Add options to dropdown with ajax
you should merge the existing vales in $form['dropdown_second']['#options'] with the one in $form_state['values']['field_name']

something like this:

$existing_options = isset($form['dropdown_second']['#options']) ? $form['dropdown_second']['#options'] : array();
$new_option = get_dropdown_options($field_val);

$form['dropdown_second'] = array(
    '#type' =&gt; 'select',
    '#title' =&gt; 'Dynamic Dropdown',    
    '#prefix' =&gt; '&lt;div id="dropdown-second-replace"&gt;',
    '#suffix' =&gt; '&lt;/div&gt;',    
    '#options' =&gt; array_merge($existing_options, $new_option),
    '#default_value' =&gt; isset($form_state['values']['dropdown_second']) ? $form_state['values']['dropdown_second'] : '',
  );