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Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon
Optics & Laser Technology 35 (2003) 1 – 4
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Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon Zhengmao Wu∗ , Guangqiong Xia, Hanqing Zhou, Jianwei Wu, Mulin Liu Department of Physics, Southwest Normal University, Chongqing 400715, China Received 22 March 2002; received in revised form 8 July 2002; accepted 29 July 2002
Abstract The transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon has been investigated. After deriving the expression of the intensity of the transmitted beam, several key characteristics of the transmitted beam have been speci7ed. The results show that the peak intensity of the transmitted beam decreases with the increase of the input angle, the position of the peak intensity of the transmitted beam is shifted, and the spot size of the transmitted beam is changed. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fabry–Perot etalon; Gaussian beam; Nonnormal incidence
(x2 + y2 ) ; exp −i 2R(z)
1. Introduction Fabry–Perot etalon has been intensively investigated [1–6] because of its wide applications in spectroscopy, laser resonators, optical communications, and so on. In most studies, in order to simplify the physical treatment, the laser beam is usually treated as a plane wave and normal incidence is supposed when its transmission through a Fabry–Perot etalon is considered [1–5]. However, in fact, a laser beam is Hermit-Gaussian in general and Gaussian in its fundamental mode. Also, normal incidence is diAcult to realize in practice. Based on the above considerations, the transmission of a Gaussian beam nonnormally through a Fabry–Perot etalon has been investigated in this paper. 2. Theory The equation that describes the propagation of a Gaussian beam along z direction can be expressed as [7] 2 !0 x + y2 E(x; y; z) = A exp − 2 exp !(z) ! (z) z −i z − arctg f ∗
Corresponding author. E-mail address: [email protected] (Z. Wu).
(1)
where A is the 7eld amplitude of the beam waist center, !0 is the beam waist, and =
2 n ;
(2a)
2 z ; !(z) = !0 1 + f 2 f f2 R(z) = z 1 + =z+ ; z z f=
n !02 ;
(2b) (2c) (2d)
where R(z) is the radius of curvature of the wave front, !(z) is the spot size which is equal to the distance in the transverse direction at which the 7eld amplitude decays to 1=e of its maximum value, n is the refractive index of the medium (in this paper, light propagation in vacuum is considered, i.e., n = 1), is the light wavelength in vacuum. Fig. 1 describes a nonnormal incidence of a Gaussian beam on a Fabry–Perot etalon, where M is the measured surface. When a Gaussian beam incident on a Fabry–Perot etalon with an angle of , the electric 7eld of the 7rst transmitted
0030-3992/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 0 2 ) 0 0 1 1 5 - 9
2
Z. Wu et al. / Optics & Laser Technology 35 (2003) 1 – 4
On the analogy of this, the electric 7eld of the (m + 1)th transmitted beam can be given by (x − mX )2 + y2 !0 exp − Em (x; y; zm ) = k(r1 r2 )m !(zm ) !2 (zm ) zm ×exp −i zm − arctg f [(x − mX )2 + y2 ] ×exp −i ; (8) 2R(zm ) where zm = z0 + 2md cos :
(9)
Thus, the total transmitted electric 7eld can be written as ∞ Em (x; y; zm ): (10) Et = m=0
Using the above expression, the total transmitted light intensity It can be derived as ∞ !2 (r1 r2 )2m 2 0 It = Et Et∗ = k 2 ! (zm ) Fig. 1. Schematic of nonnormal incidence of a Gaussian beam on a Fabry–Perot etalon.
+ k2
beam can be written as E0 = kE(x; y; z0 );
(3)
where r1 and r2 are the reJection coeAcients of the etalon at the input and output sides, respectively. Compared with the 7rst transmitted beam, x coordinate of the second transmitted beam is shifted by a distance X and X can be given by (5)
The traveled distance of the second transmitted beam in the z direction is z1 = z0 + 2d cos :
∞
2[(x − mX )2 + y2 ] !2 (zm ) ∞
(r1 r2 )m+n
m=0 n=0(m=n)
where z0 is the travelled distance of the 7rst transmitted beam, and
(4) k = A (1 − r12 )(1 − r22 );
X = 2d sin :
×exp −
m=0
(6)
Then the electric 7eld of the second transmitted beam is given by !0 (x − X )2 + y2 E1 (x; y; z1 ) = k(r1 r2 ) exp − !(z1 ) !2 (z1 ) z1 ×exp −i z1 − arctg f [(x − X )2 + y2 ] ×exp −i : (7) 2R(z1 )
!02 !(zm )!(zn )
(x − nX )2 + y2 (x − mX )2 + y2 exp − ×exp − !2 (zm ) !2 (zn ) zn zm ×cos (zn − zm ) + arctg − arctg f f (x − nX )2 + y2 (x − mX )2 + y2 + − : (11) 2 R(zn ) R(zm ) From Eq. (11), the shape of the total transmitted beam can be obtained and the characteristics of the total transmitted beam can be investigated. 3. Results and discussions Before carrying out following calculations, it should be pointed out that the selected values of the Fabry–Perot etalon length d have slight diLerences for diLerent input angle in order to satisfy the resonant condition 2d cos = k (k take a positive integer). In Fig. 2, the energy pro7le of the total transmitted beam has been plotted for (a) = 3◦ , d = 0:050068676 m, (b) =1◦ , d=0:050007675 m, (c) =0:5◦ , d=0:050001963 m, and (d) = 0:25◦ , d = 0:050000535 m. Other data used in calculations are: r12 =r22 =0:7, =632:8 nm, A=1, z0 =0:2 m, !0 = 1 mm. From this diagram, it can be seen that for a relatively large input angle (see Fig. 2(a)), the total transmitted beam is spatially separated, which can be explained as
Z. Wu et al. / Optics & Laser Technology 35 (2003) 1 – 4 0.1
3
0.3
Intensity
Intensity
0.2
0.05
0.1
0 -10
-5
0
5
10
(a)
15 20 x (mm)
25
30
35
0
40
(b)
0
5
10
5
10
x (mm) 0.5
Intensity
Intensity
0.1
0.05
0
-5
(c)
-5
0
5 x (mm)
0.25
0
10
-5
0
(d)
x (mm)
Fig. 2. Energy pro7le of the total transmitted beam for the input angle equal to: (a) 3◦ ; (b) 1◦ ; (c) 0:5◦ ; and (d) 0:25◦ , respectively. 1.0
0.8
Peak intensity
that the successive transmitted beams (i.e., E0 ; E1 ; : : : ; Em ) have been spatially separated and are not interfering appreciably with each other. For a relatively small input angle (see Fig. 2(b) – (d)), although the total transmitted beam is also a single spot, its energy pro7le has been distorted, its peak position has been shifted (vertical lines through x = 0 have been drawn in these 7gures to clarify the shift) and its spot size has been enlarged due to the interference among the successive transmitted beams. In Fig. 3, the variation of the peak intensity of the total transmitted beam with the input angle has been given. From this diagram, it can be seen that with the increase of , the peak intensity will gradually decrease and 7nally come to a constant value. This can be explained as that with the increase of , the overlapping of the successive transmitted beams will be less, which weakens the interference among these beams and results in a smaller peak intensity. When increases to a certain degree, these successive transmitted beams have been spatially separated (see Fig. 2(a)), and the peak intensity of the total transmitted beam is equal to the peak of the 7rst transmitted beam.
0.6
0.4
0.2
0.0 0
1
2
Input angle
Fig. 3. Variation of the peak intensity of the total transmitted beam with the input angle .
Fig. 4 shows the variation of the xp (x coordinate of the peak intensity of the total transmitted beam) with the input angle . From this diagram, it can be observed that with the increase of , xp gradually increases at 7rst, after passing its maximum, xp decreases and 7nally tends to zero. This can
4
Z. Wu et al. / Optics & Laser Technology 35 (2003) 1 – 4 0.5
xp (mm)
0.4
0.3
0.2
0.1
0.0 0
1
2
Input angle
Fig. 4. InJuence of the input angle on the xp (x coordinate of the peak intensity of the total transmitted beam).
Transmitted beam FWHM (mm)
3.0
2.5
2.0
1.5
However, a rapid decline of the FWHM from certain (∼ 1◦ ) can be found. It can be explained as that, in the vicinity of the determined , the energy pro7le of the total transmitted beam exhibits ripples (see Fig. 2(b)). During the calculations, if the 7rst minimum value (between the 7rst peak and the second peak in Fig. 2(b)) is bigger than a half of the peak intensity, the selected value of the FWHM will be extended to the successive peaks; if the 7rst minimum value is smaller than a half of the peak intensity, the selected value of the FWHM will be con7ned to the 7rst peak. Because the determined is a critical point of above two cases, a slight increase from the determined will result in a sharp decline of the FWHM. Finally, it should be pointed out that the Fabry–Perot etalon length d is always selected to satisfy the resonant condition during above calculations. However, this is diAcult to realize in practice. Initial calculations show that even a negligible deviation of d from the resonant condition (for example the selected values of d in this paper is changed in the 6th to 9th decimal places) will lead to the decrease of the peak intensity, the xp , and the spot size of the total transmitted beam due to the incomplete interference among the successive transmitted beams induced by deviating the resonant condition, and the further detailed works have been carrying on.
1.0 0
1
2
Input angle
Fig. 5. Dependence of the FWHM of the total transmitted beam on the input angle .
be explained as that, with the increase of , the shift distance X between two successive transmitted beams will increase, which may contribute to a larger xp . However, a larger X will weaken the interference among these successive transmitted beams, which may contribute to a smaller xp after taking into account the fact that the successive transmitted beams Em have smaller electric amplitudes with the increase of the mth. The joint reaction of the above two eLects results in the phenomena showed in this diagram. For a relatively large , the total transmitted beam has been spatially separated and the peak coordinate x = 0 of the 7rst transmitted beam is the xp of the total transmitted beam. In Fig. 5, the inJuence of on the spot size of the total transmitted beam has been shown. Because the energy pro7le of the total transmitted beam is an irregular distribution and even is split into several spatially separated light spots for a relatively large (see Fig. 2(a)), the intensity full-width at half-maximum (FWHM) of the center spot of the total transmitted beam has been used to describe the spot size of the total transmitted beam. From this diagram, the similar tendency to Fig. 4 can be observed, and the explanations to Fig. 4 can also be used to describe this diagram.
Acknowledgements The authors acknowledge the support from the Foundation for University Key Teacher by the National Ministry of Education of China, and the Commission of Science and Technology of Chongqing City of China. Moreover, the authors would like to express thanks to the referee for the helpful suggestions. References [1] Yuan S, Man W, Yu J, Gao J. Time delay properties of a Fabry–Perot interferometer. Chin Phys Lett 2001;18(3):364–6. [2] Lee J, Lee H, Hahn J. Complex traversal time for optical pulse transmission in a Fabry–Perot cavity. J Opt Soc Am B 2000;17(3): 401–6. [3] Wu Z, Xia G, Chen J, Liu L. InJuence of a Fabry–Perot etalon on a chirped optical pulse. Opt Laser Technol 2001;33(7):471–3. [4] Lawrence M, Willke B, Husman M, Gustafson E, Byer R. Dynamic response of a Fabry–Perot interferometer. J Opt Soc Am B 1999;16(4):523–32. [5] Wu Z, Xia G, Chen J. Computer model of time response of Fabry–Perot etalon 7lters to short optical pulses. J Opt Commun 1997;18(1):19–23. [6] Zhou X, Lu C, Shum P, Shalaby H, Cheng T, Ye P. A performance analysis of an all-optical clock extraction circuit based on Fabry–Perot 7lter. J Lightwave Technol 2001;19(5):603–13. [7] Zhou B, Gao Y, Chen T, Chen J. Principles of laser, 4th ed. Beijing: National Defence Industry Press, 2000. p. 69 –71.
www.elsevier.com/locate/optlastec
Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon Zhengmao Wu∗ , Guangqiong Xia, Hanqing Zhou, Jianwei Wu, Mulin Liu Department of Physics, Southwest Normal University, Chongqing 400715, China Received 22 March 2002; received in revised form 8 July 2002; accepted 29 July 2002
Abstract The transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon has been investigated. After deriving the expression of the intensity of the transmitted beam, several key characteristics of the transmitted beam have been speci7ed. The results show that the peak intensity of the transmitted beam decreases with the increase of the input angle, the position of the peak intensity of the transmitted beam is shifted, and the spot size of the transmitted beam is changed. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fabry–Perot etalon; Gaussian beam; Nonnormal incidence
(x2 + y2 ) ; exp −i 2R(z)
1. Introduction Fabry–Perot etalon has been intensively investigated [1–6] because of its wide applications in spectroscopy, laser resonators, optical communications, and so on. In most studies, in order to simplify the physical treatment, the laser beam is usually treated as a plane wave and normal incidence is supposed when its transmission through a Fabry–Perot etalon is considered [1–5]. However, in fact, a laser beam is Hermit-Gaussian in general and Gaussian in its fundamental mode. Also, normal incidence is diAcult to realize in practice. Based on the above considerations, the transmission of a Gaussian beam nonnormally through a Fabry–Perot etalon has been investigated in this paper. 2. Theory The equation that describes the propagation of a Gaussian beam along z direction can be expressed as [7] 2 !0 x + y2 E(x; y; z) = A exp − 2 exp !(z) ! (z) z −i z − arctg f ∗
Corresponding author. E-mail address: [email protected] (Z. Wu).
(1)
where A is the 7eld amplitude of the beam waist center, !0 is the beam waist, and =
2 n ;
(2a)
2 z ; !(z) = !0 1 + f 2 f f2 R(z) = z 1 + =z+ ; z z f=
n !02 ;
(2b) (2c) (2d)
where R(z) is the radius of curvature of the wave front, !(z) is the spot size which is equal to the distance in the transverse direction at which the 7eld amplitude decays to 1=e of its maximum value, n is the refractive index of the medium (in this paper, light propagation in vacuum is considered, i.e., n = 1), is the light wavelength in vacuum. Fig. 1 describes a nonnormal incidence of a Gaussian beam on a Fabry–Perot etalon, where M is the measured surface. When a Gaussian beam incident on a Fabry–Perot etalon with an angle of , the electric 7eld of the 7rst transmitted
0030-3992/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 0 2 ) 0 0 1 1 5 - 9
2
Z. Wu et al. / Optics & Laser Technology 35 (2003) 1 – 4
On the analogy of this, the electric 7eld of the (m + 1)th transmitted beam can be given by (x − mX )2 + y2 !0 exp − Em (x; y; zm ) = k(r1 r2 )m !(zm ) !2 (zm ) zm ×exp −i zm − arctg f [(x − mX )2 + y2 ] ×exp −i ; (8) 2R(zm ) where zm = z0 + 2md cos :
(9)
Thus, the total transmitted electric 7eld can be written as ∞ Em (x; y; zm ): (10) Et = m=0
Using the above expression, the total transmitted light intensity It can be derived as ∞ !2 (r1 r2 )2m 2 0 It = Et Et∗ = k 2 ! (zm ) Fig. 1. Schematic of nonnormal incidence of a Gaussian beam on a Fabry–Perot etalon.
+ k2
beam can be written as E0 = kE(x; y; z0 );
(3)
where r1 and r2 are the reJection coeAcients of the etalon at the input and output sides, respectively. Compared with the 7rst transmitted beam, x coordinate of the second transmitted beam is shifted by a distance X and X can be given by (5)
The traveled distance of the second transmitted beam in the z direction is z1 = z0 + 2d cos :
∞
2[(x − mX )2 + y2 ] !2 (zm ) ∞
(r1 r2 )m+n
m=0 n=0(m=n)
where z0 is the travelled distance of the 7rst transmitted beam, and
(4) k = A (1 − r12 )(1 − r22 );
X = 2d sin :
×exp −
m=0
(6)
Then the electric 7eld of the second transmitted beam is given by !0 (x − X )2 + y2 E1 (x; y; z1 ) = k(r1 r2 ) exp − !(z1 ) !2 (z1 ) z1 ×exp −i z1 − arctg f [(x − X )2 + y2 ] ×exp −i : (7) 2R(z1 )
!02 !(zm )!(zn )
(x − nX )2 + y2 (x − mX )2 + y2 exp − ×exp − !2 (zm ) !2 (zn ) zn zm ×cos (zn − zm ) + arctg − arctg f f (x − nX )2 + y2 (x − mX )2 + y2 + − : (11) 2 R(zn ) R(zm ) From Eq. (11), the shape of the total transmitted beam can be obtained and the characteristics of the total transmitted beam can be investigated. 3. Results and discussions Before carrying out following calculations, it should be pointed out that the selected values of the Fabry–Perot etalon length d have slight diLerences for diLerent input angle in order to satisfy the resonant condition 2d cos = k (k take a positive integer). In Fig. 2, the energy pro7le of the total transmitted beam has been plotted for (a) = 3◦ , d = 0:050068676 m, (b) =1◦ , d=0:050007675 m, (c) =0:5◦ , d=0:050001963 m, and (d) = 0:25◦ , d = 0:050000535 m. Other data used in calculations are: r12 =r22 =0:7, =632:8 nm, A=1, z0 =0:2 m, !0 = 1 mm. From this diagram, it can be seen that for a relatively large input angle (see Fig. 2(a)), the total transmitted beam is spatially separated, which can be explained as
Z. Wu et al. / Optics & Laser Technology 35 (2003) 1 – 4 0.1
3
0.3
Intensity
Intensity
0.2
0.05
0.1
0 -10
-5
0
5
10
(a)
15 20 x (mm)
25
30
35
0
40
(b)
0
5
10
5
10
x (mm) 0.5
Intensity
Intensity
0.1
0.05
0
-5
(c)
-5
0
5 x (mm)
0.25
0
10
-5
0
(d)
x (mm)
Fig. 2. Energy pro7le of the total transmitted beam for the input angle equal to: (a) 3◦ ; (b) 1◦ ; (c) 0:5◦ ; and (d) 0:25◦ , respectively. 1.0
0.8
Peak intensity
that the successive transmitted beams (i.e., E0 ; E1 ; : : : ; Em ) have been spatially separated and are not interfering appreciably with each other. For a relatively small input angle (see Fig. 2(b) – (d)), although the total transmitted beam is also a single spot, its energy pro7le has been distorted, its peak position has been shifted (vertical lines through x = 0 have been drawn in these 7gures to clarify the shift) and its spot size has been enlarged due to the interference among the successive transmitted beams. In Fig. 3, the variation of the peak intensity of the total transmitted beam with the input angle has been given. From this diagram, it can be seen that with the increase of , the peak intensity will gradually decrease and 7nally come to a constant value. This can be explained as that with the increase of , the overlapping of the successive transmitted beams will be less, which weakens the interference among these beams and results in a smaller peak intensity. When increases to a certain degree, these successive transmitted beams have been spatially separated (see Fig. 2(a)), and the peak intensity of the total transmitted beam is equal to the peak of the 7rst transmitted beam.
0.6
0.4
0.2
0.0 0
1
2
Input angle
Fig. 3. Variation of the peak intensity of the total transmitted beam with the input angle .
Fig. 4 shows the variation of the xp (x coordinate of the peak intensity of the total transmitted beam) with the input angle . From this diagram, it can be observed that with the increase of , xp gradually increases at 7rst, after passing its maximum, xp decreases and 7nally tends to zero. This can
4
Z. Wu et al. / Optics & Laser Technology 35 (2003) 1 – 4 0.5
xp (mm)
0.4
0.3
0.2
0.1
0.0 0
1
2
Input angle
Fig. 4. InJuence of the input angle on the xp (x coordinate of the peak intensity of the total transmitted beam).
Transmitted beam FWHM (mm)
3.0
2.5
2.0
1.5
However, a rapid decline of the FWHM from certain (∼ 1◦ ) can be found. It can be explained as that, in the vicinity of the determined , the energy pro7le of the total transmitted beam exhibits ripples (see Fig. 2(b)). During the calculations, if the 7rst minimum value (between the 7rst peak and the second peak in Fig. 2(b)) is bigger than a half of the peak intensity, the selected value of the FWHM will be extended to the successive peaks; if the 7rst minimum value is smaller than a half of the peak intensity, the selected value of the FWHM will be con7ned to the 7rst peak. Because the determined is a critical point of above two cases, a slight increase from the determined will result in a sharp decline of the FWHM. Finally, it should be pointed out that the Fabry–Perot etalon length d is always selected to satisfy the resonant condition during above calculations. However, this is diAcult to realize in practice. Initial calculations show that even a negligible deviation of d from the resonant condition (for example the selected values of d in this paper is changed in the 6th to 9th decimal places) will lead to the decrease of the peak intensity, the xp , and the spot size of the total transmitted beam due to the incomplete interference among the successive transmitted beams induced by deviating the resonant condition, and the further detailed works have been carrying on.
1.0 0
1
2
Input angle
Fig. 5. Dependence of the FWHM of the total transmitted beam on the input angle .
be explained as that, with the increase of , the shift distance X between two successive transmitted beams will increase, which may contribute to a larger xp . However, a larger X will weaken the interference among these successive transmitted beams, which may contribute to a smaller xp after taking into account the fact that the successive transmitted beams Em have smaller electric amplitudes with the increase of the mth. The joint reaction of the above two eLects results in the phenomena showed in this diagram. For a relatively large , the total transmitted beam has been spatially separated and the peak coordinate x = 0 of the 7rst transmitted beam is the xp of the total transmitted beam. In Fig. 5, the inJuence of on the spot size of the total transmitted beam has been shown. Because the energy pro7le of the total transmitted beam is an irregular distribution and even is split into several spatially separated light spots for a relatively large (see Fig. 2(a)), the intensity full-width at half-maximum (FWHM) of the center spot of the total transmitted beam has been used to describe the spot size of the total transmitted beam. From this diagram, the similar tendency to Fig. 4 can be observed, and the explanations to Fig. 4 can also be used to describe this diagram.
Acknowledgements The authors acknowledge the support from the Foundation for University Key Teacher by the National Ministry of Education of China, and the Commission of Science and Technology of Chongqing City of China. Moreover, the authors would like to express thanks to the referee for the helpful suggestions. References [1] Yuan S, Man W, Yu J, Gao J. Time delay properties of a Fabry–Perot interferometer. Chin Phys Lett 2001;18(3):364–6. [2] Lee J, Lee H, Hahn J. Complex traversal time for optical pulse transmission in a Fabry–Perot cavity. J Opt Soc Am B 2000;17(3): 401–6. [3] Wu Z, Xia G, Chen J, Liu L. InJuence of a Fabry–Perot etalon on a chirped optical pulse. Opt Laser Technol 2001;33(7):471–3. [4] Lawrence M, Willke B, Husman M, Gustafson E, Byer R. Dynamic response of a Fabry–Perot interferometer. J Opt Soc Am B 1999;16(4):523–32. [5] Wu Z, Xia G, Chen J. Computer model of time response of Fabry–Perot etalon 7lters to short optical pulses. J Opt Commun 1997;18(1):19–23. [6] Zhou X, Lu C, Shum P, Shalaby H, Cheng T, Ye P. A performance analysis of an all-optical clock extraction circuit based on Fabry–Perot 7lter. J Lightwave Technol 2001;19(5):603–13. [7] Zhou B, Gao Y, Chen T, Chen J. Principles of laser, 4th ed. Beijing: National Defence Industry Press, 2000. p. 69 –71.