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How many times is a laser beam diffraction-limited?
15 January
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
134 ( 1997) 503-5 13
Full length article
How many times is a laser beam diffraction-limited? S. Bollanti a, P. Di Lazzaro a, D. Murra b a Eneu. Deparrimento Inn-Fis-L.ac, P.O. Box 65,00044 Fruscari, Italy b Emu, Deparhento
Inn-Fis-Lac, P.O. Box I, 75025 Policoro, Italy
Received 22 March 1996; revised version received 5 July 19%; accepted 2 August 1996
Abstract We present an approach to the definition and estimation of the “times diffraction limit” (TDL) of highly diffracted laser beams that takes into account the near- and far-field energy distribution of the real beam instead of an ideal Gaussian beam. We have used it to find the TDL factor of two hard-edge-unstable-resonator laser beams having different quality and spatial energy distributions. The results are compared with the widely used beam quality parameter M’. The discrepancy between our ‘I’DL and the M2 coefficient becomes significant when the near-field energy distribution of a spatially coherent beam deviates from the pure Gaussian shape.
1. Introduction The problem of defining and measuring laser beam quality has been widely studied [l], mainly because users need to know the energy distribution of the beam in the artificial waist created by a focusing element and how much the actual waist dimension differs from the smallest theoretically achievable, “diffraction-limited” beam waist. There is general agreement in the literature on defining the quality of a real beam by normalising its parameters to those of a proper Gaussian beam [2-71. This approach has been widely accepted for several reasons: (a) in the paraxial approximation the lowest order transverse mode (TEM,) in stable resonators is a Gaussian beam; (b) the Gaussian beam has the lowest “diffraction-limited” divergence for the same beam waist; and (c) the introduction by Lavi et al. [2] of a quantity N, called “beam-quality parameter”, successively obtained with another formalism by Sasnett [3] and Siegman [4] and called M2; both N and M2 coefficients are minimised by the coherent 0030-4018/97/$17.00
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TEM, Gaussian beam. In fact, for a rectangular beam with coordinates x and y transverse to the propagation direction z, along the x coordinate M2 canbe calculated as [4]:
(1) where (x2) is the spatial variance (or second moment) of the energy profile at the beam waist; ( s,’> is the spatial frequency variance of the angular spread function; (( x,‘) . (~,2~)) ‘I2 = (1/49~) is the spacebandwidth product of the Gaussian-beam variances [4], and has the lowest value possible. An analogous expression can be written in the y coordinate for obtaining My’. It follows from Eq. (1) that M2 = 1 only for a Gaussian beam. The definition of MZ is based on a general and rigorous theory in terms of second moments; however, it turns out to be basically the ratio of the real beam space-bandwidth product to the ideal Gaussian space-bandwidth product.
0 1997 Elsevier Science B.V. All rights reserved.
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A recent document [8] of the “International Standardization Organization” (ISO) defines the M2 parameter as the times-diffraction-limit-factor (TDL factor). However, in the same document it is pointed out that the methods for measuring the beam parameters, based on the second-moment beam-diameter definition, “may not apply to highly diffractive beams such as those produced by unstable resonators or passing through hard-edged apertures”. A possible reason is that, in this case, the term (sf) in Eq. (1) diverges, as discussed in Refs. [4,9-l I], so the equivalence M2 = TDL factor may lead to incorrect results. On the other hand, in our opinion, it would be desirable to have a TDL definition as independent as possible of the given beam spatial energy distribution. In this paper we discuss the case of hard-edged unstable resonator laser beams that do not exhibit a Gaussian-like shape in the near-field plane, and we propose a TDL definition based on a reference beam (RB) that need not be ideal Gaussian. This TDL can be applied to Gaussian-like beams as well. The paper is organised as follows: in the next section, the RB and the TDL of real laser beams are defined. Then, a new method is given for dealing with high peakpower laser beams with super-Gaussian-like nearfield energy distributions. Finally, the theory is applied to the beams generated by two excimer laser systems equipped with unstable resonators, and a comparison is made between our TDL and the M2 values.
2. A new definition of the TDL A spatially coherent (i.e. constant-phase wavefront) beam is commonly referred to as being diffraction-limited [ 121.Hence, we define the near-field RB as having the same spatial energy distribution as the real beam and a constant phase wavefront. Thus, if a laser can generate a beam with a constant near-field phase distribution, this beam will have TDL = 1 even if its energy distribution is different from Gaussian, and then M2 > 1. We will consider the common case of the near-field plane being at the laser output, when the laser output is within the Rayleigh range from the beam waist. In the following, the term “far-field” means at the focal plane of
an ideal lens or mirror placed as close as possible to the laser output; hence, the “near-field” defined above means just before this focusing element. When the real laser beam is focused by an aberration-free optical system, the experimental far-field energy distribution (EED) is obtained in the focal plane of the optical system. Let us define the far-field theoretical energy distribution (TED) as the result of the Kirchaff diffraction integral calculated using the constant phase near-field distribution of the RB and the parameters of the focusing system. Obviously, the width of the EED will be larger than that of the TED, because of a poor mode coherence. Then, a “good” definition of the TDL factor would be the ratio of the EED to the TED widths, so we need a definition of the beam width. Two methods can be used to define the beam width: the energy content and the l/e2-diameter. The former does not tell us how the energy is distributed over the beam size; this can produce large errors on the estimation of the peak energy density value. The latter gives good information about the width of the central lobe but does not provide the corresponding energy content. Let us introduce the quantity Q = k/ e2, where k is the ratio between the energy content in the far-field beam within a given divergence full angle 6 and the total energy (here a circular symmetry is understood, but the treatment is also valid for a rectangular or elliptical symmetry, as shown in Appendix A). Once the focal length f of the relay optics is fixed, the relationship between 8 and the corresponding farfield beam diameter D is directly given by D =fO, and the average laser energy density within the circle with diameter D will be proportional to k/D2 = Q/f 2.
(2)
Conversely, Q is proportional to the timeintegrated-radiant-intensity (i.e. energy/solid angle) and, in our opinion, a beam is well characterised when both Q ,, Edand Qs6,ss are known. Here Q,,e~ refers to an angle 8 corresponding to l/e2 of the peak energy density value, and Q86,56 refers to an angle 8 corresponding to k = 0.865. The former gives the main lobe features, the latter indicates the possible presence of long energy tails. The difference between Q, , ,Z and Qss 58 reveals how far the beam is from the Gaussian ‘shape. In fact, for a pure Gaussian beam, it is obvious that Ql,e2 = Qss,sw.
S. Bollanti et al./ Optics Communications
Finally, we can define the TDL as the square root of the ratio between the reference and the real Q: lDL=($)“2=g(g)“2.
(3)
where we used D = f0. If the TED and the EED have the same shape, Eq. (3) can be reduced to the ratio of the corresponding diameters. Later, we will call this ratio the “TDL scale factor”.
3. The main steps to determine
the TDL
The procedure to determine the TDL defined in Eq. (3) can be summarised as follows: - Firstly, it is necessary to measure the near-field energy distribution (NFED) and the EED, either directly (by, e.g., a TV or a CCD camera) or indirectly (e.g., by a moving slit or knife edge). - Then, the TED is obtained by the Kirchoff diffraction integral of the constant-phase-NFED field (that is, the field of the near-field RB); this is described in the next section for the common case of a super-Gaussian (SG)-like NFED. - Once both EED(x, y) and TED( x, y> are known, their l/e*-diameters Drea’ and DRB can be directly measured, and the corresponding energy contents krra’ and kRB can be calculated as shown in Appendix A. Appendix A also shows a method to calculate the diameter corresponding to a given krea’ or kRB. It is important to underline a few considerations: (i) The Q,,,z value [and the corresponding TDL (Q,, ,?>I is much less dependent on the specific interpolation of the experimental data than the Qss,ss value. This means that the Q,,,z value can be used even in the presence of uncertain interpolations. (ii) The ratio Q/S (S = near-field beam area, defined either according to Q,,,? or to Qss.sa> is proportional to the radiance and is an invariant parameter of the related laser beam. In the case of a pure Gaussian beam with circular symmetry, we have Q,/S, = 0.865n/(4h2). (iii) Basically, the TDL defined in Eq. (3) is proportional to the ratio of the angular spread of a phase-distorted beam compared to the spread of a constant-phase beam with the same near-field ampli-
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tude profile, taking into account the respective energy contents. To some extent, we are introducing a kind of “generalised Strehl-ratio” definition. The fundamental difference between the well-known Strehl ratio [13] and our TDL is that the Strehl-ratio normalises the peak value of the phase-distorted beam energy density to the peak value of the constant-phase beam energy density, without considering their whole spatial distribution. (iv) Our TDL may suffer from a relatively arbitrary definition of the near-field plane. However, it works correctly whenever the distance between the waist of a rotationally symmetric beam and the laser output is shorter than the Rayleigh range, as well as in the case of orthogonal beams, even if they have a different waist location and Rayleigh range along the x and y transverse axes, provided that, again, the distances of both waists from the laser output are shorter than their respective Rayleigh ranges.
4. A method to determine distribution
the theoretical
energy
in the far-field
The mathematical treatment to obtain the TED consists in evaluating the Kirchoff integral of the electrical field. The Kirchoff integral reduces to a Fourier transform (FT) if the beam is focused by a perfect lens. In general, the TED calculated in this way is a numerical function, except when it is possible to have an analytical FT. Then, the TED can be compared with the EED by interpolating the numerical function with an analytical one. In order to avoid the interpolation process, it would be desirable to have an analytical TED. This is not an easy task because most laser beams have a SG-like near-field energy profile. Recently, a new analytical treatment of flattened Gaussian beams has been proposed (see Ref. [14]), which gives the propagation law of the field distribution by using Laguerre-Gauss functions. Here, we propose a simpler approach, which consists in representing a SG-like function by means of the squared convolution product (CP) of a Gaussian G( 5, WC,1with a rectangle function R( 5, We,>,with an appropriate choice of parameters: SG-like energy profile =
[w5w*,.w~,)]*
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tor, it is possible to obtain a family of curves TED(x,~;@~,) with the same shape and different widths. 2
1
xJ+C*,W&S*
9
(4)
where [= x, y, WC8 is the l/e
radius of the Gaussian function, and 2Wtr is the rectangle function length. It is well known that the FT of a CP is equal to the product of the FT of the functions processed by the convolution integral. Therefore, the FT of the CP in Eq. (4) will be expressed by the product of a Gaussian with a sine function (rectangular geometry) or with the square root of an Airy function (circular geometry) without needing either numerical approximation in the FT or interpolation. Hence, we can immediately write TED(x,O)
=IFT[CP(x,W,,,W,,)]t2 =lF-TIG(O’,p)]
TED(O,y)
FT[R(.0V,,)]t2,
=IFT([CP(y,W,,,W,,.)]Iz
=(FT[G(YJ+$~)]
+-@(~J’~r)]l’.
In the case of circular symmetry, TED(x,O) = TED(O,y). When x, y are separable variables, we have TED( x.y> = [TEN x,0)1 . [TED(O, ~11. Let us define the far-field Gaussian radius $,: @c, = V/rWcg
9
(5a)
and the characteristic parameter Qcp,, of the sine function for rectangular geometry [sin( c/Qt,,) /(s/Q*,,,>] or of the Airy function for circular geometry [J,(~/@~,)/(~/@~J2 (where J, is the first order Bessel function): @&= hf/2?rW(,.
(5b) It can be seen that the fundamental parameter of the Gaussian-rectangle CP is the ratio r = WtJWt, ‘. In fact, when r --) 0 we have a pure Gaussian function, and when r --) m we have a pure rectangle function. Once r is given, we can characterise the TED with the 2r-factor, since from Eqs. (5) we have @,/@$,,, = 2We,/Wt, = 2r. By varying the Gaussian radius Qtg and maintaining constant the 2r-fac’ Hereafter, we understand that r is a function of 5.
5. Application
to real beams: IANUS and HERCULES
the excimer
lasers
We used the method proposed in this paper to find the TDL of two XeCl excimer lasers, IANUS and HERCULES, with different beam qualities. The performances of IANUS and HERCULES are detailed in several papers [ 15- 181. Here, in Table 1 we summarise the main features of the oscillator part of IANUS, when equipped with a generalised self-filtering unstable resonator (GSFUR) [ 18,191, and of HERCULES, when equipped with a positive branch unstable resonator (PBUR). The numerical results presented in the following were obtained with a PC 486DX2/66 (16 Mbytes RAM) and Mathcad 4.0 (MathSoft) software. We chose a numerical precision within IO%, so the overall computing time for the TDL scale factor of IANUS, e.g., was about 30 minutes. The GSFUR of IANUS has a magnification 8, and the corresponding near-field laser energy distribution (measured by a TV camera, as detailed in Ref. [18]) has circular symmetry with a central hole corresponding to the shadow of the intracavity filtering aperture [ 191. The near-field energy profile was interpolated by a CP (see Eq. (4)) with an r-factor = 3.17 as shown in Fig. 1. As a consequence, the TED has a Gaussian-times-Airy shape with a 2r-factor = 6.34. As shown in Appendix B, the presence of the hole can be neglected in the following calculations. The EED( x, y) was measured using the knife-edge Table 1 Main features of IANUS and HERCULES Laser system features
IANUS (GSFUR)
HERCULES (PBUR)
Emission wavelength hm) Near-field beam size @84% energy (cm2) Output energy (J) Laser pulse FWHM (ns) Divergence @84% energy (pxad) Radiance (W/(cm2ster)) Max repetition rate (Hz)
308
308
P(O.3712 0.01 80 59 8.9x lOI 100
4.7 x 7.5 5 120 102X 169 6.2X lOI 10
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Fig. 1. Measured near-field energy profile of IANUS fitted by a convolution product of a Gaussian with a rectangle function with radii W.,,?= 1.2 mm and W,, = 3.8 mm, respectively.
(KE) method, that is, scanning a blade across the far-field beam. For example, if the KE is moved along the x-axis, the detector output is proportional to PEED(x’) = /::/_rEED(
X, y) d y dx.
(6)
When the EED and the TED have similar shapes, the previously defined family of curves TED(x,Y;@~,) can be used in place of EED( x,y) in
-0.2
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Eq. (6) to obtain the P,,,( x’;@~,,>, as detailed in Appendix C. Fig. 2 shows the experimental results obtained in the focal plane of a mirror with f= 3.16 m, together with I’,,,( x’;GXjrs= 0.265 mm>. Fig. 3 shows the corresponding function TED(x,O;@,~, = 0.265 mm) and the diffraction-limited TED. Their diameter ratio, i.e. the TDL scale factor, was 1.1. The same result was obtained by calculating the TDL(Q,,.z) and the l”DL(Qs6,& along both x and y directions. The M2 is 1.3 for this laser beam.
0.05
0.1
0.15
0.2
0.25
0.3
X AXlS(mm)
Fig. 2. Far-field energy distribution of IANUS measured by the knife-edge method along the x direction. The solid line is a plot of Eq. (Cl) = 0.265 mm. Focal length f= 3.16 m. (see Appendix C) with the best tit parameter GXxg
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Fig. 3. Comparison between IANUS diffraction-limited energy distribution TED (solid line) and the best fit function of the experimental rekts of Fig. 2, TED(x,O;@~~= 0.265 mm) (dashed line). --
HERCULES is equipped with a PBUR and has a rectangular cross section with dimensions (electrode gap X discharge height) = (9.5 X 5) cm*. The near-field energy distribution has a (1.6 X 1.6) cm* hole, corresponding to the shadow of the output coupler mirror. In this case, the presence of the hole may lead to a not negligible error in the TDL calculation when using Eq. (41, so we modified the near-field fit function. The modifications and the consequences on the TED and the TDL estimations are detailed in Appendix B. The &factor is 8.4 for the x-axis, and 4.6 for the y-axis. The HERCULES beam was focused by placing the cavity mirrors a little farther than the confocal distance, to obtain the far-field waist 26 m from the output coupler mirror. The far-field energy distribution was measured with the KE method, and Fig. 4 shows the results together with both PEED(x’;@~~= 4.4 mm) and P,,,( y’;ayg = 3 mm>. The corresponding functions TED( x,O;@~, = 4.4 mm) and TED(0, y;Qy,, = 3 mm) are shown in Fig. 5 and Fig. 6, respectively. The diffraction-limited TEDS along the x and y directions are also given for reference. The TDLs (Q,,,z) were 17.6 for the x-axis and 13.3 for the y-axis. For comparison, we calculated the M2 factor for HERCULES, according to Annex A of the IS0 guidelines [8]. The results are M2 = 19.7 for the x-axis, and M2 = 13.6 for the y-axis. It should be pointed out that the reliability of the results obtained following the IS0 normative concerning “alternative methods”
(including the KE method) “were verified for CO, gas laser beams with stable resonator geometries and power up to 10 W and with M2 up to M2 = 4, for radially symmetric beams” [8]. Since our HERCULES and IANUS laser systems are operated with different resonators and different values of wavelength, power and M2, it is not possible to estimate the confidence level of these M2 results.
6. Summary
and conclusions
We have presented a novel approach to the definition and measurement of the TDL factor of laser beams having either Gaussian-like or non-Gaussian spatial energy distribution. The main consideration is that, if laser users want to know how much the divergence value of a highly diffractive laser beam differs from its diffraction limit, they should refer to a TDL factor that mainly accounts for the spatial coherence of the beam, because the near-field energy profile has a much weaker influence than its phase distribution on the far-field energy distribution (and divergence), as discussed in Refs. [20,21]. Hence, recalling that the TDL performance of a source is strictly related to its spatial coherence and then to the Strehl-ratio (see Ref. [21]), we have suggested a TDL definition based on a kind of “generalised Strehl-ratio”, i.e. a quantitative comparison between the measured far-field energy distribution EED and a
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far-field-propagated reference beam having the same near-field shape and a constant-phase wavefront, called TED. The TDL factor can be calculated with Eq. (3) through the parameter Q. We have also presented an analytical method for determining the diffraction-limited TED for stable- and unstable-resonator beams. The proposed TDL calculation procedure can be summarised as follows: (i) Measurement of the near-field energy distribution (NFED) by e.g. a TV or a CCD camera. Interpolation of the experimental NPED by means of a squared convolution product CP of a Gaussian with a rectangle function, resulting in a SG-like function.
0
I- ~~~ -2
-__-_--+_~ ~~ -1.5
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Yq +___- _+~~~~+~~~_~ -0.5
0
509
(ii) analytical calculation of the TED in the focal plane of the focusing element by the FI of the CP. (iii> Measurement of the EED, either directly or indirectly as shown in Appendix C. (iv) Evaluation of the diameter(s) and energy content (that is the Q-factors) of both EED and TED as shown in Appendix A, and calculation of the TDL by means of Eq. (3). In the particular case the EED and the TED have the same shape, Eq. (3) can be reduced to the ratio of the respective diameters, the TDL scale factor. We used the above procedure to find the TDL of two pulsed excimer lasers equipped with different hard-edged unstable resonators and emitting beams
l
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2
2.5
,_
i
3
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-4
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-1
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1
2
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Fig. 4. (a) Far-field energy distribution of HERCULES measured along the horizontal x direction by the knife-edge method. The solid line is a plot of the Eq. (Cl), see Appendix C, with the best fit parameter GX, = 4.4 mm. (b> The same as (a) but along the vertical y direction and with eYPYR = 3 mm. Focal length f= 26 m.
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1
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with different spatial energy distribution and quality. The results are summarised in Table 2. In the case of IANUS, which emits a very good quality, roundshaped near-field energy distribution, our TDL factors have the same value, while the M* factor is 18% higher. For HERCULES, along the horizontal axis the TDLs are consistent with each other within 5%, whereas the M2 coefficient differs by 12% with respect to the TDL Q,,e2. Along the vertical axis,
1 0.9 0.9
s
OeJ
$.
0.6
c 5;
03
6
0.4
k
0.3
where the near-field energy distribution is closer to a Gaussian-like shape, the difference is less pronounced. It should be pointed out that in the HERCULES case the Qss,ssbfactor is more dependent on the specific interpolation of the experimental data than the Ql,e~. This is because the central lobes of both TED and EED of HERCULES contain less than 86.5% of the total beam energy. In our experience, if the central lobes of either TED or EED contain less
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Y AXIS (mm) Fig. 6. Comparison between HERCULES diffraction-limited energy distribution TED along the y direction (solid line) and the best fit function of the experimental results of Fig. 4b, TED(O,y,QY,,= 3 mm) (dashed line).
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Table 2 HERCULES and IANUS times-diffraction-limit-factors Laser system
IANUS HERCULES, horizontal HERCULES, vertical
TDL Q86~
TDL
TDL scale factor
M2
Q,,G
1.1 16.8 12.9
1.1 17.6 13.3
1.1 17.4 13.4
1.3 19.7 13.6
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output, it is difficult to define procedure can overestimate the there is no propagation law for Studies are in progress to include
a TDL, and our TDL number; (b) the Q parameter. such law.
Acknowledgements We thank Drs. A. Torre, G. Dattoli and T. Letardi for helpful discussions on this topic.
energy-percentage than the k value chosen, the reliability of the corresponding Qk value may be not good. In this case, we recommend using the Q,,,> (and the corresponding TDL) value; in alternative, a k value smaller than the energy percentage contained in the central lobe should be chosen. The results reported in Table 2 can be explained by considering the different meanings of the MZ and the TDL coefficients: MZ takes into account the deviation of both amplitude and phase wavefront from the ideal Gaussian-shaped, constant-phase beam, while our TDL only considers the phase distortion of the real beam, as the constant-phase NFED of the RB has the same amplitude shape. Hence, in the case of the Gaussian-like HERCULES vertical energy distribution, the M2 and TDL coefficients basically give an estimate of the beam phase distortion (NFED is Gaussian-like), so that M2 = TDL. On the contrary, in the case of IANUS and the horizontal energy distribution of HERCULES, the non-Gaussian amplitude shape gives a contribution to the M2 value, so M2 > TDL. This amplitude contribution to M2 becomes significant (compared to the phase contribution) when the non-Gaussian beam has good spatial coherence (i.e. almost constant-phase wavefront) like IANUS. In conclusion, the TDL factor proposed in this paper gives information on the spatial coherence of the real beam regardless of its amplitude shape, without requiring a direct measurement of the beam phase. Moreover, for highly diffracted laser beams it can give a realistic estimation of the margin of improvement in the laser beam quality feasibly achievable through better resonator optics and/or active-medium uniformity. The proposed method has two shortcomings: (a) in the (unlikely) case that the beam waist is intracavity and more than a Rayleigh range far from the laser
Appendix A. Calculation of the Q-factor In the Q-factor definition Q = k/O 2, the divergence 8 comes from the measurement of the beam diameter D in the focal plane, since 0 = D/f, where f is the distance between the focusing element and the focal plane. Regarding the Q,, Ed,once both TED and EED (and the corresponding l/e2 diameters of the central lobe) are known, we consider three cases for the calculation of, e.g., krea’: (i) Circular symmetry (EED( XJ) = EED(r), D = far-field laser beam l/e*-diameter). We have k real=
D’2EED( r) r dr /0 /0
(AlI
+=EED( r) r dr
(ii) Elliptical symmetry (D_,, D, = main-axis length of the laser beam in the far-field plane such that on the ellipse (2x/D.J2
+ (~Y/D,)~ = 1
(A2)
it holds EED ( x,y) = 1/e2). We have k real=
D”2 X(Y)EED(x,y) dxdy /0 /0 +m /0
+==
(A31
EED( x,y) dxdy
I0
where x(y) satisfies Eq. (A2). (iii) Rectangular symmetry CD,, D, = rectangle side-length of the laser beam in the far-field plane). We have k real=
o/2 D”2EED( x,y) dxdy /0 /0 +m /0
+= /0
EED( x,y) dxdy
W)
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Table 3 TDL results with and without the output coupler mirror shadow in the near-field energy distribution of HERCULES
By substituting EED with TED in Eqs. (Al)-(A4), we obtain the corresponding kRB. In order to calculate the Qr, defined as the Q-factor corresponding to a fixed encircled energy k, we need the diameter D(k). It is evident that Eq. (Al) can still be used to find the D(k) of a beam having circular-symmetry. In the case of Eqs. (A3) and (A4), it may happen there are many couples of D,(k) and D,(k) values corresponding to the same k; therefore, we suggest the following procedure to calculate, e.g., the Qz\,: (a) Find the diameters 0:. and Di such that:
unstable resonator lasers with diffraction coupling, like IANUS and HERCULES. In this case, Eq. (4) changes into
EED( D-;/2,0) = EED(0, D;/2) = l/e2.
SG-like energy profile with a hole
(b) Find another couple of diameters 4, and D,, satisfying the equations: (SJD,)
= (D:/D$
and D,/2 D,/2 EED( x,y) dxdy / /0 0 real = k +P
/0
+"
/0
=
Axis
TDL scale factor
TDL Q,,t2
Horizontal, no hole Horizontal, donut shape Vertical, no hole Vertical, donut shape
19 17.4 14.3 13.4
18.9 17.6 14.3 13.3
= [cp( S,w,,*W,,)]’
(Bl)
where R(,$,Weh) is a rectangle function corresponding to a shadow with a size 2Wt,,. The corresponding TED in the case of circular symmetry will be given by [22]: ‘I’ED=~[CP(~,W,,W,)]
0.865.
- R( &W[h).
- ~~~[~(~,W,,)I’,
EED( x,y) dxdy
(B2)
(A9 (c) Calculate Qg\,< 5 ) = ( krea’f2)/( 0221, where 5 = x, y.
Appendix B. TED for diffraction-coupled unstable resonator beams When the near-field energy profile has a donut shape, the mathematical treatment with the FT is more complex. A profile with a hole is typical of
where r is the radial variable in polar coordinates and p = (near-field size/shadow size). Strictly speaking, Eq. (B2) is exact only in the case of plane waves; however, it can be shown that it works well for any beam satisfying the relationship /3 + 2r > 8. Eq. (Bl) requires a longer computing time than Eq. (4) to calculate the TDL. However, (Bl) is strictly necessary only for the special case of beams having a coefficient j3 < 3. In fact, it can be demonstrated that, for /3 > 3, the difference in Q,,e~ calculated by means of Eq. (4) and Eq. (Bl) is less than 10%. As an example, Fig. 7 shows the value p =
CQcwit, hole) Qcwithout hole)]/Q(with hole)l"'S'S
Fig. 7. Relative variation of the parameter Q,,e~ with and without the hole in the near-field energy distribution versus p for different r-factors for circular symmetry.
P
for
different r-factor values, considering circular symmetry. In the case of the IANUS beam we have j3 = 8 and Fig. 7 gives a value of p < 1%; therefore, the hole in the near-field energy distribution can be neglected. The HERCULES beam has a rectangular shape, with p = 6 and r = 4.2 along the x-axis. The difference in the TDL calculation, taking into account the hole, is less than 10%; this has also been verified for the y-axis, where /3 = 4 and r = 2.3. Table 3 compares the results of the TDL calculations with Eq. (4) and Eq. (Bl).
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Appendix C. TDL with the knife-edge method After the near-field energy distribution has been measured and interpolated by using Eq. (4), and finally propagated in the far-field to obtain the TED(x,Y), the next step is to measure the far-field EED(x,y). If it can be directly measured (e.g. by using a CCD camera), we can use Eq. (3) and Appendix A to obtain the TDL. To avoid problems related to background estimation and saturation effects of the CCD [7], we can use the knife-edge (KE) technique. In this case the energy profile EED(x,y) is unknown and the measurement yields an energy distribution versus a beam dimension. For example, if the KE is moved along the x-axis, the measured quantity is P,,,(
x’) = /::/_:“EED(
x,Y) dydx.
Then, to evaluate the EED, we need a suitable function E(x,y) whose integral fits P,,,(x’). Finally, the Qrea’ parameter in Eq. (3) is calculated assuming E( x, y) - EED( x, y). We recommend using the Q,,p2 value, which is less dependent than the Qs6,ss value on the choice of the interpolating function, as mentioned in Section 3. When the near-field energy distribution can be expressed as the product of two independent functions of x and y, we suppose EED can be too, and we can write EED( x,y) = ;
[ PEED(d]
’;
[ ‘Ed
y)]’
where P E&) and PEED( Y) are measured with the KE moved along the x and y directions, respectively. In this case the computing time is strongly reduced. It can be further reduced by employing the TDL scale factor, when the TED and the EED have similar shapes. In this case, the EED(x, y) can be represented by the TED( x, y;@&,) defined in Section 4, adjusting the far-field Gaussian radius @, as a free parameter. Hence, the P,,,(x’) can be approximated by P,,,(x’;@!!)zg) = /
/+=TED( --z -72
x,y&)dydx.
(Cl)
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References [I] See, e.g., H. Weber, Opt. Quantum Electron. 24 (1992) S861. [2] S. Lavi, R. Prochaska and E. Keren, Appl. Optics 27 (1988) 3696. [3] M.W. Sasnett, in: Physics and Technology of Laser Resonators, eds. D.R. Hall and P.E. Jackson (Adam Hilger, Bristol, 1989) pp. 133-142. [4] A.E. Siegman, New Developments in Laser Resonators, Proc. SPIE 1224 (19%) 2-12. [5] F. Gori, M. Santarsiero and A. Sona, Optics Comm. 82 (1991) 197. 161 J. Sema, R. Martinez-Hetrero and P.M. Mejias, J. Opt. Sot. Am. A 8 (1991) 1094. [7] N. Reng and B. Eppich, Opt. Quantum Electron. 24 (1992) s973. [S] Document ISO/TC 172/SC 9/WGl, Terminology and test methods for lasers. Manuscript for ISO/DIS 11146. Optics and optical instruments Lasers and laser related instruments Test methods for laser beam parameters: Beam widths, divergence angle and beam propagation factor ( 15 February 1995). 191 A. Parent, M. Morin and P. Lavigne, Opt. Quantum Electron. 24 (1992) S1071. IO] R. Martinez-Herrero. On the Spatial Characterisation of Partially Coherent Beams, Proc. SPIE 1983 (1993) 513-518. I I I P.A. Behmger, Y. Champagne and C. Pare, Optics Comm. 105 (1994) 233. 121 0. Svelto, Principles of Lasers (Plenum Press, New York 1989) p. 397. [13] M. Born and E. Wolf, Principles of Optics, 6th Ed. (Pergamon Press, New York, 1993) pp. 460-462. [14] F. Gori, Optics Comm. 107 (1994) 335. [ 151 S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Hennsen, T. Letardi and C.E. Zheng, Appl. Phys. B 50 (1990) 415. [ 161 T. Letardi, S. Bollanti, P. Di Lazzaro, F. Flora, N. Lisi and C.E. Zheng, Nuovo Cimento D 14 (1992) 495. [17] T. Letardi, S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, L. Mezi, G. Schina, A. Torre and C.E. Zheng, Excimer laser development and applications at the ENEA Frascati Centre, Proc. SPIE 1810 (1993) 340-347. Its1 S. Bollanti, P. Di Lazzaro, F. Flora, T. Letardi, D. Murra, C. Petrucci and 0. Uteza, Optics Comm. 132 (19%) 565. [191 P. Di Lazzaro, T. Hermsen and C.E. Zheng, IEEE J. Quantum Electron. QE-24 (1988) 1543. ml Y.A. Ananev, Laser resonators and the beam divergence problem (Adam Hilger, Bristol, 1992) pp. 60-63. Lasers and [211M. Lurie, in: Surface Emitting Semiconductor Arrays, eds. G.A. Evans and J.M. Hammer (Academic Press, New York, 1993) pp. 435-465. [221A.I. Mahan, C.V. Bitterli and S.M. Cannon, J. Opt. Sot. Am. 54 (1964) 721.
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
134 ( 1997) 503-5 13
Full length article
How many times is a laser beam diffraction-limited? S. Bollanti a, P. Di Lazzaro a, D. Murra b a Eneu. Deparrimento Inn-Fis-L.ac, P.O. Box 65,00044 Fruscari, Italy b Emu, Deparhento
Inn-Fis-Lac, P.O. Box I, 75025 Policoro, Italy
Received 22 March 1996; revised version received 5 July 19%; accepted 2 August 1996
Abstract We present an approach to the definition and estimation of the “times diffraction limit” (TDL) of highly diffracted laser beams that takes into account the near- and far-field energy distribution of the real beam instead of an ideal Gaussian beam. We have used it to find the TDL factor of two hard-edge-unstable-resonator laser beams having different quality and spatial energy distributions. The results are compared with the widely used beam quality parameter M’. The discrepancy between our ‘I’DL and the M2 coefficient becomes significant when the near-field energy distribution of a spatially coherent beam deviates from the pure Gaussian shape.
1. Introduction The problem of defining and measuring laser beam quality has been widely studied [l], mainly because users need to know the energy distribution of the beam in the artificial waist created by a focusing element and how much the actual waist dimension differs from the smallest theoretically achievable, “diffraction-limited” beam waist. There is general agreement in the literature on defining the quality of a real beam by normalising its parameters to those of a proper Gaussian beam [2-71. This approach has been widely accepted for several reasons: (a) in the paraxial approximation the lowest order transverse mode (TEM,) in stable resonators is a Gaussian beam; (b) the Gaussian beam has the lowest “diffraction-limited” divergence for the same beam waist; and (c) the introduction by Lavi et al. [2] of a quantity N, called “beam-quality parameter”, successively obtained with another formalism by Sasnett [3] and Siegman [4] and called M2; both N and M2 coefficients are minimised by the coherent 0030-4018/97/$17.00
Copyright
PII SOO30-4018(96)00523-8
TEM, Gaussian beam. In fact, for a rectangular beam with coordinates x and y transverse to the propagation direction z, along the x coordinate M2 canbe calculated as [4]:
(1) where (x2) is the spatial variance (or second moment) of the energy profile at the beam waist; ( s,’> is the spatial frequency variance of the angular spread function; (( x,‘) . (~,2~)) ‘I2 = (1/49~) is the spacebandwidth product of the Gaussian-beam variances [4], and has the lowest value possible. An analogous expression can be written in the y coordinate for obtaining My’. It follows from Eq. (1) that M2 = 1 only for a Gaussian beam. The definition of MZ is based on a general and rigorous theory in terms of second moments; however, it turns out to be basically the ratio of the real beam space-bandwidth product to the ideal Gaussian space-bandwidth product.
0 1997 Elsevier Science B.V. All rights reserved.
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A recent document [8] of the “International Standardization Organization” (ISO) defines the M2 parameter as the times-diffraction-limit-factor (TDL factor). However, in the same document it is pointed out that the methods for measuring the beam parameters, based on the second-moment beam-diameter definition, “may not apply to highly diffractive beams such as those produced by unstable resonators or passing through hard-edged apertures”. A possible reason is that, in this case, the term (sf) in Eq. (1) diverges, as discussed in Refs. [4,9-l I], so the equivalence M2 = TDL factor may lead to incorrect results. On the other hand, in our opinion, it would be desirable to have a TDL definition as independent as possible of the given beam spatial energy distribution. In this paper we discuss the case of hard-edged unstable resonator laser beams that do not exhibit a Gaussian-like shape in the near-field plane, and we propose a TDL definition based on a reference beam (RB) that need not be ideal Gaussian. This TDL can be applied to Gaussian-like beams as well. The paper is organised as follows: in the next section, the RB and the TDL of real laser beams are defined. Then, a new method is given for dealing with high peakpower laser beams with super-Gaussian-like nearfield energy distributions. Finally, the theory is applied to the beams generated by two excimer laser systems equipped with unstable resonators, and a comparison is made between our TDL and the M2 values.
2. A new definition of the TDL A spatially coherent (i.e. constant-phase wavefront) beam is commonly referred to as being diffraction-limited [ 121.Hence, we define the near-field RB as having the same spatial energy distribution as the real beam and a constant phase wavefront. Thus, if a laser can generate a beam with a constant near-field phase distribution, this beam will have TDL = 1 even if its energy distribution is different from Gaussian, and then M2 > 1. We will consider the common case of the near-field plane being at the laser output, when the laser output is within the Rayleigh range from the beam waist. In the following, the term “far-field” means at the focal plane of
an ideal lens or mirror placed as close as possible to the laser output; hence, the “near-field” defined above means just before this focusing element. When the real laser beam is focused by an aberration-free optical system, the experimental far-field energy distribution (EED) is obtained in the focal plane of the optical system. Let us define the far-field theoretical energy distribution (TED) as the result of the Kirchaff diffraction integral calculated using the constant phase near-field distribution of the RB and the parameters of the focusing system. Obviously, the width of the EED will be larger than that of the TED, because of a poor mode coherence. Then, a “good” definition of the TDL factor would be the ratio of the EED to the TED widths, so we need a definition of the beam width. Two methods can be used to define the beam width: the energy content and the l/e2-diameter. The former does not tell us how the energy is distributed over the beam size; this can produce large errors on the estimation of the peak energy density value. The latter gives good information about the width of the central lobe but does not provide the corresponding energy content. Let us introduce the quantity Q = k/ e2, where k is the ratio between the energy content in the far-field beam within a given divergence full angle 6 and the total energy (here a circular symmetry is understood, but the treatment is also valid for a rectangular or elliptical symmetry, as shown in Appendix A). Once the focal length f of the relay optics is fixed, the relationship between 8 and the corresponding farfield beam diameter D is directly given by D =fO, and the average laser energy density within the circle with diameter D will be proportional to k/D2 = Q/f 2.
(2)
Conversely, Q is proportional to the timeintegrated-radiant-intensity (i.e. energy/solid angle) and, in our opinion, a beam is well characterised when both Q ,, Edand Qs6,ss are known. Here Q,,e~ refers to an angle 8 corresponding to l/e2 of the peak energy density value, and Q86,56 refers to an angle 8 corresponding to k = 0.865. The former gives the main lobe features, the latter indicates the possible presence of long energy tails. The difference between Q, , ,Z and Qss 58 reveals how far the beam is from the Gaussian ‘shape. In fact, for a pure Gaussian beam, it is obvious that Ql,e2 = Qss,sw.
S. Bollanti et al./ Optics Communications
Finally, we can define the TDL as the square root of the ratio between the reference and the real Q: lDL=($)“2=g(g)“2.
(3)
where we used D = f0. If the TED and the EED have the same shape, Eq. (3) can be reduced to the ratio of the corresponding diameters. Later, we will call this ratio the “TDL scale factor”.
3. The main steps to determine
the TDL
The procedure to determine the TDL defined in Eq. (3) can be summarised as follows: - Firstly, it is necessary to measure the near-field energy distribution (NFED) and the EED, either directly (by, e.g., a TV or a CCD camera) or indirectly (e.g., by a moving slit or knife edge). - Then, the TED is obtained by the Kirchoff diffraction integral of the constant-phase-NFED field (that is, the field of the near-field RB); this is described in the next section for the common case of a super-Gaussian (SG)-like NFED. - Once both EED(x, y) and TED( x, y> are known, their l/e*-diameters Drea’ and DRB can be directly measured, and the corresponding energy contents krra’ and kRB can be calculated as shown in Appendix A. Appendix A also shows a method to calculate the diameter corresponding to a given krea’ or kRB. It is important to underline a few considerations: (i) The Q,,,z value [and the corresponding TDL (Q,, ,?>I is much less dependent on the specific interpolation of the experimental data than the Qss,ss value. This means that the Q,,,z value can be used even in the presence of uncertain interpolations. (ii) The ratio Q/S (S = near-field beam area, defined either according to Q,,,? or to Qss.sa> is proportional to the radiance and is an invariant parameter of the related laser beam. In the case of a pure Gaussian beam with circular symmetry, we have Q,/S, = 0.865n/(4h2). (iii) Basically, the TDL defined in Eq. (3) is proportional to the ratio of the angular spread of a phase-distorted beam compared to the spread of a constant-phase beam with the same near-field ampli-
134 (1997) 503-513
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tude profile, taking into account the respective energy contents. To some extent, we are introducing a kind of “generalised Strehl-ratio” definition. The fundamental difference between the well-known Strehl ratio [13] and our TDL is that the Strehl-ratio normalises the peak value of the phase-distorted beam energy density to the peak value of the constant-phase beam energy density, without considering their whole spatial distribution. (iv) Our TDL may suffer from a relatively arbitrary definition of the near-field plane. However, it works correctly whenever the distance between the waist of a rotationally symmetric beam and the laser output is shorter than the Rayleigh range, as well as in the case of orthogonal beams, even if they have a different waist location and Rayleigh range along the x and y transverse axes, provided that, again, the distances of both waists from the laser output are shorter than their respective Rayleigh ranges.
4. A method to determine distribution
the theoretical
energy
in the far-field
The mathematical treatment to obtain the TED consists in evaluating the Kirchoff integral of the electrical field. The Kirchoff integral reduces to a Fourier transform (FT) if the beam is focused by a perfect lens. In general, the TED calculated in this way is a numerical function, except when it is possible to have an analytical FT. Then, the TED can be compared with the EED by interpolating the numerical function with an analytical one. In order to avoid the interpolation process, it would be desirable to have an analytical TED. This is not an easy task because most laser beams have a SG-like near-field energy profile. Recently, a new analytical treatment of flattened Gaussian beams has been proposed (see Ref. [14]), which gives the propagation law of the field distribution by using Laguerre-Gauss functions. Here, we propose a simpler approach, which consists in representing a SG-like function by means of the squared convolution product (CP) of a Gaussian G( 5, WC,1with a rectangle function R( 5, We,>,with an appropriate choice of parameters: SG-like energy profile =
[w5w*,.w~,)]*
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tor, it is possible to obtain a family of curves TED(x,~;@~,) with the same shape and different widths. 2
1
xJ+C*,W&S*
9
(4)
where [= x, y, WC8 is the l/e
radius of the Gaussian function, and 2Wtr is the rectangle function length. It is well known that the FT of a CP is equal to the product of the FT of the functions processed by the convolution integral. Therefore, the FT of the CP in Eq. (4) will be expressed by the product of a Gaussian with a sine function (rectangular geometry) or with the square root of an Airy function (circular geometry) without needing either numerical approximation in the FT or interpolation. Hence, we can immediately write TED(x,O)
=IFT[CP(x,W,,,W,,)]t2 =lF-TIG(O’,p)]
TED(O,y)
FT[R(.0V,,)]t2,
=IFT([CP(y,W,,,W,,.)]Iz
=(FT[G(YJ+$~)]
+-@(~J’~r)]l’.
In the case of circular symmetry, TED(x,O) = TED(O,y). When x, y are separable variables, we have TED( x.y> = [TEN x,0)1 . [TED(O, ~11. Let us define the far-field Gaussian radius $,: @c, = V/rWcg
9
(5a)
and the characteristic parameter Qcp,, of the sine function for rectangular geometry [sin( c/Qt,,) /(s/Q*,,,>] or of the Airy function for circular geometry [J,(~/@~,)/(~/@~J2 (where J, is the first order Bessel function): @&= hf/2?rW(,.
(5b) It can be seen that the fundamental parameter of the Gaussian-rectangle CP is the ratio r = WtJWt, ‘. In fact, when r --) 0 we have a pure Gaussian function, and when r --) m we have a pure rectangle function. Once r is given, we can characterise the TED with the 2r-factor, since from Eqs. (5) we have @,/@$,,, = 2We,/Wt, = 2r. By varying the Gaussian radius Qtg and maintaining constant the 2r-fac’ Hereafter, we understand that r is a function of 5.
5. Application
to real beams: IANUS and HERCULES
the excimer
lasers
We used the method proposed in this paper to find the TDL of two XeCl excimer lasers, IANUS and HERCULES, with different beam qualities. The performances of IANUS and HERCULES are detailed in several papers [ 15- 181. Here, in Table 1 we summarise the main features of the oscillator part of IANUS, when equipped with a generalised self-filtering unstable resonator (GSFUR) [ 18,191, and of HERCULES, when equipped with a positive branch unstable resonator (PBUR). The numerical results presented in the following were obtained with a PC 486DX2/66 (16 Mbytes RAM) and Mathcad 4.0 (MathSoft) software. We chose a numerical precision within IO%, so the overall computing time for the TDL scale factor of IANUS, e.g., was about 30 minutes. The GSFUR of IANUS has a magnification 8, and the corresponding near-field laser energy distribution (measured by a TV camera, as detailed in Ref. [18]) has circular symmetry with a central hole corresponding to the shadow of the intracavity filtering aperture [ 191. The near-field energy profile was interpolated by a CP (see Eq. (4)) with an r-factor = 3.17 as shown in Fig. 1. As a consequence, the TED has a Gaussian-times-Airy shape with a 2r-factor = 6.34. As shown in Appendix B, the presence of the hole can be neglected in the following calculations. The EED( x, y) was measured using the knife-edge Table 1 Main features of IANUS and HERCULES Laser system features
IANUS (GSFUR)
HERCULES (PBUR)
Emission wavelength hm) Near-field beam size @84% energy (cm2) Output energy (J) Laser pulse FWHM (ns) Divergence @84% energy (pxad) Radiance (W/(cm2ster)) Max repetition rate (Hz)
308
308
P(O.3712 0.01 80 59 8.9x lOI 100
4.7 x 7.5 5 120 102X 169 6.2X lOI 10
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2
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.
1
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Fig. 1. Measured near-field energy profile of IANUS fitted by a convolution product of a Gaussian with a rectangle function with radii W.,,?= 1.2 mm and W,, = 3.8 mm, respectively.
(KE) method, that is, scanning a blade across the far-field beam. For example, if the KE is moved along the x-axis, the detector output is proportional to PEED(x’) = /::/_rEED(
X, y) d y dx.
(6)
When the EED and the TED have similar shapes, the previously defined family of curves TED(x,Y;@~,) can be used in place of EED( x,y) in
-0.2
-0.15
-0.1
-0.05
0
Eq. (6) to obtain the P,,,( x’;@~,,>, as detailed in Appendix C. Fig. 2 shows the experimental results obtained in the focal plane of a mirror with f= 3.16 m, together with I’,,,( x’;GXjrs= 0.265 mm>. Fig. 3 shows the corresponding function TED(x,O;@,~, = 0.265 mm) and the diffraction-limited TED. Their diameter ratio, i.e. the TDL scale factor, was 1.1. The same result was obtained by calculating the TDL(Q,,.z) and the l”DL(Qs6,& along both x and y directions. The M2 is 1.3 for this laser beam.
0.05
0.1
0.15
0.2
0.25
0.3
X AXlS(mm)
Fig. 2. Far-field energy distribution of IANUS measured by the knife-edge method along the x direction. The solid line is a plot of Eq. (Cl) = 0.265 mm. Focal length f= 3.16 m. (see Appendix C) with the best tit parameter GXxg
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0.05
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Fig. 3. Comparison between IANUS diffraction-limited energy distribution TED (solid line) and the best fit function of the experimental rekts of Fig. 2, TED(x,O;@~~= 0.265 mm) (dashed line). --
HERCULES is equipped with a PBUR and has a rectangular cross section with dimensions (electrode gap X discharge height) = (9.5 X 5) cm*. The near-field energy distribution has a (1.6 X 1.6) cm* hole, corresponding to the shadow of the output coupler mirror. In this case, the presence of the hole may lead to a not negligible error in the TDL calculation when using Eq. (41, so we modified the near-field fit function. The modifications and the consequences on the TED and the TDL estimations are detailed in Appendix B. The &factor is 8.4 for the x-axis, and 4.6 for the y-axis. The HERCULES beam was focused by placing the cavity mirrors a little farther than the confocal distance, to obtain the far-field waist 26 m from the output coupler mirror. The far-field energy distribution was measured with the KE method, and Fig. 4 shows the results together with both PEED(x’;@~~= 4.4 mm) and P,,,( y’;ayg = 3 mm>. The corresponding functions TED( x,O;@~, = 4.4 mm) and TED(0, y;Qy,, = 3 mm) are shown in Fig. 5 and Fig. 6, respectively. The diffraction-limited TEDS along the x and y directions are also given for reference. The TDLs (Q,,,z) were 17.6 for the x-axis and 13.3 for the y-axis. For comparison, we calculated the M2 factor for HERCULES, according to Annex A of the IS0 guidelines [8]. The results are M2 = 19.7 for the x-axis, and M2 = 13.6 for the y-axis. It should be pointed out that the reliability of the results obtained following the IS0 normative concerning “alternative methods”
(including the KE method) “were verified for CO, gas laser beams with stable resonator geometries and power up to 10 W and with M2 up to M2 = 4, for radially symmetric beams” [8]. Since our HERCULES and IANUS laser systems are operated with different resonators and different values of wavelength, power and M2, it is not possible to estimate the confidence level of these M2 results.
6. Summary
and conclusions
We have presented a novel approach to the definition and measurement of the TDL factor of laser beams having either Gaussian-like or non-Gaussian spatial energy distribution. The main consideration is that, if laser users want to know how much the divergence value of a highly diffractive laser beam differs from its diffraction limit, they should refer to a TDL factor that mainly accounts for the spatial coherence of the beam, because the near-field energy profile has a much weaker influence than its phase distribution on the far-field energy distribution (and divergence), as discussed in Refs. [20,21]. Hence, recalling that the TDL performance of a source is strictly related to its spatial coherence and then to the Strehl-ratio (see Ref. [21]), we have suggested a TDL definition based on a kind of “generalised Strehl-ratio”, i.e. a quantitative comparison between the measured far-field energy distribution EED and a
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far-field-propagated reference beam having the same near-field shape and a constant-phase wavefront, called TED. The TDL factor can be calculated with Eq. (3) through the parameter Q. We have also presented an analytical method for determining the diffraction-limited TED for stable- and unstable-resonator beams. The proposed TDL calculation procedure can be summarised as follows: (i) Measurement of the near-field energy distribution (NFED) by e.g. a TV or a CCD camera. Interpolation of the experimental NPED by means of a squared convolution product CP of a Gaussian with a rectangle function, resulting in a SG-like function.
0
I- ~~~ -2
-__-_--+_~ ~~ -1.5
-1
Yq +___- _+~~~~+~~~_~ -0.5
0
509
(ii) analytical calculation of the TED in the focal plane of the focusing element by the FI of the CP. (iii> Measurement of the EED, either directly or indirectly as shown in Appendix C. (iv) Evaluation of the diameter(s) and energy content (that is the Q-factors) of both EED and TED as shown in Appendix A, and calculation of the TDL by means of Eq. (3). In the particular case the EED and the TED have the same shape, Eq. (3) can be reduced to the ratio of the respective diameters, the TDL scale factor. We used the above procedure to find the TDL of two pulsed excimer lasers equipped with different hard-edged unstable resonators and emitting beams
l
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;
;
2
2.5
,_
i
3
3.5
X AXIS (mm)
-4
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-2
-1
0
1
2
Y AXIS (mm)
Fig. 4. (a) Far-field energy distribution of HERCULES measured along the horizontal x direction by the knife-edge method. The solid line is a plot of the Eq. (Cl), see Appendix C, with the best fit parameter GX, = 4.4 mm. (b> The same as (a) but along the vertical y direction and with eYPYR = 3 mm. Focal length f= 26 m.
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/I
A;A -1
-2
\ \\ \\ \
\\
.r,
\
1
0
_---__
-_ 3
2
X AXIS (mm) Fig. 5. Comparison between HERCULES diffraction-limited energy distribution TED along the x direction (solid line) and the best fit function of the experimental results of Fig. 4a, TED(x,O;@~,,= 4.4 mm) (dashed line).
with different spatial energy distribution and quality. The results are summarised in Table 2. In the case of IANUS, which emits a very good quality, roundshaped near-field energy distribution, our TDL factors have the same value, while the M* factor is 18% higher. For HERCULES, along the horizontal axis the TDLs are consistent with each other within 5%, whereas the M2 coefficient differs by 12% with respect to the TDL Q,,e2. Along the vertical axis,
1 0.9 0.9
s
OeJ
$.
0.6
c 5;
03
6
0.4
k
0.3
where the near-field energy distribution is closer to a Gaussian-like shape, the difference is less pronounced. It should be pointed out that in the HERCULES case the Qss,ssbfactor is more dependent on the specific interpolation of the experimental data than the Ql,e~. This is because the central lobes of both TED and EED of HERCULES contain less than 86.5% of the total beam energy. In our experience, if the central lobes of either TED or EED contain less
,** ‘\ \
;
1
I’
‘\ \
/
I
\
I
:
t
\
,’
\ \\ \
I
I \
/
\ \
:
\
I’
J
\
1
Ii
\ \
I
0.2
T
0.1
1---__
0
-3
\
I
I) _d
.2
/ I’
\\ ‘\,
Ark -1
\\
0
1
_____^ 2
i 3
Y AXIS (mm) Fig. 6. Comparison between HERCULES diffraction-limited energy distribution TED along the y direction (solid line) and the best fit function of the experimental results of Fig. 4b, TED(O,y,QY,,= 3 mm) (dashed line).
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Table 2 HERCULES and IANUS times-diffraction-limit-factors Laser system
IANUS HERCULES, horizontal HERCULES, vertical
TDL Q86~
TDL
TDL scale factor
M2
Q,,G
1.1 16.8 12.9
1.1 17.6 13.3
1.1 17.4 13.4
1.3 19.7 13.6
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134 (1997) 503-513
output, it is difficult to define procedure can overestimate the there is no propagation law for Studies are in progress to include
a TDL, and our TDL number; (b) the Q parameter. such law.
Acknowledgements We thank Drs. A. Torre, G. Dattoli and T. Letardi for helpful discussions on this topic.
energy-percentage than the k value chosen, the reliability of the corresponding Qk value may be not good. In this case, we recommend using the Q,,,> (and the corresponding TDL) value; in alternative, a k value smaller than the energy percentage contained in the central lobe should be chosen. The results reported in Table 2 can be explained by considering the different meanings of the MZ and the TDL coefficients: MZ takes into account the deviation of both amplitude and phase wavefront from the ideal Gaussian-shaped, constant-phase beam, while our TDL only considers the phase distortion of the real beam, as the constant-phase NFED of the RB has the same amplitude shape. Hence, in the case of the Gaussian-like HERCULES vertical energy distribution, the M2 and TDL coefficients basically give an estimate of the beam phase distortion (NFED is Gaussian-like), so that M2 = TDL. On the contrary, in the case of IANUS and the horizontal energy distribution of HERCULES, the non-Gaussian amplitude shape gives a contribution to the M2 value, so M2 > TDL. This amplitude contribution to M2 becomes significant (compared to the phase contribution) when the non-Gaussian beam has good spatial coherence (i.e. almost constant-phase wavefront) like IANUS. In conclusion, the TDL factor proposed in this paper gives information on the spatial coherence of the real beam regardless of its amplitude shape, without requiring a direct measurement of the beam phase. Moreover, for highly diffracted laser beams it can give a realistic estimation of the margin of improvement in the laser beam quality feasibly achievable through better resonator optics and/or active-medium uniformity. The proposed method has two shortcomings: (a) in the (unlikely) case that the beam waist is intracavity and more than a Rayleigh range far from the laser
Appendix A. Calculation of the Q-factor In the Q-factor definition Q = k/O 2, the divergence 8 comes from the measurement of the beam diameter D in the focal plane, since 0 = D/f, where f is the distance between the focusing element and the focal plane. Regarding the Q,, Ed,once both TED and EED (and the corresponding l/e2 diameters of the central lobe) are known, we consider three cases for the calculation of, e.g., krea’: (i) Circular symmetry (EED( XJ) = EED(r), D = far-field laser beam l/e*-diameter). We have k real=
D’2EED( r) r dr /0 /0
(AlI
+=EED( r) r dr
(ii) Elliptical symmetry (D_,, D, = main-axis length of the laser beam in the far-field plane such that on the ellipse (2x/D.J2
+ (~Y/D,)~ = 1
(A2)
it holds EED ( x,y) = 1/e2). We have k real=
D”2 X(Y)EED(x,y) dxdy /0 /0 +m /0
+==
(A31
EED( x,y) dxdy
I0
where x(y) satisfies Eq. (A2). (iii) Rectangular symmetry CD,, D, = rectangle side-length of the laser beam in the far-field plane). We have k real=
o/2 D”2EED( x,y) dxdy /0 /0 +m /0
+= /0
EED( x,y) dxdy
W)
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Table 3 TDL results with and without the output coupler mirror shadow in the near-field energy distribution of HERCULES
By substituting EED with TED in Eqs. (Al)-(A4), we obtain the corresponding kRB. In order to calculate the Qr, defined as the Q-factor corresponding to a fixed encircled energy k, we need the diameter D(k). It is evident that Eq. (Al) can still be used to find the D(k) of a beam having circular-symmetry. In the case of Eqs. (A3) and (A4), it may happen there are many couples of D,(k) and D,(k) values corresponding to the same k; therefore, we suggest the following procedure to calculate, e.g., the Qz\,: (a) Find the diameters 0:. and Di such that:
unstable resonator lasers with diffraction coupling, like IANUS and HERCULES. In this case, Eq. (4) changes into
EED( D-;/2,0) = EED(0, D;/2) = l/e2.
SG-like energy profile with a hole
(b) Find another couple of diameters 4, and D,, satisfying the equations: (SJD,)
= (D:/D$
and D,/2 D,/2 EED( x,y) dxdy / /0 0 real = k +P
/0
+"
/0
=
Axis
TDL scale factor
TDL Q,,t2
Horizontal, no hole Horizontal, donut shape Vertical, no hole Vertical, donut shape
19 17.4 14.3 13.4
18.9 17.6 14.3 13.3
= [cp( S,w,,*W,,)]’
(Bl)
where R(,$,Weh) is a rectangle function corresponding to a shadow with a size 2Wt,,. The corresponding TED in the case of circular symmetry will be given by [22]: ‘I’ED=~[CP(~,W,,W,)]
0.865.
- R( &W[h).
- ~~~[~(~,W,,)I’,
EED( x,y) dxdy
(B2)
(A9 (c) Calculate Qg\,< 5 ) = ( krea’f2)/( 0221, where 5 = x, y.
Appendix B. TED for diffraction-coupled unstable resonator beams When the near-field energy profile has a donut shape, the mathematical treatment with the FT is more complex. A profile with a hole is typical of
where r is the radial variable in polar coordinates and p = (near-field size/shadow size). Strictly speaking, Eq. (B2) is exact only in the case of plane waves; however, it can be shown that it works well for any beam satisfying the relationship /3 + 2r > 8. Eq. (Bl) requires a longer computing time than Eq. (4) to calculate the TDL. However, (Bl) is strictly necessary only for the special case of beams having a coefficient j3 < 3. In fact, it can be demonstrated that, for /3 > 3, the difference in Q,,e~ calculated by means of Eq. (4) and Eq. (Bl) is less than 10%. As an example, Fig. 7 shows the value p =
CQcwit, hole) Qcwithout hole)]/Q(with hole)l"'S'S
Fig. 7. Relative variation of the parameter Q,,e~ with and without the hole in the near-field energy distribution versus p for different r-factors for circular symmetry.
P
for
different r-factor values, considering circular symmetry. In the case of the IANUS beam we have j3 = 8 and Fig. 7 gives a value of p < 1%; therefore, the hole in the near-field energy distribution can be neglected. The HERCULES beam has a rectangular shape, with p = 6 and r = 4.2 along the x-axis. The difference in the TDL calculation, taking into account the hole, is less than 10%; this has also been verified for the y-axis, where /3 = 4 and r = 2.3. Table 3 compares the results of the TDL calculations with Eq. (4) and Eq. (Bl).
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Appendix C. TDL with the knife-edge method After the near-field energy distribution has been measured and interpolated by using Eq. (4), and finally propagated in the far-field to obtain the TED(x,Y), the next step is to measure the far-field EED(x,y). If it can be directly measured (e.g. by using a CCD camera), we can use Eq. (3) and Appendix A to obtain the TDL. To avoid problems related to background estimation and saturation effects of the CCD [7], we can use the knife-edge (KE) technique. In this case the energy profile EED(x,y) is unknown and the measurement yields an energy distribution versus a beam dimension. For example, if the KE is moved along the x-axis, the measured quantity is P,,,(
x’) = /::/_:“EED(
x,Y) dydx.
Then, to evaluate the EED, we need a suitable function E(x,y) whose integral fits P,,,(x’). Finally, the Qrea’ parameter in Eq. (3) is calculated assuming E( x, y) - EED( x, y). We recommend using the Q,,p2 value, which is less dependent than the Qs6,ss value on the choice of the interpolating function, as mentioned in Section 3. When the near-field energy distribution can be expressed as the product of two independent functions of x and y, we suppose EED can be too, and we can write EED( x,y) = ;
[ PEED(d]
’;
[ ‘Ed
y)]’
where P E&) and PEED( Y) are measured with the KE moved along the x and y directions, respectively. In this case the computing time is strongly reduced. It can be further reduced by employing the TDL scale factor, when the TED and the EED have similar shapes. In this case, the EED(x, y) can be represented by the TED( x, y;@&,) defined in Section 4, adjusting the far-field Gaussian radius @, as a free parameter. Hence, the P,,,(x’) can be approximated by P,,,(x’;@!!)zg) = /
/+=TED( --z -72
x,y&)dydx.
(Cl)
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513
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