Abstract

We report on the production and characterisation of optical microring resonators and optical channel waveguides by using fluorine-ion implantation and planar structuring in lithium niobate. We demonstrate the production of single-mode planar waveguides by low fluence fluorine-ion implantation (ϕ=2.5·1014 ions/cm2) into lithium niobate wafers. The waveguides are strongly confined by the amorphous 2-µm wide optical barrier induced by the implantation process. A refractive index contrast of Δno=0.17 at the telecom wavelength λ=1.5 µm has been determined between the waveguide and the barrier. Planar structuring with ridge height of up to 1.2 µm has been achieved by laser lithography masking and Ar+ sputtering. For TE waves, the channel waveguides exhibit propagation losses lower than 8 dB/cm. First ring resonators with 80-µm radius have been fabricated by planar structuring in fluorine-ion implanted lithium niobate. The measured resonance curves show an extinction ratio of 14 dB, a free spectral range of 2.0 nm and a finesse of 4.

© 2008 Optical Society of America

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    [CrossRef]
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2007 (7)

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Günter, "Electro-optically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407-410 (2007).
[CrossRef]

T-J. Wang, C-H. Chu, and C-Y. Lin, "Electro-optically tunable microring resonators on lithium niobate," Opt. Lett. 32, 2777-2779 (2007).
[CrossRef] [PubMed]

T-J. Wang and C-H. Chu, "Wavelength-Tunable Microring Resonator on Lithium Niobate," IEEE Photon. Technol. Lett. 19, 1904-1906 (2007).
[CrossRef]

F. Chen, X-L. Wang, and K-M. Wang, "Development of ion-implanted optical waveguides in optical materials: A review," Opt. Mat. 29,1 523-1542 (2007).
[CrossRef]

J. Olivares, A. García-Navarro, G. García, F. Agulló-López, Agulló-Rueda, A. García-Cabañes, and M. Carrascosa, "Buried amorphous layers by electronic excitation in ion-beam irradiated lithium niobate: Structure and kinetics," J. Appl. Phys. 101, 033512 (2007).
[CrossRef]

A. Majki??, G. Poberaj, R. Degl???Innocenti, M. Döbeli, and P. Günter, "Cr:LiSrAlF6 channel waveguides as broadband fluorescence sources," Appl. Phys. B 88, 205-209 (2007).
[CrossRef]

R. Degl???Innocenti, A. Majkic, P. Vorburger, G. Poberaj, P. Günter, M. Döbeli, "Ultraviolet electro-optic amplitude modulation in ?-BaB2O4 waveguides," Appl. Phys. Lett. 91, 051105 (2007).
[CrossRef]

2006 (1)

H. Tazawa and H. Steier, "Analysis of Ring Resonator-Based Travelling-Wave Modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
[CrossRef]

2005 (2)

J. Olivares, G. García, A. García-Navarro, F. Agulló-López, O. Caballero, and A. García-Cabañes, "Generation of high-confinement step-like optical waveguides in LiNbO3 by swift heavy ion-beam irradiation," Appl. Phys. Lett. 86, 183501 (2005).
[CrossRef]

W. Sohler, B. K. Das, D. Dey, S. Reza, H. Suche, R. Ricken, "Erbium-Doped Lithium Niobate Waveguide Lasers," IEICE Trans. Electron. E 88-C, 990-996 (2005).
[CrossRef]

2003 (2)

C-Y. Chao and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Appl. Phys. Lett. 83, 1527-1529 (2003).
[CrossRef]

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

2002 (2)

P. Rabiei, W. H. Steier, C. Zhang, and L. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002).
[CrossRef]

M. Jazbinšek and M. Zgonik, "Material tensor parameters of LiNbO3 relevant for electro- and elasto-optics," Appl. Phys. B 74, 407-414 (2002).
[CrossRef]

1999 (1)

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

1985 (2)

R. Regener and W. Sohler, "Loss in Low-Finesse Ti:LiNbO3 Optical Waveguide Resonators," Appl.Phys. B 36, 143-147 (1985).
[CrossRef]

A. Mahapatra and W. C. Robinson, "Integrated-optic ring resonators made by proton exchange in lithium niobate," Appl. Opt. 24, 2285-2286 (1985).
[CrossRef] [PubMed]

Agulló-López, F.

J. Olivares, A. García-Navarro, G. García, F. Agulló-López, Agulló-Rueda, A. García-Cabañes, and M. Carrascosa, "Buried amorphous layers by electronic excitation in ion-beam irradiated lithium niobate: Structure and kinetics," J. Appl. Phys. 101, 033512 (2007).
[CrossRef]

J. Olivares, G. García, A. García-Navarro, F. Agulló-López, O. Caballero, and A. García-Cabañes, "Generation of high-confinement step-like optical waveguides in LiNbO3 by swift heavy ion-beam irradiation," Appl. Phys. Lett. 86, 183501 (2005).
[CrossRef]

Agulló-Rueda, F.

J. Olivares, A. García-Navarro, G. García, F. Agulló-López, Agulló-Rueda, A. García-Cabañes, and M. Carrascosa, "Buried amorphous layers by electronic excitation in ion-beam irradiated lithium niobate: Structure and kinetics," J. Appl. Phys. 101, 033512 (2007).
[CrossRef]

Armenise, M. N.

M. N. Armenise, "Fabrication techniques of lithium niobate waveguides," IEE Proceedings 135, Pt. J, No 2, 85-91 (1988).

Bakhru, R.

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

Caballero, O.

J. Olivares, G. García, A. García-Navarro, F. Agulló-López, O. Caballero, and A. García-Cabañes, "Generation of high-confinement step-like optical waveguides in LiNbO3 by swift heavy ion-beam irradiation," Appl. Phys. Lett. 86, 183501 (2005).
[CrossRef]

Chao, C-Y.

C-Y. Chao and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Appl. Phys. Lett. 83, 1527-1529 (2003).
[CrossRef]

Chen, F.

F. Chen, X-L. Wang, and K-M. Wang, "Development of ion-implanted optical waveguides in optical materials: A review," Opt. Mat. 29,1 523-1542 (2007).
[CrossRef]

Chu, C-H.

T-J. Wang and C-H. Chu, "Wavelength-Tunable Microring Resonator on Lithium Niobate," IEEE Photon. Technol. Lett. 19, 1904-1906 (2007).
[CrossRef]

T-J. Wang, C-H. Chu, and C-Y. Lin, "Electro-optically tunable microring resonators on lithium niobate," Opt. Lett. 32, 2777-2779 (2007).
[CrossRef] [PubMed]

Dalton, L.

Das, B. K.

W. Sohler, B. K. Das, D. Dey, S. Reza, H. Suche, R. Ricken, "Erbium-Doped Lithium Niobate Waveguide Lasers," IEICE Trans. Electron. E 88-C, 990-996 (2005).
[CrossRef]

Dey, D.

W. Sohler, B. K. Das, D. Dey, S. Reza, H. Suche, R. Ricken, "Erbium-Doped Lithium Niobate Waveguide Lasers," IEICE Trans. Electron. E 88-C, 990-996 (2005).
[CrossRef]

Evans, C.

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

García, G.

J. Olivares, A. García-Navarro, G. García, F. Agulló-López, Agulló-Rueda, A. García-Cabañes, and M. Carrascosa, "Buried amorphous layers by electronic excitation in ion-beam irradiated lithium niobate: Structure and kinetics," J. Appl. Phys. 101, 033512 (2007).
[CrossRef]

J. Olivares, G. García, A. García-Navarro, F. Agulló-López, O. Caballero, and A. García-Cabañes, "Generation of high-confinement step-like optical waveguides in LiNbO3 by swift heavy ion-beam irradiation," Appl. Phys. Lett. 86, 183501 (2005).
[CrossRef]

García-Cabañes, A.

J. Olivares, G. García, A. García-Navarro, F. Agulló-López, O. Caballero, and A. García-Cabañes, "Generation of high-confinement step-like optical waveguides in LiNbO3 by swift heavy ion-beam irradiation," Appl. Phys. Lett. 86, 183501 (2005).
[CrossRef]

García-Navarro, A.

J. Olivares, A. García-Navarro, G. García, F. Agulló-López, Agulló-Rueda, A. García-Cabañes, and M. Carrascosa, "Buried amorphous layers by electronic excitation in ion-beam irradiated lithium niobate: Structure and kinetics," J. Appl. Phys. 101, 033512 (2007).
[CrossRef]

J. Olivares, G. García, A. García-Navarro, F. Agulló-López, O. Caballero, and A. García-Cabañes, "Generation of high-confinement step-like optical waveguides in LiNbO3 by swift heavy ion-beam irradiation," Appl. Phys. Lett. 86, 183501 (2005).
[CrossRef]

Guarino, A.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Günter, "Electro-optically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407-410 (2007).
[CrossRef]

Guo, L. J.

C-Y. Chao and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Appl. Phys. Lett. 83, 1527-1529 (2003).
[CrossRef]

Jazbinšek, M.

M. Jazbinšek and M. Zgonik, "Material tensor parameters of LiNbO3 relevant for electro- and elasto-optics," Appl. Phys. B 74, 407-414 (2002).
[CrossRef]

Kumar, A.

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

Levy, M.

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

Lin, C-Y.

Mahapatra, A.

Majki??, A.

A. Majki??, G. Poberaj, R. Degl???Innocenti, M. Döbeli, and P. Günter, "Cr:LiSrAlF6 channel waveguides as broadband fluorescence sources," Appl. Phys. B 88, 205-209 (2007).
[CrossRef]

Olivares, J.

J. Olivares, A. García-Navarro, G. García, F. Agulló-López, Agulló-Rueda, A. García-Cabañes, and M. Carrascosa, "Buried amorphous layers by electronic excitation in ion-beam irradiated lithium niobate: Structure and kinetics," J. Appl. Phys. 101, 033512 (2007).
[CrossRef]

J. Olivares, G. García, A. García-Navarro, F. Agulló-López, O. Caballero, and A. García-Cabañes, "Generation of high-confinement step-like optical waveguides in LiNbO3 by swift heavy ion-beam irradiation," Appl. Phys. Lett. 86, 183501 (2005).
[CrossRef]

Osgood, R. M.

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

Poberaj, G.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Günter, "Electro-optically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407-410 (2007).
[CrossRef]

A. Majki??, G. Poberaj, R. Degl???Innocenti, M. Döbeli, and P. Günter, "Cr:LiSrAlF6 channel waveguides as broadband fluorescence sources," Appl. Phys. B 88, 205-209 (2007).
[CrossRef]

Rabiei, P.

Radojevic, A. M.

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

Regener, R.

R. Regener and W. Sohler, "Loss in Low-Finesse Ti:LiNbO3 Optical Waveguide Resonators," Appl.Phys. B 36, 143-147 (1985).
[CrossRef]

Reza, S.

W. Sohler, B. K. Das, D. Dey, S. Reza, H. Suche, R. Ricken, "Erbium-Doped Lithium Niobate Waveguide Lasers," IEICE Trans. Electron. E 88-C, 990-996 (2005).
[CrossRef]

Rezzonico, D.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Günter, "Electro-optically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407-410 (2007).
[CrossRef]

Ricken, R.

W. Sohler, B. K. Das, D. Dey, S. Reza, H. Suche, R. Ricken, "Erbium-Doped Lithium Niobate Waveguide Lasers," IEICE Trans. Electron. E 88-C, 990-996 (2005).
[CrossRef]

Robinson, W. C.

Sohler, W.

W. Sohler, B. K. Das, D. Dey, S. Reza, H. Suche, R. Ricken, "Erbium-Doped Lithium Niobate Waveguide Lasers," IEICE Trans. Electron. E 88-C, 990-996 (2005).
[CrossRef]

R. Regener and W. Sohler, "Loss in Low-Finesse Ti:LiNbO3 Optical Waveguide Resonators," Appl.Phys. B 36, 143-147 (1985).
[CrossRef]

Steier, H.

H. Tazawa and H. Steier, "Analysis of Ring Resonator-Based Travelling-Wave Modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
[CrossRef]

Steier, W. H.

Suche, H.

W. Sohler, B. K. Das, D. Dey, S. Reza, H. Suche, R. Ricken, "Erbium-Doped Lithium Niobate Waveguide Lasers," IEICE Trans. Electron. E 88-C, 990-996 (2005).
[CrossRef]

Tazawa, H.

H. Tazawa and H. Steier, "Analysis of Ring Resonator-Based Travelling-Wave Modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
[CrossRef]

Tian, C.

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

Vahala, K. J.

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

Wang, K-M.

F. Chen, X-L. Wang, and K-M. Wang, "Development of ion-implanted optical waveguides in optical materials: A review," Opt. Mat. 29,1 523-1542 (2007).
[CrossRef]

Wang, T-J.

T-J. Wang and C-H. Chu, "Wavelength-Tunable Microring Resonator on Lithium Niobate," IEEE Photon. Technol. Lett. 19, 1904-1906 (2007).
[CrossRef]

T-J. Wang, C-H. Chu, and C-Y. Lin, "Electro-optically tunable microring resonators on lithium niobate," Opt. Lett. 32, 2777-2779 (2007).
[CrossRef] [PubMed]

Wang, X-L.

F. Chen, X-L. Wang, and K-M. Wang, "Development of ion-implanted optical waveguides in optical materials: A review," Opt. Mat. 29,1 523-1542 (2007).
[CrossRef]

Zgonik, M.

M. Jazbinšek and M. Zgonik, "Material tensor parameters of LiNbO3 relevant for electro- and elasto-optics," Appl. Phys. B 74, 407-414 (2002).
[CrossRef]

Zhang, C.

Appl. Opt. (1)

Appl. Phys. B (2)

M. Jazbinšek and M. Zgonik, "Material tensor parameters of LiNbO3 relevant for electro- and elasto-optics," Appl. Phys. B 74, 407-414 (2002).
[CrossRef]

A. Majki??, G. Poberaj, R. Degl???Innocenti, M. Döbeli, and P. Günter, "Cr:LiSrAlF6 channel waveguides as broadband fluorescence sources," Appl. Phys. B 88, 205-209 (2007).
[CrossRef]

Appl. Phys. Lett. (4)

R. Degl???Innocenti, A. Majkic, P. Vorburger, G. Poberaj, P. Günter, M. Döbeli, "Ultraviolet electro-optic amplitude modulation in ?-BaB2O4 waveguides," Appl. Phys. Lett. 91, 051105 (2007).
[CrossRef]

A. M. Radojevic, M. Levy, R. M. Osgood, A. Kumar, R. Bakhru, C. Tian, and C. Evans, "Large etch-selectivity enhancement in the epitaxial liftoff of single-crystal LiNbO3 films," Appl. Phys. Lett. 74, 3197-3199 (1999).
[CrossRef]

C-Y. Chao and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Appl. Phys. Lett. 83, 1527-1529 (2003).
[CrossRef]

J. Olivares, G. García, A. García-Navarro, F. Agulló-López, O. Caballero, and A. García-Cabañes, "Generation of high-confinement step-like optical waveguides in LiNbO3 by swift heavy ion-beam irradiation," Appl. Phys. Lett. 86, 183501 (2005).
[CrossRef]

Appl.Phys. B (1)

R. Regener and W. Sohler, "Loss in Low-Finesse Ti:LiNbO3 Optical Waveguide Resonators," Appl.Phys. B 36, 143-147 (1985).
[CrossRef]

E (1)

W. Sohler, B. K. Das, D. Dey, S. Reza, H. Suche, R. Ricken, "Erbium-Doped Lithium Niobate Waveguide Lasers," IEICE Trans. Electron. E 88-C, 990-996 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

T-J. Wang and C-H. Chu, "Wavelength-Tunable Microring Resonator on Lithium Niobate," IEEE Photon. Technol. Lett. 19, 1904-1906 (2007).
[CrossRef]

H. Tazawa and H. Steier, "Analysis of Ring Resonator-Based Travelling-Wave Modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
[CrossRef]

J. Appl. Phys. (1)

J. Olivares, A. García-Navarro, G. García, F. Agulló-López, Agulló-Rueda, A. García-Cabañes, and M. Carrascosa, "Buried amorphous layers by electronic excitation in ion-beam irradiated lithium niobate: Structure and kinetics," J. Appl. Phys. 101, 033512 (2007).
[CrossRef]

J. Lightwave Technol. (1)

Na. Photonics (1)

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Günter, "Electro-optically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407-410 (2007).
[CrossRef]

Nature (1)

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

Opt. Lett. (1)

Opt. Mat. (1)

F. Chen, X-L. Wang, and K-M. Wang, "Development of ion-implanted optical waveguides in optical materials: A review," Opt. Mat. 29,1 523-1542 (2007).
[CrossRef]

Other (4)

M. N. Armenise, "Fabrication techniques of lithium niobate waveguides," IEE Proceedings 135, Pt. J, No 2, 85-91 (1988).

M. Hammer, K. R. Hiremath, and R. Stoffer, "Analytical approaches to the description of Optical Microresonator Devices," in Microresonators as Building Blocks for VLSI Photonics, F. Michelotti, A. Driessen, and M. Bertolotti, eds., (AIP Conference Proceedings, Melville, New York, 2004).

J. F. Ziegler, J. P. Biersack, and U. Littmark, The Stopping and Range of Ions in Solids (Pergamon Press, New York, 1985).

Concept to Volume (C2V), "OlympIOs Integrated Optics Software," http://www.c2v.nl

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Figures (8)

Fig. 1.
Fig. 1.

(a). Depth position of the high-energy (closer to surface, open circles) and low-energy (closed circles) crystalline-amorphous boundaries as a function of implantation fluence. Measurements were performed on samples irradiated with 22 MeV F4+ ions (del =4.7 µm): our measurements (open circles), data from [13] (full circles). The barrier width dB for fluence ϕ=2.5·1014 ions/cm2 is shown. The lines connecting the points are guides to the eye. (b). Simulated depth position of the maximal electronic stopping power del in F-implanted LiNbO3 as a function of ion energy. The line is the polynomial approximation discussed in the text. dB and the estimated waveguide thickness for ϕ=2.5·1014 ions/cm2 and E=14.5 MeV are shown.

Fig. 2.
Fig. 2.

Calculated combined bending and tunnelling losses (left scale) and effective mode index Neff at ring’s outer rim (right scale) as a function of the waveguide bend radius. Neff of the straight waveguide with the same cross-section is 2.159. Calculation is for the first TE optical mode and wavelength 1.55 µm. Waveguide cross-section dimensions as in Fig. 3.

Fig. 3.
Fig. 3.

Simulation of light propagation in the waveguide of interest. (a) Vertical index profile of the waveguide structure (at X=0). (b) Simulated electric field profile of the first TE optical mode at λ=1.55 µm in the waveguide with 80 µm bend radius. Optical barrier width is 2 µm. Waveguide cross-section is trapezoidal with waveguide height 1.35 um, ridge height 1.2 µm, base width 3.7 µm and top width 2.7 µm.

Fig. 4.
Fig. 4.

Scanning electron micrograph of a microring resonator and a bus waveguide, structured in LiNbO3. (a). The whole ring and the bus waveguide. Ring radius is 80 µm, ridge height is 1.2 µm. (b). Enlarged coupling region, the gap size is ~0.2 µm.

Fig. 5.
Fig. 5.

SEM image of the fabricated ridge structure. The waveguide cross-section is trapezoidal with the base width 3.7 µm, top width 2.7 µm and ridge height 1.2 µm. The amorphised barrier layer beneath the ridge is visible.

Fig. 6.
Fig. 6.

Measured TE wave transmission spectrum of the bus waveguide coupled to a microring resonator. Small oscillations seen at the regions outside of resonances stem from Fabry-Perot resonances of the 5.6 mm long waveguide.

Fig. 7.
Fig. 7.

Simulated electric potential in the waveguide cross-section upon the application of a voltage V=100 V between the electrodes. Equipotential contours separate a 5 V potential drop. About 55% of the potential drop occurs in the SiO2 buffer layer.

Fig. 8.
Fig. 8.

(red) Measured light transmission as a function of the roundtrip phase θ. The line connecting the points is a guide to the eye. (blue) Transmission curve (T=cos2(φ/2)) of an equivalent Mach-Zehnder modulator, for phase φ=θ - π. Points of maximum transmission slopes |dT/|max are marked with arrows.

Tables (1)

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Table 1. Properties of demonstrated microring resonators in LiNbO3

Equations (4)

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Δ λ FSR = λ 2 2 π R · N g = 2.03 n m .
δ ( N eff ( λ ) L λ ) = 0 gives ( 1 λ N eff λ N eff λ 2 ) δ λ + 1 λ δ N eff = 0 ,
hence δ λ = λ N g δ N eff .
V π eq = π 2 ( d T d V max ) 1 = π 2 ( d T d θ d θ d V max ) 1 = V π 0 2 d T d θ max ,

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