- Email: [email protected]
Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon: a nonresonant case
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Optics & Laser Technology 35 (2003) 123 – 126 www.elsevier.com/locate/optlastec
Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon: a nonresonant case Zhengmao Wu∗ , Guangqiong Xia Department of Physics, Southwest Normal University, Chongqing 400715, People’s Republic of China Received 15 August 2002; received in revised form 18 October 2002; accepted 27 October 2002
Abstract Under the nonresonant case where the carrier frequency of a Gaussian beam deviates from the resonant frequency of a Fabry–Perot etalon, the transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon has been investigated. The results show that under the nonresonant case, variations of the peak intensity, the position of the peak intensity and the spot size of the transmitted beam with the input angle behave di6erently and even with a reversed tendency compared with those obtained under the resonant case. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fabry–Perot etalon; Gaussian beam; Nonnormal incidence; Nonresonant case
1. Introduction
It can be expressed as [1]
Since Fabry–Perot etalons are used commonly in conjunction with laser beams, the transmission of a Gaussian beam through a Fabry–Perot etalon has received considerable attention [1,2]. In an earlier paper, the transmission of a Gaussian beam nonnormally through a Fabry–Perot etalon has been investigated under the resonant case that the carrier frequency of the Gaussian beam is supposed to conform to the resonant frequency of the etalon [1]. However, the resonant case is di@cult to attain perfectly in practice because the etalon length is easily to be a6ected by external causes such as temperature, etc. And even a slight change of the etalon length will make the resonant frequency of the etalon deviate from the carrier frequency of the Gaussian beam, which initiates the nonresonant case. In this paper, after focusing on the nonresonant case, the transmission of a Gaussian beam nonnormally through a Fabry–Perot etalon has been investigated.
It = Et Et∗ = k 2
∞
(r1 r2 )2m
m=0
!02 2 ! (zm )
2[(x − mX )2 + y2 ] × exp − !2 (zm ) + k2
∞
∞
(r1 r2 )m+n
m=0 n=0(m=n)
!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 + − ; (1) 2 R(zn ) R(zm )
2. Theory
where Et is the total transmitted electric Feld, r1 and r2 are the reGection coe@cients of the etalon at the input and output mirrors, respectively, !0 is the beam waist, and
For a Gaussian beam passing through a Fabry–Perot etalon with an angle of , the total transmitted light intensity
Et =
∗
Corresponding author. E-mail address: [email protected] (Z. Wu).
∞
Em (x; y; zm );
(2a)
m=0
k =A
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 6 2 - 7
(1 − r12 )(1 − r22 );
(2b)
=
Z. Wu, G. Xia / Optics & Laser Technology 35 (2003) 123 – 126
2n ;
(2c)
2 z !(z) = !0 1 + ; f
2 f f2 ; R(z) = z 1 + =z+ z z f=
n!02 ;
(2d) 3. Results and discussions (2e) (2f)
X = 2d sin ;
(2g)
zm = z0 + 2md cos ;
(2h)
where Em is the electric Feld of the (m + 1)th transmitted beam, A is the Feld amplitude of the beam waist center, !(z) is the spot size which is equal to the distance in the transverse direction at which the Feld amplitude decays to 1=e of its maximum value, R(z) is the radius of curvature of the wave front, n is the refractive index of the medium (in this paper, light propagation in vacuum is considered, i.e., n = 1), is the carrier wavelength, d is the etalon length, X is the x coordinate shift between two transmitted beams in succession, z0 is the traveled distance of the Frst transmitted beam, zm is the traveled distance of the (m+1)th transmitted beam. From Eq. (1), it can be seen that the total transmitted light intensity It is composed of two terms: the Frst describes the successive transmitted beams; the second represents the interference among these successive transmitted beams, which is of most importance to the results obtained in this paper. For a Gaussian beam with the carrier wavelength passing through a Fabry–Perot etalon at an angle of , the resonant condition is 2d cos = k
(k is a positive integer):
Based on the above analysis, the shape and the characteristics of the total transmitted beam can be speciFed under di6erent nonresonant cases.
In Fig. 1, under various , the energy proFle of the total transmitted beam has been plotted for (a) = 1◦ , and (b) = 0:25◦ , where curves a, b, c and d are for = 0; =8; =4; =2, respectively. 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. 1(a)), even though some distinctions (such as shape, peak intensity, etc.) among these curves can be found, the inGuence of on the energy proFle of the total transmitted beam is also small, which is because of the spatial overlapping of the successive transmitted beams being small and so the interference among these beams is weak. For a relatively small input angle (see Fig. 1(b)), the e6ect
0.1
Intensity
124
c
d 0 -5
5
10
5
10
X (mm) 0.5
a
(4)
Based on this expression, it can be seen that for a given and , every d corresponds to a Fxed . Theoretically, the etalon length d can be selected from zero to inFnity, and the corresponding varies only between zero and =2 after taking into account the quantity k. Therefore, it may be more reasonable to investigate the inGuence of the on the transmission of a Gaussian beam passing nonnormally through a Fabry–Perot etalon. It should be pointed out that, in the following discussions, as the variations of the characteristics of the total transmitted beam with the input angle are investigated for a given in this paper, the selected values of d in the following calculations have slight di6erences in order to satisfy Eq. (4).
Intensity
(0 6 6 =2):
0
(3)
As mentioned above, the etalon length d is di@cult to be Fxed accurately and a slight change of the d always exists in practice. Under this circumstance, one has 2d cos = k +
a b
0.25
b
c d 0 -5
0 X (mm)
Fig. 1. Energy proFle of the total transmitted beam for the input angle equal to (a) 1◦ and (b) 0:25◦ , where curves a, b, c and d are for = 0; =8; =4; =2, respectively.
Z. Wu, G. Xia / Optics & Laser Technology 35 (2003) 123 – 126 1
125
0.5 a 0.4 b
0.2 a
Xp (mm)
Peak Intensity
0.3
0.5
0.1
c
0 -0.1
b
-0.2
c
d
d -0.3
0 0
0.5
1 Input Angle
1.5
2
0
0.5
1
1.5
2
Input Angle
Fig. 2. Variation of the peak intensity of the total transmitted beam with the input angle for di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively.
Fig. 3. xp (x coordinate of the peak intensity of the total transmitted beam) vs. the input angle under di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively.
of the is obvious due to the fact that there exists an appreciable spatial overlapping of the successive transmitted beams and the interference among these beams is violent. For =0 (see curve a, which is related to the resonant case), the interference among these successive transmitted beams is constructive and so a highest peak intensity can be observed. With the increase of from 0 to =2 (see curve d, which is related to the anti-resonant case), the interference initiates a change from constructive, partially constructive, partially destructive and Fnally to destructive, which results in the decrease of the peak intensity of the total transmitted beam. In Fig. 2, the variation of the peak intensity of the total transmitted beam with the input angle has been given for di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively. For = 0 (see curve a), with the increase of , the peak intensity will gradually decrease and Fnally come to a constant value, which has been explained as that with the increase of , the overlapping of the successive transmitted beams will be less, which weakens the constructive 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, which makes the interference among these beams disappear (i.e., the second term of Eq. (1) tends to zero), and the peak intensity of the total transmitted beam is equal to the peak of the Frst transmitted beam. However, for = =2 (see curve d), there exists an opposite change tendency with the increase of . This can been explained as that for = =2, the destructive interference leads to a lower peak intensity of the total transmitted beam than that of the Frst transmitted beam; with the increase of , the less overlapping of the successive transmitted beams weakens the destructive interference, which results in an increase of the power intensity. When increases to a cer-
tain degree, these successive transmitted beams have been spatially separated and the interference e6ect gradually disappears, and the peak intensity of the total transmitted beam is equal to that of the Frst transmitted beam just as that for = 0. Fig. 3 shows the variation of xp (x coordinate of the peak intensity of the total transmitted beam) with the input angle for di6erent , where curves a, b, c and d are for =0; =8; =4; =2, respectively. For =0 (see curve a), with the increase of , xp gradually increases along the +x direction at Frst due to the constructive interference, after passing its maximum, xp decreases and Fnally tends to zero, which has been explained in Ref. [1]. However, for = =2 (see curve d), compared with that obtained for = 0, an opposite shift of xp along the −x direction can be observed, which is brought about by the destructive interference among the successive transmitted beams. For = =2, with the increase of , xp gradually decreases along the −x direction at Frst, after passing its minimum, xp increases and Fnally tends to zero. This can be explained as with the increase of , the shift distance X between two successive transmitted beams will increase, and then xp will have a tendency of shifting a larger value towards the −x direction due to the destructive interference among the successive transmitted beams; in the meantime, a larger X will result in a less overlapping of the successive transmitted beams and then weaken the destructive interference e6ect among these successive transmitted beams, which make xp have a shift tendency to zero 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 e6ects results in the phenomena shown in curve d. For a relatively large , the total transmitted beam has been spatially separated and the peak coordinate x = 0 of the Frst transmitted beam is the xp of the total transmitted beam.
126
Z. Wu, G. Xia / Optics & Laser Technology 35 (2003) 123 – 126 3
a
Transmitted Beam FWHM (mm)
2.5 b 2 c 1.5
d
1
0.5 0
0.5
1 Input Angle
1.5
2
Fig. 4. FWHM of the total transmitted beam against the input angle for di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively.
In Fig. 4, the inGuence of on the spot size of the total transmitted beam has been shown for di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively. The intensity full-width-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, a similar tendency to Fig. 3 can be observed. For = 0 (see curve a), with the increase of , the spot size gradually increases at Frst, after passing its maximum, decreases and Fnally tends to a constant (is equal to the intensity FWHM of the Frst transmitted beam). However, for = =2 (see curve d), there appears a di6erent case. With the increase of , the spot size gradually decreases at Frst, after passing its minimum, increases and Fnally tends to the same constant as that for = 0. This can be explained as, for = =2, the destructive interference causes a smaller spot size of the total transmitted beam than that of the Frst transmitted beam, with the increase of , the shift distance X between two successive transmitted beams will increase, and the destructive interference among the successive transmitted beams may contribute to a smaller spot size. Meanwhile, less overlapping of the successive transmitted beams due
to a larger X will weaken the destructive interference e6ect and make the spot size near the spot size of the Frst transmitted beam after taking into account the successive transmitted beams Em being smaller electric amplitudes with the increase of the mth. The joint reaction of the above two effects leads to the changed curve shown in curve d. For a relatively large , the total transmitted beam has been spatially separated and the spot size of the Frst transmitted beam is that of the total transmitted beam. By the way, it should be pointed out that for curves a, b and c, a rapid decline of the FWHM from certain can be observed. This is because the energy proFle of the total transmitted beam is an irregular distribution and is even split into several spatially separated light spots for a relatively large , and then a special method [1] has been used to determine the FWHM, which induces the sharp decline. Finally, it should be pointed out that in practice, one usually encounters the case that a Gaussian beam passes through a Fabry–Perot etalon at a Fxed and the etalon length d may vary due to external factors. Under this circumstance, the transmission of a Gaussian beam can also be speciFed from the above diagrams after taking into consideration the fact that the cases for di6erent have been shown in these diagrams. Acknowledgements The authors acknowledge the support from the National Ministry of Education of China, and the Commission of Science and Technology of Chongqing City of China. The authors acknowledge the referee to Ref. [1] because this work is partly drawn inspiration from his review report. References [1] Wu Z, Xia G, Zhou H, Wu J, Liu M. Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon. Opt Laser Technol 2003; 35(1): 1–4. [2] Moreno F, Gonzalez F. Transmission of a Gaussian beam of low divergence through a high-Fnesse Fabry–Perot device. J Opt Soc Am A 1992;9(12):2173–5.
Optics & Laser Technology 35 (2003) 123 – 126 www.elsevier.com/locate/optlastec
Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon: a nonresonant case Zhengmao Wu∗ , Guangqiong Xia Department of Physics, Southwest Normal University, Chongqing 400715, People’s Republic of China Received 15 August 2002; received in revised form 18 October 2002; accepted 27 October 2002
Abstract Under the nonresonant case where the carrier frequency of a Gaussian beam deviates from the resonant frequency of a Fabry–Perot etalon, the transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon has been investigated. The results show that under the nonresonant case, variations of the peak intensity, the position of the peak intensity and the spot size of the transmitted beam with the input angle behave di6erently and even with a reversed tendency compared with those obtained under the resonant case. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fabry–Perot etalon; Gaussian beam; Nonnormal incidence; Nonresonant case
1. Introduction
It can be expressed as [1]
Since Fabry–Perot etalons are used commonly in conjunction with laser beams, the transmission of a Gaussian beam through a Fabry–Perot etalon has received considerable attention [1,2]. In an earlier paper, the transmission of a Gaussian beam nonnormally through a Fabry–Perot etalon has been investigated under the resonant case that the carrier frequency of the Gaussian beam is supposed to conform to the resonant frequency of the etalon [1]. However, the resonant case is di@cult to attain perfectly in practice because the etalon length is easily to be a6ected by external causes such as temperature, etc. And even a slight change of the etalon length will make the resonant frequency of the etalon deviate from the carrier frequency of the Gaussian beam, which initiates the nonresonant case. In this paper, after focusing on the nonresonant case, the transmission of a Gaussian beam nonnormally through a Fabry–Perot etalon has been investigated.
It = Et Et∗ = k 2
∞
(r1 r2 )2m
m=0
!02 2 ! (zm )
2[(x − mX )2 + y2 ] × exp − !2 (zm ) + k2
∞
∞
(r1 r2 )m+n
m=0 n=0(m=n)
!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 + − ; (1) 2 R(zn ) R(zm )
2. Theory
where Et is the total transmitted electric Feld, r1 and r2 are the reGection coe@cients of the etalon at the input and output mirrors, respectively, !0 is the beam waist, and
For a Gaussian beam passing through a Fabry–Perot etalon with an angle of , the total transmitted light intensity
Et =
∗
Corresponding author. E-mail address: [email protected] (Z. Wu).
∞
Em (x; y; zm );
(2a)
m=0
k =A
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 6 2 - 7
(1 − r12 )(1 − r22 );
(2b)
=
Z. Wu, G. Xia / Optics & Laser Technology 35 (2003) 123 – 126
2n ;
(2c)
2 z !(z) = !0 1 + ; f
2 f f2 ; R(z) = z 1 + =z+ z z f=
n!02 ;
(2d) 3. Results and discussions (2e) (2f)
X = 2d sin ;
(2g)
zm = z0 + 2md cos ;
(2h)
where Em is the electric Feld of the (m + 1)th transmitted beam, A is the Feld amplitude of the beam waist center, !(z) is the spot size which is equal to the distance in the transverse direction at which the Feld amplitude decays to 1=e of its maximum value, R(z) is the radius of curvature of the wave front, n is the refractive index of the medium (in this paper, light propagation in vacuum is considered, i.e., n = 1), is the carrier wavelength, d is the etalon length, X is the x coordinate shift between two transmitted beams in succession, z0 is the traveled distance of the Frst transmitted beam, zm is the traveled distance of the (m+1)th transmitted beam. From Eq. (1), it can be seen that the total transmitted light intensity It is composed of two terms: the Frst describes the successive transmitted beams; the second represents the interference among these successive transmitted beams, which is of most importance to the results obtained in this paper. For a Gaussian beam with the carrier wavelength passing through a Fabry–Perot etalon at an angle of , the resonant condition is 2d cos = k
(k is a positive integer):
Based on the above analysis, the shape and the characteristics of the total transmitted beam can be speciFed under di6erent nonresonant cases.
In Fig. 1, under various , the energy proFle of the total transmitted beam has been plotted for (a) = 1◦ , and (b) = 0:25◦ , where curves a, b, c and d are for = 0; =8; =4; =2, respectively. 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. 1(a)), even though some distinctions (such as shape, peak intensity, etc.) among these curves can be found, the inGuence of on the energy proFle of the total transmitted beam is also small, which is because of the spatial overlapping of the successive transmitted beams being small and so the interference among these beams is weak. For a relatively small input angle (see Fig. 1(b)), the e6ect
0.1
Intensity
124
c
d 0 -5
5
10
5
10
X (mm) 0.5
a
(4)
Based on this expression, it can be seen that for a given and , every d corresponds to a Fxed . Theoretically, the etalon length d can be selected from zero to inFnity, and the corresponding varies only between zero and =2 after taking into account the quantity k. Therefore, it may be more reasonable to investigate the inGuence of the on the transmission of a Gaussian beam passing nonnormally through a Fabry–Perot etalon. It should be pointed out that, in the following discussions, as the variations of the characteristics of the total transmitted beam with the input angle are investigated for a given in this paper, the selected values of d in the following calculations have slight di6erences in order to satisfy Eq. (4).
Intensity
(0 6 6 =2):
0
(3)
As mentioned above, the etalon length d is di@cult to be Fxed accurately and a slight change of the d always exists in practice. Under this circumstance, one has 2d cos = k +
a b
0.25
b
c d 0 -5
0 X (mm)
Fig. 1. Energy proFle of the total transmitted beam for the input angle equal to (a) 1◦ and (b) 0:25◦ , where curves a, b, c and d are for = 0; =8; =4; =2, respectively.
Z. Wu, G. Xia / Optics & Laser Technology 35 (2003) 123 – 126 1
125
0.5 a 0.4 b
0.2 a
Xp (mm)
Peak Intensity
0.3
0.5
0.1
c
0 -0.1
b
-0.2
c
d
d -0.3
0 0
0.5
1 Input Angle
1.5
2
0
0.5
1
1.5
2
Input Angle
Fig. 2. Variation of the peak intensity of the total transmitted beam with the input angle for di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively.
Fig. 3. xp (x coordinate of the peak intensity of the total transmitted beam) vs. the input angle under di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively.
of the is obvious due to the fact that there exists an appreciable spatial overlapping of the successive transmitted beams and the interference among these beams is violent. For =0 (see curve a, which is related to the resonant case), the interference among these successive transmitted beams is constructive and so a highest peak intensity can be observed. With the increase of from 0 to =2 (see curve d, which is related to the anti-resonant case), the interference initiates a change from constructive, partially constructive, partially destructive and Fnally to destructive, which results in the decrease of the peak intensity of the total transmitted beam. In Fig. 2, the variation of the peak intensity of the total transmitted beam with the input angle has been given for di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively. For = 0 (see curve a), with the increase of , the peak intensity will gradually decrease and Fnally come to a constant value, which has been explained as that with the increase of , the overlapping of the successive transmitted beams will be less, which weakens the constructive 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, which makes the interference among these beams disappear (i.e., the second term of Eq. (1) tends to zero), and the peak intensity of the total transmitted beam is equal to the peak of the Frst transmitted beam. However, for = =2 (see curve d), there exists an opposite change tendency with the increase of . This can been explained as that for = =2, the destructive interference leads to a lower peak intensity of the total transmitted beam than that of the Frst transmitted beam; with the increase of , the less overlapping of the successive transmitted beams weakens the destructive interference, which results in an increase of the power intensity. When increases to a cer-
tain degree, these successive transmitted beams have been spatially separated and the interference e6ect gradually disappears, and the peak intensity of the total transmitted beam is equal to that of the Frst transmitted beam just as that for = 0. Fig. 3 shows the variation of xp (x coordinate of the peak intensity of the total transmitted beam) with the input angle for di6erent , where curves a, b, c and d are for =0; =8; =4; =2, respectively. For =0 (see curve a), with the increase of , xp gradually increases along the +x direction at Frst due to the constructive interference, after passing its maximum, xp decreases and Fnally tends to zero, which has been explained in Ref. [1]. However, for = =2 (see curve d), compared with that obtained for = 0, an opposite shift of xp along the −x direction can be observed, which is brought about by the destructive interference among the successive transmitted beams. For = =2, with the increase of , xp gradually decreases along the −x direction at Frst, after passing its minimum, xp increases and Fnally tends to zero. This can be explained as with the increase of , the shift distance X between two successive transmitted beams will increase, and then xp will have a tendency of shifting a larger value towards the −x direction due to the destructive interference among the successive transmitted beams; in the meantime, a larger X will result in a less overlapping of the successive transmitted beams and then weaken the destructive interference e6ect among these successive transmitted beams, which make xp have a shift tendency to zero 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 e6ects results in the phenomena shown in curve d. For a relatively large , the total transmitted beam has been spatially separated and the peak coordinate x = 0 of the Frst transmitted beam is the xp of the total transmitted beam.
126
Z. Wu, G. Xia / Optics & Laser Technology 35 (2003) 123 – 126 3
a
Transmitted Beam FWHM (mm)
2.5 b 2 c 1.5
d
1
0.5 0
0.5
1 Input Angle
1.5
2
Fig. 4. FWHM of the total transmitted beam against the input angle for di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively.
In Fig. 4, the inGuence of on the spot size of the total transmitted beam has been shown for di6erent , where curves a, b, c and d are for = 0; =8; =4; =2, respectively. The intensity full-width-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, a similar tendency to Fig. 3 can be observed. For = 0 (see curve a), with the increase of , the spot size gradually increases at Frst, after passing its maximum, decreases and Fnally tends to a constant (is equal to the intensity FWHM of the Frst transmitted beam). However, for = =2 (see curve d), there appears a di6erent case. With the increase of , the spot size gradually decreases at Frst, after passing its minimum, increases and Fnally tends to the same constant as that for = 0. This can be explained as, for = =2, the destructive interference causes a smaller spot size of the total transmitted beam than that of the Frst transmitted beam, with the increase of , the shift distance X between two successive transmitted beams will increase, and the destructive interference among the successive transmitted beams may contribute to a smaller spot size. Meanwhile, less overlapping of the successive transmitted beams due
to a larger X will weaken the destructive interference e6ect and make the spot size near the spot size of the Frst transmitted beam after taking into account the successive transmitted beams Em being smaller electric amplitudes with the increase of the mth. The joint reaction of the above two effects leads to the changed curve shown in curve d. For a relatively large , the total transmitted beam has been spatially separated and the spot size of the Frst transmitted beam is that of the total transmitted beam. By the way, it should be pointed out that for curves a, b and c, a rapid decline of the FWHM from certain can be observed. This is because the energy proFle of the total transmitted beam is an irregular distribution and is even split into several spatially separated light spots for a relatively large , and then a special method [1] has been used to determine the FWHM, which induces the sharp decline. Finally, it should be pointed out that in practice, one usually encounters the case that a Gaussian beam passes through a Fabry–Perot etalon at a Fxed and the etalon length d may vary due to external factors. Under this circumstance, the transmission of a Gaussian beam can also be speciFed from the above diagrams after taking into consideration the fact that the cases for di6erent have been shown in these diagrams. Acknowledgements The authors acknowledge the support from the National Ministry of Education of China, and the Commission of Science and Technology of Chongqing City of China. The authors acknowledge the referee to Ref. [1] because this work is partly drawn inspiration from his review report. References [1] Wu Z, Xia G, Zhou H, Wu J, Liu M. Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon. Opt Laser Technol 2003; 35(1): 1–4. [2] Moreno F, Gonzalez F. Transmission of a Gaussian beam of low divergence through a high-Fnesse Fabry–Perot device. J Opt Soc Am A 1992;9(12):2173–5.