In the past several years, decentered laser beamshave attracted more and more attention because oftheir unique characteristics.1–7 Decentered Gauss-ian beams were first proposed by Casperson in1973.1 It is shown that when an ordinary Gaussianbeam is reflected from a spherical mirror whosecenter is offset from the beam axis, the reflectedbeam is a decentered Gaussian beam. The propa-gation properties of a decentered Gaussian beam(DGB) and its transmission through optical sys-tems have been studied by Al-Rashed and Saleh.2Palma3 developed the concept of ray parameters forcharacterizing a decentered Gaussian beam, and amodel of DGB bundles for characterizing Bessel–Gaussian beams. The coherent and incoherent com-binations of off-axis Gaussian beams and off-axisHermite–Gaussian beams were studied by Lü andMa,4 and Luis et al.5 The diffraction properties of ascalar, off-axis Gaussian beam were also studied by
Cronin et al.6 More recently, we introduced the de-centered elliptical Gaussian beam (DEGB) and thedecentered elliptical Hermite–Gaussian beam (DE-HGB) to describe decentered astigmatic high-orderlaser beams, and studied the propagations of aDEGB and a DEHGB through unapertured complexoptical systems studied in detail.7,8 In this paper,for the more general case, we study the propagationof a DEGB and a DEHGB through apertured parax-ial optical systems. By expanding the hard aperturefunction into a finite sum of complex Gaussian func-tions,9,10 we obtain approximate analytical formu-las for a DEGB and a DEHGB passing throughapertured paraxial optical systems. Some numeri-cal examples and application examples are given.
2. Propagation of a Decentered EllipticalHermite–Gaussian Beam Through AperturedParaxial ABCD Optical Systems
The electric field of a DEGB at z � 0 is defined intensor form as follows7:
E�r1� � exp��ik2 �r1 � r0�T Q1
�1�r1 � r0��, (1)
where k � 2��� is the wavenumber, � is the wave-length, r1 is position vector given by r1
T � �x1 y1�, r0 isa complex vector called the decentered parametergiven by r0
T � �xd � ix0 yd � iy0�; here xd and yd denotethe displacements (in the x and y directions) of the
Y. Cai ([email protected]) is with the Joint ResearchCenter of Photonics of the Royal Institute of Technology and Zhe-jiang University, East Building No. 5, Zijingang Campus, ZhejiangUniversity, Hangzhou 310058, China. L. Zhang is with the Schoolof Science, Hangzhou Dianzi University, Hangzhou 310018, China.
Received 6 September 2005; revised 17 February 2006; accepted17 February 2006; posted 17 February 2006 (Doc. ID 64528).
center of the amplitude distribution, and x0 and y0denote the displacements (in the x and y directions) ofthe center of the phase front.1 Q1
�1 is the complexcurvature tensor for the generalized elliptical Gauss-ian beam given by11,12
Q1�1 � �q1xx
�1 q1xy�1
q1xy�1 q1yy
�1�. (2)
where 1�q0x � �2i�kw0x2, 1�q0xy � �2i�kw0xy
2, 1�q0y ��2i�kw0y
2, w0x and w0y are the beam waist interceptsin the x and y directions, respectively, and w0xy is thecrossed element related to the orientation of the beamspot.
First, we study the propagation of a DEGB throughan apertured nonsymmetrical optical system. More
specifically, a nonsymmetrical optical system denotesan astigmatic optical system and a symmetric opti-cal system denotes a stigmatic optical system13; weuse the terminology “astigmatic” and “stigmatic” in-stead of “nonsymmetrical” and “symmetrical,” respec-tively. Within the framework of paraxial approx-imation, the propagation of a DEGB through anapertured astigmatic optical system can be treated bythe generalized Collins formula, which can be writtenin tensor form as follows12,14:
E2�r2� � �in1
��det�B��1�2 exp��ikl0���E1�r1�H�r1�
� exp��ikl1�dr1, (3)
Fig. 1. Normalized irradiance distribution of a DEGB and its cross irradiance profile (y � 0) at several propagation distances after passingthrough a circular aperture located at z � 0: (a) z � 20 mm and (b) z � 500 mm.
where H�r� is the transmission function of the hard-edged aperture; for simplicity, we call H�r� the hardaperture function in the following text. l0 is theeikonal along the propagation axis. l1 is the eikonalgiven by
l1 �12�r1
r2�T� n1B�1A �n1B�1
n2�C � DB�1A� n2DB�1��r1
r2�, (4)
where n1 and n2 are the refractive indices of the inputand output spaces, respectively. For the sake of sim-plicity, we assume n1 � n2 � 1 in the following text.r1 and r2 are the position vectors in the input andoutput planes, respectively. A, B, C, and D are thesubmatrices of the optical system defined by
� r2
r2��� �A B
C D�� r1
r1��. (5)
If the hard-edged aperture is circular, H�r1� can beexpressed as follows:
H�r1� �1, |r1| � a1
0, |r1| � a1, (6)
where a1 denotes the radius of the aperture. Then thehard-edged aperture function can be expanded asthe sum of complex Gaussian functions with finitenumbers as follows9,10:
H�r1� � m�1
M
Am exp��Bm
a12 r1
2�, (7)
where Am and Bm are the expansion and Gaussian
coefficients, which can be obtained by optimizationcomputation directly.9,10 Numerical results show thatthe simulation accuracy increases as M increases.
After rearrangement, the hard-edged aperturefunction given by Eq. (7) can be expressed in tensorform as follows:
H�r� � m�1
M
Am exp��ik2 rTRmr�, (8)
with
Rm �2Bm
ika12 �1 0
0 1�. (9)
Substituting Eqs. (1) and (8) into Eq. (3), after tediousbut straightforward tensor integration, we obtain
where I is a 2 � 2 unit matrix, and
Q2m�1 � �C � D�Q1
�1 � Rm���A � B�Q1�1 � Rm���1.
(11)
In the above derivation, we used the following inte-gral formula15:
Equation (10) is the approximate analytical propaga-tion formula for a DEGB passing through a circularapertured aligned astigmatic optical system. Whenthe radius of the aperture a1 → , Eq. (10) reduces toEq. (9) of Ref. 7, which denotes the propagation for-mula for a DEGB passing through an unaperturedaligned astigmatic optical system.
If the hard-edged aperture is rectangular, thenH(r1) can be expressed as follows:
H�r1� �1, x1 � a1, y1 � b1,0, x1 � a1, y1 � b1,
(14)
where 2a1 and 2b1 denote the aperture widths in thex and y directions, respectively. Equation (14) alsocan be expanded as the sum of complex Gaussianfunctions with finite numbers as follows:
H�x1, y1� � m�1
M
Am exp��Bm
a12 x1
2�j�1
J
Aj exp��Bj
b12 y1
2�,
(15)
Am, Bm, Aj, and Bj are the expansion and Gaussian
Fig. 2. Normalized irradiance distribution of a DEGB at distance z � 100 mm after passing through a circular aperture for differentvalues of radius located at z � 0: (a) a1 � 0.5 mm, (b) a1 � 1 mm, (c) a1 � 2 mm, (d) a1 � 10 mm.
coefficients. After rearrangement, Eq. (15) can be ex-pressed in tensor form as follows:
H�r1� � m�1
M
j�1
J
AmAj exp��ik2 r1
TRmjr1�, (16)
with
Rmj � �2Bm
ika12 0
02Bj
ikb12�. (17)
Substituting Eqs. (1) and (16) into Eq. (3), after somevector integration, we obtain
E�r2� � exp��ikl0� m�1
M
j�1
J
AmAj�det�A � BQ1�1
� BRmj���1�2 exp��ik2 r2
TQ2mj�1r2�
� exp��ik2 r0
T�B�1AQ1 � RmjQ1 � I��1
� �B�1A � Rm�r0�exp�ikr0T�AQ1 � B
� BRmjQ1��1r2�, (18)
where Q2�1 is related to Q1
�1,
Q2mj�1 � �C � D�Q1
�1 � Rmj���A � B�Q1�1 � Rmj���1.
(19)
Equation (18) is the approximate analytical propaga-tion formula for a DEGB passing through a rectan-gular apertured aligned astigmatic optical system.Under the conditions of a1 → and b1 → , Eq. (18)also reduces to Eq. (9) of Ref. 7.
To compare the results obtained by using the ap-proximate analytical expressions with those obtainedby numerically integrating the Collins formula di-rectly, we calculated the normalized irradiance distri-bution of a DEGB and corresponding cross irradianceprofile �y � 0� at several propagation distancesafter passing through a circular aperture located atz � 0, the results are shown in Fig. 1. In the calcu-lation, the parameters were chosen as
A � �1 00 1�, B � �z 0
0 z�, C � �0 00 0�, D � �1 0
0 1�,(20)
r0T � �0.5 0.5�, � � 632.8 nm, a1 � 1 mm, w0x �
1 mm, w0y � 1.5 mm, w0xy � 2 mm, and the coeffi-cients Am and Bm of the hard aperture function arechosen as the same as those in Refs. 9 and 10 (whereM � 10). In Fig. 1, the solid curves denote the resultscalculated by the approximate analytical formula
Eq. (10), and the dotted curves denote the resultscalculated by integrating Collins formula Eq. (3) nu-merically. From Fig. 1, we can find that the simula-tion results obtained by using the approximateanalytical expression are in good agreement withthose obtained by using the numerical integralcalculation. In Fig. 2, we calculate the normalizedirradiance distribution of a DEGB at distance z �100 mm after passing through a circular aperture fordifferent values of radius. It is clear from Fig. 2 thatthe influence of the aperture on the irradiance distri-bution of the DEGB gradually disappears as the ra-dius of the aperture increases.
Now, we study the propagation of a DEGB throughan apertured misaligned stigmatic optical system.The propagation of a DEGB through an aperturedmisaligned stigmatic optical system can be treated bythe following generalized integral formula16,17:
E�r2� � �ik
2��det�B��1�2 exp��ikl0���E�r1�H�r1�
� exp��ik2 �r1
TB�1Ar1 � 2r1TB�1r2
� r2TD B�1r2��
� exp��ik2 �r1
TB�1ef � r2TB�1gh��dr1, (21)
where r1T � �x1 y1�, r2
T � �x2 y2�, ef � �e f�,gh � �g h�, A, B, C, and D take the following form:
A��a 00 a�, B��b 0
0 b�, C � �c 00 c�, D��d 0
0 d�,(22)
and a, b, c, and d are the transfer matrix elements ofthe aligned stigmatic optical system. A, B, C, and Dsatisfy the following relations:
�B�1A�T� B�1A, ��B�1�T
� �C � D B�1A�,
�D B�1�T� D B�1. (23)
The parameters e, f, g, and h take the following form:
e � 2��T�x � T�x��, (24)
f � 2��T�y � T�y��, (25)
g � 2�b�T � d�T��x � 2�b�T � d T��x�, (26)
h � 2�b�T � d�T��y � 2�b�T � d T��y�, (27)
where �x, �x�, �y, and �y� denote the two-dimensionalmisalignment parameters, �x and �y are the displace-ments in the x and y directions, respectively, and �x�
and �y� are the tilting angles of the element in the xand y directions, respectively. �T, T, �T, and �T rep-resent the misaligned matrix elements defined by17
�T � 1 � a, T � 1 � b, �T � �c, �T � �1 � d.(28)
For forward-going optical elements, �T is chosen forthe “�” sign; for backward-going ones, �T is chosen forthe “�” sign.
Substituting Eqs. (1) and (8) into Eq. (21), andapplying Eqs. (12) and (23), after some vector inte-gration and tensor operation, we obtain
E�r2� � exp��ikl0� m�1
M
Am
� �det�A � BQ1�1 � BRm���1�2
� exp��ik2 r2
TB�1gh �ik2 r2
TQ2m�1r2�
� exp��ik2 r0
T�B�1AQ1 � RmQ1 � I��1
� �B�1A � Rm�r0 � ikr0T
� �AQ1 � B � BRmQ1��1r2�� exp��
ik2 r2
TB�1T�A � BQ1�1 � BRm��1ef
�ik2 r0
T�AQ1 � B � BRmQ1��1ef�� exp�ik
8 efTB�1T�A � BQ1
�1 � BRm��1ef�,(29)
where
Q2m�1 � �C � D�Q1
�1 � Rm���A � B�Q1�1 � Rm���1.
(30)
Equation (29) is the approximate analytical propaga-tion formula for a DEGB passing through a circularapertured misaligned stigmatic optical system. Whenthe radius of the aperture a1 → and r0
T � 0, Eq. (29)reduces to Eq. (12) of Ref. 14, which denotes the prop-agation formula for an elliptical Gaussian beam(EGB) passing through an unapertured misalignedstigmatic optical system.
Similarly, substituting Eqs. (1) and (16) into Eq.(21), and applying Eqs. (12) and (23), after vector in-tegration, we can obtain the approximate analyticalpropagation formula for a DEGB passing through arectangular apertured misaligned stigmatic opticalsystem as follows:
E�r2� � exp��ikl0� m�1
M
j�1
J
AmAj
� �det�A � BQ1�1 � BRmj���1�2
� exp��ik2 r2
TB�1gh �ik2 r2
TQ2mj�1r2�
� exp��ik2 r0
T�B�1AQ1 � RmjQ1 � I��1
� �B�1A � Rmj�r0 � ikr0T
� �AQ1 � B � BRmjQ1��1r2�� exp��
ik2 r2
TB�1T�A � BQ1�1 � BRmj��1ef
�ik2 r0
T�AQ1 � B � BRmjQ1��1ef�� exp�ik
8 efTB�1T�A � BQ1
�1 � BRmj��1ef�,(31)
with
Q2mj�1 � �C � D�Q1
�1 � Rmj���A � B�Q1�1 � Rmj���1.
(32)
3. Propagation of a Decentered EllipticalHermite–Gaussian Beam Through Apertured ParaxialABCD Optical Systems
EHGB was proposed by Cai and Lin18 to describe theastigmatic higher-order Gaussian beam. More re-cently, we extended the EHGB to the decentered caseand studied the propagation of a DEHGB throughunapertured paraxial optical systems in detail.8 Inthis section, for the more general case, we study thepropagation of a DEHGB through apertured paraxialoptical systems.
The electric field of the DEHGB at z � 0 is definedin tensor form as follows8:
Ep�r1� � exp��ik2 �r1 � r0�TQe
�1�r1 � r0��� Hp� ik�r1 � r0�TQh
�1�r1 � r0��,
p � 0, 1, 2, 3, . . . (33)
When the decentered parameter vanishes, i.e.,r0
T � �0 0�, Eq. (33) reduces to an EHGB. Whenp � 0, Eq. (33) is reduced to the form of a DEGB.
Substituting Eqs. (33) and (8) into Eq. (3), andapplying Eq. (13) and the following integral formu-la15:
��
exp���a1�2x � a1�2b�2
2 �Hp�x�dx � 2�a
� �1 � 2a�p�2Hp� b
1 � 2a�, (34)
we obtain (after vector integration and tensor operation)
Equation (35) is the approximate analytical propaga-tion formula for a DEHGB passing through a circularapertured aligned astigmatic optical system. Whenp � 0, Eq. (35) can easily reduce to Eq. (10). When theradius of the aperture a1 → , Eq. (35) reduces to Eq.(9) of Ref. 8, which denotes the propagation formulafor a DEHGB passing through an unaperturedaligned astigmatic optical system.
Fig. 3. Normalized irradiance distribution of an EFGB at z � 80 after passing through the circular aperture for different values of radius:(a) a1 � 0.5 mm, (b) a1 � 1 mm, (c) a1 � 2 mm, (d) a1 � 10 mm.
Substituting Eqs. (33) and (16) into Eq. (3) andapplying Eqs. (13) and (34), after tedious integrationwe obtain the approximate analytical propagationformula for a DEHGB passing through a rectangularapertured aligned astigmatic optical system as fol-lows:
E2p�r2� � m�1
M
j�1
J
AmAj�det�A � BQe1�1 � BRmj���1�2
� �1 � 2�det�B�1AQh1 � Qe1�1Qh1
� RmjQh1��p�2 exp��ikl0�
� exp��ik2 r2
TQe2mj�1r2�
� exp��ik2 r0
T�B�1AQe1 � RmjQe1 � I��1
� �B�1A � Rmj�r0�Hp��1 � 2�det
� �B�1AQh1 � Qe1�1Qh1 � RmjQh1���1�2
� ik�r2 � r0�TQh2mj�1�r2 � r0��
� exp�ikr0T�AQe1 � B � BRmjQe1��1r2�,
(38)
with
Qe2mj�1 � �C � D�Qe1
�1 � Rmj��� �A � B�Qe1
�1 � Rmj���1, (39)
Qh2mj�1 � �A � B�Qe1
�1 � Rmj���1�1T
� �AQh1 � BQe1�1Qh1 � BRmjQh1��1.
(40)
Equations (10), (18), (29), (31), (35), and (38) are themain results of this paper. They provide a convenientand powerful way to study the propagation and trans-formation of a DEGB and a DEHGB passing throughapertured paraxial optical systems and avoid time-consuming numerical integration.
4. Application Example
As an application example of the derived results inSections 2 and 3, in this section we study the pro-pagation of decentered elliptical flattened Gaussianbeam (DEFGB) through apertured optical systems byexpressing the DEFGB as a superposition of a seriesof DEGBs or DEHGBs.
The electric field of the elliptical flattened Gauss-ian beam (EFGB) at z � 0 can be expressed as a finiteseries of generalized elliptical Gaussian beams withdifferent parameters as follows19:
EN�r� � n�1
N ��1�n�1
N �Nn�exp��ik2 rTQ1n
�1r�, (41)
where �Nn� denotes a binomial coefficient, N is the
order of EFGB, and Qln�1 � nQl
�1 with Q1�1 being the
complex curvature tensor for the EGB.The DEFGB can be easily defined as follows:
EN�r� � n�1
N ��1�n�1
N �Nn�� exp��
ik2 �r � r0�TQ1n
�1�r � r0��. (42)
Then by applying Eqs. (10), we can obtain the follow-ing analytical propagation formula for a DEFGBpassing through a circular apertured astigmaticparaxial optical system:
EN�r2� � n�1
N
m�1
M ��1�n�1
N �Nn�Am
� �det�A � BQln�1 � BRm���1�2
� exp��ik2 r2
TQ2nm�1r2�
� exp��ik2 r0
T�B�1AQ1n � RmQ1nI��1
� �B�1A � Rm�r0�� exp�ikr0
T�AQ1n � B � BRmQ1n��1r2�,(43)
where
Q2nm�1 � �C � D�Q1n
�1 � Rm���A � B�Q1n�1 � Rm���1.
(44)
Similarly, we can easily obtain the approximate an-alytical propagation formulas for a DEFGB passingthrough a rectangular apertured astigmatic paraxialoptical system, and through circular or rectangularapertured stigmatic paraxial optical systems by ap-plying Eqs. (18), (29), and (31) (not presented here).
There is another definition of EFGB. The electricfield of the EFGB is expressed as8,18
Eq. (46) can be expanded as a superposition of aseries of DEHGB as follows:
E�r1� � n�0
N 1
�2n�! m�n
N 1
23m
�2m�!
�m � n� ! m!
� exp��ik2 �r1 � r0�TQ1,N
�1�r1 � r0��� H2m��ikQ1,N
�1�1�2�r1 � r0��. (48)
Then by applying Eqs. (35) and (38), we can easilyobtain the approximate analytical propagation for-mulas for a DEFGB passing through aperturedparaxial optical systems (not presented here).
As a numerical example, we study the propagationproperties of an EFGB through a circular aperturelocated at z � 0. By applying Eqs. (20), (43), and (44),we calculate in Fig. 3 the normalized irradiance dis-tribution of an EFGB at z � 80 after passing throughthe circular aperture for different values of ra-dius with r0
T � �0 0�, � � 632.8 nm, w0x � 1 mm,w0y � 2 mm, and w0xy � 1.8 mm. One sees from Fig.3 that the circular aperture has a strong influence onthe irradiance distribution of the EFGB. Under theinfluence of the circular aperture, the flat-toppedbeam profile disappears [see Figs. 3(a)–3(c)]. Whenthe radius of the aperture is small, the focused beamspot is of circular symmetry, which is much differentfrom the elliptical beam spot in the input plane at z� 0 [see Figs. 3(a) and 3(b)]. When the radius of theaperture is large enough, the influence of the aper-ture on the irradiance distribution disappears [seeFig. 3(d)].
5. Conclusion
In conclusion, by using a tensor method, we havederived the approximate formulas for the propaga-tion of a DEGB passing through apertured complexoptical systems based on the fact that the hard aper-ture function can be expanded into a finite sum ofcomplex Gaussian functions. Numerical results showthat the results calculated by use of the approximateanalytical expression are in good agreement withthose obtained by use of the numerical integral cal-culation. Furthermore, for the more general case, wehave derived the approximate analytical formulas fora DEHGB passing through apertured complex opticalsystems. As an application example, we study thepropagation of two kinds of DEFGB through aper-tured optical systems by expressing the DEFGB as asuperposition of a series of DEGBs or DEHGBs. Our
results can avoid time-consuming numerical integra-tion and provide a convenient way for studying thepropagations and transformations of a DEGB and aDEHGB through apertured paraxial optical systems.Our formulas can be applied to study the propagationof many other complex astigmatic beams such as anastigmatic laser beam array and hollow beam, forexample, whose electric fields can be expressed as asuperposition of a finite sum of DEGs or DEHGBsthrough apertured paraxial optical systems.
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