1. Introduction

The Fresnel zone structure and associated central intensity core of a laser with an unstable resonator is now considered for the same M =2 half-symmetric unstable cavity geometry whose passive cavity mode structure properties were presented in Part I. The spatially extended gain medium of the laser is described here by the homogeneously broadened gain coefficient [1]

where g ₀ is the small-signal gain coefficient, I_S is the saturation intensity, and where I(r,z) is the local two-way intensity of the cavity field, given by the incoherent sum I(r,z) = I ₊ (r, z)+ I _-(r,z), where I ₊(r,z) is the intensity of the cavity field incident upon the transverse plane at z from the left, and where I _-(r,z) is the intensity of the cavity field incident upon the transverse plane at z from the right, as depicted in Fig. 1. For simplicity, it is assumed here that the spatially extended saturable gain medium fills the entire cavity volume. This simple form of the saturable gain coefficient has found considerable application in the description of continuous wave (cw) CO₂ lasers [2, 3].

 

Fig. 1. Unfolded cavity geometry for a positive branch half-symmetric unstable resonator.

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2. The Thin-Sheet Gain-Phase Approximation

The continuous distribution of the nonlinear gain medium throughout the cavity volume is typically modeled as a discrete series of longitudinally uniform gain-phase segments in which the properties of the medium are assumed constant within each axial segment [2, 3], so that

The medium properties in each axial segment are then accounted for on a single transverse plane situated within that segment, with free-space propagation assumed between each neighboring gain-phase plane. This simplification is known as the thin-sheet gain-phase approximation and criteria for its validity have been derived by Rensch [4] and Milonni [5]. With the axial cavity length partitioned into N_g sections of length 2∆_z and the gain phase sheet taken at the mid-plane of each section, as depicted in Fig. 1, this approximation will be valid provided that the typical Fresnel number

for the propagation between two adjacent gain-phase sheets is very large (typically on the order of 100) and that the inequality

which imposes a limit on both the magnitude and the variation of the gain within each gain-phase segment, is satisfied.

 

Fig. 2. Equivalent Fresnel number dependence of (a) the total intracavity power P_in incident upon the outcoupling aperture, (b) the outcoupled cavity power P_out, and (c) the flux eigenvalue for 1, 5, and 40

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The dependence of the active cavity mode structure on the number of gain-phase sheets is illustrated in Figs. 2–3 for 1, 5, and 40 gain-phase sheets evenly distributed over the cavity length z_T. In general, the total intracavity power incident upon the outcoupling aperture-feedback mirror is found [6] to decrease as the number of gain-phase sheets increases, as seen in Fig. 2(a), as does the outcoupled cavity power depicted in Fig. 2(b). However, the flux eigenvalue, given by

remains essentially unchanged as the number of gain-phase sheets is increased, as seen in Fig. 2(c). The dependence of the converged mode structure that is incident upon the outcoupling aperturefeedback mirror of the cavity on the number of gain-phase sheets, illustrated in Fig. 3 for the M = 2, N_eq = 2.5 half-symmetric unstable resonator, is also seen to be very weak.

 

Fig. 3. Relative intensity and phase of the intracavity mode structure incident upon the outcoupling aperture-feedback mirror of an M = 2, N_eq = 2.5 half-symmetric unstable resonator with 1, 5, and 40 gain-phase sheets.

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These results are in agreement with that described by Siegman [1] who concludes that “transverse gain variations and gain saturation have only minor effects on the mode patterns…” However, Siegman′s conclusion is based solely on results obtained when the entire gain medium is “pasted” on the back mirror of the unstable open cavity. The detailed results presented in the present paper are for the more physically realistic situation of a spatially extended, saturable gain medium that fills the entire cavity volume. The results presented in Figs. 2–4 show the characteristic manner in which the numerically determined active mode structure properties approach the actual, continuously distributed gain medium properties as the number of gain-phase sheets is increased, as evidenced in Fig. 4 for the intracavity power. A close approximation to the continuously extended medium case is obtained when N_g = 40 , in which case the typical Fresnel number for the propagation between two adjacent gain-phase sheets is given by N_F = 40a ² /(λz_T), and has the value N_F = 53.33 for the N_eq = 0.5 cavity, while it has the value N_F = 693.3 for the N_eq = 6.5 cavity. Consequently, the first condition for the validity of the thin-sheet gain-phase approximation is satisfied for each value of the Fresnel number considered in this study when N_g = 40 , but it is not satisfied for the smaller Fresnel cavity cases when N_g = 5 (for example, for the N_g =5, N_eq = 0.5 case, N_F = 6.67 is too small to justify use of the approximation).

3. Three-Dimensional Field Structure in an Active, Positive Branch Half-Symmetric Unstable Cavity with a Spatially Extended, Homogeneously Broadened Gain Medium

The numerically determined three-dimensional intensity structure of the intracavity field of a laser with an unstable resonator, modeled with 40 equally spaced gain-phase sheets, is presented in Figs. 5–14 for the same M =2 passive cavity cases depicted in Figs. 2–11 in Part I, respectively. For the passive cavity cases depicted in Part I, the peak in the intensity structure typically appears in the feedback field and the relative intensity decreases as the field propagates away from the feedback mirror due to the effect of the geometric magnification. The exact opposite occurs for the active cavity cases depicted in Figs. 5–14, where now the peak in the intensity structure typically appears in the field incident upon the outcoupling aperture and the intensity increases as the field propagates away from the feedback mirror because of the amplification due to the gain medium which dominates the decrease due to the geometric magnification when the physical state is above the threshold value for laser operation.

 

Fig. 4. Convergence of the thin-sheet gain-phase approximation of the intracavity power P_in incident upon in the outcoupling aperture to the continuously extended limit (obtained in the limit as N → ∞) at several values of the cavity Fresnel number.

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Fig. 5. Active three-dimensional intracavity field distribution in an M=2 half-symmetric unstable cavity with N_eq = 0.5 .

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Fig. 6. Active three-dimensional intracavity field distribution in an M=2 half-symmetric unstable cavity with N_eq = 0.75.

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Fig. 7. Active three-dimensional intracavity field distribution in an M = 2 half-symmetric unstable cavity with N_eq = 1.0.

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Fig. 8. Active three-dimensional intracavity field distribution in an M = 2 half-symmetric unstable cavity with N_eq =1.25.

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Fig. 9. Active three-dimensional intracavity field distribution in an M = 2 half-symmetric unstable cavity with N_eq = 1.5.

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Fig. 10. Active three-dimensional intracavity field distribution in an M = 2 half-symmetric unstable cavity with N_eq =1.75.

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Fig. 11. Active three-dimensional intracavity field distribution in an M = 2 half-symmetric unstable cavity with N_eq = 6.0.

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Fig. 12. Active three-dimensional intracavity field distribution in an M = 2 half-symmetric unstable cavity with N_eq = 6.25.

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Fig. 13. Active three-dimensional intracavity field distribution in an M = 2 half-symmetric unstable cavity with N_eq = 6.5.

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Fig. 14. Active three-dimensional intracavity field distribution in an M = 2 half-symmetric unstable cavity with N_eq = 6.75.

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Figures 5–14 clearly show the dependence of the central intensity core or oscillator filament of the cavity mode field with the equivalent Fresnel number of the cavity. This oscillator filament is strongest when N_eq = n + 1/2, while it is weakest when N_eq = n, where n is an integer. This is clearly evident from a comparison of the three-dimensional intracavity field structure presented in Fig. 5 with that in Fig. 7, Fig. 7 with that in Fig. 9 and Fig. 11 with that in Fig. 13. Comparison of the active intracavity field distributions presented in Figs. 5–14 with their passive cavity counterparts in Figs. 1–10 of Part I of this pair of papers shows that the oscillator filament is better defined in each active case than in its’ corresponding passive case. This is simply due to the amplifying action of the uniformly extended gain medium

4. Discussion

These results clearly show the importance of the Fresnel zone structure on the intracavity mode structure properties of an unstable resonator in both the passive (purely optical) and active (laser) cavity cases. The numerical results presented both here and in Part I of this pair of papers has demonstrated the applicability of Anan'ev's analogy [7] that a laser with an unstable cavity corresponds to an optical system comprised of a driving generator and an amplifier with a matching telescope between them, particularly for the active cavity case. The role of the generator is played by the central intensity core that is defined by the central Fresnel zone of the cavity and the role of the amplifier by the remaining peripheral zone of the cavity, with the edge-diffracted field at the feedback aperture edge providing the controlling feedback to the central intensity core. It is this mechanism of diffractive feedback into a converging wave field and its interaction with the magnifying or diverging cavity field that produces the central intensity core and determines the diffractive properties of the cavity mode structure.

The results presented here have also detailed the manner in which the total intracavity and outcoupled powers vary as the number of gain-phase segments is increased so as to properly model the effects produced by a spatially-extended, homogeneously broadened saturable gain medium that fills the useful cavity volume of the laser. Starting with just a single gain-phase sheet situated midway between the cavity end mirrors, both the total intracavity and outcoupled powers rapidly decrease as the number N_g of equally-spaced gain-phase sheets is increased while keeping the total small signal gain-length product a constant. For example, for the M = 2, N_eq =1.5 half-symmetric unstable resonator, doubling N_g from 1 to 2 decreases the total intracavity power from 386 watts to 283 watts, a decrease of 26.7%, while further doubling N_g from 2 to 4 decreases that power from 283 watts to 245 watts, a decrease of 13.4%, and further doubling N_g from 4 to 8 gain-phase sheets decreases the intracavity power from 245 watts to 228 watts, a decrease of only 6.9%. The approximation of only using a single gain-phase sheet, as is routinely done, is then seen to be unacceptable when the detailed structure of the intracavity field is desired, particularly when considering peak power loading on intracavity elements.