## Three-dimensional field structure in open unstable resonators Part I: Passive cavity results

Optics Express, Vol. 4, Issue 10, pp. 388-399 (1999)

http://dx.doi.org/10.1364/OE.4.000388

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### Abstract

The three-dimensional field distribution of the diffractive cavity mode structure in a passive, open, unstable resonator is presented as a function of the equivalent Fresnel number of the cavity. The qualitative structure of this intracavity field distribution, including the central intensity core (or oscillator filament), is characterized in terms of the Fresnel zone structure that is defined over the cavity feedback aperture. Previous related research is reviewed.

© Optical Society of America

## 1. Introduction

*M*and equivalent Fresnel number

*N*

_{eq}. These properties are embodied in the appropriate diffractive transverse mode eigenvalue equation [1–3]; for a cylindrical cavity, each azimuthal component radial mode satisfies the integral equation

*l*= 0,±1,±2,… is the azimuthal mode index and

*n*= 1,2,3,… is the radial mode index for the total cavity mode field

*u*(

*r, ϕ*) =

*u*

_{nl}(

*r*) exp (

*ilϕ*). The radial mode indices are chosen such that

*γ*

_{nl}≥

*γ*

_{n+1,l}, where

*γ*

_{nl}= |

*γ̃*

_{nl}| denotes the magnitude of the complex eigenvalue of the mode. The integration domain in Eq. (1) extends over the normalized transverse radial extent of the cavity feedback aperture. Here

*a*is the feedback aperture dimension,

*B*is the equivalent collimated cavity length, and

*λ*is the wavelength of the cavity wave field.

*M*determines the geometric optical properties of the transverse mode structure, as described by the geometrical mode equation of Siegman and Arrathoon [2

2. A. E. Siegman and R. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quant. Electron. **QE-3**, 156–163 (1967). [CrossRef]

*f*

_{l}= 1 for a positive branch (

*M*> 1) cavity, while

*f*

_{l}= (-1)

^{l+1}for a negative branch (

*M*< -1) cavity. This relation expresses the conservation of energy in a single geometrical magnification of the cavity field and follows from the asymptotic behavior of the integral equation (1) in the limit as the collimated Fresnel number

*N*

_{c}approaches infinity. Superimposed on this geometrical optics contribution is the diffractive edge-scattered wave from the cavity feedback aperture whose first-order contribution modifies the geometric mode equation (3) into the form [3

3. K. E. Oughstun, “Unstable resonator modes,” in *Progress in Optics*, vol. xxiv, E. Wolf, ed. (North-Holland, 1987) pp. 165–387. [CrossRef]

*r*< |

*M*| asymptotically as

*N*

_{c}→ ∞. The transverse mode structure for a large Fresnel number cavity is then seen to be dominated by the geometric optics mode solution plus the secondary edge diffracted wave that is a characteristic of the cavity magnification and collimated Fresnel number.

*N*

_{c}becomes small and approaches zero in the limit, the transverse mode equation (1) may be approximated as

*N*

_{c}of the unstable cavity represents the number of Fresnel half-period zones for a plane wave field filling the magnified feedback aperture when viewed from the center of that aperture at a distance of one equivalent collimated cavity length away [3

3. K. E. Oughstun, “Unstable resonator modes,” in *Progress in Optics*, vol. xxiv, E. Wolf, ed. (North-Holland, 1987) pp. 165–387. [CrossRef]

3. K. E. Oughstun, “Unstable resonator modes,” in *Progress in Optics*, vol. xxiv, E. Wolf, ed. (North-Holland, 1987) pp. 165–387. [CrossRef]

4. E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” IEEE Proc. **62**, 410–412 (1974). [CrossRef]

*N*

_{eq}of an unstable cavity is defined such that the quantity

*N*

_{eq}

*λ*is equal to the sagittal distance between the expanding geometrical optics mode phase front and the corresponding converging wave front at the edge of the cavity feedback aperture [5

5. Yu. A. Anan’ev, “Unstable resonators and their applications (Review),” Sov. J. Quant. Electron. **1**, 565–586 (1972). [CrossRef]

## 2. Fresnel Zone Structure of an Open Unstable Cavity

5. Yu. A. Anan’ev, “Unstable resonators and their applications (Review),” Sov. J. Quant. Electron. **1**, 565–586 (1972). [CrossRef]

*L*

_{n}= 1- |

^{2}after each round-trip propagation through the cavity, while the field scattered along other directions rapidly escapes from the unstable cavity. As a consequence, the intensity of the converging wave, which is negligible near the edge of the cavity feedback aperture, is geometrically amplified as it propagates inward toward the cavity optical axis to such an extent that it has a significant influence on the entire structure of the cavity field [5

5. Yu. A. Anan’ev, “Unstable resonators and their applications (Review),” Sov. J. Quant. Electron. **1**, 565–586 (1972). [CrossRef]

*N*

_{eq}

*λ*between these two wavefronts. As a consequence, when the equivalent Fresnel number of the cavity is changed by unity, the phase shift between the diverging and converging waves changes by 2

*π*and the coherent interaction between those two wave fields is essentially unchanged [5

**1**, 565–586 (1972). [CrossRef]

*Progress in Optics*, vol. xxiv, E. Wolf, ed. (North-Holland, 1987) pp. 165–387. [CrossRef]

7. W. H. Steier and G. L. McAllister, “A simplified method for predicting unstable resonator mode profiles,” IEEE J. Quant. Electron. **QE-11**, 725–728 (1975). [CrossRef]

8. K. E. Oughstun, “Passive cavity transverse mode stability and its influence on the active cavity mode properties for unstable optical resonators,” in *Optical Resonators*, SPIE Proceedingsvol. 1224, D. A. Holmes, ed. (SPIE, 1990) pp. 80–98. [CrossRef]

*Progress in Optics*, vol. xxiv, E. Wolf, ed. (North-Holland, 1987) pp. 165–387. [CrossRef]

8. K. E. Oughstun, “Passive cavity transverse mode stability and its influence on the active cavity mode properties for unstable optical resonators,” in *Optical Resonators*, SPIE Proceedingsvol. 1224, D. A. Holmes, ed. (SPIE, 1990) pp. 80–98. [CrossRef]

*r*≤

*a*. At the feedback mirror edge (

*r*=

*a*) this function is equal to the equivalent Fresnel number of the cavity. The associated cavity Fresnel zones over the circular feedback aperture are then defined by the set of concentric circles whose radii

*r*

_{n}≤

*a*satisfy the condition

*f*< 1 such that the fractional number

*f*is set by the numerical value of

*N*

_{eq}. The radii of the cavity Fresnel zones at the feedback aperture are then given by

**1**, 565–586 (1972). [CrossRef]

## 3. Three-Dimensional Field Structure in a Positive Branch Half-Symmetric Unstable Cavity

*M*= 2 and the intracavity field structure was numerically determined as a function of the equivalent Fresnel number of the cavity. The diffractive field calculations are based on the angular spectrum of plane waves representation utilizing the Fast Fourier Transform (FFT) as described, along with the associated sampling criteria, in Ref. 3

*Progress in Optics*, vol. xxiv, E. Wolf, ed. (North-Holland, 1987) pp. 165–387. [CrossRef]

*l*= 0) mode field structure incident upon the cavity feedback mirror was obtained in a Fox and Li type iteration procedure, the three dimensional intracavity field distribution was obtained by calculating the diffractive feedback field at 80 transverse planes evenly spaced through the unfolded cavity [11].

*N*

_{eq}= 0.5 so that

*n*= 0 and

*f*= 0.5; the feedback mirror (which extends in the transverse dimension from -

*a*to +

*a*indicated at the left of this and subsequent figures) then encompasses one-half of the central Fresnel zone and a well-defined central intensity core is seen to emanate from this region into the central volume of the cavity with a transverse mode discrimination ratio

*γ*

_{1,0}/

*γ*

_{2,0}= 2.34 that is at (or very near to) the global maximum for this

*M*= 2 cavity, as well as a near maximum eigenvalue magnitude

*γ*

_{0,1}= 0.746 and minimal outcoupling loss

*L*= 1 -

*N*

_{eq}= 0.75 , so that

*f*= 0.75 , the feedback mirror encompasses three-quarters of the central Fresnel zone and the transverse mode discrimination ratio has decreased in value to

*γ*

_{1,0}/

*γ*

_{2,0}= 1.38; the central intensity core illustrated in Fig. 3 is not as well-defined about the optic axis as that depicted in Fig. 2. When

*N*

_{eq}= 1.0, so that

*n*= 1 and

*f*= 0, the feedback mirror occupies a full Fresnel zone, the transverse mode discrimination ratio is near minimal at

*γ*

_{1,0}/

*γ*

_{2,0}= 1.06, and the central intensity core is now poorly defined about the optic axis due to destructive interference between the converging and diverging wave fields, as seen in Fig. 4. The eigenvalue magnitude

*γ*

_{0,1}= 0.661 of the dominant cavity mode is now very near to a local minimum with an associated locally maximum outcoupling loss of

*L*= 0.563 due to the poorly defined central intensity core. This marks the beginning of the next cycle with

*n*= 1. When

*N*

_{eq}= 1 25, as illustrated in Fig. 5, so that

*n*= 1 and

*f*= 0.25, the central Fresnel zone occupies (radially) the inner 20% of the feedback mirror and the definition of the central intensity core of the cavity mode field about the optic axis has increased from that depicted in Fig. 4. Locally optimal behavior is obtained when

*N*

_{eq}= 1.5, so that

*n*= 1 and

*f*= 0.5, as seen in Fig. 6. The eigenvalue magnitude

*γ*

_{0,1}= 0.689 is now at (or very near to) a local maximum, as is the transverse mode discrimination ratio with a value of 1.26, with an associated locally minimal outcoupling loss of

*L*= 0.525 due to the well-defined central intensity core that extends from the feedback aperture past the end mirror of the cavity. At

*N*

_{eq}= 1.75, as illustrated in Fig. 7, so that

*n*= 1 and

*f*= 0.75, the central Fresnel zone occupies (radially) the inner 42.86% of the feedback mirror and the definition of the central intensity core of the cavity mode field about the optic axis has decreased from that depicted in Fig. 6, with an associated decrease in the eigenvalue magnitude.

*N*

_{eq}= 6.0, the central intensity core is weakly defined, as seen in Fig. 8, has improved definition at

*N*

_{eq}= 6.25 , as seen in Fig. 9, achieves a local optimum at

*N*

_{eq}= 6.5 , as seen in Fig. 10, and decreases in definition at

*N*= 6.75, as seen in Fig. 11.

## 4. Discussion

9. Yu. A. Anan’ev, “Angular divergence of radiation of solid-state lasers,” Sov. Phys. Usp. **14**, 197–215 (1971). [CrossRef]

## Acknowledgments

## References and Links

1. | A. E. Siegman, Lasers (University Science Books, 1986) Chapters 21–23. |

2. | A. E. Siegman and R. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quant. Electron. |

3. | K. E. Oughstun, “Unstable resonator modes,” in |

4. | E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” IEEE Proc. |

5. | Yu. A. Anan’ev, “Unstable resonators and their applications (Review),” Sov. J. Quant. Electron. |

6. | Yu. A. Anan’ev, |

7. | W. H. Steier and G. L. McAllister, “A simplified method for predicting unstable resonator mode profiles,” IEEE J. Quant. Electron. |

8. | K. E. Oughstun, “Passive cavity transverse mode stability and its influence on the active cavity mode properties for unstable optical resonators,” in |

9. | Yu. A. Anan’ev, “Angular divergence of radiation of solid-state lasers,” Sov. Phys. Usp. |

10. | Yu. A. Anan’ev, “Establishment of oscillations in unstable resonators,” Sov. J. Quant. Electron. |

11. | C. C. Khamnei, |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(140.4780) Lasers and laser optics : Optical resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 5, 1999

Published: May 10, 1999

**Citation**

Kurt Oughstun and Chris Khamnei, "Three-dimensional field structure
in open unstable resonators Part I: Passive cavity results," Opt. Express **4**, 388-399 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-10-388

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### References

- A. E. Siegman, Lasers (University Science Books, 1986) Chapters 21-23.
- A. E. Siegman and R. Arrathoon, "Modes in unstable optical resonators and lens waveguides," IEEE J. Quant. Electron. QE-3, 156-163 (1967). [CrossRef]
- K. E. Oughstun, "Unstable resonator modes," in Progress in Optics, vol. xxiv, E. Wolf, ed. (North-Holland, 1987) pp. 165-387. [CrossRef]
- E. A. Sziklas and A. E. Siegman, "Diffraction calculations using fast Fourier transform methods," IEEE Proc. 62, 410-412 (1974). [CrossRef]
- Yu A. Ananev, "Unstable resonators and their applications (Review)," Sov. J. Quant. Electron. 1, 565-586 (1972). [CrossRef]
- Yu. A. Ananev, Laser Resonators and the Beam Divergence Problem (Adam Hilger, 1992).
- W. H. Steier and G. L. McAllister, "A simplified method for predicting unstable resonator mode profiles," IEEE J. Quant. Electron. QE-11, 725-728 (1975). [CrossRef]
- K. E. Oughstun, "Passive cavity transverse mode stability and its influence on the active cavity mode properties for unstable optical resonators," in Optical Resonators, SPIE Proceedings vol. 1224, D. A. Holmes, ed. (SPIE, 1990) pp. 80-98. [CrossRef]
- Yu. A. Ananev, "Angular divergence of radiation of solid-state lasers," Sov. Phys. Usp. 14, 197-215 (1971). [CrossRef]
- Yu. A. Ananev, "Establishment of oscillations in unstable resonators," Sov. J. Quant. Electron. 5, 615-617 (1975). [CrossRef]
- C. C. Khamnei, Open Unstable Optical Resonator Mode Field Theory (Ph.D. dissertation, University of Vermont, 1998).

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