That's the same principle as the hard drive actuator system, extended. It is not a solenoid actuator in the conventional sense.Wikipedia even has an image of such a motor
You could use a halbach array to avoid the need for the heavy external iron backplates, which link the magnets in that diagram.
There are numerous forms of linear motor. Many use three phase drive, rather than quadrature.
The type Buk suggested would be better with a coil either side of the permanent magnet, that gives more torque; a single side version suffers from inverse square strength reduction as the gap increases.
Adding a "C" frame to connect the outer ends of the coils with a direct magnetic path would give another major improvement.