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When a wave reaches a hypothetical border line in air, the temperature increases, and the density of air decreases, so according to the law of refraction and Snell's law, the angle of refractions are bigger than the angle of incidences. But I can only imagine that it will lead to a horizontal line, how can it be lifted again towards the perceiver's eyes? Intuitive Explanation

Apologies for my bad drawing

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Indeed it is a thoughtful question but Snell's law admits its own limitations. As you say light bends away from normal in successively layers and its angle of refraction increases however it is the angle of incidence for next layer and since light is going from denser to rarer medium , angle of incidence reaches a critical value after which it is not refracted but reflected and hence reverses its direction of travel.

PS : suppose you assume upper layer to be air and lower denser layer to be water , this critical angle happens to be 48.6 degrees.Since successive layers of air differ slightly in refractive index , angle will be close to 90 degree.enter image description here

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Your own drawing shows what's happening. Look at wavefronts (you'd better draw them less curved, almost plane). You know that speed of light increases downwards. Concentrate on an almost vertical wavefront (the ones nearest to ground): how will it proceed? Its highest part will go on more slowly than its lowest part. As a result, the wavefront will tilt upwards.

It' also possible a full mathematical treatment (using rays it's simpler than with wavefronts). But I'm not sure you want to see it.

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