The popliteal artery [a. poplitea], at the back of the knee-joint, extends along the lower third of the thigh and the upper part of the leg, reaching from the opening in the great adductor to the lower border of the popliteus muscle. It is continuous above with the femoral, and divides at the lower end into the anterior and posterior tibial arteries.

This artery at first inclines from the inner side of the limb to reach the middle of the knee-joint, and thence continues vertically to its lower end. Lying deeply in its whole course, it is covered for some distance at its upper end by the semi-membranosus muscle; lower down, a little above the knee, it is placed in the intermuscular interval named the popliteal space, where it iscovered by the fascia, and overlaid by the muscles which bound that space. The lower part of the artery is covered for a considerable distance by the gastrocnemius muscle, and at its termination by the upper margin of the soleus muscle.

At first this artery lies close to the inner side of the femur ; in descending the vessel gets behind the bone, and as this is here curved forwards, while the course of the artery is straight, there is an interval between the two. The popliteal artery then rests against the posterior ligament of the knee-joint, and afterwards on the popliteus muscle.

Vein

The popliteal vein lies close to the artery, behind and somewhat to its outer side. The vein is frequently double along the lower part of the artery, and, more rarely, at the upper part also. The short saphenous vein, ascending into the popliteal space over the gastroenemius muscle, approaches the artery, as it is about to terminate in the popliteal vein.

Nerve

The inner division of the sciatic nerve lies at first to the outer side of the artery, but much nearer to the surface than the vessel : the nerve afterwards crosses over the artery, and is then placed to its inner side.

Popliteal space.

Behind the lower end of the femur the flexor muscles of the leg (called also the hamstring muscles) diverge to their places of attachment at each side of the limb — the biceps to the fibula, the semi-membranosus and semi-tendinosus to the inner side of the tibia. In this way there is formed an interval, which is bounded laterally by those muscles above, and by the head of the gastrocnemius below. This is the popliteal space.
Whilst passing through the popliteal space, the artery is surrounded by a quantity of fat, in which a few small lymphatic glands will be found. The fascia lata, it is to be observed, holds the muscles bounding the popliteal space so closely together that the line of separation between them is marked only by a slight depression on the surface of the membrane. By removing the fascia and the fat (which in some cases is abundant), the popliteal artery with its accompanying vein, and the internal popliteal nerve, will be brought into view ; placed, as regards the surface, in the opposite order to that in which they have just been mentioned. Thus, the artery lies deepest, and midway between the sides of the limb ; the vein is in contact with the artery, but superficial to it and to its outer side ; whilst the nerve is removed to some distance from the vessels, lying much nearer to the surface, and still further to the outer side of the popliteal space.

Peculiarities of the popliteal artery

Deviations from the ordinary condition of the popliteal artery are not frequently met with. The principal departure from the ordinary arrangement consists in its premature division into terminal branches. Such an early division has been found to take place most frequently opposite the flexure of the knee-joint, and not higher.
In a few instances, the popliteal artery has been seen to divide into the anterior tibial and peroneal arteries — the posterior tibial being small or absent. In a single case, the popliteal artery was found to furnish at its end the peroneal artery as well as its two usual branches, the anterior and posterior tibials.

Branches of the popliteal artery.

The popliteal artery gives off five articular branches, two above and two below the joint, and one which passes forwards into it ; also some large muscular branches to the hamstring muscles, and to the gastrocnemius.
The muscular branches may be divided into a superior and an inferior set.
The superior branches, three or four in number, arise from the upper part of the popliteal artery, and are distributed to the lower ends of the flexor muscles of the leg, reaching also to the vasti muscles. They anastomose with the lower perforating arteries, with the terminal branches of the deep femoral artery, and with some of the articular arteries.
The inferior muscular branches, sural arteries, usually two in number, and of considerable size, arise from the back of the popliteal artery, opposite the knee-joint, and enter, one the outer and the other the inner head of the gastrocnemius muscle, which they supply, as well as the fleshy part of the plantaris muscle.
Over the surface of the gastrocnemius will be found at each side, and in the middle of the limb, slender branches, which descend a considerable distance along the calf of the leg, beneath the integument. These small vessels arise separately from the popliteal artery, or from some of its branches.
The articular arteries. — Two of these pass off, nearly at right angles from the popliteal artery, one to each side, above the flexure of the joint, whilst two have a similar arrangement below it ; hence they are named the upper internal and external, and the lower internal and external. Besides these, there is a fifth articular artery, called the middle articular, because it enters the middle of the back of the joint.
The upper articular arteries, [a. articulares genu superiores,]. — That of the inner side turns over the femur just above the condyle ; and, passing under the tendon of the great adductor and the vastus internus, divides into two branches.
Of these, one, comparatively superficial, enters the substance of the vastus, which it supplies, and inosculates with the anastomotic branch of the femoral, and with the lower internal articular artery. The other branch runs close to the femur, ramifies upon it, and also on the knee-joint, and communicates with the upper external articular artery.
The upper external articular artery passes outwards, a little above the outer condyle of the femur, under cover of the biceps muscle, and, after perforating the intermuscular septum, divides into a superficial and a deep branch. The latter, lying close upon the femur, spreads branches upon it and the articulation, and anastomoses with the preceding vessel, with the anastomotic of the femoral, and with the lower external articular artery ; the superficial branch descends through the vastus to the patella, anastomosing with other branches and assisting in the supply of the joint.
The lower articular arteries, [a. articulares genu inferiores.] — The internal artery passes downwards below the corresponding tuberosity of the tibia, lying between the bone and the internal lateral ligament; its branches ramify on the front and inner part of the joint, as far as the patella and its ligament. The external artery takes its course outwards, under cover of the outer head of the gastroenemius in the first instance, and afterwards under the external lateral ligament of the knee, and the tendon of the biceps muscle, passing above the head of the fibula and along the border of the external semilunar cartilage. Having reached the fore part of the joint, it divides near the patella into branches, which communicate with the lower articular artery of the opposite side, and with the recurrent branch from the anterior tibial ; whilst others ascend, and anastomose with the upper articular arteries.
In this manner the four articular branches form at the front and sides of the knee-joint a network of vessels.
The remaining articular artery, called, from its position, the middle articular, and from its being a single vessel, azygos, [a. articulationis genu media s. azygos,] is a small branch which arises from the popliteal artery, opposite the flexure of the joint. It pierces the posterior ligament, and supplies the crucial ligaments and the other structures, within the articulation. This small artery frequently arises from one of the other articular branches, especially from the upper and external branch.

From Quain's anatomy.