OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 20 — Oct. 1, 2007
  • pp: 12692–12698
« Show journal navigation

Generating a geometric mode for clarifying differences between an operator method and SU(2) wave representation

Ching-Hsu Chen and Chi-Feng Chiu  »View Author Affiliations


Optics Express, Vol. 15, Issue 20, pp. 12692-12698 (2007)
http://dx.doi.org/10.1364/OE.15.012692


View Full Text Article

Acrobat PDF (419 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We study both numerically and experimentally a specific geometric mode, named VW mode, in an end-pumped Nd:YVO4 laser with a plano-concave cavity near the 1/3-degeneracy. Three theoretical methods are used to analyze the transverse profiles, the operator method [J. Opt. Soc. Am. A 22, 1559 (2005)], the SU(2) wave representation [Phys. Rev. A 69, 053807 (2004)], and the Fox-Li approach. Some differences among them are addressed and clarified. Moreover, the generating conditions for VW mode are found peculiar and its propagation character is demonstrated. By comparing the experimental mode patterns with the three numerical results, we conclude that the field of a geometric mode in the operator method should be extended to include those of the reverse directional trajectories and the SU(2) coherent state representation is found too specific to produce the fringes within some transverse patterns.

© 2007 Optical Society of America

1. Introduction

When we apply the two wave descriptions of [5

5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004). [CrossRef]

] and [7

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

] for the W mode, M mode, and the so-called N mode (see Fig. 3(b) of [7

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

]) in this paper, respectively, nearly the same trajectories and interference fringes can be obtained from the two. For another geometric mode (see Fig. 3(a) of [7

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

]), named VW mode, the interference fringes are absent on the end mirrors of the three-fold degenerate symmetric cavity. However, we found that the fringes should appear there according to SU(2) representation. The difference of these two descriptions needs to be further clarified via observing the VW mode. In this paper, we numerically study the VW mode by using three methods, the aforementioned two theoretical representations and the Fox-Li approach based on Collins diffraction integral together with a rate equation to compare with experimentally generated VW mode. We will show that the VW mode can be generated by three pumping beams in an end-pumped Nd:YVO4 laser operated near the 1/3-degeneracy with a plano-concave cavity. By observing the transverse patterns of the VW mode, we conclude the operator method in [7

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

] can be extended so that the field of a geometric mode should include simultaneously those of the reverse directional trajectories. On the other hand, the SU(2) representation in [5

5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004). [CrossRef]

] is a special one that can not produce the fringes within some transverse patterns. In section 2, we compare the three numerical results for the VW mode. In section 3, the experimental result is compared with our simulations. The conclusions are stated in section 4.

2. The numerical results

According to ray tracing in a plano-concave cavity with three-fold degeneracy or g1g2 = 1/4, six-bounce or three round-trip rays will construct a closed trajectory. We plot a specific one that is symmetric with the optical axis in Fig. 1(a), in which z = 0 is the position of the flat end mirror. In Fig. 1(a), the initial ray is indicated starting at (x, z) = (0, 0) with a slope of 2a/L, then it is reflected back to (x, z) = (a, 0) and subsequently retraces itself after two round trips. Here L is the cavity length and x is the transverse coordinate. Using the operator method [7

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

] and choosing the transverse momentum divided by the total momentum q/k = π/180 and the position vector a = 0, we plot the mode trajectories in Fig. 1(b). This is equivalent to considering a fundamental Gaussian beam with its center at (x, z) = (0, 0) and with a slope q/k that retraces itself after three round trips. We can see that the mode trajectories in Fig. 1(b) are the same as the closed rays in Fig. 1(a), which are like overlapped V and W shapes, so that we call it the VW mode in this paper. The VW mode is similar to Fig. 3(a) of [7

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

] except that the distance between the neighboring spots on z = 0 is half of that on z = 6 cm because the plano-concave cavity is used here. In order to compare with the numerical results below, we have implemented the laser wavelength of 1064 nm, the cavity length of 6 cm, the radius of curvature of the curved mirror of 8 cm, and the fundamental Gaussian beam waist is 108 μm .

On the other hand, using the SU(2) representation [5

5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004). [CrossRef]

] according to the Schwinger representation [8

8. J. Banerji and G. S. Agarwal, “Non-linear wave packet dynamics of coherent states of various symmetry groups,” Opt. Express 5, 220–229 (1999). [CrossRef] [PubMed]

,9

9. V. Buzek and T. Quang, “Generalized coherent state for bosonic realization of SU(2) Lie algebra,” J. Opt. Soc. Am. B 6, 2447–2449 (1989). [CrossRef]

] for a family of the Hermite-Gaussian (HG) modes, the coherent state is given by,

Φn(x,y,z;τ)=1(1+τ2)n2p=0n(np)12τpΦ3p,0(HG)xyz,
(1)

where τ = exp(), Φ(HG) 3p,0(x,y,z) is HG mode with the transverse mode index 3p in x direction and p = 0, 1, 2, ..., n. We depict the trajectory of the VW mode with the total wave function being Φn(x,y,z;ϕ=π/2)+Φn(x,y,z;ϕ=-π/2) and n = 20 in Fig. 1(c). We can see that Fig. 1(c) and 1(b) have the same geometric trajectories. However, in Fig. 1(b) only two regions where two beams propagate in the same direction have the interference fringes but the fringes on the end mirrors preserve for a long distance in Fig. 1(c) which is absent in Fig. 1(b).

Fig. 1. (a). Closed ray trajectories symmetric with the optical axis in a three-fold degenerate plano-concave cavity. (b) VW mode trajectories using the multibounce Gaussian beams model in [7]. (c) VW mode trajectories using the SU(2) representation in [5]. (d) VW mode trajectories using the Collins integral together with a rate equation.

The numerical model, by using the Collins integral together with a rate equation, can be found in [12

12. C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, “Multibeam-waist modes in an end-pumped Nd-YVO4 laser,” J. Opt. Soc. Am. B 20, 1220–1226 (2003). There is lack of γ in the last term of equation (3) in this reference. [CrossRef]

]. Here, the Collins integral [13

13. S. A. Collins, “Lens-system diffraction-integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1174 (1970). [CrossRef]

] is written as

Em+1(x)=iBλwwexp(ik2L)Em+(x)exp{(iπ/Bλ)(Ax′2+Bxx′+Dx2)}dx′,
(2)

where [ABCD] is the round trip transmission matrix, E+ m(x’) and E- m+1(x) are the electric fields of the m-th and the (m+1)-th round trips, respectively, at the planes immediately after and before the gain medium; x’ and x are the corresponding transverse coordinates, λ is the wavelength of the laser, 2w is the aperture width on the reference plane that is set on the flat mirror end. To implement the integral by the Romberg method, we divided the 2 mm aperture width into 2048 segments. The used parameters of the gain medium and the cavity configuration are the same as in [12

12. C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, “Multibeam-waist modes in an end-pumped Nd-YVO4 laser,” J. Opt. Soc. Am. B 20, 1220–1226 (2003). There is lack of γ in the last term of equation (3) in this reference. [CrossRef]

].

Since the spot centers on z = 0 in Fig. 1(c) are at x = -507, 0, +507 μm, we choose the three transverse positions as the pump beam centers. When the three Gaussian pump radii are chosen 108 μm and the pump power ratio are 1: 0.92: 1, the numerical trajectory is shown in Fig. 1(d), in which the scale of x is from -1 mm to +1 mm. We found besides the fringes occurring at the crossings of beams that is similar to Fig. 1(c), they are obvious over all the trajectories in Fig. 1(d). Notice that the fringe spacing of the side spots are 60 μm which are double of the middle spot one of 30μm on z = 0. When the three spots on z = 6 cm are compared with those on z = 0, both the fringe spacing and the spot size are doubled correspondingly. The results are the same as Fig. 1(c) and match with the deduction from the picture of ray and wave optics. As shown in Fig. 1(a), the three angles indicated between two rays intersection at z = 0 are θ,2θ ,θ and those at z = 6 cm are θ/2 ,θ,θ/2. When two beams interfere by a small angle the fringe spacing is inversely proportional to this angle, which explains the data of fringe spacing.

3. Experimental setup and results

The experimental setup is shown in Fig. 2. This laser contains a 1mm-thick a cut 2.0 at% microchip crystal and an output coupler with radius of curvature of 20 cm having 10% transmission at the lasing wavelength of 1064 nm. One face of the crystal facing the pumping beam had a dichroic coating with greater than 99.8% reflection at 1064 nm and greater than 99.5% transmission at the pump wavelength of 808 nm; the other surface was made of antireflection layer at 1064 nm. The pump source was a 1-W fiber-coupled laser diode with a 200μm of core diameter and a numerical aperture of 0.22. A convergent lens with 50 mm focal length and a 20x objective lens were added after the fiber output a distance of 5 cm and 85 cm, respectively to focus the pump beam into the laser crystal. Between the two lenses, the pump beam was split into three beams by two 50/50 beam splitters and two reflective mirrors.

The pump radii were estimated to be ∼150 μm and the neighboring beam centers were separated ∼900 μm apart. After the output coupler we added a transform lens to monitor the beam profile variation with the propagation distance. The laser output was detected with a charge coupled device (CCD).

Fig. 2. Schematic diagram of the experimental setup: LD, laser diode; BS, beam splitter; M, reflective mirror; OC, output coupler; LF, line filter; TL, transform lens; other abbreviations defined in text.

When the cavity length is tuned near the 1/3-degeneration point of 15 cm and the power of the three pump beams are about 120, 70, and 100 mW, the observed VW mode patterns with the propagation distance are shown in Fig. 3(a). First, we see that the right largest 3-spots pattern has obvious fringe structure and the fringe spacing of the middle spot is half of the other two, which agree with the numerical result in the last paragraph of section 2. Second, the 3-spots patterns indicated with arrows in Fig. 3(a) have asymmetric spot sizes. This is because the two pump sizes are not completely the same, resulting from their different propagation distances before going into the gain medium. The little asymmetric pump sizes do not influence the fringe spacing within the spots. Third, we intentionally adjusted the pumping power a little asymmetric so that we can see the 3-spots patterns have asymmetric intensities. Fourth, we see the 3-spots patterns from left to right exhibit upside down twice.

In order to compare with the experimental results, we simulated the laser patterns by placing a transform lens with a focal length of 5.2 cm a distance of 10.5 cm from the curved mirror (the output coupler in experiment). This is equivalent to propagating the field solution a distance of 16.5 cm and then through the transform lens. Here we neglect the refraction from the finite thickness of the output coupler. The trajectory after the transform lens is shown in Fig. 3(b), in which the pattern variation coincides very well with the experiments. Note that here Z’ = 0 is the position of the transform lens. Due to the design of three pump beam paths, a wider transversal pump region and a longer cavity length were used in experiment. This choice does not influence the coincidence between the experiment and the simulation.

We see that the three spots on z = 0 of Fig. 1(d) were transmitted to Z’ = 7.6 cm. The intra-cavity beam trajectories in Fig. 1(d) were transformed to the respective positions between Z’ = 7.6 and 10.3 cm that follows the Gaussian lens law. Moreover, as compared with the images between Z’ = 7.6 and 10.3 cm, the imaged beam trajectories between Z’ = 6.7 and 7.6 cm shrink and are reversed along the propagation direction. According the Gaussian lens law, the inverse image comes from the objects that are located between 16.5 cm and 22.5 cm before the transform lens, where is just the position of the flat end mirror and the virtual image of the curved end mirror with respect to the position of the flat mirror. Under the situation of asymmetric pump power, our simulation reveals that the two transverse profiles on the two end mirrors in Fig. 1(d) are upside down and then they exhibit upside down twice after the transform lens, which matches with the experiment. This phenomenon can be understood that the VW mode follows the Gaussian lens law not only for their imaging positions but also the reciprocal real images.

Fig. 3. The experimental mode patterns (a) and the numerical mode profile variation (b) after the transform lens.

Furthermore, when the middle pump power is increased to surpass the side pump power, the laser can not generate the VW mode. Contrarily, when we decreased the middle pump, the VW mode patterns exist but blurred with some interference fringes. The best pumping power condition is about 1: 0.5: 1 to 1: 0.95: 1. If one side pump is blocked and the middle pump power is half of the other side pump, one side of the three spots is large and the profile variation is difficult to be identified the VW mode. All the experimental observations match with our simulations.

Finally, we discuss the extension of the operator method and the insufficiency of SU(2) representation for geometric modes. The fringes within 3-spots pattern can be obtained when the six fields of the reversal trajectories are included in the operator method. Since there are six rays leaving the flat mirror end in Fig. 1(b), the total laser field on any transverse plane within the cavity is the accumulated electric fields from the six initial rays, or equivalently both the co-propagating and counter-propagating beams should be considered. This is Eq. (46) of [7

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

] must include twelve fields, six for the rays beginning at (0, 0) with positive slope and six for negative slope. On the other hand, the 6-spots patterns in Fig. 3(a) and the corresponding patterns in Figs. 3(b) and 1(d) have fringes within them. However, in Fig. 1(c) there are no fringes for the two inner trajectories from z = 0.6 to 1.2 cm and from z = 3.2 to 4.2 cm and we know that the absence of fringe is independent of the order n. Hence the inner two spots of the 6-spots pattern implies the insufficiency of the SU(2) representation. The square root of the binomial distribution in Eq. (1) is too special to produce the fringes within all trajectories of the VW mode. Besides, the series of HG function in Eq. (5) of [5

5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004). [CrossRef]

] does not satisfy the condition of Eq. (2.13) of [9

9. V. Buzek and T. Quang, “Generalized coherent state for bosonic realization of SU(2) Lie algebra,” J. Opt. Soc. Am. B 6, 2447–2449 (1989). [CrossRef]

], in which the total numbers of bosons is constant.

4. Conclusion

We have successfully generated a specific mode, called VW mode, in an end-pumped Nd:YVO4 laser with a plano-concave cavity near the degeneration point of g1g2 = 1/4. The generating condition for the VW mode is somewhat peculiar that is the three pumping beam power must have a proper ratio. Our simulations match with the experiments very well. By observing the 3-spots pattern of the VW mode, we conclude the operator method in [7

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

] can be extended to include simultaneously those of the reverse directional trajectories. Moreover, the SU(2) coherent state representation in [5

5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004). [CrossRef]

] is insufficient because it is unable to describe the fringes within the 6-spots pattern of the VW mode.

Acknowledgments

The research was partially supported by the National Science Council of the Republic of China under Grant NSC 95-2112-M-415-003.

References and links

1.

I. A. Ramsay and J. J. Degnan, “A ray analysis of optical resonators formed by two spherical mirrors,” Appl. Opt. 9, 385–398 (1970). [CrossRef] [PubMed]

2.

B. Sterman, A. Gabay, S. Yatsiv, and E. Dagan, “Off-axis folded laser beam trajectories in a strip-line CO2 laser,” Opt. Lett. 14, 1309–1311 (1989). [CrossRef] [PubMed]

3.

D. Dick and F. Hanson, “M modes in a diode side-pumped Nd:glass slab laser,” Opt. Lett. 16, 476–477 (1991). [CrossRef] [PubMed]

4.

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001). [CrossRef]

5.

Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A 69, 053807 (2004). [CrossRef]

6.

R. Akis and D. K. Ferry, “Ballistic transport and scarring effects in coupled quantum dots,” Phys. Rev. B 59, 7529–7536 (1999). [CrossRef]

7.

J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A 22, 1559–1566 (2005). [CrossRef]

8.

J. Banerji and G. S. Agarwal, “Non-linear wave packet dynamics of coherent states of various symmetry groups,” Opt. Express 5, 220–229 (1999). [CrossRef] [PubMed]

9.

V. Buzek and T. Quang, “Generalized coherent state for bosonic realization of SU(2) Lie algebra,” J. Opt. Soc. Am. B 6, 2447–2449 (1989). [CrossRef]

10.

P. T. Tai, C. H. Chen, and W. F. Hsieh, “Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser,” Opt. Express 12, 5827–5833 (2004). [CrossRef] [PubMed]

11.

C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. 43, 6001–6006 (2004). [CrossRef] [PubMed]

12.

C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, “Multibeam-waist modes in an end-pumped Nd-YVO4 laser,” J. Opt. Soc. Am. B 20, 1220–1226 (2003). There is lack of γ in the last term of equation (3) in this reference. [CrossRef]

13.

S. A. Collins, “Lens-system diffraction-integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1174 (1970). [CrossRef]

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.3410) Lasers and laser optics : Laser resonators
(140.3580) Lasers and laser optics : Lasers, solid-state
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: August 1, 2007
Revised Manuscript: September 10, 2007
Manuscript Accepted: September 13, 2007
Published: September 19, 2007

Citation
Ching-Hsu Chen and Chi-Feng Chiu, "Generating a geometric mode for clarifying differences between an operator method and SU(2) wave representation," Opt. Express 15, 12692-12698 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12692


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. I. A. Ramsay and J. J. Degnan, "A ray analysis of optical resonators formed by two spherical mirrors," Appl. Opt. 9, 385-398 (1970). [CrossRef] [PubMed]
  2. B. Sterman, A. Gabay, S. Yatsiv, and E. Dagan, "Off-axis folded laser beam trajectories in a strip-line CO2 laser," Opt. Lett. 14,1309-1311 (1989). [CrossRef] [PubMed]
  3. D. Dick and F. Hanson, "M modes in a diode side-pumped Nd:glass slab laser," Opt. Lett. 16,476-477 (1991). [CrossRef] [PubMed]
  4. J. Dingjan, M. P. van Exter, and J. P. Woerdman, "Geometric modes in a single-frequency Nd:YVO4 laser,’’Opt. Commun. 188, 345-351 (2001). [CrossRef]
  5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, "Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity," Phys. Rev. A 69, 053807 (2004). [CrossRef]
  6. R. Akis and D. K. Ferry, "Ballistic transport and scarring effects in coupled quantum dots," Phys. Rev. B 59, 7529-7536 (1999). [CrossRef]
  7. J. Visser, N. J. Zelders, and G. Nienhuis, "Wave description of geometric modes of a resonator," J. Opt. Soc. Am. A 22, 1559-1566 (2005). [CrossRef]
  8. J. Banerji and G. S. Agarwal, "Non-linear wave packet dynamics of coherent states of various symmetry groups," Opt. Express 5,220-229 (1999). [CrossRef] [PubMed]
  9. V. Buzek and T. Quang, "Generalized coherent state for bosonic realization of SU(2) Lie algebra," J. Opt. Soc. Am. B 6, 2447-2449 (1989). [CrossRef]
  10. P. T. Tai, C. H. Chen, and W. F. Hsieh, "Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser," Opt. Express 12, 5827-5833 (2004). [CrossRef] [PubMed]
  11. C. H. Chen, P. T. Tai, and W. F. Hsieh, "Bottle beam from a bare laser for single-beam trapping," Appl. Opt. 43, 6001-6006 (2004). [CrossRef] [PubMed]
  12. C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, "Multibeam-waist modes in an end-pumped Nd-YVO4 laser," J. Opt. Soc. Am. B 20, 1220-1226 (2003). There is lack of in the last term of equation (3) in this reference. [CrossRef]
  13. S. A. Collins, "Lens-system diffraction-integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1174 (1970). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited