Referee 1's Review:

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N.B. These are the comments made by the referee when reviewing an earlier version of this paper. Prior to publication, the manuscript has been revised in light of these comments and to address other editorial requirements.
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GENERAL COMMENTS:

I warmly congratulate the Authors on this nice and elegant paper! I enjoyed very much the manuscript, which presents one of the most important discoveries in the field of polarization vision in the last decades: The Authors gave the first evidence of the possibility of circular polarization vision in an animal. Earlier, circular polarization vision has been only hypothesized, especially among the scarab beetles, which possess metallic shiny cuticles, reflecting left-handed circularly polarized light. Since in Nature, apart from these metallic shiny scarabs, circularly polarized light is rare, until now it has been believed that animals do not perceive the circular polarization of light. The Authors demonstrated that this is not true, at least in the case of mantis shrimps.

In my opinion, in the near future this paper could induce a paradigm change in the study and interpretation of polarization vision. Although the exact role of circular polarization vision in mantis shrimps is still not fully clear, the Authors showed that Gonodactylus smithii have the anatomical and receptor-physiological prerequisites of circular polarization vision. Since the carapace of these shrimps does not polarize the reflected light, they may use their linear and circular polarization vision to detect their transparent, and thus otherwise (i.e., without linear-circular polarization vision) almost invisible pelagic prey. In the future it would be worth demonstrating also behaviourally that these shrimps indeed use circular polarization vision for prey detection.

The experimental, theoretical and computational methods used by the Authors seem to be sound, and their conclusions are correct in my opinion.

On the basis of the above I strongly recommend the publication of this paper. I suggest only some minor changes in the manuscript.

SPECIAL COMMENTS:

page 2, Abstract, row 2 : "in optical intensity" should be replaced by "in light intensity"

page 2, Abstract, row 9 : I suggest to change the term "optimal polarisation vision" to "true" or "full" or "ideal polarisation vision", because 'optimal' usually refers to the rivalry of at least two antagonistic effects/phenomena. Here circular polarization vision does not rival with linear polarization vision, but both cooperate or function parallel.

page 2, Abstract, row 14 : "We emphasise that" should be replaced by "We suggest that", because the strong statement in this sentence is not fully proven yet.

page 2 : "if the plane changes predictably with time the light is said to be fully polarised". This statement is wrong, because elliptically polarized light, for instance, is not fully polarized, and its electric field vector changes predictably: its end-point rotates predictably along the polarization ellipse.

pages 2-3 : "natural light tends to be partially-polarised". This seems again to be an incorrect statement, because in the literature under 'natural light' generally the direct sunlight is meant, which however, is unpolarized (both the degree of linear and circular polarization are zero). This sentence could be changed as follows, for example: "in nature/in the natural optical environment the ambient (scattered and/or reflected) light tends to be partially polarised"

page 3 and throughout the text : "Poincare sphere", "Stokes' parameter", "Stokes' vector" etc. The Authors should be consequent with the use of ' in the genitive case: either "Poincare's sphere" and "Stokes' vector", or "Poincare sphere" and "Stokes vector"

page 3 : Note that there are 4 Stokes parameters, rather than only 3; the first one is the light intensity I itself. The Authors normalized all Stokes parameters with the first one, thus remaining practically only three parameters, because thus the first equals to 1. This should be clearly said in the text.

page 3 and throughout the text : "right-and left-hand" could be replaced by "right- and left-hand"

page 3, row 3 from the bottom : "any change in the degree and type of polarisation" should be replaced by "any change in the type, degree, direction and handedness of polarisation"

page 5, last paragraph : "hemispheres (DH and VH) each sense" should be replaced by "hemisphere (DH and VH) each senses", or similar

page 6, rows 3-4 from the bottom : "the thickness is consistent with that required for quarter-wave retardance". Here the reader expects some quantitative data both on the thickness and the satisfaction of the condition of quarter-wave retardance.

page 8, row 3 : "Figures 4b,c" should be replaced by "Figures 4c,d"

page 9, row 5 from the bottom : "left-than" should be replaced by "left- than"

page 12, rows 3-4 : "requiring three-channel camera systems" should be replaced by "requiring four-channel camera systems", because there are 4 Stokes parameters, thus 4 independent measurements are necessary (e.g., with the use of 3 linear polarizers with 3 different directions of transmission, and one circular polarizer).