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Novel unitary optical mode converter
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 282–284 www.elsevier.com/locate/optlastec
Novel unitary optical mode converter Hongzhou Dong, Shunxiang Shi, Jiali Li School of Technical Physics, Xidian University, P.O. Box 203, No.2 Tai Bai South Road, Xi’an 710071, PR China Received 21 April 2005; received in revised form 16 July 2005; accepted 3 August 2005 Available online 22 September 2005
Abstract A new unitary optical mode converter for transforming Hermite–Gaussian beams into Laguerre–Gaussian beams is proposed, which is made up of gradient index medium. Being unitary cylinder with only two reflecting surfaces, the mode converter is easily adjusted and has very low reflecting loss. r 2005 Elsevier Ltd. All rights reserved. Keywords: Matrix optics; Optical mode converter; Gradient index
1. Introduction Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams are the eigen solutions of the paraxial wave equation in Cartesian and cylindrical coordinates, respectively. In some practical case, such as laser diode (LD)-to-fibre coupling, it is necessary to transform the output beams of LD, regarded as TEMm,n-mode HG beam, into LG beams which is of rotational symmetry for improving coupling efficiency. 2. Principles of mode transforming It has been proved theoretically that a LG mode can be decomposed into a set of HG modes [1–3]: uLG mn ðx; y; zÞ ¼
N X
ig bðm; n; kÞuHG Ng;g ðx; y; zÞ,
(1)
g¼0
where N ¼ m þ n; b ¼ ðm; n; gÞ denotes a polynomial about n; m; g. The factor ig in Eq. (1) corresponds to a p=2 relative phase difference between successive components. A HG mode whose principal axes make an angle of Corresponding author. Tel.: +86 29 88207311; fax: +86 29 88202551.
E-mail addresses: [email protected] (H. Dong), [email protected] (S. Shi). 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.08.007
451 with the ðx; yÞ axes can also be decomposed into the same constituent set: X N xþy xy p ffiffi ffi p ffiffi ffi bðm; n; gÞ uHG (2) ; ; z ¼ uHG mn Ng;g ðx; y; zÞ, 2 2 g¼0 with the same real coefficient bðm; n; gÞ as above. In this expansion, however, the successive components are in phase. Therefore, if an astigmatic optical system is introduced and the phase of the successive components in Eq. (2) is rectified in the astigmatic area, so that a p=2 relative phase difference is imported among successive components, the diagonal HG mode can be converted into LG mode. On the basis of second-order moments principle, Ref. [4] put forward an optical system composed of three cylindrical lens, which can realize the transformation from the waist plane of HG beams to that of LG beams. The system can be depicted by the following matrix: 2 3 1 1 zR zR 6 7 1 1 zR zR 7 16 6 7, B¼ 6 (3) 2 4 1=zR 1=zR 1 1 7 5 1=zR 1=zR 1 1 where zR is the Rayleigh range of HG beams in two principal planes. Moreover, matrix B may be denoted by
ARTICLE IN PRESS H. Dong et al. / Optics & Laser Technology 39 (2007) 282–284
B ¼ P1 MP, where 2
cosðp=4Þ
3. The configuration of the novel optical converter
sinðp=4Þ
6 6 sinðp=4Þ cosðp=4Þ P¼6 6 0 0 4 0 0 2
0
6 0 6 M¼6 4 1=zR 0
283
0 0 cosðp=4Þ sinðp=4Þ
0
zR
0
3
1 0
0 0
0 0
7 7 7, 5
0
0
1
0
3
7 7 7, sinðp=4Þ 7 5 cosðp=4Þ 0
(4)
(5)
P1 is the inverse matrix of P. M is the transform matrix of three cylindrical lens system. That M multiplied by P1 and P means that the optical system described by matrix M should be rotated 451 relative to the principal axis of input beams. By using the second-order moments method and generalized Huygens–Fresmel diffraction integral, the transformation properties of the Hermite– Gaussian beam with real and complex argument passing through three cylindrical lens system are studied in detail in Refs. [5–7]. Because the system contains three discrete components, it is difficult to adjust the relative position of three cylindrical lens, and the reflecting loss is large owing to six reflecting surfaces of the system. Consequently, the application of the system is restricted in practice. A different mode converter system composed of two cylindrical lens and one lens is put forward in Ref. [8], but the system have the same disadvantage as three cylindrical lens system due to discrete components. According to matrix optics, matrix M can denote different optical systems, so an optimal converter meeting practical requirements can be designed. In this paper, a novel unitary optical mode converter made up of gradient index medium is proposed, which can realize the transformation from HG to LG mode.
x H2
The novel optical converter is shown in Fig. 1. It is formed by gradient index medium with two cylindrical end faces whose curvatures are R1 and R2 respectively, and the refractive index distribution of the medium is nðrÞ ¼ n0 ð1 br2 2Þ, where n0 is the on-axis value of refractive index, b indicates the distribution constant, and r is the radial position. In Fig. 1, H 1 stands for the waist plane of HG beams in two principal planes and H 2 for that of LG beams, Thus the whole transform matrix from H 1 to H 2 plane may be denoted by 32 3 2 1 0 0 0 1 0 d 0 76 7 6 1 0 07 6 0 1 0 d 76 0 76 7 6 0 M ¼6 76 7 0 n0 0 7 6 0 0 1 0 76 0 54 5 4 2
0 0 0 pffiffiffi cos bl
x′
H1 d
y′
y Fig. 1. A novel mode converter.
1n0 R2
0
0
6 6 6 pffiffiffi 6 0 cos bl 6 6 6 pffiffiffi pffiffiffi 6 0 6 b sin bl 4 pffiffiffi pffiffiffi 0 b sin bl 3 2 1 0 0 0 21 0 d 76 6 60 1 0 07 0 1 0 76 6 76 6 1 76 60 0 0 n0 0 0 1 76 6 54 4 n0 1 1 0 nR1 0 n0 0 0 0
n0 pffiffi sin bl pffiffi b
0 cos
pffiffiffi bl 0
0
0
3
7 7 7 7 7 b 7 7 7 0 7 pffiffiffi 5 cos bl pffiffi sin bl pffiffi
3
7 d7 7 7. 07 5 1
ð6Þ
In the expression, the first and the fifth matrix show the transform characteristics of free space having a length of d, and the third matrix indicates that of the gradient index medium having a length of l. The second and the fourth matrix show that of cylindrical end faces, where R1 o0; R2 o0, and nðrÞ n0 under the paraxial approximation. Multiplying five matrices together, one can find that M ¼ M 0 when the parameters of M0 satisfy the following relations: zR d ¼ pffiffiffi , (7) 2þ1
y x
0
pffiffiffi 2zR ðn0 1Þ jR1 j ¼ R2 ¼ R ¼ pffiffiffi , 2þ1
z
d
1
b¼
n20 z2R
h
1 pffiffiffi pffiffiffi 2 i , 1 1 ð 2=ð 2 þ 1ÞÞ
pffiffiffi pffiffiffi arccos 1 ð 2=ð 2 þ 1ÞÞ pffiffiffi l¼ . b
(8)
(9)
(10)
In other words, the transform characteristic of optical device shown in Fig. 1 is equivalent to that of three
ARTICLE IN PRESS 284
H. Dong et al. / Optics & Laser Technology 39 (2007) 282–284
cylindrical lens system proposed in Ref. [4] when all parameters of the device satisfy Eqs. (7)–(10). In addition, the waist radius w of the output LG beams on the H 2 plane can be expressed in terms of the Rayleigh range zR of input HG beams on the plane H 1 as [5] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4lzR ðm þ n þ 1Þ , (11) w¼ p
x H2 x′
z
x H1 d
y
d y′ y
where m and n are the mode orders of the input HG beams, l is the wavelength.
Fig. 2. The operation sketch map of the converter.
5. Conclusion 4. Practical example When zR of input beams is given, all parameters of the converter and the value of w can be determined according to Eqs. (7)–(11). For example, if the input beams are assumed to TEM5,0-mode HG beams with the wavelength of 0.8 mm, and zR ¼ 1:6 mm, n0 ¼ 1:63, then we can obtain: d ¼ 678 mm, R ¼ 604 mm, l ¼ 2:78 mm, b ¼ 0:17 mm2 and w ¼ 100 mm. In practical operation, the principal axis (x0 , y0 ) of converter should be rotated 451 relative to the principal axis (x, y) of input HG beams, then the transform matrix from H 1 to H 2 plane is B as is shown in Fig. 2. According to the above discussion, the transform of the beams from HG mode to LG mode can be realized. It should be pointed out that the distance from H 1 plane to the converter, according to expression (7), is less than the Rayleigh range of input beams, so the input beams can be considered as parallel beams. They can pass the converter completely without any restriction caused by numerical aperture. When the Rayleigh range and the waist position of input HG beams are different in two principal planes, a waist adjuster composed usually of two vertical cylindrical lens is needed. Such HG beams can also be transformed into LG beams by combining the adjuster and the converter.
A novel unitary optical mode converter with two cylindrical end faces is proposed in this paper. It is made up of gradient index medium. Being unitary and having only two reflecting surfaces, the converter has very low reflecting loss and is easily adjusted, so the mode converter could be useful for practical work. References [1] Allen L, Beijerbergen MW, Spreeuw RJC, Woerdman JP. Orbital angular momentum of light and the transformation of Laguerre– Gaussian laser modes. Phys Rev 1992;A45:8185–9. [2] Abramochkin E, Volosrnikov V. Beam transformations and nontransformed beams. Opt Commun 1991;83:123–5. [3] Beijersbergen MW, Allen L. Astigmatic laser mode converters and transfer of orbital angular momentum. Opt Commun 1993;96:123–32. [4] Eppich B, Fiberg AT. Twist of coherent fields and bram quality. SPIE 1996;2870:260–7. [5] Ping Wu, Baida Lu¨. Propagation properties of three-dimensional TEMm,n-mode Hermite–Gaussian beams through an optical symmetric system. Laser Technol 2003;27(2):126–9 [in Chinese]. [6] Ping Wu, Baida Lu¨. Symmetrization of astigmatic Hermite–Gaussian beams and related problems. High Power Laser Particle Beams 2002;4(14):546–50 [in Chinese]. [7] Ping Wu, Baida Lu¨. Transformation of astigmatic Hermite Gaussian beams with complex argument by an optical symmetrizing system. High Power Laser Particle Beams 2002;5(14):697–701 [in Chinese]. [8] Tingrong Zhang, Baida Lu¨. Novel types of optical system for symmetrizing three dimensional astigmatic Hermite Gaussian beams. Laser Technol 2003;2(27):147–9 [in Chinese].
Optics & Laser Technology 39 (2007) 282–284 www.elsevier.com/locate/optlastec
Novel unitary optical mode converter Hongzhou Dong, Shunxiang Shi, Jiali Li School of Technical Physics, Xidian University, P.O. Box 203, No.2 Tai Bai South Road, Xi’an 710071, PR China Received 21 April 2005; received in revised form 16 July 2005; accepted 3 August 2005 Available online 22 September 2005
Abstract A new unitary optical mode converter for transforming Hermite–Gaussian beams into Laguerre–Gaussian beams is proposed, which is made up of gradient index medium. Being unitary cylinder with only two reflecting surfaces, the mode converter is easily adjusted and has very low reflecting loss. r 2005 Elsevier Ltd. All rights reserved. Keywords: Matrix optics; Optical mode converter; Gradient index
1. Introduction Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams are the eigen solutions of the paraxial wave equation in Cartesian and cylindrical coordinates, respectively. In some practical case, such as laser diode (LD)-to-fibre coupling, it is necessary to transform the output beams of LD, regarded as TEMm,n-mode HG beam, into LG beams which is of rotational symmetry for improving coupling efficiency. 2. Principles of mode transforming It has been proved theoretically that a LG mode can be decomposed into a set of HG modes [1–3]: uLG mn ðx; y; zÞ ¼
N X
ig bðm; n; kÞuHG Ng;g ðx; y; zÞ,
(1)
g¼0
where N ¼ m þ n; b ¼ ðm; n; gÞ denotes a polynomial about n; m; g. The factor ig in Eq. (1) corresponds to a p=2 relative phase difference between successive components. A HG mode whose principal axes make an angle of Corresponding author. Tel.: +86 29 88207311; fax: +86 29 88202551.
E-mail addresses: [email protected] (H. Dong), [email protected] (S. Shi). 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.08.007
451 with the ðx; yÞ axes can also be decomposed into the same constituent set: X N xþy xy p ffiffi ffi p ffiffi ffi bðm; n; gÞ uHG (2) ; ; z ¼ uHG mn Ng;g ðx; y; zÞ, 2 2 g¼0 with the same real coefficient bðm; n; gÞ as above. In this expansion, however, the successive components are in phase. Therefore, if an astigmatic optical system is introduced and the phase of the successive components in Eq. (2) is rectified in the astigmatic area, so that a p=2 relative phase difference is imported among successive components, the diagonal HG mode can be converted into LG mode. On the basis of second-order moments principle, Ref. [4] put forward an optical system composed of three cylindrical lens, which can realize the transformation from the waist plane of HG beams to that of LG beams. The system can be depicted by the following matrix: 2 3 1 1 zR zR 6 7 1 1 zR zR 7 16 6 7, B¼ 6 (3) 2 4 1=zR 1=zR 1 1 7 5 1=zR 1=zR 1 1 where zR is the Rayleigh range of HG beams in two principal planes. Moreover, matrix B may be denoted by
ARTICLE IN PRESS H. Dong et al. / Optics & Laser Technology 39 (2007) 282–284
B ¼ P1 MP, where 2
cosðp=4Þ
3. The configuration of the novel optical converter
sinðp=4Þ
6 6 sinðp=4Þ cosðp=4Þ P¼6 6 0 0 4 0 0 2
0
6 0 6 M¼6 4 1=zR 0
283
0 0 cosðp=4Þ sinðp=4Þ
0
zR
0
3
1 0
0 0
0 0
7 7 7, 5
0
0
1
0
3
7 7 7, sinðp=4Þ 7 5 cosðp=4Þ 0
(4)
(5)
P1 is the inverse matrix of P. M is the transform matrix of three cylindrical lens system. That M multiplied by P1 and P means that the optical system described by matrix M should be rotated 451 relative to the principal axis of input beams. By using the second-order moments method and generalized Huygens–Fresmel diffraction integral, the transformation properties of the Hermite– Gaussian beam with real and complex argument passing through three cylindrical lens system are studied in detail in Refs. [5–7]. Because the system contains three discrete components, it is difficult to adjust the relative position of three cylindrical lens, and the reflecting loss is large owing to six reflecting surfaces of the system. Consequently, the application of the system is restricted in practice. A different mode converter system composed of two cylindrical lens and one lens is put forward in Ref. [8], but the system have the same disadvantage as three cylindrical lens system due to discrete components. According to matrix optics, matrix M can denote different optical systems, so an optimal converter meeting practical requirements can be designed. In this paper, a novel unitary optical mode converter made up of gradient index medium is proposed, which can realize the transformation from HG to LG mode.
x H2
The novel optical converter is shown in Fig. 1. It is formed by gradient index medium with two cylindrical end faces whose curvatures are R1 and R2 respectively, and the refractive index distribution of the medium is nðrÞ ¼ n0 ð1 br2 2Þ, where n0 is the on-axis value of refractive index, b indicates the distribution constant, and r is the radial position. In Fig. 1, H 1 stands for the waist plane of HG beams in two principal planes and H 2 for that of LG beams, Thus the whole transform matrix from H 1 to H 2 plane may be denoted by 32 3 2 1 0 0 0 1 0 d 0 76 7 6 1 0 07 6 0 1 0 d 76 0 76 7 6 0 M ¼6 76 7 0 n0 0 7 6 0 0 1 0 76 0 54 5 4 2
0 0 0 pffiffiffi cos bl
x′
H1 d
y′
y Fig. 1. A novel mode converter.
1n0 R2
0
0
6 6 6 pffiffiffi 6 0 cos bl 6 6 6 pffiffiffi pffiffiffi 6 0 6 b sin bl 4 pffiffiffi pffiffiffi 0 b sin bl 3 2 1 0 0 0 21 0 d 76 6 60 1 0 07 0 1 0 76 6 76 6 1 76 60 0 0 n0 0 0 1 76 6 54 4 n0 1 1 0 nR1 0 n0 0 0 0
n0 pffiffi sin bl pffiffi b
0 cos
pffiffiffi bl 0
0
0
3
7 7 7 7 7 b 7 7 7 0 7 pffiffiffi 5 cos bl pffiffi sin bl pffiffi
3
7 d7 7 7. 07 5 1
ð6Þ
In the expression, the first and the fifth matrix show the transform characteristics of free space having a length of d, and the third matrix indicates that of the gradient index medium having a length of l. The second and the fourth matrix show that of cylindrical end faces, where R1 o0; R2 o0, and nðrÞ n0 under the paraxial approximation. Multiplying five matrices together, one can find that M ¼ M 0 when the parameters of M0 satisfy the following relations: zR d ¼ pffiffiffi , (7) 2þ1
y x
0
pffiffiffi 2zR ðn0 1Þ jR1 j ¼ R2 ¼ R ¼ pffiffiffi , 2þ1
z
d
1
b¼
n20 z2R
h
1 pffiffiffi pffiffiffi 2 i , 1 1 ð 2=ð 2 þ 1ÞÞ
pffiffiffi pffiffiffi arccos 1 ð 2=ð 2 þ 1ÞÞ pffiffiffi l¼ . b
(8)
(9)
(10)
In other words, the transform characteristic of optical device shown in Fig. 1 is equivalent to that of three
ARTICLE IN PRESS 284
H. Dong et al. / Optics & Laser Technology 39 (2007) 282–284
cylindrical lens system proposed in Ref. [4] when all parameters of the device satisfy Eqs. (7)–(10). In addition, the waist radius w of the output LG beams on the H 2 plane can be expressed in terms of the Rayleigh range zR of input HG beams on the plane H 1 as [5] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4lzR ðm þ n þ 1Þ , (11) w¼ p
x H2 x′
z
x H1 d
y
d y′ y
where m and n are the mode orders of the input HG beams, l is the wavelength.
Fig. 2. The operation sketch map of the converter.
5. Conclusion 4. Practical example When zR of input beams is given, all parameters of the converter and the value of w can be determined according to Eqs. (7)–(11). For example, if the input beams are assumed to TEM5,0-mode HG beams with the wavelength of 0.8 mm, and zR ¼ 1:6 mm, n0 ¼ 1:63, then we can obtain: d ¼ 678 mm, R ¼ 604 mm, l ¼ 2:78 mm, b ¼ 0:17 mm2 and w ¼ 100 mm. In practical operation, the principal axis (x0 , y0 ) of converter should be rotated 451 relative to the principal axis (x, y) of input HG beams, then the transform matrix from H 1 to H 2 plane is B as is shown in Fig. 2. According to the above discussion, the transform of the beams from HG mode to LG mode can be realized. It should be pointed out that the distance from H 1 plane to the converter, according to expression (7), is less than the Rayleigh range of input beams, so the input beams can be considered as parallel beams. They can pass the converter completely without any restriction caused by numerical aperture. When the Rayleigh range and the waist position of input HG beams are different in two principal planes, a waist adjuster composed usually of two vertical cylindrical lens is needed. Such HG beams can also be transformed into LG beams by combining the adjuster and the converter.
A novel unitary optical mode converter with two cylindrical end faces is proposed in this paper. It is made up of gradient index medium. Being unitary and having only two reflecting surfaces, the converter has very low reflecting loss and is easily adjusted, so the mode converter could be useful for practical work. References [1] Allen L, Beijerbergen MW, Spreeuw RJC, Woerdman JP. Orbital angular momentum of light and the transformation of Laguerre– Gaussian laser modes. Phys Rev 1992;A45:8185–9. [2] Abramochkin E, Volosrnikov V. Beam transformations and nontransformed beams. Opt Commun 1991;83:123–5. [3] Beijersbergen MW, Allen L. Astigmatic laser mode converters and transfer of orbital angular momentum. Opt Commun 1993;96:123–32. [4] Eppich B, Fiberg AT. Twist of coherent fields and bram quality. SPIE 1996;2870:260–7. [5] Ping Wu, Baida Lu¨. Propagation properties of three-dimensional TEMm,n-mode Hermite–Gaussian beams through an optical symmetric system. Laser Technol 2003;27(2):126–9 [in Chinese]. [6] Ping Wu, Baida Lu¨. Symmetrization of astigmatic Hermite–Gaussian beams and related problems. High Power Laser Particle Beams 2002;4(14):546–50 [in Chinese]. [7] Ping Wu, Baida Lu¨. Transformation of astigmatic Hermite Gaussian beams with complex argument by an optical symmetrizing system. High Power Laser Particle Beams 2002;5(14):697–701 [in Chinese]. [8] Tingrong Zhang, Baida Lu¨. Novel types of optical system for symmetrizing three dimensional astigmatic Hermite Gaussian beams. Laser Technol 2003;2(27):147–9 [in Chinese].