Abstract

Unstable resonators show an intense discrimination of undesired higher order modes but a high beam quality cannot be obtained with conventional resonators composed of spherical mirrors. In the present paper we demonstrate the possibility to tailor the fundamental mode by inserting diffractive elements into the resonator to generate a desirable output beam profile even for unstable resonators. We show a concept to design such surface structured elements for customizing the amplitude shape of the outcoupled beam. Further we demonstrate the first experimental realization of an unstable diffractive resonator with a Gaussian shaped amplitude profile of the laser beam for a vertical external cavity surface emitting laser (VECSEL).

©2009 Optical Society of America

1. Introduction

Geometrical stable laser resonators consisting of spherical mirrors generate a high beam quality when operating in the fundamental mode. However the discrimination of high order modes in stable resonators is limited, especially for high gain laser systems. Such higher order modes must be suppressed by internal loss mechanisms. Commonly hard apertures, such as pinholes, are implemented into the resonator to cause diffraction losses to these higher order modes. An aperture however also affects the edge regions of the desired Gaussian fundamental mode and degrades the beam quality and efficiency of the laser. Furthermore, the achievable ratio in the round trip losses between modes of adjacent order is limited as well. An improved method to achieve diffraction losses for high order modes is the use of surface structured mirrors [1].

A much better discrimination of undesired higher order modes is intrinsically obtained in unstable resonator geometries, where the limiting aperture not only acts as modal filter but is also used for coupling the laser beam out of the resonator. The aperture extracts energy from the 0th order mode inside the resonator but not as an absorptive loss. Thus, this energy drop of the 0th order will not decrease the efficiency because it is used as the laser beam, whereas the resonator internal diffraction loss of higher modes will exceed the laser threshold.

In case of hard edged spherical mirrors the beams coupled out from unstable resonators cannot obtain a high beam quality [2]. This can be improved with resonator mirrors having a variable reflectivity [3] as experimentally realized in unstable resonators by Belanger et al. [4]. The difficulties in fabrication and limited performance of dielectric coatings constrain the application of those resonator types. As has been shown theoretically in [5] it is possible to combine the unstable resonator geometry and its desirable mode selecting properties with the design freedom given by using surface structured resonator mirrors. This approach makes possible the construction of a laser resonator with a Gaussian shaped beam profile and an extremely strong discrimination of higher order modes. A first experimental realization of this resonator type was demonstrated in [6] for a Nd:YAG laser where a Gaussian amplitude distribution in the outcoupling region could be shown. Because of high nonlinear effects in the active medium, the designed mode could be achieved at the laser’s threshold only, but not for higher pump powers.

In the present paper we report the realization of an unstable resonator with a surface structured mirror and a hard-edged outcoupling region for a semiconductor disk laser as schematically shown in Fig. 1 . This special active medium is ideally suited for the realization of such a diffractive laser system because of its comparably weak thermal induced aberrations.

 

Fig. 1 Schematic view of the diffractive unstable resonator. (1) Backcoupling mirror (BCM) with amplifying VECSEL disk and optical pumped region (2), (3) Outcoupling mirror (OCM) with diffractive surface profile (4) and eccentric outcoupling of a Gaussian beam (5).

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For this work the optical design procedure, which is used for the calculation of the diffractive mirrors has been adapted from [5] to the case of a single diffractive mirror to reduce the alignment effort. The new design algorithm is described in section 2. Section 3 describes the resonator configuration used in the experiment and in section 4 the fabrication for the diffractive mirror is described. The experimental characterization and the assembly of the realized laser system are treated in the last section.

2. Design approach

The aim of the design is to calculate the surface profile of the outcoupling mirror of an unstable laser resonator with a predefined output beam. An external cavity semiconductor disk laser which is grown on a distributed Bragg reflector (DBR) is used as the amplifying medium and as the backcoupling mirror.

To explain the design algorithm we use the algebraic formulation for the resonator theory as described for example in [7]. The resonator modes can be considered as the eigenfunctions of the round-trip operator Z^ which determines the transformation of the field during a complete pass though the resonator. The equation

Z^Un(x,y)=λnUn(x,y)
provides the relation for the eigenvalues λn of the modeUn(x,y). Comparing the energies of the field before and after one roundtrip, a relative loss of v=1λn2 is found. For a geometrically stable resonator the amplitude distribution of the outcoupled field is proportional to that of the resonator’s internal field in front of the outcoupling mirror. Neglecting other losses than the outcoupling, the eigenvalues do not depend on n and can be left out for the modal analysis. To calculate the mode Un(x,y) only eigenvalues of λ=±1 must be considered and only positive ones can exist physically because of the destructive interference condition otherwise. Equation (1) is now describing the identity operation of the resonator’s eigenspace. In this special case of a lossless stable resonator the roundtrip operator is Hermitian and modes travelling in different directions must be phase conjugated in each xy-plane. Thus in resonators, which consist of two mirrors without spatial dependant losses each mirrorR^i causes a phase conjugation of the field [5]. We have
R^iU(x,y,zRi)=U(x,y,zRi)=U(x,y,zRi+).
In this equation the “-” or “+” sign upon the z-coordinate denotes the direction of propagation. For the position of the mirror, this sign indicates the field directly before (“-”) or after (“+“) the reflection.

For the first design step of the structured mirror, the roundtrip operator for an unstable resonator is considered to be a small change to a Hermitian one. We use Eq. (2) to obtain the phase functions of the structured mirror what leads to

R^i=2·arg(U(x,y,zRi)).

With this assumption it is possible to start an iterative propagation algorithm which approximates a first phase function of the mirror. In this type of iterative algorithm a roundtrip operator is applied to a starting field repeatedly while the field is manipulated by certain auxiliary operators. The iteration causes a convergence of the field similar to a Fox-Li iteration [9] but the properties of the field can be customized by the auxiliary operators.

As a starting field we use a flat top amplitude profile with a plane phase function at the position of the amplifying medium to have a high overlap of the designed mode with the gain region of the amplifier. To obtain a certain energy part of the mode in the later outcoupling region, a spatial dependent attenuation at the position of the structured mirror is introduced as an auxiliary operator in the iterative algorithm. The phase function of the structured mirror is calculated using Eq. (3) in each step of the iteration. The algorithm is stopped if a predefined part of the mode energy lies within the outcoupling region. The result is a phase function of the structured element which is used for further optimization.

The spatial dependent attenuation which was used to adjust the energy ratio of out- and backcoupling is now left out and replaced by an original hard edged aperture. In case of a moderate outcoupling the modal behavior of the resulting roundtrip operator does not change severely. Obviously, this configuration does not initially lead to the desired amplitude profile of the outcoupled beam and the diffraction losses are higher then desired. To obtain a proper surface profile for the outcoupling mirror (OCM) of an unstable resonator the conjugation term in Eq. (2) must be dropped in the following design step, because the roundtrip operator will not be Hermitian anymore [8]. Thus, a second iterative propagation algorithm is needed to further optimize the phase function of the outcoupling mirror. The two steps of the design algorithm are illustrated in Fig. 2 , The used symbols are explained in Table 1 .

 

Fig. 2 A schematic comprehension of the two step design algorithm

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Tables Icon

Table 1. Description of symbols used in the algebraic formulation

In the second step of the design, a different auxiliary operator is used in a second iterative propagation algorithm. In this auxiliary operation, the desired amplitude profile of the outcoupling beam is inserted into the resonator internal field at the position of the structured mirror (zOCM) directly before the outcoupling. When inserting the desired amplitude profile the phase function of the resonator internal mode is remained and just the amplitudes are exchanged. This field is now propagated backwards through the resonator to the position zOCM+of the structured mirror directly after the outcoupling. The phase function of the outcoupling mirror has to transform the field at zOCM intto the one at zOCM+with the desired outcoupling amplitude profile inserted to obtain the desired outcoupling beam profile.

Obviously this transformation will only be possible, if the amplitude functions of these two fields were nearly equivalent. In this case the phase function of the outcoupling mirror can easily be calculated by dividing the fields. In the first steps of the iteration the condition of equivalent amplitudes is only fulfilled approximately. For this reason a Fox-Li Iteration [9] is included into the iterative algorithm to recalculate the mode for each phase function of the outcoupling mirror.

During the iterations diffraction losses are decreasing and the resulting outcoupled field better fits the target. The aborting condition of the algorithm is reached, when no changes in the amplitude profiles appear, and the diffraction losses remain constant.

3. Designed resonator

The active medium which is used in the design is a 1052nm emitting VECSEL (vertical external cavity surface emitting laser) device pumped by an 808nm diode laser like used in [11]. The amplifying layers of the disk are grown on a distributed Bragg reflector (DBR), which is used as the back coupling mirror. The amplifying region is defined by the spot diameter of the optical pump beam which has an approximated diameter of 150µm. The highest possible gain is about 12% at an injection current of 2A of the pump-diode. The resonator is designed to have an outcoupled energy of 4% per roundtrip. The first higher order mode has a roundtrip loss of 27% which exceeds the amplification of the gain medium. The outcoupling loss is consciously kept lower then the maximum performance of the amplifier to test the modal behavior at a higher gain for proving the modal discrimination. Furthermore there will be additional losses by the coating and imperfections of the surface profile due to fabrication constraints.

As input parameters for our algorithm a mirror diameter of 500µm and a resonator length of 7mm are chosen. The beam will be coupled out from the resonator by an eccentric hole of 200µm diameter. Figure 3 shows the designed diffractive outcoupling mirror and Fig. 4 shows the intensity distribution of the zero order mode at this position.

 

Fig. 3 Calculated phase function of the outcoupling mirror (OCM). (a) phase function as calculated by the design algorithm, (b) spherical part of the appropriate substrate curvature subtracted. (1) circular outcoupling region.

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Fig. 4 Intensity distribution of the resonator mode on the diffractive mirror (b). (a) Normalized intensity in the outcoupling region (eccentric hole marked red)

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4. Fabrication of the diffractive mirror

The designed phase function contains a considerable spherical part which causes a large number of 2π phase jumps as illustrated in Fig. 3. The slopes of such jumps are difficult to fabricate lithographically and the coating may not reflect properly in those regions. For this reason we decided to subtract this spherical part by using a concave substrate with a proper curvature for the lithography. The curvature of the used substrates has a radius of 10.374mm. In this case there are no phase jumps remaining in regions where a high intensity of the resonator internal mode will be present on the mirror. Figure 4 shows the intensity profile of the resonator mode on the diffractive mirror.

The fabrication is performed with a DWL400 (Heidelberg Instruments) laser lithography system on a concave substrate using a special technology as described in [10]. This element is used to build a convex molding stamp for replication of the structure in an optically transparent polymer on both a plane and a concave substrate. Afterwards a high reflective dielectric coating is added to the surface. The process technology for structured coatings on complex surfaces is still under development. Therefore the coating had to be removed in the outcoupling window after the coating process. A short CO2 – Laser pulse was used to ablate the coating by thermal tensions. This technique was first performed on plane surfaces leading to very accurate circular regions in which the coating was fully removed. The laser beam which is used for the ablation is slightly defocused on concave substrates and mechanical tensions of the coating impair the geometrical accuracy of the outcoupling hole. However some usable samples could be created with this method as shown in the microscopic images in Fig. 5 .

 

Fig. 5 Microscopic images of two mirror samples. The simulated intensity in the outcoupling region is painted into the graphics. The same false color representation than in Fig. 4 is used. (1) diffractive mirror structure, (2) outcoupling region, (3) diffractive lens structure

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To obtain the designed aperture diameter of the resonator mirror, a defocusing diffractive lens structure surrounds the mirror structure shown in Fig. 3b which deflects disturbing field parts out of the resonator. This lens structure is also used to align the diffractive resonator mirror.

5. Assembly, alignment and results

The disk laser, which was used as active medium, is mounted on a water-cooled copper heat sink. The diffractive mirror and the disk are pre-aligned by a He-Ne-Laser by observing the structure surrounding the OCM-region in an image processing setup. Because of the narrow tolerances a piezo-controlled two-axis stage was used to accomplish the fine tuning of the resonator alignment for proper laser operation.

The intensity distributions of the outcoupled laser beam shown in Fig. 6 were measured by a CCD-camera. The measurement of the near field shows a nearly Gaussian shaped intensity distribution in the outcoupling region and the far field also shows a Gaussian shaped spot. The evaluation of waist size and divergence angle confirms that there are no significant phase aberrations in the beam. The beam quality was measured up to the maximum possible gain of the disc amplifier while a beam parameter product M2 of 1.2 was not exceeded.

 

Fig. 6 Near-field (a) and far-field (b) images of the laser mode. The amplifier is operating at the highest possible gain and a high beam quality of the outcoupled field is observed.

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In addition to the almost Gaussian beam in the outcoupling plane a secondary spot is visible on the right side of the near-field intensity peak in Fig. 6. This effect is a result of the finite reflectivity of the dielectric mirror coating which has residual transmission of 0.5-1%.

The stability of the modal behavior at higher pump currents approves the high modal discrimination of the unstable diffractive resonator. The measured intensity distribution is in a very good agreement with the simulation.

6. Conclusions

We have developed a design procedure for unstable laser resonators with a single diffractive mirror and a predefined output beam. This concept was applied to a semiconductor disc laser. The diffractive element was fabricated lithographically on a concave substrate. Finally, we demonstrated this unstable resonator in lasing operation with a Gaussian beam coupled out of the none-reflective hard edged aperture of the diffractive resonator mirror. A high modal discrimination was theoretically expected by the design and experimentally shown by a stable mode even at a high gain of the active medium.

Acknowledgements

The authors would like to thank D. Radtke for the lithographic fabrication. We also acknowledge support of the Institute of Materials Science and Technology (IMT) of the FSU Jena for structuring the outcoupling zones and OSRAM OS for providing the disc laser.

References and links

1. P. A. Bélanger and C. Pare, “Optical resonators using graded phase mirrors,” Opt. Lett. 16(14), 1057–1059 (1991). [CrossRef]   [PubMed]  

2. A. Siegman, Lasers(University Science Books, Mill Valley, California, 1986).

3. H. Zucker, “Optical resonators with variable reflective mirrors,” Bell Syst. J. 49, 2349–2376 (1970).

4. P. A. Belanger and C. Pare, “Unstable laser resonators with a specified output profile using a graded-reflectivity mirror: Geometrical optics limit,” Opt. Commun. 109, 553–555 (1985).

5. U. D. Zeitner and F. Wyrowski, “Design of Unstable Laser Resonators with User-Defined Mode Shape,” IEEE J. Quantum Electron. 37,(12), 1594–1599 (2001). [CrossRef]  

6. A. Büttner, Untersuchung Experimenteller Verfahren zur resonatorinternen Modenformung. PhD Thesis, Friedrich Schiller Universität Jena, 2005.

7. U. Zeitner, F. Wyrowski, and H. Zellmer, “External Design Freedom for Optimization of Resonator Originated Beam Shaping,” IEEE J. Quantum Electron. 36(10), 1105–1109 (2000). [CrossRef]  

8. U. Zeitner, Optimierung von Lasereigenschaften durch generalisierte Konzepte im Resonatordesign. PhD Thesis, Friedrich Schiller Universität Jena, 1999.

9. A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. 40, 453–488 (1961).

10. D. Radtke and U. D. Zeitner, “Laser-lithography on non-planar surfaces,” Opt. Express 15(3), 1167–1174 (2007). [CrossRef]   [PubMed]  

11. R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007). [CrossRef]  

References

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  1. P. A. Bélanger and C. Pare, “Optical resonators using graded phase mirrors,” Opt. Lett. 16(14), 1057–1059 (1991).
    [Crossref] [PubMed]
  2. A. Siegman, Lasers(University Science Books, Mill Valley, California, 1986).
  3. H. Zucker, “Optical resonators with variable reflective mirrors,” Bell Syst. J. 49, 2349–2376 (1970).
  4. P. A. Belanger and C. Pare, “Unstable laser resonators with a specified output profile using a graded-reflectivity mirror: Geometrical optics limit,” Opt. Commun. 109, 553–555 (1985).
  5. U. D. Zeitner and F. Wyrowski, “Design of Unstable Laser Resonators with User-Defined Mode Shape,” IEEE J. Quantum Electron. 37,(12), 1594–1599 (2001).
    [Crossref]
  6. A. Büttner, Untersuchung Experimenteller Verfahren zur resonatorinternen Modenformung. PhD Thesis, Friedrich Schiller Universität Jena, 2005.
  7. U. Zeitner, F. Wyrowski, and H. Zellmer, “External Design Freedom for Optimization of Resonator Originated Beam Shaping,” IEEE J. Quantum Electron. 36(10), 1105–1109 (2000).
    [Crossref]
  8. U. Zeitner, Optimierung von Lasereigenschaften durch generalisierte Konzepte im Resonatordesign. PhD Thesis, Friedrich Schiller Universität Jena, 1999.
  9. A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. 40, 453–488 (1961).
  10. D. Radtke and U. D. Zeitner, “Laser-lithography on non-planar surfaces,” Opt. Express 15(3), 1167–1174 (2007).
    [Crossref] [PubMed]
  11. R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007).
    [Crossref]

2007 (2)

D. Radtke and U. D. Zeitner, “Laser-lithography on non-planar surfaces,” Opt. Express 15(3), 1167–1174 (2007).
[Crossref] [PubMed]

R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007).
[Crossref]

2001 (1)

U. D. Zeitner and F. Wyrowski, “Design of Unstable Laser Resonators with User-Defined Mode Shape,” IEEE J. Quantum Electron. 37,(12), 1594–1599 (2001).
[Crossref]

2000 (1)

U. Zeitner, F. Wyrowski, and H. Zellmer, “External Design Freedom for Optimization of Resonator Originated Beam Shaping,” IEEE J. Quantum Electron. 36(10), 1105–1109 (2000).
[Crossref]

1991 (1)

1985 (1)

P. A. Belanger and C. Pare, “Unstable laser resonators with a specified output profile using a graded-reflectivity mirror: Geometrical optics limit,” Opt. Commun. 109, 553–555 (1985).

1970 (1)

H. Zucker, “Optical resonators with variable reflective mirrors,” Bell Syst. J. 49, 2349–2376 (1970).

Belanger, P. A.

P. A. Belanger and C. Pare, “Unstable laser resonators with a specified output profile using a graded-reflectivity mirror: Geometrical optics limit,” Opt. Commun. 109, 553–555 (1985).

Bélanger, P. A.

Fox, A.

A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. 40, 453–488 (1961).

Hartke, R.

R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007).
[Crossref]

Heumann, E.

R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007).
[Crossref]

Huber, G.

R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007).
[Crossref]

Kühnelt, M.

R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007).
[Crossref]

L, T.

A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. 40, 453–488 (1961).

Pare, C.

P. A. Bélanger and C. Pare, “Optical resonators using graded phase mirrors,” Opt. Lett. 16(14), 1057–1059 (1991).
[Crossref] [PubMed]

P. A. Belanger and C. Pare, “Unstable laser resonators with a specified output profile using a graded-reflectivity mirror: Geometrical optics limit,” Opt. Commun. 109, 553–555 (1985).

Radtke, D.

Steegmüller, U.

R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007).
[Crossref]

Wyrowski, F.

U. D. Zeitner and F. Wyrowski, “Design of Unstable Laser Resonators with User-Defined Mode Shape,” IEEE J. Quantum Electron. 37,(12), 1594–1599 (2001).
[Crossref]

U. Zeitner, F. Wyrowski, and H. Zellmer, “External Design Freedom for Optimization of Resonator Originated Beam Shaping,” IEEE J. Quantum Electron. 36(10), 1105–1109 (2000).
[Crossref]

Zeitner, U.

U. Zeitner, F. Wyrowski, and H. Zellmer, “External Design Freedom for Optimization of Resonator Originated Beam Shaping,” IEEE J. Quantum Electron. 36(10), 1105–1109 (2000).
[Crossref]

Zeitner, U. D.

D. Radtke and U. D. Zeitner, “Laser-lithography on non-planar surfaces,” Opt. Express 15(3), 1167–1174 (2007).
[Crossref] [PubMed]

U. D. Zeitner and F. Wyrowski, “Design of Unstable Laser Resonators with User-Defined Mode Shape,” IEEE J. Quantum Electron. 37,(12), 1594–1599 (2001).
[Crossref]

Zellmer, H.

U. Zeitner, F. Wyrowski, and H. Zellmer, “External Design Freedom for Optimization of Resonator Originated Beam Shaping,” IEEE J. Quantum Electron. 36(10), 1105–1109 (2000).
[Crossref]

Zucker, H.

H. Zucker, “Optical resonators with variable reflective mirrors,” Bell Syst. J. 49, 2349–2376 (1970).

Appl. Phys. B (1)

R. Hartke, E. Heumann, G. Huber, M. Kühnelt, and U. Steegmüller, “Efficient green generation by intracavity frequency doubling of an optical pumped semiconductor disk laser,” Appl. Phys. B 87(1), 95–99 (2007).
[Crossref]

Bell Syst. J. (1)

H. Zucker, “Optical resonators with variable reflective mirrors,” Bell Syst. J. 49, 2349–2376 (1970).

Bell Syst. Tech. (1)

A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. 40, 453–488 (1961).

IEEE J. Quantum Electron. (2)

U. D. Zeitner and F. Wyrowski, “Design of Unstable Laser Resonators with User-Defined Mode Shape,” IEEE J. Quantum Electron. 37,(12), 1594–1599 (2001).
[Crossref]

U. Zeitner, F. Wyrowski, and H. Zellmer, “External Design Freedom for Optimization of Resonator Originated Beam Shaping,” IEEE J. Quantum Electron. 36(10), 1105–1109 (2000).
[Crossref]

Opt. Commun. (1)

P. A. Belanger and C. Pare, “Unstable laser resonators with a specified output profile using a graded-reflectivity mirror: Geometrical optics limit,” Opt. Commun. 109, 553–555 (1985).

Opt. Express (1)

Opt. Lett. (1)

Other (3)

A. Siegman, Lasers(University Science Books, Mill Valley, California, 1986).

U. Zeitner, Optimierung von Lasereigenschaften durch generalisierte Konzepte im Resonatordesign. PhD Thesis, Friedrich Schiller Universität Jena, 1999.

A. Büttner, Untersuchung Experimenteller Verfahren zur resonatorinternen Modenformung. PhD Thesis, Friedrich Schiller Universität Jena, 2005.

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Figures (6)

Fig. 1
Fig. 1 Schematic view of the diffractive unstable resonator. (1) Backcoupling mirror (BCM) with amplifying VECSEL disk and optical pumped region (2), (3) Outcoupling mirror (OCM) with diffractive surface profile (4) and eccentric outcoupling of a Gaussian beam (5).
Fig. 2
Fig. 2 A schematic comprehension of the two step design algorithm
Fig. 3
Fig. 3 Calculated phase function of the outcoupling mirror (OCM). (a) phase function as calculated by the design algorithm, (b) spherical part of the appropriate substrate curvature subtracted. (1) circular outcoupling region.
Fig. 4
Fig. 4 Intensity distribution of the resonator mode on the diffractive mirror (b). (a) Normalized intensity in the outcoupling region (eccentric hole marked red)
Fig. 5
Fig. 5 Microscopic images of two mirror samples. The simulated intensity in the outcoupling region is painted into the graphics. The same false color representation than in Fig. 4 is used. (1) diffractive mirror structure, (2) outcoupling region, (3) diffractive lens structure
Fig. 6
Fig. 6 Near-field (a) and far-field (b) images of the laser mode. The amplifier is operating at the highest possible gain and a high beam quality of the outcoupled field is observed.

Tables (1)

Tables Icon

Table 1 Description of symbols used in the algebraic formulation

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

Z^Un(x,y)=λnUn(x,y)
R^iU(x,y,zRi)=U(x,y,zRi)=U(x,y,zRi+)
R^i=2·arg(U(x,y,zRi))

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