (iii) any Gaussian monochromatic wave component p( x,v) satisfies the relation v W,‘< v) = const, where W,(v) is the beam waist width of the Gaussian monochromatic wave component q( X, v). Fortunately, using matrix optics, one can prove that the polychromatic light beams made up of a superposition of the different fundamental transverse modes (which correspond to different frequencies) of a spherical mirror resonator, can fulfil the three conditions. This property implies that the polychromatic light beams emitted from an ultrabroadband gain medium laser device with a spherical mirror resonator are diffraction-limited provided that only the fundamental transverse modes can oscillate in the laser device. In order to understand the polychromatic diffraction-
I-11
Q. Cao, X. Deng / Optics Communications 142 (1997) 135-145
limited light beams more clearly, let us consider a frequency-independent Gaussian beam which has the form
(60) at its beam waist plane. where on is the beam waist width which corresponds to any monochromatic wave component cp(x,v). One can easily prove that the M* factor of a polychromatic light beam of this kind is always greater than I. That is to say, this kind of polychromatic light beams are not yet diffraction-limited light beams though any of their monochromatic wave component cp(x,v) has a Gaussian distribution along the transverse coordinate at the beam waist plane. 5.3. Polychromotic light beams whose spatial field distributions are independent of frequency at their beam waist plunes In order to understand the influences on the characterization width Wc and the generalized M* factor of the frequency spectral density P(v) more clearly, let us investigate the propagation of polychromatic light beams whose field distributions have the form cp(x,v> =A(v)u(x)
(6’)
at their beam waist planes, where for simplicity A(v) and u(x) are both assumed to satisfy the condition /“,IA(v)1*dv= 1 and l”,lu(x)l’dx= I. For this kind of polychromatic light beams, the average squared wavelength p and the second moment of the spatial-frequency distribution z can be defined as h2= ?C;r;P(v)dv,
(62)
f,’ = 1:” f$/G)i’df,, -x
(63)
where $(f,) VU>
In terms of 2 and 2,
rr-
M4=
(W2
I I 1+-
(q2
M4
O’
In summary, we have thoroughly generalized the ABCD propagation law and the M* factor for general polychromatic light beams in terms of the Fourier transform method, we have shown that the characterization width Vc and the M* factor are appropriate for describing the propagation capability and the beam quality of a general polychromatic light beam respectively, we presented a new approach for interpreting the physical meaning of the effective radius of curvature, and analysed several special examples such as polychromatic diffraction-limited light beams and multiwavelength laser beams. The results obtained in this paper can be used to study the spatial propagation characteristics of general polychromatic light beams through first-order optical systems.
Acknowledgements This research was supported by the National Hi-Tech (863-416) Foundation of China. The authors thank the reviewer for suggesting a clear comparison between the results obtained in this paper and those obtained in Ref.
(64) Appendix A. Derivations of some equations in Section 2
one can obtain
A* w,‘= ,M$,
6. Conclusions
[181.
is the Fourier transform of u(x), namely
= 1X u(x)exp(-i2rf,x)dx. Pm
The factors 2 and (AA>2/(h)2 reflect the influences on the characterization width ?Yc and the generalized M’ factor of the frequency spectral distribution respectively, and show that W, increases with increasing 2 and M2 increases with increasing (A A)'/m*. That is to say, for a polychromatic light beam of this kind, when its spatial field distribution U(X) at its beam waist plane has been determined, the greater A’, the weaker the propagation capability; and the greater (A A)*/m*, the bigger the M2 factor.
(65)
A.1. Deriution
(66)
For the monochromatic [2] has obtained
where (AA)’ =? - (a*, Mt = 4Wtz and Wz = 4/Y,x*lu(x)l dx. Because (AA)* is always greater than zero, the generalized M* factor for a light beam of this kind is always greater than M,‘. Obviously, the relation M,j = 4W0’z is just the expression of the M2 factor of monochromatic stationary light beams [I].
of equations (8)~(101 wave component
w;(v)=A~w~(~)+~ABL~,(v)+B*u,(v), L.*(v) =ACw;(
cp(x, v 1, Ref.
(A.1)
v) + (AD + BC)L.,( v) + BDu,( v), (‘4.2)
u2( v) = C*w:( v) + 2DCc,(
v) + D’u,( v),
(A.3)
133
Q. Cao, X. Deng/Optics Communications 142 (19971 13.5-145 where
A.3. Derivation of equation (17) From Eq. (l4), the parameter
i+J( rJ> = 4/= x%p(x,v)12dx, -z
U, can be expressed
as
V,2_tW2 u, =
(A.9)
w,2
Employing v,
Eqs. (8), (9) and (A.9), one can obtain
= (CW,2 + DV,)(
AW,2 + BV,) + BDSF; , (A.10)
W,’
( AW,2 + BV,)* + B’%f;
In the above expressions, the wavelength A used in Ref. [2] has been replaced by C,/u. Multiplying Eqs. (A.l)-(A.3) by a constant 2, and then integrating them, one can immediately obtain
WC -=
~=2w;(v)du=A2jc~2~;(v)d~+2ABjaxZu,(v)dv
In terms of expression (l5), the relation AD - BC = 1 and Eqs. (A.lO), (A.1 1), one can prove that
+B’
W,’
C+
X2+)dv, /0
(A.4)
VCW, ( A W,2 + BV, )’+ B’T2
(A.1 I) ’
D/Q,
A + B/Q, CW? + DV, - iD‘Zfc
jam2C2(V)du=ACjom2i+)dv
= AW,‘+BV,-iBYc +(AD+BC)im2il,(v)dv
=
(CWf
+ DV,)(
AWf
+ BV,) + BEYZ
- iFcW,
(AW,2 + BV,)* + B’;yTc’ + BD[2u,(v)dv,
(A.5)
V, =_--=_ W2’
c2u2(v)dv=C2c2Wj(r)dy+2DCja=2,:,(r)dY
iYYc
1
W,’
Qz’
(A.12)
It is apparent that Eq. (A.12) is an equivalent sion of Eq. (17). +D2[2U,(v)dy. Obviously,
Eqs. (A.4)-(A.6)
expres-
(A.6)
are just Eqs. (8)-(10).
Appendix B. Comparison between the results obtained in Section 2 and those obtained in Ref. US1
A.2. Derivation of equation (14) From Eqs. (8)-(lo), W;U2 =A’C*WP
Let us start from the original definition transform [25]
one can obtain
+ 4ABCDV,2 + B2D21Jf
H(f,)
= jm h(x)exp(-iTf,x)dx, TI
+ 2AC( AD + BC)W,2V,
h(x) =
+( ~*~~+~*c*)wfCi,
(B.‘)
&j: ff(fx)exp(i~fxx)dfx, 3c
+2BD(AD+BC)V,U,, v,~=A~c*w~+(AD+Bc)~v~+
(‘4.7) B~D*U,
+2AC(AD+BC)W,2V,+2ABCDW,2lJ, + 2BD( AD + BC)V,U,.
(A.81
Then, from Eqs. (A.7), (A.8), Eq. (14) can be easily derived, namely
w,'u,- v; = (AD - BC)‘W,~U, = w,‘u,
P.2)
where h(x) can be an arbitrary function provided that its integral /?,lh(x)(‘dx is finite, and r is the Fourier transform coefficient. For the Fourier transform with transform coefficient r, the Parseval’s theorem (a special case of theorem IV in Ref. [25]) and the Fourier transform of the differential operator can be expressed as /I
--P
IH(f_J*
df, = T/i
Ih(x)I* zc
dx,
03.3)
- (AD - BC)‘V, f,H(f,)=
-v,‘.
In the above derivation procedure, = 1 has been employed.
of the Fourier
the relation
AD - BC
-j:zG$$exp(-irf;*)dx.
(E.4)
Let us now investigate the relation between the overall intensity-moments defined in Section 2 and those defined
13‘4
Q. Cm, X. Deng/Optics
Communications
in Ref. [ 181. For simplicity, in the following, we shall ignore the difference 2~ between the angular frequency w and the linear frequency v, because it is very apparent that this small difference does not have any influence on the problem concerned here. In terms of the linear frequency v, the overall squared beam width (x2> and the overall squared far-field divergence ( u2> defined in Ref. [I 81 can be expressed as
142 (1997) 135-145
Substituting Eqs. (B. 13) and (B.14) into Eqs. (B. IO) and (B.6), G, and (u2> can be expressed as zz
G,=2C,
1
+(x,v)l’dnix, mccv
z jj0
jmjx v-31ap(x,v)/h12dvdx
c;
O -=
(“2)=G
I
z x’lF(x,v)/2dxdv 1 F” jj --z
(_r2)=
2
=-
(u’)
//0
(B.16) ~~(F(x,v)/~dvdx,
F 0 lI0
p=
= 6_i
P.9
u21G(U,v)12dUdv
U21G(u,v)12dvdu, --cc
(B.6)
where F(x,v)
=_/= 4(x,t)exp(-i2nvt)dt, --a
P.7)
G(u,v)
= jm F(x,v)exp --r
03.8)
F,, =
;DDIF(x,v)I’dxdv
z 2mr jj0
z
__v+(x,v)12dvdx
rx
= -$ j ja
!I‘
(B.15)
IF(x,v)12dvdx,
(B.9)
--5
From Eqs. (B.16) and Eq. (12), one can find that the squared divergence angle defined in Ref. [18] is completely different from that given in Section 2. As claimed by Ref. 1181, the squared divergence angle defined there does not obey the ABCD law. That is to say, the squared divergence angle defined in Ref. [ 181 is not appropriate to be used to extend the ABCD law to polychromatic pulsed beams. From Eqs. (B.8), (B.15) and (B.l6), one can deduce that the mathematical reason which leads to the unsuitable definition of the divergence angle in Ref. [ 181 is that the influence of the Fourier transform coefficient r= 25-v/C, which is just related to the frequency has not been taken into account. It can be easily proved that, for a quasi-monochromatic pulsed beam such as a SCLQ field [18], U = 4(u2>, namely, in this special case, the divergence angles defined in Section 2 and defined in Ref. [18] are approximately equivalent. This property is just consistent with the conclusion that the overall spatial second-moments of paraxial SCLQ fields follow the ABCD law [18].
G, = j j= IG(u,v)12dudv -Ti = 2 * jj0
= IG(u,v)l’dv
Comparing F(x,v)
du.
(B.lO) Appendix
-rx
C. Derivation of inequality
(24)
Eq. (B.7) with Eq. (.5), one can deduce that
= cp(x,-
By employing F, = 1 and
v) = cp*(x,v).
(B.11)
this relation, one can easily prove that
(B.12)
w2=4(,?),
where W* is given by Eq. (6). Expression (B.12) shows that the squared beam width defined in Section 2 is equivalent to that defined in Ref. 1181. Employing Eqs. (B.3), (B.4) and (B.l I), one can obtain
jrIG(u,v)12du=~ jrbp(x,v)12dx, --P
--co
j:XlaG(u,v)i2du=
From Eq. (I 3), one can prove that
2
(B.13)
2
I
%4(x,v) 2 ax ’ [
(C.1)
where the equal-sign can be used only when c?L(x,v)/ax = 0 in the whole transverse plane. Substituting inequality (C.1) into Eq. (12), the following result can be derived,
j;&lFi’dx. (B.14)
dvdx.
(C.2)
Q. Cao, X. Deng / Optics Communications
In terms of the Schwarz inequality,
one can obtain
CC.31 where we have assumed that x&x,v) 00, and we used the relation
vanishes for x +
-P(v)
(-9 In expression (C.31, the equal-sign can be used only when -C,xv~.Jx,v) = +a(x,v)/dx, where C, is an arbitrary positive constant. The reason why C, is not a negative constant but a positive constant is that the integral lY,&x,v)dx is finite. By employing expressions (C.21, (C.3) and Eq. (6), one can immediately obtain inequality (24). In expression (241, when the conditions only aL(x,v)/ax = 0 and - C,xvcp,(x,v) = &$x,v)/~x are both satisfied, the equal-sign can be used.
References [ll A.E. Siegman, Proc. Sot. Photo-Opt. Instrum. Eng. 1224 (1990) 2.
142 (1997) 135-145
145
[2] P.A. B&anger, Optics Lett. 16 (1991) 196. [3] A.E. Siegman, IEEE J. Quantum Electron. 27 (1991) 1146. [4] M. Morin, P. Bernard, P. Galarneall, Optics Lett. 19 (19941 1379. [5] Y. Champagne, J. Opt. Sot. Am. A 12 (19951 1707. ii H. Weber, Opt. Quantum Electron. 24 (1992) S1027. 171 H. Weber, Opt. Quantum Electron. 24 (1992) S861. 181M.J. Bastiaans, J. Opt. Sot. Am. A 3 (19861 1227. 191 M.J. Bastiaans, Optik 88 (1991) 163. 1101R. Simon, N. Mukunda, E.C.G. Sudarshan, Optics Comm. 65 (1988) 322. 1111R. Martinez-Herrero, P.M. Mejias, M. Sanchez, J.L.H. Neira, Opt. Quantum Electron. 24 (1992) S1021. 1121 R. Martinez-Herrero, P.M. Mejias, H. Weber, Opt. Quantum Electron. 25 (1993) 423. 1131Q. Lin, S. Wang, J. Alda, E. Bernabeu, Optics Lett. 18 (1993) 669. 1141A.E. Siegman, Lasers (University Science, Mill Valley, CA, 1986) Ch. 9. 1151 S.P. Dijaili, A. Dienes, J.S. Smith, IEEE J. Quantum Electron 26 (1990) 1158. [I61 A.G. Kostenbauder, IEEE J. Quantum Electron, 26 (1990) 1148. 1171 See for example, J. Zhou, G. Taft, C.P. Huang, M.M. Mumane, H. Kapteyn, I.P. Christov, Optics Len. 19 (1994) 1149. Optics Len. 20 (1995) 1181P.M. Mejias, R. Martinez-Herrero. 660. 1191M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975) Ch. 10. 1201S.A. Collins, J. Opt. Sot. Am. 60 (1970) 1168. 1211J. Paye, A. Migus, J. Opt. Sot. Am. B 12 (1995) 1480. 1221B.O. Maier, N.G. Preobrazhenskii, Opt. Spectr. (USSR) 66 (1989) 643. 1231J.D. Jackson, Classical electrodynamics (Wiley, New York, 1975) Ch. 6. 1241M. Lax, W.H. Louisell, W.B. McKnight, Phys. Rev. A 11 (19751 1365. 1251 See for example, E. Lalor, J. Opt. Sot. Am. 58 (1968) 1235.