This is really cool, is there anything similar for hydrogen bombs? I've always been curious how they are different from fission bombs (although obviously a lot of their true workings are still classified).
Probably not. This appears to be the Mark 3 "Fat Man" fission bomb developed by the U.S. during WWII and represents one of the few actual nuclear weapons whose designs are known in reasonable detail (due to the age of the design and some impressive historical detective work[1]).
The detailed layout of components in a real hydrogen bomb is still officially secret and a matter of much speculation. Journalist and activist Howard Morland pieced together a rough outline from unclassified sources and the government sued to block publication of his article.[2]
That suit had the practical effect of confirming at least some of Morland's deductions, and much additional information has come to light since then including at least one alleged cutaway of a real weapon[3].
IANANP (I an not a nuclear physicist) but I have read a great deal, and your best bet is to read http://nuclearweaponarchive.org/ which is chock full of interesting information, including a comprehensive list of every nuclear detonation, ever. Go read there about how Hydrogen bombs work.
TLDR for that site; there are two major factors in play with an H-bomb. The first is the addition of a fair amount of deuterium (Hydrogen with an extra neutron) around the explosive core. This acts as a large source of potential energy, and after the initial fission reaction is kicked off, the deuterium ignites and fussion occurs, similar to that which happens on the sun: 2 Hydrogens become 1 Helium, and 2 neutrons go flying off.
The Fission reaction has to be managed differently, however, as there needs to be a secondary reflector system in use. This secondary core reflector system captures the X-rays after the initial fission begins. These x-rays are reflected back into the core, causing the implosion to have a second stage. This "second stage" ignites the third stage fuel, and boom. The documentation will call it only 2 stages, though that x-ray reflection device is an essential part of kickstarting fusion.
The first H-Bomb detonation was essentially a A-bomb surrounded by liquid deuterium in a massive cooling system. The cooling system was actually the size of a building. http://nuclearweaponarchive.org/Usa/Tests/Ivy.html
Of interesting side note is the second H-bomb Castle Bravo: http://nuclearweaponarchive.org/Usa/Tests/Castle.html ) included a Lithium-deuteride solid fuel around the detonation device. The detonation was accidentally the largest ever conducted by the United States. It irradiated a huge swath of the Pacific ocean, and it's follow-up test had similar results.
Devices which used Lithium-6 or enriched Lithium instead of Deuterium would also be considered H-bombs, despite their not having Hydrogen. That's why we call them thermonuclear.
This was because of the added Lithium in the solid fuel. You're basically tripling the amount of neutrons available to the fusion reaction, and thus, the detonation was completely out of hand and exceeded calculations by an exponential factor.
Your description also leaves out the strictly optional, but common 3rd stage of a thermonuclear device: the thick Uranium blanket surrounding the entire assembly. Even "depleted" U-238 will undergo fast neutron fission, which the deuterium / tritium fusion reaction produces in extreme abundance. In most TN weapons, most of the explosive yield results from fission of the Uranium blanket, not the fusion reaction. In a "neutron bomb", or "enhanced radiation warhead", neutrons are the desired product rather than explosive yield, so these typically omit the U casing.
Richard Rhodes' "Dark Sun: The Making of the Hydrogen Bomb" provides an excellent description and paper bibliography.
Hydrogen bombs are basically this plus extra stuff nearby that undergoes fusion when exposed to the fission explosion. Wikipedia has pretty good illustrations and explanations, although not quite to the same level: https://en.wikipedia.org/wiki/Thermonuclear_weapon
I'll second the recommendation for http://nuclearweaponarchive.org/ , especially the "nuclear weapons faq" section, it's a great site though some of the material can be heavy going.
Thermonuclear bombs use a multi-stage design. A nuclear bomb (such as a fission bomb) can be thought of a bit like a light-bulb, only an extremely bright one that is so hot it shines mainly in the x-ray spectrum. A multi-stage thermonuclear bomb takes the light from a fission bomb and traps it inside of a "hohlraum" which is a fancy german word for a box or chamber that happens to be made out of very heavy metals (often depleted Uranium) that is reasonably opaque to the soft x-rays emitted by the nuclear explosion.
Of course, no mere metal container is going to contain the energy of a nuclear explosion for long, the inside surface layers of the hohlraum will absorb some fraction of the x-rays and be vaporized (this is called ablation). This ablation of surface material is so energetic it creates a thrust, like a rocket exhaust coming off the surface of the material it pushes the material away, blowing up the hohlraum like a balloon. The same forces play out on the other occupant of the hohlraum, a capsule containing fusion fuel (often lithium deuteride paste) with a shell made out of similar "high-z" x-ray opaque material. The ablation of the surface of the fusion fuel capsule creates incredible forces, powered as it is by the energy of a nuclear explosion. This causes the fusion fuel capsule to implode, creating conditions of extraordinarily high matter density, pressure, and temperature (due to adiabatic heating). At the center of the fusion fuel capsule there is a fission bomb core, a chunk of Plutonium or Uranium. Because of the speed and strength of the "radiation implosion" (though strictly speaking it's a radiation powered ablation implosion) the amount of fissile material can be small compared to a chemically powered implosion design (keeping in mind that critical mass goes down as density goes up). This "spark plug" then kicks off a super critical fission chain reaction which starts releasing energy that heats up the fusion fuel and kicks off a fusion chain reaction. The fusion reaction then releases a metric crap-ton of very high energy neutrons which facilitate fission reactions including fissioning of depleted Uranium (U-238) in the bomb components (such as the hohlraum, the tamper around the fission primary, the fusion fuel capsule casing, etc.)
In principle, much is the same. The main difference is that the explosive force to implode the main bomb (or the "secondary") comes from a nuclear explosion.
H-bomb works like this: You need to initiate fusion. For this you actually don't need Uranium or Plutonium, you just want fusion to occur in Hydrogen. However fusion only occurs when temperatures are extremely high and kept that way for small slice of time. This can be achieved by - guess what - firing fission bombs (i.e. regular atomic bomb)! However you can't start fusion by firing just one atomic bomb because fusion material will just fly off. You want fusion material to experience extreme pressures and temps. So - again guess what - we fire two or more nuclear bombs precisely at the same time with fusion material in the center. Another way is to use reflector so that energy from fission can be concentrated on fusion material. That's pretty much it. So each H-bomb may carry one or more regular A-bombs.
A side story, which is the most amazing part, is that this whole process was worked out as mathematical theory and huge amount of time and money spent on the faith that math would work out. The math was so complicated that physicists needed computers to do that. That was back in late 40s and computers used valves and almost never worked. The story is that they finally gave up on computer and physicists would sat through many many days and nights calculating everything by hand and double checking. And all that finally worked exactly what math had predicated.