#### References:



[Multiplane gravitational lensing. I. Morse theory and image](https://aip.scitation.org/doi/10.1063/1.530961)

[Multiplane gravitational lensing. II. Global geometry of caustics](https://aip.scitation.org/doi/citedby/10.1063/1.530962)

[Multiplane gravitational lensing. III. Upper bound on number of](https://phy.duke.edu/multiplane-gravitational-lensing-iii-upper-bound-number-images)





[Shude Mao](https://aip.scitation.org/author/Mao%2C+Shude), [Hans J. Witt](https://aip.scitation.org/author/Witt%2C+Hans+J), [Jin H. An](https://aip.scitation.org/author/An%2C+Jin+H). (2014) Three-dimensional microlensing. *Monthly Notices of the Royal Astronomical Society* **437**:2, 1554-1560.

https://arxiv.org/pdf/1310.5486.pdf

Notice also that the second lensing galaxy in this case should in principle be “microlensed” by the
first galaxy, although in practice, due to the extended size nature of the galaxy, the effect may be small unless there are bright compact sources such as a new supernovae or an active galactic nucleus at the centre of the second galaxy.

Critical lens, caustics, mag and light curves

Also consider second  gravlens (with bright compact centre) lensed by the first lens



2 point mass on the same plane

On the Minimum Magnification Between Caustic Crossings for Microlensing by Binary and Multiple Stars

http://articles.adsabs.harvard.edu/pdf/1995ApJ...447L.105W



Investigation of high amplification events in light curves of gravitationally lensed quasars.

https://ui.adsabs.harvard.edu/abs/1990A%26A...236..311W/abstract , http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1990A%26A...236..311W&defaultprint=YES&page_ind=0&filetype=.pdf





Images for a binary gravitational lens from a single real algebraic equation

https://www.aanda.org/articles/aa/pdf/2002/28/aaee022.pdf , or http://cds.cern.ch/record/560759/files/0206267.pdf



https://arxiv.org/pdf/astro-ph/9408064.pdf

The complex theory of gravitational lensing: Beltrami equation and cluster lensing



[Statistics of Gravitational Microlensing Magnification. II. Three-Dimensional Lens Distribution](https://arxiv.org/abs/astro-ph/9608139)

https://arxiv.org/pdf/astro-ph/9608139.pdf



Astrometric Properties of Gravitational Binary-Microlens Events and Their Applications

https://iopscience.iop.org/article/10.1086/307996/fulltext/40430.text.html













CRITICAL CURVES AND CAUSTICS OF TRIPLE-LENS MODELS

https://iopscience.iop.org/article/10.1088/0004-637X/806/1/99/meta



[Perturbation theory of N point-mass gravitational lens systems without
symmetry: small mass-ratio approximation](https://watermark.silverchair.com/mnras0394-0818.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAnIwggJuBgkqhkiG9w0BBwagggJfMIICWwIBADCCAlQGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQM8-R3fpZ-vKcveQmLAgEQgIICJbFuLl2ng1Wx97f3NyF7rkUMjKFAZHLWeWTGWSqygvaqb-ne_dioFDymqEHvdlCl0HaHShS84Nvd10ikKrPU_JuUNdMCRJmaBlAOLJay7mpy_qhTLjV8GxCe_hWxzmqSdG9QoHWKDv3I4nh1_Lovz111jg0SmowCPQcGNUF3SnqoueVbIGefDWfBdcrngjYVFkQxd-emzl39AmuJLJ3YpbqQgJUMntP4Z7iTl3qS3QkFVM_XWc2xSVrM2pA9uCTAiWsIz5LRLcwWVCdbfei7_DIg06vHFBcJvI1iC6YXoqDF1-rauigYGAoznL3VGiR3J133--HNASlGogpiA54cmvvfkuAcTG7KUCcscuf00ud7SNW28tzGBT1mFH27tV4ZFSy8w1pJPn8VBIF73Fnt8cA-CUkB7MdBf8TxOL3pFxBXNTzyFQTT4CAX4dZB1ykv83AQyO3lkSRntDBLGpeSKXNnLjAAS2PxiAa7xg2zvk4PUFmVydNAuJ30M9oBb4l02jmVtNWg1sgmmFguW_LmnGRbuTI2g-_GCkYbzXVqNzTIvxN3CLy4raAt8JIykKne0n_u7RSXbVV5oDw5DuM9gf_14UPDeI6AC3SmdBdTbsCgY9dKja-zDPxALgcS5ntG7TP-m-B9GXsifcq5T6Go0oowzetx_Et7B0Xc8akoqcPo4HqEEL8r3gr1qlE9Jw8VNNC7BLvMf06i_pRuuOrp9i_ydX_nMA)