There are three types of acoustic transients generates by the laser.

First thre is the (bipolar) thermoelastic stress transent caused by thermal expansion of the material.

Second ther is a kind of recoil momentum due to the repulsion of the vaporized material (Is this what you call back pressure wave?)

If you are processing thin films you may also see a tensile wave caused by the reflection of the laser generated wave at the opposite side of your probe.  

Depending on laser parameter and material properties these transients can have different shapes and amplitudes.

It will be better to see picture of your experimental observations. At the present, it is difficult to understand what do you mean.

A strong shock wave can be due to recoil pressure induced by the ablated material. To answer more definitely it would be better, indeed, to see details of your experiment.

The material evaporated from the surface by absorption of the laser radiation is in a high pressure high temperature state inducing a shock-wave and transferring momentum omnidirectional to the ambience. So also in backward direction into the bulk of the material.

This is maybe what you call the "original thermoelastic wave".

As you use ns-pulsed laser radiation you have a second effect which maybe the answer to your question could be related to. As the plasma forms above the surface the laser light is still present and is being absorbed within the plasma. By the free electrons coming from the material plasma also a dielectric breakdown in the ambient air is induced. A so called Laser Supported Detonation (LSD (-:) is initiated in the ambient air which in consequence may be responsible for your second shock wave.

Best regards,

Martin

My laser damage is induced on the rear surface, so the laser is from the bulk to the exit surface ,that is to say, laser will not induce a lsdw. @Martin Hermans 

so, does the early plasma and subsequent ejecta both induce  corresponding shock waves in the bulk?  

I think that you observed usual effect, characteristic for all wave phenomena.

Compression is followed by depression, for example as at a tsunami. Better than all tsunami Japanese and Chinese as the tsunami at coast of the countries often witnessed know. Usually during a tsunami at first will be bared a sea bottom (i.e. depression and to all people is cheerful) and then the high wave (i.e. compression and to all people is sad) comes, there is and vice versa, at first a compression, and then depression.

The similar history is observed at interaction of laser radiation with substance. We investigated in due time interaction of a laser beam of a cylindrical form with water and other environments (environment heating, electrostriction etc.) . In the environment excessive pressure (within the cylinder) was created. Outside the wave of excessive pressure extended, and in the cylinder the depression wave extended that as a result led to emergence of bubbles and cavitation on a cylinder axis.

This idea belongs to my research supervisor G. A. Askaryan. Please, our work: [G. A. Askar’yan, A. G. Krasnovskii and A. V. Yurkin, Sov. Phys. JETP 70 (6), June 1990].

could you offer the full text of your work? I searched it but without any results. thank you!@Alexander Yurkin 

Please my paper. And, please, best wishers to your Chinese Husband and Si-Dzin-Pin's Chinese Wife. :-)

You can use a 532nm laser to see how the effect is.

Your question is not explicit. When the laser hits the glass area, it vaporizes part of the glass into a hot plasma which quickly expands, heating the local atmosphere. What is the form of the glass?. Does it break? If so, the so-called "back-pressure" wave may be due to Wallner-Line generation, a phenomenon endemic to all glass fracturing. See my book "Handbook of Glass Fractography", available from Amazon.com or authorhouse.com -RC Ropp

 I'm not sure if you are asking why the back surface is where the damage is taking place, but this is generally true for such tests.  If you look at the electric field profile in the material, the reflection from the second surface interferes with the incident beam inside the material, leading to an enhancement (constructive interference) in the electric-field profile inside the material, right near the exit surface.  If the material is tested in a configuration with a reflection from the exit surface, this will generally occur.