- Email: [email protected]
Matrix factorization of the mode converter
ARTICLE IN PRESS
Optics & Laser Technology 37 (2005) 265–270 www.elsevier.com/locate/optlastec
Matrix factorization of the mode converter Ping Wua,*, Xiaofei Songa, Baida Lu. a,b a
Department of Applied Physics, Southwest Jiaotong University, Chengdu 610031, China Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, China
b
Received 9 February 2004; accepted 20 April 2004 Available online 25 June 2004
Abstract Based on the basic properties of the reciprocal symmetrical matrix, a method is proposed, which permits us to factorize the transfer matrix of the mode converter into a product of reciprocal symmetrical matrices. The procedure is simple and is suited for computer calculations. Some factorization examples are given and illustrated. The results are useful for the design of different types of the mode converter to satisfy practical applications. r 2004 Elsevier Ltd. All rights reserved. Keywords: Matrix factorization; Reciprocal symmetrical matrix; Mode converter
1. Introduction
2. Reciprocal symmetrical matrix
The transformation of a general astigmatic beam into a stigmatic one is a topic of practical interest. Eppich and Laabs et al. showed [1,2] that any three-dimensional astigmatic Hermite–Gaussian beam can be transformed into a symmetric Lagueree–Gaussian beam with twist by using a mode converter consisting of three cylindrical lenses. The theoretical predication was confirmed by the experiments of TEMm0 -mode Nd:YAG lasers and diode laser stacks [1–4]. It was also found [4] that the threecylindrical-lens mode converter is equivalent to a two lens optics, or a single thick cylindrical lens. Therefore, an interesting question arises: Can the transfer matrix of the mode converter be factorized according to a certain rule? The aim of the present paper is to study the matrix factorization of the mode converter. In Section 2 the definition and some basic properties of the 4 4 reciprocal symmetrical matrix are studied. It is shown that the transfer matrix of the mode converter satisfies the condition of the 4 4 reciprocal symmetrical matrix. Then, Section 3 presents the factorization procedure of the transfer matrix of the mode converter into a product of reciprocal symmetrical matrices. Finally, Section 4 summarizes the main results obtained in this paper.
It is well known that a general non-symmetrical optical system can be expressed in terms of a 4 4 matrix [5,6] 2 3 a11 a12 b11 b12 6 7 A B 6 a21 a22 b21 b22 7 ~ ¼ M ¼6 ð1Þ 7; 4 c11 c12 d11 d12 5 C D c21
0030-3992/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2004.04.005
d21
d22
where ‘‘-’’ denotes the forward matrix (from the left input reference plane to the right output reference plane). Assume that the input and output planes are in the same type of optical medium (usually free space), thus we have At D Ct B ¼ E; ð2Þ 1 0 where E ¼ ; and t denotes the transposition of 0 1 the matrix. t ’ D Bt ~ be M Let the backward matrix of M ¼ [7], Ct At ~ is defined as the reciprocal symmetrical matrix if the M condition ’
~ ¼M M *Corresponding author. E-mail address: ping w [email protected] (P. Wu).
c22
is fulfilled.
ð3Þ
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
266
From Eq. (3) it turns out that the sub-matrices of the reciprocal symmetrical matrix satisfy the condition
reciprocal symmetrical. Thus, we have ~ n1 yM ~ 1: ~ ¼M ~n M M
ð7Þ
A ¼ Dt ;
The backward matrix of Eq. (7) are
B ¼ Bt ;
M ¼ M 1 M2 y Mn :
ð8Þ
C ¼ Ct ;
Eqs. (7) and (8) can be rewritten as ~ 1; ~ ¼M ~n ~ PM M
ð9Þ
’
D ¼ At :
ð4Þ
In addition, we have ’
~ 1 ¼ M1 ; M
ð5Þ
where ‘‘1’’ denotes the reverse of the matrix, i.e., t ’ 1 Bt A ~ 1 ¼ D B and M ¼ M : Eq. (5) C A Ct Dt indicates that the reverse matrix of the reciprocal symmetrical matrix retains its reciprocal symmetry unchanged. In accordance with Ref. [1] the transfer matrix of the mode converter in the principal axes representation reads as 3 2 0 0 f 0 7 6 6 0 1 0 0 7 7 6 1 7 ~ ¼6 0 0 0 7: ð6Þ M 6 7 6 f 7 6 5 4 1 0 1 0 f It is readily shown that Eq. (6) satisfies condition (3). In practice the beam converter consists of spherical and cylindrical lenses and their separation (free-space propagation) whose matrices are all reciprocal symmetrical. As a result, the matrix factorization of the beam converter is reduced to factorizing Eq. (6) by means of reciprocal symmetrical matrices.
3. Factorization procedure of the transfer matrix of the mode converter As shown in Fig. 1, assume that a reciprocal symmetrical optical system expressed in terms of the ~ is composed of n optical systems, whose matrix M ~ 1; M ~ 2 yM ~ n respectively, and are all matrices are M
’
’
’
’
’
’ ’
M ¼ M 1 P Mn ;
ð10Þ
where ~ n2 yM ~ 2; ~¼M ~ n1 M P
ð11Þ
’
’
’
’
P ¼ M2 M3 y Mn1 :
ð12Þ
Now we factorize Eq. (7) by using reciprocal symme’ ’ ~1 ¼ M ~ n; M trical matrices in pairs, i.e., M 1 ¼ Mn ; Eqs. (9) and (10) become ~ 1; ~ ¼M ~1 ~ PM ð13Þ M ’
’
’ ’
M ¼ M 1 P M1 ;
ð14Þ
respectively. Because of the reciprocal symmetrical ~ 1 and M ~ ; the substitution from property of matrices M ’ ~1 ¼ M M into Eqs. (13) and (14) yields 1 ’
~ P¼ P:
ð15Þ
Eq. (15) implies that after the first factorization the remaining matrix is still reciprocal symmetrical. Therefore, the further factorization can be performed continuously in a similar way, till only a simple reciprocal symmetrical matrix remains.
4. Application examples To illustrate the application of the matrix factorization procedure described in Section 3, some typical examples are given as follows. Example 1. Firstly, we choose a pair of the spherical lens with focal length f whose matrix is expressed as 3 2 1 0 0 0 7 6 1 0 07 6 0 7 6 1 7 ~1 ¼ 6 0 1 0 7: ð16Þ M 6 7 6 f 7 6 5 4 1 0 1 0 f From Eq. (13) it follows that
Fig. 1. Notation relating to the optical system.
~ ~ 1 M ~ M ~ 1 : P1 ¼ M 1 1
ð17Þ
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
Secondly, we with distance 2 1 0 60 1 ~2 ¼ 6 M 6 40 0 0
0
267
use a pair of the free-spare propagation z whose matrix is 3 z 0 0 z7 7 ð18Þ 7: 1 05 0
1
Thus ~ ~ 1 ~ ~ 1 : P2 ¼ M P1 M 2 2
ð19Þ
Thirdly, we use a pair of the cylindrical lens with focal length F in the y direction whose matrix reads as 3 2 1 0 0 0 60 1 0 07 7 6 ~3 ¼ 6 7 ð20Þ M 6 0 0 1 0 7: 5 4 1 0 1 0 F Finally, we have
Fig. 2. Example 1 of the mode converter.
pffiffi
2
Þf in the y direction. The separations between S1 pffiffi 2 and C1 ; S2 and C2 are equal and are ð1 2 Þf ; and the
ð1
2
~ 1 ~ ~ 1 ~ P2 M P3 ¼ M 3 3 3 2 1 0 f 2z 0
7 6 2z 2z z zð1 þ f Þ 2z 7 6 7 60 0 z z 1 þ 1 þ þ 7 6 f f F 7 6 ¼60 7: 0 1 0 7 6 7 6 2z zzð1þ f Þ 6 2z 2z 2z 7 5 4 1 þ 1 þ þ z zð1 þ Þ 2 2z f f f F 0 þ þ 0 1 þ þ f f F F F Assume that P3 is the matrix of the free-spare propagation, the following relations should be fulfilled: 1 þ
2z 2z z zð1 þ f Þ þ ¼ 1; f F
ð22Þ
2z f 2z ¼ z z 1 þ ; f 2 1 þ þ f F
2z f
þ
ð23Þ
2z 1 þ 2z f þ z zð1 þ f Þ
F
¼ 0:
ð24Þ
Eqs. (22)–(24) lead to
z¼
pffiffiffi! 2 1 f 2
ð25Þ
and F ¼ z:
ð26Þ
Therefore, the mode converter shown in Fig. 2 consists of a pair of the spherical lens S1 ; S2 with focal length f ; and a pair of the cylindrical lens C1 ; C2 with focal length
ð21Þ
pffiffiffi separation between C1 and C2 is ð 2 1Þf : The transfer matrix factorization of the system is expressed as 2
0 6 6 0 6 1 ~ ¼6 M 6 6 f 6 4 0 2 1 6 6 0 6 1 6 ¼6 6 f 6 4 0 2
0 1 0 1 f 0 1 0 1 f 0
3 0 7 0 7 7 7 0 0 7 7 7 5 0 1 3 0 0 7 0 07 7 7 1 07 7 7 5 0 1 pffiffiffi! 2 1 f 2 f 0
3
61 6 6 6 6 60 6 6 6 40
1
0
0
1
0
0
0
0
1
0 pffiffiffi! 2 f 1 2
7 7 7 7 7 7 7 7 7 5
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
268
2
3
1
0
0
0
60 6 6 6 60 6 60 4
1
0
07 7 7 07 7 7 17 5
2
1
6 60 6 6 40 2
0 1
60 6 6 0 6 6 6 40 2
0 1 1 pffiffiffi 0 2 Þf ð1 pffiffiffi2 0 ð 2 1Þf 1
0
0 0
1 0 0
0
pffiffiffi 7 ð 2 1Þf 7 7 7 5 0 0
1 3
1 0 07 7 7 0 1 07 7 1 7 pffiffi 0 15 2 ð1 2 Þf pffiffiffi! 2 0 1 f 2
61 6 6 6 6 60 1 6 6 6 40 0 0 0 2 1 0 6 1 6 0 6 1 6 0 6 6 f 6 4 1 0 f
0 1 0 0 0 1 0
0
3
7 07 7 7 0 7: 7 7 5 1
3
0
Fig. 3. Example 2 of the mode converter.
3 0
7 7 pffiffiffi! 7 7 2 7 f7 1 7 2 7 7 5 0 1 Fig. 4. Example 3 of the mode converter.
ð27Þ
Following the above factorization procedure, we can find a variety of optical system which consist of different reciprocal symmetrical optical systems, but have the same transfer matrix as Eq. (6). Example 2. The mode converter shown in Fig. 3 is composed of a pair of the cylindrical lens C1 ; C2 with focal length f in the y direction and a bifocal lens B with focal length f2 and f in the x and y directions, respectively. The separation between C1 and B; C2 and B is equal to f : The transfer matrix of the system is expressed as 2
0 6 6 0 6 1 ~ ¼6 M 6 6 f 6 4 0
0 1
f 0
0
0
1 f
0
3 0 7 0 7 7 7 0 7 7 7 5 1
2
3
2 1 0 7 6 60 1 0 07 60 1 7 6 6 ¼60 0 1 07 6 7 40 0 6 5 4 1 0 0 1 0 0 f 3 2 1 0 0 0 2 7 6 1 1 0 07 6 0 7 60 6 1 7 6 6 0 1 076 6 7 40 6 f 7 6 5 4 2 0 0 1 0 f 3 2 1 0 0 0 7 6 60 1 0 07 7 6 6 0 0 1 0 7: 7 6 5 4 1 0 0 1 f 1
0
0
0
3
f
0
0
f7 7 7 05
1 0
1
0 1
f 0
0
1
3 0 f7 7 7 05
0
0
1
ð28Þ
Example 3. Fig. 4 gives the third example. The mode converter consists of a pair of the free-space propagapffiffiffi tion with distances ð 2 1Þf and a pair of the bifocal
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
pffiffiffi pffiffiffi lens B1 ; B2 with focal lengths 2f and ð 2 1Þf in the x and y directions, The separation between pffiffirespectively. ffi B1 and B2 is 2ð 2 1Þf : The transfer matrix of the system is written as 3 2 0 0 f 0 7 6 6 0 1 0 0 7 7 6 1 7 ~ ¼6 0 0 0 7 M 6 7 6 f 7 6 5 4 1 0 1 0 f 3 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 7 6 60 1 0 ð 2 1Þf 7 7 ¼6 7 6 5 40 0 1 0 0 0 2
0
6 0 6 6 1 6 6 pffiffiffi 6 2f 6 4 0 2
1 0
6 60 1 6 6 40 0 0 0 2 1 6 0 6 6 1 6 6 pffiffiffi 6 2f 6 4 0 2
1
1
1 0
6 60 1 6 6 40 0 0 0
3
0
0
0
1
0
0
1
07 7 7 07 7 7 7 5 1
1 pffiffiffi ð 2 1Þf pffiffiffi 2ð 2 1Þf 0 1
0
0 pffiffiffi 2ð 2 1Þf 0
0 0
0
1
0
0
1
7 7 7 7 5
1 3 0 07 7 7 07 7 7 7 5 1
1 pffiffiffi 0 ð 2 1Þf pffiffiffi ð 2 1Þf 0 pffiffiffiffiffiffiffiffiffiffiffi 0 ð 2 1Þf 1 0 0
3
3 7 7 7: 7 5
1
Fig. 5. Example 4 of the mode converter.
ð29Þ
269
Example 4. The fourth mode converter is shown in Fig. 5, which consists of appair of the free-space ffiffiffi propagation with distance ð 2 1Þf ; a pair pffiffiffi of the cylindrical lens C1 ; C2 with focal length ð2 2Þf in the y direction and a spherical lens S with focal length f : The separation between C1 and S; C2 and S is equal pffiffiffi to ð2 2Þf : The transfer matrix of the system reads as
2
0
0
f
0
3
7 6 6 0 1 0 0 7 7 6 1 7 ~ ¼6 0 0 0 7 M 6 7 6 f 7 6 5 4 1 0 1 0 f 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 6 60 1 0 ð 2 1Þf ¼6 6 40 0 1 0
3 7 7 7 7 5
0 0 0 1 3 2 1 0 0 0 60 1 0 07 7 6 7 6 60 7 0 1 0 7 6 5 4 1 pffiffiffi 0 0 1 ð2 2Þf 2 pffiffiffi 1 0 ð2 2Þf 0 pffiffiffi 6 60 1 0 ð2 2Þf 6 6 40 0 1 0 0 0 0 1 3 2 1 0 0 0 7 6 1 0 07 6 0 7 6 1 7 6 0 1 07 6 7 6 f 7 6 5 4 1 0 1 0 f 2 pffiffiffi 1 0 ð2 2Þf 0 pffiffiffi 6 60 1 0 ð2 2Þf 6 6 40 0 1 0 0 0 0 1 3 2 1 0 0 0 60 1 0 07 7 6 7 6 60 7 0 1 0 7 6 5 4 1 pffiffiffi 0 0 1 ð2 21Þf 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 6 60 1 0 ð 2 1Þf 6 6 40 0 1 0 0
0
0
1
3 7 7 7 7 5
3 7 7 7 7 5
3 7 7 7: 7 5
ð30Þ
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
270
The factorization technique in combination with the matrix optics method can be also used to synthesize a compact mode converter for the miniaturization and simplification of the transformation. For example, by multiplying the matrices of the spherical lens S and its separations with the cylindrical lenses C1 and C2 ; Eq. (30) becomes 2
0
0
f
3
0
7 6 6 0 1 0 0 7 7 6 1 7 ~ ¼6 0 0 0 7 M 6 7 6 f 7 6 5 4 1 0 1 0 f 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 6 60 1 0 ð 2 1Þf ¼6 6 40 0 1 0 0 0 2 1 60 6 6 60 6 4 0 2 6 6 6 6 6 6 6 6 4
1 þ
0 0 1
0 0
0 1 pffiffiffi ð2 2Þf
pffiffiffi 2
0 1 f
0 1 þ
1 60 6 6 60 6 4 0
0
pffiffiffi 2
3
pffiffiffi 2Þf
1 þ
1 f
3
pffiffiffi 7 2Þf 7 7 7 7 0 7 7 pffiffiffi 7 5 1 þ 2
pffiffiffi 2
0
3
0
0
0
1
0
07 7 7 07 7 5 1
0
0
2ð1 þ
0
0 1 1 pffiffiffi 0 ð2 2Þf 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 6 60 1 0 ð 2 1Þf 6 6 40 0 1 0 0 0
5. Concluding remarks
7 7 7 7 5
1 3 0 07 7 7 07 7 5 1 2ð1 þ
0
0
2
1
Fig. 6. Synthesis of the optical system shown in Fig. 5.
In this study the method of the reciprocal symmetrical matrix factorization in pairs has been proposed, by means of which the transfer matrix of the mode converter can be factorized into a product of reciprocal symmetrical matrices. The illustrative examples have shown that the results are applicable to the design of different types of mode converter to meet practical applications. The factorization procedure is simple and well suited for computer calculations. The method can be further extended to factorizing other types of reciprocal symmetrical matrices and synthesizing optical systems, and is of practical interest.
Acknowledgements This work was supported by the Foundation of Science and Technology Development of Southwest Jiaotong University.
References
3 7 7 7: 7 5
ð31Þ
1
Eq. (31) indicates that the mode converter consists of a single spherical thick lens S extended with two cylindrical lenses C1 and C2 : The input and output reference planes of the system are the planes z ¼ 0 and z ¼ 2f ; respectively. The focal F and thickpLffiffiffi of plength ffiffiffi the spherical lens S are F ¼ 2f and L ¼ 2ð2 2Þf ; respectively (Fig. 6). The other parameters are the same as those in Fig. 5.
[1] Eppich B, Friberg AT, Gao C, Weber H. Twist of coherent field and beam quality. SPIE 1996;2870:260–7. [2] Laabs H, Gao C, Weber H. Twisting of three-dimensional Hermite–Gaussian beams. J Mod Opt 1999;46:709–19. [3] Gao C, Laabs H, Weber H, Brand T, Kugler N. Symmetrization of astigmatic high power diode laser stacks. Opt Quant Electron 1999;34:1207–18. [4] Laabs H, Gao C, Weber H, et al. Transformation of Hermite– Gaussian-beams into complex Hermite–Gaussian and Laguerre– Gaussian beams. SPIE 1999;3611:258–68. [5] Arsenault HH. A matrix representation for nonsymmetrical optical system. J Opt 1980;11:87–91. [6] Collins SA. Lens-system diffraction integral written in terms of matrix optical. J Opt Soc Am A 1970;60:1168–77. [7] Lu. B. Laser optics, 3rd ed. Beijing: China Higher Education Press; 2003.
Optics & Laser Technology 37 (2005) 265–270 www.elsevier.com/locate/optlastec
Matrix factorization of the mode converter Ping Wua,*, Xiaofei Songa, Baida Lu. a,b a
Department of Applied Physics, Southwest Jiaotong University, Chengdu 610031, China Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, China
b
Received 9 February 2004; accepted 20 April 2004 Available online 25 June 2004
Abstract Based on the basic properties of the reciprocal symmetrical matrix, a method is proposed, which permits us to factorize the transfer matrix of the mode converter into a product of reciprocal symmetrical matrices. The procedure is simple and is suited for computer calculations. Some factorization examples are given and illustrated. The results are useful for the design of different types of the mode converter to satisfy practical applications. r 2004 Elsevier Ltd. All rights reserved. Keywords: Matrix factorization; Reciprocal symmetrical matrix; Mode converter
1. Introduction
2. Reciprocal symmetrical matrix
The transformation of a general astigmatic beam into a stigmatic one is a topic of practical interest. Eppich and Laabs et al. showed [1,2] that any three-dimensional astigmatic Hermite–Gaussian beam can be transformed into a symmetric Lagueree–Gaussian beam with twist by using a mode converter consisting of three cylindrical lenses. The theoretical predication was confirmed by the experiments of TEMm0 -mode Nd:YAG lasers and diode laser stacks [1–4]. It was also found [4] that the threecylindrical-lens mode converter is equivalent to a two lens optics, or a single thick cylindrical lens. Therefore, an interesting question arises: Can the transfer matrix of the mode converter be factorized according to a certain rule? The aim of the present paper is to study the matrix factorization of the mode converter. In Section 2 the definition and some basic properties of the 4 4 reciprocal symmetrical matrix are studied. It is shown that the transfer matrix of the mode converter satisfies the condition of the 4 4 reciprocal symmetrical matrix. Then, Section 3 presents the factorization procedure of the transfer matrix of the mode converter into a product of reciprocal symmetrical matrices. Finally, Section 4 summarizes the main results obtained in this paper.
It is well known that a general non-symmetrical optical system can be expressed in terms of a 4 4 matrix [5,6] 2 3 a11 a12 b11 b12 6 7 A B 6 a21 a22 b21 b22 7 ~ ¼ M ¼6 ð1Þ 7; 4 c11 c12 d11 d12 5 C D c21
0030-3992/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2004.04.005
d21
d22
where ‘‘-’’ denotes the forward matrix (from the left input reference plane to the right output reference plane). Assume that the input and output planes are in the same type of optical medium (usually free space), thus we have At D Ct B ¼ E; ð2Þ 1 0 where E ¼ ; and t denotes the transposition of 0 1 the matrix. t ’ D Bt ~ be M Let the backward matrix of M ¼ [7], Ct At ~ is defined as the reciprocal symmetrical matrix if the M condition ’
~ ¼M M *Corresponding author. E-mail address: ping w [email protected] (P. Wu).
c22
is fulfilled.
ð3Þ
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
266
From Eq. (3) it turns out that the sub-matrices of the reciprocal symmetrical matrix satisfy the condition
reciprocal symmetrical. Thus, we have ~ n1 yM ~ 1: ~ ¼M ~n M M
ð7Þ
A ¼ Dt ;
The backward matrix of Eq. (7) are
B ¼ Bt ;
M ¼ M 1 M2 y Mn :
ð8Þ
C ¼ Ct ;
Eqs. (7) and (8) can be rewritten as ~ 1; ~ ¼M ~n ~ PM M
ð9Þ
’
D ¼ At :
ð4Þ
In addition, we have ’
~ 1 ¼ M1 ; M
ð5Þ
where ‘‘1’’ denotes the reverse of the matrix, i.e., t ’ 1 Bt A ~ 1 ¼ D B and M ¼ M : Eq. (5) C A Ct Dt indicates that the reverse matrix of the reciprocal symmetrical matrix retains its reciprocal symmetry unchanged. In accordance with Ref. [1] the transfer matrix of the mode converter in the principal axes representation reads as 3 2 0 0 f 0 7 6 6 0 1 0 0 7 7 6 1 7 ~ ¼6 0 0 0 7: ð6Þ M 6 7 6 f 7 6 5 4 1 0 1 0 f It is readily shown that Eq. (6) satisfies condition (3). In practice the beam converter consists of spherical and cylindrical lenses and their separation (free-space propagation) whose matrices are all reciprocal symmetrical. As a result, the matrix factorization of the beam converter is reduced to factorizing Eq. (6) by means of reciprocal symmetrical matrices.
3. Factorization procedure of the transfer matrix of the mode converter As shown in Fig. 1, assume that a reciprocal symmetrical optical system expressed in terms of the ~ is composed of n optical systems, whose matrix M ~ 1; M ~ 2 yM ~ n respectively, and are all matrices are M
’
’
’
’
’
’ ’
M ¼ M 1 P Mn ;
ð10Þ
where ~ n2 yM ~ 2; ~¼M ~ n1 M P
ð11Þ
’
’
’
’
P ¼ M2 M3 y Mn1 :
ð12Þ
Now we factorize Eq. (7) by using reciprocal symme’ ’ ~1 ¼ M ~ n; M trical matrices in pairs, i.e., M 1 ¼ Mn ; Eqs. (9) and (10) become ~ 1; ~ ¼M ~1 ~ PM ð13Þ M ’
’
’ ’
M ¼ M 1 P M1 ;
ð14Þ
respectively. Because of the reciprocal symmetrical ~ 1 and M ~ ; the substitution from property of matrices M ’ ~1 ¼ M M into Eqs. (13) and (14) yields 1 ’
~ P¼ P:
ð15Þ
Eq. (15) implies that after the first factorization the remaining matrix is still reciprocal symmetrical. Therefore, the further factorization can be performed continuously in a similar way, till only a simple reciprocal symmetrical matrix remains.
4. Application examples To illustrate the application of the matrix factorization procedure described in Section 3, some typical examples are given as follows. Example 1. Firstly, we choose a pair of the spherical lens with focal length f whose matrix is expressed as 3 2 1 0 0 0 7 6 1 0 07 6 0 7 6 1 7 ~1 ¼ 6 0 1 0 7: ð16Þ M 6 7 6 f 7 6 5 4 1 0 1 0 f From Eq. (13) it follows that
Fig. 1. Notation relating to the optical system.
~ ~ 1 M ~ M ~ 1 : P1 ¼ M 1 1
ð17Þ
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
Secondly, we with distance 2 1 0 60 1 ~2 ¼ 6 M 6 40 0 0
0
267
use a pair of the free-spare propagation z whose matrix is 3 z 0 0 z7 7 ð18Þ 7: 1 05 0
1
Thus ~ ~ 1 ~ ~ 1 : P2 ¼ M P1 M 2 2
ð19Þ
Thirdly, we use a pair of the cylindrical lens with focal length F in the y direction whose matrix reads as 3 2 1 0 0 0 60 1 0 07 7 6 ~3 ¼ 6 7 ð20Þ M 6 0 0 1 0 7: 5 4 1 0 1 0 F Finally, we have
Fig. 2. Example 1 of the mode converter.
pffiffi
2
Þf in the y direction. The separations between S1 pffiffi 2 and C1 ; S2 and C2 are equal and are ð1 2 Þf ; and the
ð1
2
~ 1 ~ ~ 1 ~ P2 M P3 ¼ M 3 3 3 2 1 0 f 2z 0
7 6 2z 2z z zð1 þ f Þ 2z 7 6 7 60 0 z z 1 þ 1 þ þ 7 6 f f F 7 6 ¼60 7: 0 1 0 7 6 7 6 2z zzð1þ f Þ 6 2z 2z 2z 7 5 4 1 þ 1 þ þ z zð1 þ Þ 2 2z f f f F 0 þ þ 0 1 þ þ f f F F F Assume that P3 is the matrix of the free-spare propagation, the following relations should be fulfilled: 1 þ
2z 2z z zð1 þ f Þ þ ¼ 1; f F
ð22Þ
2z f 2z ¼ z z 1 þ ; f 2 1 þ þ f F
2z f
þ
ð23Þ
2z 1 þ 2z f þ z zð1 þ f Þ
F
¼ 0:
ð24Þ
Eqs. (22)–(24) lead to
z¼
pffiffiffi! 2 1 f 2
ð25Þ
and F ¼ z:
ð26Þ
Therefore, the mode converter shown in Fig. 2 consists of a pair of the spherical lens S1 ; S2 with focal length f ; and a pair of the cylindrical lens C1 ; C2 with focal length
ð21Þ
pffiffiffi separation between C1 and C2 is ð 2 1Þf : The transfer matrix factorization of the system is expressed as 2
0 6 6 0 6 1 ~ ¼6 M 6 6 f 6 4 0 2 1 6 6 0 6 1 6 ¼6 6 f 6 4 0 2
0 1 0 1 f 0 1 0 1 f 0
3 0 7 0 7 7 7 0 0 7 7 7 5 0 1 3 0 0 7 0 07 7 7 1 07 7 7 5 0 1 pffiffiffi! 2 1 f 2 f 0
3
61 6 6 6 6 60 6 6 6 40
1
0
0
1
0
0
0
0
1
0 pffiffiffi! 2 f 1 2
7 7 7 7 7 7 7 7 7 5
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
268
2
3
1
0
0
0
60 6 6 6 60 6 60 4
1
0
07 7 7 07 7 7 17 5
2
1
6 60 6 6 40 2
0 1
60 6 6 0 6 6 6 40 2
0 1 1 pffiffiffi 0 2 Þf ð1 pffiffiffi2 0 ð 2 1Þf 1
0
0 0
1 0 0
0
pffiffiffi 7 ð 2 1Þf 7 7 7 5 0 0
1 3
1 0 07 7 7 0 1 07 7 1 7 pffiffi 0 15 2 ð1 2 Þf pffiffiffi! 2 0 1 f 2
61 6 6 6 6 60 1 6 6 6 40 0 0 0 2 1 0 6 1 6 0 6 1 6 0 6 6 f 6 4 1 0 f
0 1 0 0 0 1 0
0
3
7 07 7 7 0 7: 7 7 5 1
3
0
Fig. 3. Example 2 of the mode converter.
3 0
7 7 pffiffiffi! 7 7 2 7 f7 1 7 2 7 7 5 0 1 Fig. 4. Example 3 of the mode converter.
ð27Þ
Following the above factorization procedure, we can find a variety of optical system which consist of different reciprocal symmetrical optical systems, but have the same transfer matrix as Eq. (6). Example 2. The mode converter shown in Fig. 3 is composed of a pair of the cylindrical lens C1 ; C2 with focal length f in the y direction and a bifocal lens B with focal length f2 and f in the x and y directions, respectively. The separation between C1 and B; C2 and B is equal to f : The transfer matrix of the system is expressed as 2
0 6 6 0 6 1 ~ ¼6 M 6 6 f 6 4 0
0 1
f 0
0
0
1 f
0
3 0 7 0 7 7 7 0 7 7 7 5 1
2
3
2 1 0 7 6 60 1 0 07 60 1 7 6 6 ¼60 0 1 07 6 7 40 0 6 5 4 1 0 0 1 0 0 f 3 2 1 0 0 0 2 7 6 1 1 0 07 6 0 7 60 6 1 7 6 6 0 1 076 6 7 40 6 f 7 6 5 4 2 0 0 1 0 f 3 2 1 0 0 0 7 6 60 1 0 07 7 6 6 0 0 1 0 7: 7 6 5 4 1 0 0 1 f 1
0
0
0
3
f
0
0
f7 7 7 05
1 0
1
0 1
f 0
0
1
3 0 f7 7 7 05
0
0
1
ð28Þ
Example 3. Fig. 4 gives the third example. The mode converter consists of a pair of the free-space propagapffiffiffi tion with distances ð 2 1Þf and a pair of the bifocal
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
pffiffiffi pffiffiffi lens B1 ; B2 with focal lengths 2f and ð 2 1Þf in the x and y directions, The separation between pffiffirespectively. ffi B1 and B2 is 2ð 2 1Þf : The transfer matrix of the system is written as 3 2 0 0 f 0 7 6 6 0 1 0 0 7 7 6 1 7 ~ ¼6 0 0 0 7 M 6 7 6 f 7 6 5 4 1 0 1 0 f 3 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 7 6 60 1 0 ð 2 1Þf 7 7 ¼6 7 6 5 40 0 1 0 0 0 2
0
6 0 6 6 1 6 6 pffiffiffi 6 2f 6 4 0 2
1 0
6 60 1 6 6 40 0 0 0 2 1 6 0 6 6 1 6 6 pffiffiffi 6 2f 6 4 0 2
1
1
1 0
6 60 1 6 6 40 0 0 0
3
0
0
0
1
0
0
1
07 7 7 07 7 7 7 5 1
1 pffiffiffi ð 2 1Þf pffiffiffi 2ð 2 1Þf 0 1
0
0 pffiffiffi 2ð 2 1Þf 0
0 0
0
1
0
0
1
7 7 7 7 5
1 3 0 07 7 7 07 7 7 7 5 1
1 pffiffiffi 0 ð 2 1Þf pffiffiffi ð 2 1Þf 0 pffiffiffiffiffiffiffiffiffiffiffi 0 ð 2 1Þf 1 0 0
3
3 7 7 7: 7 5
1
Fig. 5. Example 4 of the mode converter.
ð29Þ
269
Example 4. The fourth mode converter is shown in Fig. 5, which consists of appair of the free-space ffiffiffi propagation with distance ð 2 1Þf ; a pair pffiffiffi of the cylindrical lens C1 ; C2 with focal length ð2 2Þf in the y direction and a spherical lens S with focal length f : The separation between C1 and S; C2 and S is equal pffiffiffi to ð2 2Þf : The transfer matrix of the system reads as
2
0
0
f
0
3
7 6 6 0 1 0 0 7 7 6 1 7 ~ ¼6 0 0 0 7 M 6 7 6 f 7 6 5 4 1 0 1 0 f 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 6 60 1 0 ð 2 1Þf ¼6 6 40 0 1 0
3 7 7 7 7 5
0 0 0 1 3 2 1 0 0 0 60 1 0 07 7 6 7 6 60 7 0 1 0 7 6 5 4 1 pffiffiffi 0 0 1 ð2 2Þf 2 pffiffiffi 1 0 ð2 2Þf 0 pffiffiffi 6 60 1 0 ð2 2Þf 6 6 40 0 1 0 0 0 0 1 3 2 1 0 0 0 7 6 1 0 07 6 0 7 6 1 7 6 0 1 07 6 7 6 f 7 6 5 4 1 0 1 0 f 2 pffiffiffi 1 0 ð2 2Þf 0 pffiffiffi 6 60 1 0 ð2 2Þf 6 6 40 0 1 0 0 0 0 1 3 2 1 0 0 0 60 1 0 07 7 6 7 6 60 7 0 1 0 7 6 5 4 1 pffiffiffi 0 0 1 ð2 21Þf 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 6 60 1 0 ð 2 1Þf 6 6 40 0 1 0 0
0
0
1
3 7 7 7 7 5
3 7 7 7 7 5
3 7 7 7: 7 5
ð30Þ
ARTICLE IN PRESS P. Wu et al. / Optics & Laser Technology 37 (2005) 265–270
270
The factorization technique in combination with the matrix optics method can be also used to synthesize a compact mode converter for the miniaturization and simplification of the transformation. For example, by multiplying the matrices of the spherical lens S and its separations with the cylindrical lenses C1 and C2 ; Eq. (30) becomes 2
0
0
f
3
0
7 6 6 0 1 0 0 7 7 6 1 7 ~ ¼6 0 0 0 7 M 6 7 6 f 7 6 5 4 1 0 1 0 f 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 6 60 1 0 ð 2 1Þf ¼6 6 40 0 1 0 0 0 2 1 60 6 6 60 6 4 0 2 6 6 6 6 6 6 6 6 4
1 þ
0 0 1
0 0
0 1 pffiffiffi ð2 2Þf
pffiffiffi 2
0 1 f
0 1 þ
1 60 6 6 60 6 4 0
0
pffiffiffi 2
3
pffiffiffi 2Þf
1 þ
1 f
3
pffiffiffi 7 2Þf 7 7 7 7 0 7 7 pffiffiffi 7 5 1 þ 2
pffiffiffi 2
0
3
0
0
0
1
0
07 7 7 07 7 5 1
0
0
2ð1 þ
0
0 1 1 pffiffiffi 0 ð2 2Þf 2 pffiffiffi 1 0 ð 2 1Þf 0 pffiffiffi 6 60 1 0 ð 2 1Þf 6 6 40 0 1 0 0 0
5. Concluding remarks
7 7 7 7 5
1 3 0 07 7 7 07 7 5 1 2ð1 þ
0
0
2
1
Fig. 6. Synthesis of the optical system shown in Fig. 5.
In this study the method of the reciprocal symmetrical matrix factorization in pairs has been proposed, by means of which the transfer matrix of the mode converter can be factorized into a product of reciprocal symmetrical matrices. The illustrative examples have shown that the results are applicable to the design of different types of mode converter to meet practical applications. The factorization procedure is simple and well suited for computer calculations. The method can be further extended to factorizing other types of reciprocal symmetrical matrices and synthesizing optical systems, and is of practical interest.
Acknowledgements This work was supported by the Foundation of Science and Technology Development of Southwest Jiaotong University.
References
3 7 7 7: 7 5
ð31Þ
1
Eq. (31) indicates that the mode converter consists of a single spherical thick lens S extended with two cylindrical lenses C1 and C2 : The input and output reference planes of the system are the planes z ¼ 0 and z ¼ 2f ; respectively. The focal F and thickpLffiffiffi of plength ffiffiffi the spherical lens S are F ¼ 2f and L ¼ 2ð2 2Þf ; respectively (Fig. 6). The other parameters are the same as those in Fig. 5.
[1] Eppich B, Friberg AT, Gao C, Weber H. Twist of coherent field and beam quality. SPIE 1996;2870:260–7. [2] Laabs H, Gao C, Weber H. Twisting of three-dimensional Hermite–Gaussian beams. J Mod Opt 1999;46:709–19. [3] Gao C, Laabs H, Weber H, Brand T, Kugler N. Symmetrization of astigmatic high power diode laser stacks. Opt Quant Electron 1999;34:1207–18. [4] Laabs H, Gao C, Weber H, et al. Transformation of Hermite– Gaussian-beams into complex Hermite–Gaussian and Laguerre– Gaussian beams. SPIE 1999;3611:258–68. [5] Arsenault HH. A matrix representation for nonsymmetrical optical system. J Opt 1980;11:87–91. [6] Collins SA. Lens-system diffraction integral written in terms of matrix optical. J Opt Soc Am A 1970;60:1168–77. [7] Lu. B. Laser optics, 3rd ed. Beijing: China Higher Education Press; 2003.